Research ArticleConsensus of Multi-Integral Fractional-Order MultiagentSystems with Nonuniform Time-Delays

Jun Liu 12 Wei Chen 1 Kaiyu Qin 2 and Ping Li 1

1School of Control Engineering Chengdu University of Information Technology Chengdu 610225 China2School of Aeronautics and Astronautics University of Electronic Science and Technology of ChinaChengdu 611731 China

Correspondence should be addressed to Jun Liu liujuncuiteducn

Received 2 June 2018 Revised 15 August 2018 Accepted 17 October 2018 Published 1 November 2018

Guest Editor Christos K Volos

Copyright copy 2018 Jun Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Consensus of fractional-order multiagent systems (FOMASs) with single integral has been wildly studied However the dynamicswith multiple integral (especially double integral to sextuple integral) also exist in FOMASs and they are rarely studied at presentIn this paper consensus problems for multi-integral fractional-order multiagent systems (MIFOMASs) with nonuniform time-delays are addressedThe consensus conditions for MIFOMASs are obtained by a novel frequency-domain method which properlyeliminates consensus problems of the systems associated with nonuniform time-delays Besides the method revealed in this paperis applicable to classical high-ordermultiagent systems which is a special case ofMIFOMASs Finally several numerical simulationswith different parameters are performed to validate the correctness of the results

1 Introduction

The research related to multiagent systems (MASs) has beengoing on for decades due to its many meaningful appli-cations eg sweep coverage control of MASs [1] flockingbehavior of mobile robots [2] and coordinated attitude con-trol of a formation of satellites [3] Consensus is an agreementon the quality of certain concerns about the specific states ofall agents which is one of themost fundamental requirementsfor the research on MASs

Up to now numerous studies have been conducted toresolve the problems about consensus of MASs with differentdynamics During the past decades a lot of results have beenaccomplished about consensus of first-order MASs [4ndash11] In[4] a simple model was presented for the phase transitionof a set of self-driven particle and it was demonstrated thatthe headings of all agents in MASs converged to a commonvalue by simulation In [5] authors provided some theoreticalexplanations for Vicsekrsquos linearized model and analyzedthe alignment of undirected switching topologies of agentsthat were regularly connected Based on the research worksof [5] more relaxed consensus conditions over dynamic

switching topologies were given in [6 7] Authors in [8] putforward a framework about consensus theory of MASs withdirected information flow linknode failures time-delaysand so on Robust119867infin control about consensus problems forMASs with parameter uncertainties external disturbancesnonidentical state and time-delays was discussed in [9] Inrecent years more and more researchers have paid moreattention to consensus of second-order MASs [12ndash16] Forinstance authors in [12] investigated consensus problemsfor second-order continuous-time MASs in the presence ofjointly connected topologies and time-delay In [13] twotypes of consensus problems for second-orderMASswith andwithout delay over switching topology and directed topologywere studied In [14] the local consensus problem for second-orderMASswhose dynamicswere nonlinear dynamics underdirected and switching random topology was discussedand several sufficient conditions were derived to ensure theMASs reach consensus Furthermore taking into accountthe fact that high-order MASs were widely used consensusproblems for high-order MASs have been studied in [17ndash21] In particular the output consensus problem in [17] wasaddressed for high-order MASs with external disturbances

HindawiComplexityVolume 2018 Article ID 8154230 24 pageshttpsdoiorg10115520188154230

2 Complexity

and some conditionswere derived to ensure consensus for theMASs The consensus problems in [18] were considered fora class of high-order MASs with time-delays and switchingnetworks and a nearest-neighbour rule was designed andsome conditions were derived to guarantee consensus forthe systems with time-delays The conditions of consensusin [21] for high-order MASs with nonuniform time-delayswere proposed by a novel frequency-domain approach whichproperly resolved the challenges associated with multipletime-delays

It is worth noting that many results above about MASswere based on the integer-order dynamics In fact manyscholars have declared that the essential characteristic orbehavior of an object in the complex environment couldbe better revealed by adopting fractional-order dynamicsExamples include unmanned aerial vehicles operating inan environment with the impacts of rain and wind [22]food searching with the help of the individual secretionsand microbial [23] and submarine robots in the bottomof the sea with large amounts of microorganisms and vis-cous substances [24] Compared to integer-order dynamicsfractional-order dynamics provided an excellent tool in thedescription of memory and hereditary properties [25 26]Moreover authors in [27 28] indicated that the integer-ordersystems were only the special examples of the fractional-order systems Based on these facts the research resultson consensus of FOMASs with single integral in [29ndash34]have been continuously springing up in recent years As weknow consensus problem of FOMASs was first proposedand investigated by Cao et al [29] Next consensus controlof FOMASs with time-delays was studied by Yang et al[30 31] where homogeneous dynamics and heterogeneousdynamics were used to illustrate the agent of system In[32] consensus problem of linear FOMASs with input time-delay and the consensus problem of nonlinear FOMASswith input time-delay were investigated respectively In [33]consensus problems were studied for FOMASs with nonuni-form time-delays Meanwhile by means of matrix theorytool Laplace transform and graph theory tool two delaymargins were obtained as the consensus conditions Latelyconsensus of FOMASs with double integral was proposedin [35ndash39] The consensus problem of FOMASs with doubleintegral over fixed topology was studied in [35] By applyingMittag-Leffler function Laplace transform and dwell timetechnique consensus for FOMASs with double integral overswitching topology was investigated in [36] Based on theslidingmode estimator consensus problem for FOMASswithdouble integral was studied in [37] Bymeans ofmatrix theorytool Laplace transform and graph theory tool consensusproblems for a FOMAS with double integral and time-delaywere studied in [38] Nevertheless the above research resultson the consensus problems of FOMASs with or withouttime-delays were based on the single-integral fractional-order or double-integral fractional-order dynamics To thisday there is almost no research on consensus problems ofMIFOMASs with time-delays especially nonuniform time-delays

Motivated by above analysis we extend FOMASs fromsingle-integral fractional-order dynamics to multi-integral

fractional-order ones in this paper Consensus problemsof FOMASs with multiple integral in the presence ofnonuniform time-delays are studied The main idea ofthis paper is to first obtain the characteristic polynomialof a MIFOMAS with imaginary eigenvalues through themodel transformation of the system and then determinethe stability conditions of the system according to thischaracteristic polynomial so as to determine the consensusconditions of the system according to the stability conditionsof the system The consensus conditions of the MIFOMASwith nonuniform time-delays can be obtained by inequali-ties

The main contributions of this paper are as followsFirstly we consider multi-integral fractional-order dynamicsAs far as we know this paper is the first paper that stud-ies consensus of MIFOMASs Just as integer-order MASshave first-order (single-integral) MASs [4ndash11] second-order(double-integral) MASs [12ndash16] and high-order (multi-integral) MASs [17ndash21] FOMASs also have single-integralFOMASs [29ndash34] double-integral FOMASs [35ndash39] andMIFOMASs which makes the overall theory of FOMASsperfect from single-integral to multi-integral FOMASs Inaddition single-integral and double-integral FOMASs arethe special cases of MIFOMASs Secondly we consider sym-metric and asymmetric time-delays The symmetric time-delays contain up to 119899(119899 minus 1)2 different values and theasymmetric time-delays contain up to 119899(119899 minus 1) differentvalues when the MIFOMAS consists of n agents Thirdlywe consider the dynamics of each agent containing mul-tiple state variables with different fractional orders TheMIFOMAS with nonuniform time-delays consists of someagents and each agent contains multiple state variables withdifferent fractional orders Finally we derive the consen-sus conditions for a MIFOMAS with nonuniform time-de-lays

The remainder of this article is organized as followsIn Section 2 fractional calculus and its Laplace transformare given In Section 3 the knowledge about graph theoryis shown out In Sections 4 and 5 consensus algorithmsfor a MIFOMAS in the presence of nonuniform time-delays are studied In Section 6 some numerical exampleswith different parameters are simulated to verify the resultsFinally conclusions are drawn out in Section 7

2 Fractional Calculus

In [40] several different definitions of fractional calcu-lus have been proposed in which the Caputo fractionalderivative played an important role in fractional-ordersystems Because the initial value of Caputo fractionalderivative has practical signification in many problemswhich is commonly used in the variety of physical fieldsErgo this paper will model the system dynamical char-acteristics by using Caputo derivative which is definedby

119862

119886119863120572119905 119909 (119905) = 1Γ (119898 minus 120572) int119905119886

119909(119898) (120578)(119905 minus 120578)1+120572minus119898 119889120578 (1)

Complexity 3

where 119886 isin R denotes the initial value 120572 represents the orderof the Caputo derivative and 119898 minus 1 lt 120572 le 119898(119898 isin 119911+) Γ(sdot) isgiven by

Γ (119910) = int+infin0

119890minus119905119905119910minus1119889119905 (2)

If 119862119886119863120572119905 119909(119905) is replaced by 119909(120572)(119905) and the Laplace trans-form of 119909(119905) is represented by 119883(119904) = L119909(119905) =intinfin0minus

119890minus119904119905119909(119905)119889119905 then the following equation can be used todenote Laplace transform of the Caputo derivative

L 119909(120572) (119905)=

119904120572119883 (119904) minus 119904120572minus1119909 (0minus) 120572 isin (0 1]119904120572119883 (119904) minus 119904120572minus1119909 (0minus) minus 119904120572minus21199091015840 (0minus) 120572 isin (1 2]

(3)

where 119909(0minus) = lim119905997888rarr0minus119909(119905) and 1199091015840(0minus) = lim119905997888rarr0minus1199091015840(119905)3 Graph Theory

For a MAS with 119899 agents the network topology can bedenoted by a graph G = (VE) where V = 1199041 119904119899and E sube V2 respectively represent the set of nodes andthe set of edges The node indices belong to a finite index setI = 1 2 119899 The weighted adjacency matrix is denotedby A = [119886119894119895]119899times119899 The element of the 119894-th row and the 119895-thcolumn in matrix A indicates the connection state betweenagents 119904119894 and 119904119895 If nodes 119904119894 and 119904119895 are connected ie 119890119894119895 isinE then 119886119894119895 gt 0 and 119904119895 is called a neighbor of node 119904119894119873119894 = 119895 isin I 119895 = 119894 denotes the index set of all neighborsof agent 119894 If nodes 119904119894 and 119904119895 are connected and 119886119894119895 = 119886119895119894then G is an undirected graph otherwise the G is a directedgraph In a directed graph a directed path is a sequence ofedges by (1199041 1199042) (1199042 1199043) where (119904119895 119904119894) isin E The directedgraph has a directed spanning tree if all other nodes havedirectional paths from the same node The Laplacian matrixof the graph G is defined by 119871 = Δ minus A isin R119899times119899 where Δ ≜diag119889119890119892119900119906119905(1199041) 119889119890119892119900119906119905(1199042) 119889119890119892119900119906119905(119904119894) 119889119890119892119900119906119905(119904119899) is adiagonal matrix with 119889119890119892119900119906119905(119904119894) = sum119899119895=1 119886119894119895 Supposing somegraphs G1G2 G119872 and graph G consist of the samenodes and the edge set of graphG is the sum of the edge setsof other graphs G1G2 G119872 then there is 119871 = sum119872119898=1 119871119898which means the Laplacian matrix of graph G is the sum ofother graphsrsquo Laplacian matrix

4 Problem Statement

There are two lemmas [41] for the later analysis

Lemma 1 If graph G is an undirected and connected graphthen its Laplacian matrix 119871 has one singleton zero eigenvalueand other eigenvalues are all positive ie 0 = 1205821 lt 1205822 le 1205823 lesdot sdot sdot le 120582119899Lemma 2 If graphG is a directed graph with a spanning treethen its Laplacian matrix 119871 has one singleton zero eigenvalue

and other eigenvalues have a positive real part ie 1205821 =0Re(120582119894) gt 0 (119894 = 2 3 119899)Consider a MIFOMAS composed of 119899 agents Each node

in graph G corresponds to each agent of the MIFOMASIf 119886119894119895 gt 0 we can think that the 119894-th agent can get stateinformation from the 119895-th agent The dynamics of the 119894-thagent of the MIFOMAS are represented by

119909(1205721)1198941 (119905) = 1199091198942 (119905) 119909(1205722)1198942 (119905) = 1199091198943 (119905)

119909(120572119897minus1)119894(119897minus1) (119905) = 119909119894119897 (119905) 119909(120572119897)119894119897 (119905) = 119906119894 (119905)

119894 isin I

(4)

where 1199091198941(119905) 1199091198942(119905) 119909119894119897(119905) isin R respectively represent119897 different states of the 119894-th agent 119909(1205721)1198941 (119905) is the 1205721-orderCaputo derivative of 1199091198941(119905) 119909(1205722)1198942 (119905) is the 1205722-order Caputoderivative of 1199091198942(119905) 119909(120572119897)119894119897 (119905) is the 120572119897-order Caputo deriva-tive of 119909119894119897(119905) (1205721 1205722 120572119897 isin (0 1]) and 119906119894(119905) isin R is thecontrol input

Definition 3 If and only if the states of all agents in MIFO-MAS (4) satisfy

lim119905997888rarr+infin

(1199091198941 (119905) minus 1199091198951 (119905)) = 0lim119905997888rarr+infin

(1199091198942 (119905) minus 1199091198952 (119905)) = 0

lim119905997888rarr+infin

(119909119894119897 (119905) minus 119909119895119897 (119905)) = 0119894 119895 isin I

(5)

then MIFOMAS (4) can reach consensus In (5)1199091198941(119905) 1199091198942(119905) 119909119894119897(119905) isin R respectively represent 119897 differentstates of the 119894-th agent and 1199091198951(119905) 1199091198952(119905) 119909119895119897(119905) isin Rrespectively represent 119897 different states of the 119895-th agent

In this paper the control protocol for MIFOMAS (4) willbe given by

119906119894 (119905) = minus 119897minus1sum119896=1

119875119896+1119909119894119896+1 (119905)+ sum119895isin119873119894

119886119894119895 [1199091198951 (119905 minus 120591119894119895) minus 1199091198941 (119905 minus 120591119894119895)] (6)

4 Complexity

where119873119894 denotes the neighbors index collection of the agent119894 119886119894119895 is the (119894 119895)-th element of A 120591119894119895 gt 0 is the time-delay which is from the 119895-th agent to the 119894-th agent and119875119896+1 gt 0 are scale coefficients If all 120591119894119895 = 120591119895119894 then the time-delays are symmetrical else the time-delays are asymmet-rical The symmetric time-delays and the asymmetric time-delays are two different forms of nonuniform time-delaysFor ease of analysis we define that 120591119898 denote 119872 differenttime-delays of MIFOMAS (4) ie 120591119898 isin 120591119894119895 119894 119895 isinI (119898 = 1 2 119872) Then the following control protocolis provided to resolve consensus problems of MIFOMAS(4)

119906119894 (119905) = minus 119897minus1sum119896=1

119875119896+1119909119894119896+1 (119905)+ sum119895isin119873119894

119886119894119895 [1199091198951 (119905 minus 120591119898) minus 1199091198941 (119905 minus 120591119898)] (7)

Assume the state vector of the 119894-th agent is 120585119894(119905) =[1199091198941(119905) 1199091198942(119905) 119909119894119897(119905)]119879 isin R119897 and the joint state vec-tor of MIFOMAS (4) consisting of 119899 agents is 120595(119905) =[1205851198791 (119905) 1205851198792 (119905) 120585119879119899 (119905)]119879 isin R119897119899

If we define two matrices as follows

119860 =[[[[[[[[[[[[[

0 1 0 sdot sdot sdot 0 00 0 1 sdot sdot sdot 0 0 d

0 0 0 sdot sdot sdot 1 00 0 0 sdot sdot sdot 0 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897minus1 minus119875119897

]]]]]]]]]]]]]

isin R119897times119897 (8)

and

119861 =[[[[[[[

0 0 sdot sdot sdot 0 d

0 0 sdot sdot sdot 01 0 sdot sdot sdot 0

]]]]]]]

isin R119897times119897 (9)

then under the control protocol given by (7) the closed-loopdynamics of MIFOMAS (4) can be described as

[119909(1205721)11 (119905) 119909(1205722)12 (119905) 119909(120572119897)1119897 (119905) 119909(1205721)21 (119905) 119909(1205722)22 (119905) 119909(120572119897)2119897 (119905) 119909(1205721)1198991 (119905) 119909(1205722)1198992 (119905) 119909(120572119897)119899119897 (119905)]119879 = (119868119899otimes 119860)120595 (119905) minus 119872sum

119898=1

(119871119898 otimes 119861) sdot 120595 (119905 minus 120591119898) (10)

5 Main Results

Theorem 4 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 1 Define the following functions

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(11)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectivelydenote the real part and the imaginary part of119865119897(120596)

If all 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) for the MIFOMAS(10) with symmetric time-delays then the control protocol(7) can resolve the consensus problem of the MIFOMAS (10)with symmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem ofthe MIFOMAS (10) with symmetric time-delays e value of120596 corresponding to 119897 in 119879119897(120596) is determined by the followingequation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (12)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Proof We shall apply the frequency-domain method toanalyze the MIFOMAS (10) with symmetric time-delays andwe can get

119868119899 otimes[[[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]]]

Ψ (119904)

minus

119868119899 otimes[[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]]

120595(0minus)

Complexity 5

minus (119868119899 otimes 119860)Ψ (119904) + 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898Ψ (119904) = 0Ψ (119904) = R120595 (0minus)119866minus1120591119898 (119904)

(13)

whereΨ(119904) is the Laplace transformof120595(119905)120595(0minus) is the initialvalue of 120595(119905)

R = 119868119899 otimes[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]

(14)

and

119866120591119898 (119904) =

119868119899 otimes

[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]

minus[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

(15)

Motivated by the stability analysis of a fractional-ordersystem in [42] we can study consensus of the MIFOMAS(10) with symmetric time-delays by analyzing the character-istic eigenvaluesrsquo position of the characteristic polynomialdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays Specifically consensus of the MIFOMAS (10) withoutdelays (all 120591119898 = 0) is necessary for consensus of theMIFOMAS (10) with symmetric time-delays that is to say inthis case the characteristic eigenvalues of det[119866120591119898(119904)] of the

MIFOMAS (10) with symmetric time-delays are all situatedin the left half plane (LHP) of the complex plane and as 120591119898increases continuously from zero the characteristic eigen-value of det[119866120591119898(119904)] of the MIFOMAS (10) with symmetrictime-delays will change continuously from the LHP to theright half plane (RHP) of the complex plane Once thecharacteristic eigenvalue of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays reaches the RHP of thecomplex plane through the imaginary axis the MIFOMAS(10) with symmetric time-delays will be unstable and can notachieve consensus Ergo we only need to consider the criticaltime-delay when the nonzero characteristic eigenvalue ofdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays is just situated on the imaginary axis for the first time as120591119898 increases continuously from zero and the correspondingtime-delay is just the delay margin 120591 of the MIFOMAS (10)with symmetric time-delays

Assume 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 =1 1199061 1199062 119906119897minus1 119906119897 isin C119899 we have the following equa-tions

119868119899

otimes

[[[[[[[[[[[[[

(minus119895120596)1205721 0 sdot sdot sdot 0 00 (minus119895120596)1205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot (minus119895120596)120572119897minus1 00 0 sdot sdot sdot 0 (minus119895120596)120572119897

]]]]]]]]]]]]]

minus[[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

6 Complexity

119868119899

otimes

[[[[[[[[[[[

(minus119895120596)1205721 minus1 0 sdot sdot sdot 00 (minus119895120596)1205722 minus1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot minus10 1198752 1198753 sdot sdot sdot 119875119897 + (minus119895120596)120572119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

119868119899 otimes Ω 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(1)

+ 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2)

= 0(16)

(1) = 119868119899 otimes Ω 119906 = 1198681198991199061 otimes (Ω [1 0 0]119879) + 1198681198991199062otimes (Ω [0 1 0 0]119879) + 1198681198991199063otimes (Ω [0 0 1 0 0]119879) + sdot sdot sdot + 119868119899119906119897minus1otimes (Ω [0 0 sdot sdot sdot 0 1 0]119879) + 119868119899119906119897otimes (Ω [0 0 1]119879) = 1198681198991199061 otimes [(minus119895120596)1205721 0 0]119879+ 1198681198991199062 otimes [minus1 (minus119895120596)1205722 0 0 1198752]119879 + 1198681198991199063 otimes [0minus 1 (minus119895120596)1205723 0 0 1198753]119879 + sdot sdot sdot + 119868119899119906119897minus1 otimes [0 0 minus1 (minus119895120596)120572119897minus1 119875119897minus1]119879 + 119868119899119906119897 otimes [0 0 minus1(minus119895120596)120572119897 + 119875119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061 0 0]119879 + 119868119899otimes [minus1199062 (minus119895120596)1205722 1199062 0 0 11987521199062]119879 + 119868119899 otimes [0minus 1199063 (minus119895120596)1205723 1199063 0 0 11987531199063]119879 + sdot sdot sdot + 119868119899 otimes [0 0 minus119906119897minus1 (minus119895120596)120572119897minus1 119906119897minus1 119875119897minus1119906119897minus1]119879 + 119868119899 otimes [0

0 minus119906119897 ((minus119895120596)120572119897 + 119875119897) 119906119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061minus 1199062 (minus119895120596)1205722 1199062 minus 1199063 (minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897 11987521199062+ 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897]119879 = [119868119899 ((minus119895120596)1205721 1199061 minus 1199062) 119868119899 ((minus119895120596)1205722 1199062 minus 1199063) 119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) 119868119899 (11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897)]119879

(17)

(2) = 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906 = 119872sum119898=1

(119871119898 otimes 119861) 119906119890119895120596120591119898

= 119872sum119898=1

(1198711198981199061) otimes (119861 [1 0 0]119879) + (1198711198981199062)otimes (119861 [0 1 0 0]119879) + sdot sdot sdot + (119871119898119906119897minus1)otimes (119861 [0 0 0 1 0]119879) + (119871119898119906119897)otimes (119861 [0 0 1]119879) 119890119895120596120591119898 = 119872sum

119898=1

(1198711198981199061)otimes ([0 0 0 1]119879) + (1198711198981199062) otimes ([0 0 0 0]119879)+ sdot sdot sdot + (119871119898119906119897minus1) otimes ([0 0 0 0 0]119879) + (119871119898119906119897)otimes ([0 0 0]119879) 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897) [(1199061otimes [0 0 0 1]119879) + (1199062 otimes [0 0 0 0]119879) + sdot sdot sdot+ (119906119897minus1 otimes [0 0 0 0]119879) + (119906119897otimes [0 0 0 0]119879)] 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897)

sdot [0 0 0 1199061]119879 119890119895120596120591119898 = [0 0 0119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898]119879

(18)

Because of (1) + (2) = 0 we have119868119899 ((minus119895120596)1205721 1199061 minus 1199062) = 0119868119899 ((minus119895120596)1205722 1199062 minus 1199063) = 0

Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

Complexity 3

where 119886 isin R denotes the initial value 120572 represents the orderof the Caputo derivative and 119898 minus 1 lt 120572 le 119898(119898 isin 119911+) Γ(sdot) isgiven by

Γ (119910) = int+infin0

119890minus119905119905119910minus1119889119905 (2)

If 119862119886119863120572119905 119909(119905) is replaced by 119909(120572)(119905) and the Laplace trans-form of 119909(119905) is represented by 119883(119904) = L119909(119905) =intinfin0minus

119890minus119904119905119909(119905)119889119905 then the following equation can be used todenote Laplace transform of the Caputo derivative

L 119909(120572) (119905)=

119904120572119883 (119904) minus 119904120572minus1119909 (0minus) 120572 isin (0 1]119904120572119883 (119904) minus 119904120572minus1119909 (0minus) minus 119904120572minus21199091015840 (0minus) 120572 isin (1 2]

(3)

where 119909(0minus) = lim119905997888rarr0minus119909(119905) and 1199091015840(0minus) = lim119905997888rarr0minus1199091015840(119905)3 Graph Theory

For a MAS with 119899 agents the network topology can bedenoted by a graph G = (VE) where V = 1199041 119904119899and E sube V2 respectively represent the set of nodes andthe set of edges The node indices belong to a finite index setI = 1 2 119899 The weighted adjacency matrix is denotedby A = [119886119894119895]119899times119899 The element of the 119894-th row and the 119895-thcolumn in matrix A indicates the connection state betweenagents 119904119894 and 119904119895 If nodes 119904119894 and 119904119895 are connected ie 119890119894119895 isinE then 119886119894119895 gt 0 and 119904119895 is called a neighbor of node 119904119894119873119894 = 119895 isin I 119895 = 119894 denotes the index set of all neighborsof agent 119894 If nodes 119904119894 and 119904119895 are connected and 119886119894119895 = 119886119895119894then G is an undirected graph otherwise the G is a directedgraph In a directed graph a directed path is a sequence ofedges by (1199041 1199042) (1199042 1199043) where (119904119895 119904119894) isin E The directedgraph has a directed spanning tree if all other nodes havedirectional paths from the same node The Laplacian matrixof the graph G is defined by 119871 = Δ minus A isin R119899times119899 where Δ ≜diag119889119890119892119900119906119905(1199041) 119889119890119892119900119906119905(1199042) 119889119890119892119900119906119905(119904119894) 119889119890119892119900119906119905(119904119899) is adiagonal matrix with 119889119890119892119900119906119905(119904119894) = sum119899119895=1 119886119894119895 Supposing somegraphs G1G2 G119872 and graph G consist of the samenodes and the edge set of graphG is the sum of the edge setsof other graphs G1G2 G119872 then there is 119871 = sum119872119898=1 119871119898which means the Laplacian matrix of graph G is the sum ofother graphsrsquo Laplacian matrix

4 Problem Statement

There are two lemmas [41] for the later analysis

Lemma 1 If graph G is an undirected and connected graphthen its Laplacian matrix 119871 has one singleton zero eigenvalueand other eigenvalues are all positive ie 0 = 1205821 lt 1205822 le 1205823 lesdot sdot sdot le 120582119899Lemma 2 If graphG is a directed graph with a spanning treethen its Laplacian matrix 119871 has one singleton zero eigenvalue

and other eigenvalues have a positive real part ie 1205821 =0Re(120582119894) gt 0 (119894 = 2 3 119899)Consider a MIFOMAS composed of 119899 agents Each node

in graph G corresponds to each agent of the MIFOMASIf 119886119894119895 gt 0 we can think that the 119894-th agent can get stateinformation from the 119895-th agent The dynamics of the 119894-thagent of the MIFOMAS are represented by

119909(1205721)1198941 (119905) = 1199091198942 (119905) 119909(1205722)1198942 (119905) = 1199091198943 (119905)

119909(120572119897minus1)119894(119897minus1) (119905) = 119909119894119897 (119905) 119909(120572119897)119894119897 (119905) = 119906119894 (119905)

119894 isin I

(4)

where 1199091198941(119905) 1199091198942(119905) 119909119894119897(119905) isin R respectively represent119897 different states of the 119894-th agent 119909(1205721)1198941 (119905) is the 1205721-orderCaputo derivative of 1199091198941(119905) 119909(1205722)1198942 (119905) is the 1205722-order Caputoderivative of 1199091198942(119905) 119909(120572119897)119894119897 (119905) is the 120572119897-order Caputo deriva-tive of 119909119894119897(119905) (1205721 1205722 120572119897 isin (0 1]) and 119906119894(119905) isin R is thecontrol input

Definition 3 If and only if the states of all agents in MIFO-MAS (4) satisfy

lim119905997888rarr+infin

(1199091198941 (119905) minus 1199091198951 (119905)) = 0lim119905997888rarr+infin

(1199091198942 (119905) minus 1199091198952 (119905)) = 0

lim119905997888rarr+infin

(119909119894119897 (119905) minus 119909119895119897 (119905)) = 0119894 119895 isin I

(5)

then MIFOMAS (4) can reach consensus In (5)1199091198941(119905) 1199091198942(119905) 119909119894119897(119905) isin R respectively represent 119897 differentstates of the 119894-th agent and 1199091198951(119905) 1199091198952(119905) 119909119895119897(119905) isin Rrespectively represent 119897 different states of the 119895-th agent

In this paper the control protocol for MIFOMAS (4) willbe given by

119906119894 (119905) = minus 119897minus1sum119896=1

119875119896+1119909119894119896+1 (119905)+ sum119895isin119873119894

119886119894119895 [1199091198951 (119905 minus 120591119894119895) minus 1199091198941 (119905 minus 120591119894119895)] (6)

4 Complexity

where119873119894 denotes the neighbors index collection of the agent119894 119886119894119895 is the (119894 119895)-th element of A 120591119894119895 gt 0 is the time-delay which is from the 119895-th agent to the 119894-th agent and119875119896+1 gt 0 are scale coefficients If all 120591119894119895 = 120591119895119894 then the time-delays are symmetrical else the time-delays are asymmet-rical The symmetric time-delays and the asymmetric time-delays are two different forms of nonuniform time-delaysFor ease of analysis we define that 120591119898 denote 119872 differenttime-delays of MIFOMAS (4) ie 120591119898 isin 120591119894119895 119894 119895 isinI (119898 = 1 2 119872) Then the following control protocolis provided to resolve consensus problems of MIFOMAS(4)

119906119894 (119905) = minus 119897minus1sum119896=1

119875119896+1119909119894119896+1 (119905)+ sum119895isin119873119894

119886119894119895 [1199091198951 (119905 minus 120591119898) minus 1199091198941 (119905 minus 120591119898)] (7)

Assume the state vector of the 119894-th agent is 120585119894(119905) =[1199091198941(119905) 1199091198942(119905) 119909119894119897(119905)]119879 isin R119897 and the joint state vec-tor of MIFOMAS (4) consisting of 119899 agents is 120595(119905) =[1205851198791 (119905) 1205851198792 (119905) 120585119879119899 (119905)]119879 isin R119897119899

If we define two matrices as follows

119860 =[[[[[[[[[[[[[

0 1 0 sdot sdot sdot 0 00 0 1 sdot sdot sdot 0 0 d

0 0 0 sdot sdot sdot 1 00 0 0 sdot sdot sdot 0 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897minus1 minus119875119897

]]]]]]]]]]]]]

isin R119897times119897 (8)

and

119861 =[[[[[[[

0 0 sdot sdot sdot 0 d

0 0 sdot sdot sdot 01 0 sdot sdot sdot 0

]]]]]]]

isin R119897times119897 (9)

then under the control protocol given by (7) the closed-loopdynamics of MIFOMAS (4) can be described as

[119909(1205721)11 (119905) 119909(1205722)12 (119905) 119909(120572119897)1119897 (119905) 119909(1205721)21 (119905) 119909(1205722)22 (119905) 119909(120572119897)2119897 (119905) 119909(1205721)1198991 (119905) 119909(1205722)1198992 (119905) 119909(120572119897)119899119897 (119905)]119879 = (119868119899otimes 119860)120595 (119905) minus 119872sum

119898=1

(119871119898 otimes 119861) sdot 120595 (119905 minus 120591119898) (10)

5 Main Results

Theorem 4 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 1 Define the following functions

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(11)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectivelydenote the real part and the imaginary part of119865119897(120596)

If all 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) for the MIFOMAS(10) with symmetric time-delays then the control protocol(7) can resolve the consensus problem of the MIFOMAS (10)with symmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem ofthe MIFOMAS (10) with symmetric time-delays e value of120596 corresponding to 119897 in 119879119897(120596) is determined by the followingequation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (12)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Proof We shall apply the frequency-domain method toanalyze the MIFOMAS (10) with symmetric time-delays andwe can get

119868119899 otimes[[[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]]]

Ψ (119904)

minus

119868119899 otimes[[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]]

120595(0minus)

Complexity 5

minus (119868119899 otimes 119860)Ψ (119904) + 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898Ψ (119904) = 0Ψ (119904) = R120595 (0minus)119866minus1120591119898 (119904)

(13)

whereΨ(119904) is the Laplace transformof120595(119905)120595(0minus) is the initialvalue of 120595(119905)

R = 119868119899 otimes[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]

(14)

and

119866120591119898 (119904) =

119868119899 otimes

[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]

minus[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

(15)

Motivated by the stability analysis of a fractional-ordersystem in [42] we can study consensus of the MIFOMAS(10) with symmetric time-delays by analyzing the character-istic eigenvaluesrsquo position of the characteristic polynomialdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays Specifically consensus of the MIFOMAS (10) withoutdelays (all 120591119898 = 0) is necessary for consensus of theMIFOMAS (10) with symmetric time-delays that is to say inthis case the characteristic eigenvalues of det[119866120591119898(119904)] of the

MIFOMAS (10) with symmetric time-delays are all situatedin the left half plane (LHP) of the complex plane and as 120591119898increases continuously from zero the characteristic eigen-value of det[119866120591119898(119904)] of the MIFOMAS (10) with symmetrictime-delays will change continuously from the LHP to theright half plane (RHP) of the complex plane Once thecharacteristic eigenvalue of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays reaches the RHP of thecomplex plane through the imaginary axis the MIFOMAS(10) with symmetric time-delays will be unstable and can notachieve consensus Ergo we only need to consider the criticaltime-delay when the nonzero characteristic eigenvalue ofdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays is just situated on the imaginary axis for the first time as120591119898 increases continuously from zero and the correspondingtime-delay is just the delay margin 120591 of the MIFOMAS (10)with symmetric time-delays

Assume 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 =1 1199061 1199062 119906119897minus1 119906119897 isin C119899 we have the following equa-tions

119868119899

otimes

[[[[[[[[[[[[[

(minus119895120596)1205721 0 sdot sdot sdot 0 00 (minus119895120596)1205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot (minus119895120596)120572119897minus1 00 0 sdot sdot sdot 0 (minus119895120596)120572119897

]]]]]]]]]]]]]

minus[[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

6 Complexity

119868119899

otimes

[[[[[[[[[[[

(minus119895120596)1205721 minus1 0 sdot sdot sdot 00 (minus119895120596)1205722 minus1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot minus10 1198752 1198753 sdot sdot sdot 119875119897 + (minus119895120596)120572119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

119868119899 otimes Ω 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(1)

+ 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2)

= 0(16)

(1) = 119868119899 otimes Ω 119906 = 1198681198991199061 otimes (Ω [1 0 0]119879) + 1198681198991199062otimes (Ω [0 1 0 0]119879) + 1198681198991199063otimes (Ω [0 0 1 0 0]119879) + sdot sdot sdot + 119868119899119906119897minus1otimes (Ω [0 0 sdot sdot sdot 0 1 0]119879) + 119868119899119906119897otimes (Ω [0 0 1]119879) = 1198681198991199061 otimes [(minus119895120596)1205721 0 0]119879+ 1198681198991199062 otimes [minus1 (minus119895120596)1205722 0 0 1198752]119879 + 1198681198991199063 otimes [0minus 1 (minus119895120596)1205723 0 0 1198753]119879 + sdot sdot sdot + 119868119899119906119897minus1 otimes [0 0 minus1 (minus119895120596)120572119897minus1 119875119897minus1]119879 + 119868119899119906119897 otimes [0 0 minus1(minus119895120596)120572119897 + 119875119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061 0 0]119879 + 119868119899otimes [minus1199062 (minus119895120596)1205722 1199062 0 0 11987521199062]119879 + 119868119899 otimes [0minus 1199063 (minus119895120596)1205723 1199063 0 0 11987531199063]119879 + sdot sdot sdot + 119868119899 otimes [0 0 minus119906119897minus1 (minus119895120596)120572119897minus1 119906119897minus1 119875119897minus1119906119897minus1]119879 + 119868119899 otimes [0

0 minus119906119897 ((minus119895120596)120572119897 + 119875119897) 119906119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061minus 1199062 (minus119895120596)1205722 1199062 minus 1199063 (minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897 11987521199062+ 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897]119879 = [119868119899 ((minus119895120596)1205721 1199061 minus 1199062) 119868119899 ((minus119895120596)1205722 1199062 minus 1199063) 119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) 119868119899 (11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897)]119879

(17)

(2) = 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906 = 119872sum119898=1

(119871119898 otimes 119861) 119906119890119895120596120591119898

= 119872sum119898=1

(1198711198981199061) otimes (119861 [1 0 0]119879) + (1198711198981199062)otimes (119861 [0 1 0 0]119879) + sdot sdot sdot + (119871119898119906119897minus1)otimes (119861 [0 0 0 1 0]119879) + (119871119898119906119897)otimes (119861 [0 0 1]119879) 119890119895120596120591119898 = 119872sum

119898=1

(1198711198981199061)otimes ([0 0 0 1]119879) + (1198711198981199062) otimes ([0 0 0 0]119879)+ sdot sdot sdot + (119871119898119906119897minus1) otimes ([0 0 0 0 0]119879) + (119871119898119906119897)otimes ([0 0 0]119879) 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897) [(1199061otimes [0 0 0 1]119879) + (1199062 otimes [0 0 0 0]119879) + sdot sdot sdot+ (119906119897minus1 otimes [0 0 0 0]119879) + (119906119897otimes [0 0 0 0]119879)] 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897)

sdot [0 0 0 1199061]119879 119890119895120596120591119898 = [0 0 0119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898]119879

(18)

Because of (1) + (2) = 0 we have119868119899 ((minus119895120596)1205721 1199061 minus 1199062) = 0119868119899 ((minus119895120596)1205722 1199062 minus 1199063) = 0

Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

4 Complexity

where119873119894 denotes the neighbors index collection of the agent119894 119886119894119895 is the (119894 119895)-th element of A 120591119894119895 gt 0 is the time-delay which is from the 119895-th agent to the 119894-th agent and119875119896+1 gt 0 are scale coefficients If all 120591119894119895 = 120591119895119894 then the time-delays are symmetrical else the time-delays are asymmet-rical The symmetric time-delays and the asymmetric time-delays are two different forms of nonuniform time-delaysFor ease of analysis we define that 120591119898 denote 119872 differenttime-delays of MIFOMAS (4) ie 120591119898 isin 120591119894119895 119894 119895 isinI (119898 = 1 2 119872) Then the following control protocolis provided to resolve consensus problems of MIFOMAS(4)

119906119894 (119905) = minus 119897minus1sum119896=1

119875119896+1119909119894119896+1 (119905)+ sum119895isin119873119894

119886119894119895 [1199091198951 (119905 minus 120591119898) minus 1199091198941 (119905 minus 120591119898)] (7)

Assume the state vector of the 119894-th agent is 120585119894(119905) =[1199091198941(119905) 1199091198942(119905) 119909119894119897(119905)]119879 isin R119897 and the joint state vec-tor of MIFOMAS (4) consisting of 119899 agents is 120595(119905) =[1205851198791 (119905) 1205851198792 (119905) 120585119879119899 (119905)]119879 isin R119897119899

If we define two matrices as follows

119860 =[[[[[[[[[[[[[

0 1 0 sdot sdot sdot 0 00 0 1 sdot sdot sdot 0 0 d

0 0 0 sdot sdot sdot 1 00 0 0 sdot sdot sdot 0 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897minus1 minus119875119897

]]]]]]]]]]]]]

isin R119897times119897 (8)

and

119861 =[[[[[[[

0 0 sdot sdot sdot 0 d

0 0 sdot sdot sdot 01 0 sdot sdot sdot 0

]]]]]]]

isin R119897times119897 (9)

then under the control protocol given by (7) the closed-loopdynamics of MIFOMAS (4) can be described as

[119909(1205721)11 (119905) 119909(1205722)12 (119905) 119909(120572119897)1119897 (119905) 119909(1205721)21 (119905) 119909(1205722)22 (119905) 119909(120572119897)2119897 (119905) 119909(1205721)1198991 (119905) 119909(1205722)1198992 (119905) 119909(120572119897)119899119897 (119905)]119879 = (119868119899otimes 119860)120595 (119905) minus 119872sum

119898=1

(119871119898 otimes 119861) sdot 120595 (119905 minus 120591119898) (10)

5 Main Results

Theorem 4 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 1 Define the following functions

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(11)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectivelydenote the real part and the imaginary part of119865119897(120596)

If all 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) for the MIFOMAS(10) with symmetric time-delays then the control protocol(7) can resolve the consensus problem of the MIFOMAS (10)with symmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem ofthe MIFOMAS (10) with symmetric time-delays e value of120596 corresponding to 119897 in 119879119897(120596) is determined by the followingequation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (12)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Proof We shall apply the frequency-domain method toanalyze the MIFOMAS (10) with symmetric time-delays andwe can get

119868119899 otimes[[[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]]]

Ψ (119904)

minus

119868119899 otimes[[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]]

120595(0minus)

Complexity 5

minus (119868119899 otimes 119860)Ψ (119904) + 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898Ψ (119904) = 0Ψ (119904) = R120595 (0minus)119866minus1120591119898 (119904)

(13)

whereΨ(119904) is the Laplace transformof120595(119905)120595(0minus) is the initialvalue of 120595(119905)

R = 119868119899 otimes[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]

(14)

and

119866120591119898 (119904) =

119868119899 otimes

[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]

minus[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

(15)

Motivated by the stability analysis of a fractional-ordersystem in [42] we can study consensus of the MIFOMAS(10) with symmetric time-delays by analyzing the character-istic eigenvaluesrsquo position of the characteristic polynomialdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays Specifically consensus of the MIFOMAS (10) withoutdelays (all 120591119898 = 0) is necessary for consensus of theMIFOMAS (10) with symmetric time-delays that is to say inthis case the characteristic eigenvalues of det[119866120591119898(119904)] of the

MIFOMAS (10) with symmetric time-delays are all situatedin the left half plane (LHP) of the complex plane and as 120591119898increases continuously from zero the characteristic eigen-value of det[119866120591119898(119904)] of the MIFOMAS (10) with symmetrictime-delays will change continuously from the LHP to theright half plane (RHP) of the complex plane Once thecharacteristic eigenvalue of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays reaches the RHP of thecomplex plane through the imaginary axis the MIFOMAS(10) with symmetric time-delays will be unstable and can notachieve consensus Ergo we only need to consider the criticaltime-delay when the nonzero characteristic eigenvalue ofdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays is just situated on the imaginary axis for the first time as120591119898 increases continuously from zero and the correspondingtime-delay is just the delay margin 120591 of the MIFOMAS (10)with symmetric time-delays

Assume 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 =1 1199061 1199062 119906119897minus1 119906119897 isin C119899 we have the following equa-tions

119868119899

otimes

[[[[[[[[[[[[[

(minus119895120596)1205721 0 sdot sdot sdot 0 00 (minus119895120596)1205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot (minus119895120596)120572119897minus1 00 0 sdot sdot sdot 0 (minus119895120596)120572119897

]]]]]]]]]]]]]

minus[[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

6 Complexity

119868119899

otimes

[[[[[[[[[[[

(minus119895120596)1205721 minus1 0 sdot sdot sdot 00 (minus119895120596)1205722 minus1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot minus10 1198752 1198753 sdot sdot sdot 119875119897 + (minus119895120596)120572119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

119868119899 otimes Ω 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(1)

+ 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2)

= 0(16)

(1) = 119868119899 otimes Ω 119906 = 1198681198991199061 otimes (Ω [1 0 0]119879) + 1198681198991199062otimes (Ω [0 1 0 0]119879) + 1198681198991199063otimes (Ω [0 0 1 0 0]119879) + sdot sdot sdot + 119868119899119906119897minus1otimes (Ω [0 0 sdot sdot sdot 0 1 0]119879) + 119868119899119906119897otimes (Ω [0 0 1]119879) = 1198681198991199061 otimes [(minus119895120596)1205721 0 0]119879+ 1198681198991199062 otimes [minus1 (minus119895120596)1205722 0 0 1198752]119879 + 1198681198991199063 otimes [0minus 1 (minus119895120596)1205723 0 0 1198753]119879 + sdot sdot sdot + 119868119899119906119897minus1 otimes [0 0 minus1 (minus119895120596)120572119897minus1 119875119897minus1]119879 + 119868119899119906119897 otimes [0 0 minus1(minus119895120596)120572119897 + 119875119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061 0 0]119879 + 119868119899otimes [minus1199062 (minus119895120596)1205722 1199062 0 0 11987521199062]119879 + 119868119899 otimes [0minus 1199063 (minus119895120596)1205723 1199063 0 0 11987531199063]119879 + sdot sdot sdot + 119868119899 otimes [0 0 minus119906119897minus1 (minus119895120596)120572119897minus1 119906119897minus1 119875119897minus1119906119897minus1]119879 + 119868119899 otimes [0

0 minus119906119897 ((minus119895120596)120572119897 + 119875119897) 119906119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061minus 1199062 (minus119895120596)1205722 1199062 minus 1199063 (minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897 11987521199062+ 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897]119879 = [119868119899 ((minus119895120596)1205721 1199061 minus 1199062) 119868119899 ((minus119895120596)1205722 1199062 minus 1199063) 119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) 119868119899 (11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897)]119879

(17)

(2) = 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906 = 119872sum119898=1

(119871119898 otimes 119861) 119906119890119895120596120591119898

= 119872sum119898=1

(1198711198981199061) otimes (119861 [1 0 0]119879) + (1198711198981199062)otimes (119861 [0 1 0 0]119879) + sdot sdot sdot + (119871119898119906119897minus1)otimes (119861 [0 0 0 1 0]119879) + (119871119898119906119897)otimes (119861 [0 0 1]119879) 119890119895120596120591119898 = 119872sum

119898=1

(1198711198981199061)otimes ([0 0 0 1]119879) + (1198711198981199062) otimes ([0 0 0 0]119879)+ sdot sdot sdot + (119871119898119906119897minus1) otimes ([0 0 0 0 0]119879) + (119871119898119906119897)otimes ([0 0 0]119879) 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897) [(1199061otimes [0 0 0 1]119879) + (1199062 otimes [0 0 0 0]119879) + sdot sdot sdot+ (119906119897minus1 otimes [0 0 0 0]119879) + (119906119897otimes [0 0 0 0]119879)] 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897)

sdot [0 0 0 1199061]119879 119890119895120596120591119898 = [0 0 0119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898]119879

(18)

Because of (1) + (2) = 0 we have119868119899 ((minus119895120596)1205721 1199061 minus 1199062) = 0119868119899 ((minus119895120596)1205722 1199062 minus 1199063) = 0

Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

Complexity 5

minus (119868119899 otimes 119860)Ψ (119904) + 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898Ψ (119904) = 0Ψ (119904) = R120595 (0minus)119866minus1120591119898 (119904)

(13)

whereΨ(119904) is the Laplace transformof120595(119905)120595(0minus) is the initialvalue of 120595(119905)

R = 119868119899 otimes[[[[[[[[[[[

1199041205721minus1 0 sdot sdot sdot 0 00 1199041205722minus1 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1minus1 00 0 sdot sdot sdot 0 119904120572119897minus1

]]]]]]]]]]]

(14)

and

119866120591119898 (119904) =

119868119899 otimes

[[[[[[[[[[

1199041205721 0 sdot sdot sdot 0 00 1199041205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot 119904120572119897minus1 00 0 sdot sdot sdot 0 119904120572119897

]]]]]]]]]]

minus[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

(15)

Motivated by the stability analysis of a fractional-ordersystem in [42] we can study consensus of the MIFOMAS(10) with symmetric time-delays by analyzing the character-istic eigenvaluesrsquo position of the characteristic polynomialdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays Specifically consensus of the MIFOMAS (10) withoutdelays (all 120591119898 = 0) is necessary for consensus of theMIFOMAS (10) with symmetric time-delays that is to say inthis case the characteristic eigenvalues of det[119866120591119898(119904)] of the

MIFOMAS (10) with symmetric time-delays are all situatedin the left half plane (LHP) of the complex plane and as 120591119898increases continuously from zero the characteristic eigen-value of det[119866120591119898(119904)] of the MIFOMAS (10) with symmetrictime-delays will change continuously from the LHP to theright half plane (RHP) of the complex plane Once thecharacteristic eigenvalue of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays reaches the RHP of thecomplex plane through the imaginary axis the MIFOMAS(10) with symmetric time-delays will be unstable and can notachieve consensus Ergo we only need to consider the criticaltime-delay when the nonzero characteristic eigenvalue ofdet[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays is just situated on the imaginary axis for the first time as120591119898 increases continuously from zero and the correspondingtime-delay is just the delay margin 120591 of the MIFOMAS (10)with symmetric time-delays

Assume 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with symmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 =1 1199061 1199062 119906119897minus1 119906119897 isin C119899 we have the following equa-tions

119868119899

otimes

[[[[[[[[[[[[[

(minus119895120596)1205721 0 sdot sdot sdot 0 00 (minus119895120596)1205722 sdot sdot sdot 0 0 d

0 0 sdot sdot sdot (minus119895120596)120572119897minus1 00 0 sdot sdot sdot 0 (minus119895120596)120572119897

]]]]]]]]]]]]]

minus[[[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 10 minus1198752 minus1198753 sdot sdot sdot minus119875119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

6 Complexity

119868119899

otimes

[[[[[[[[[[[

(minus119895120596)1205721 minus1 0 sdot sdot sdot 00 (minus119895120596)1205722 minus1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot minus10 1198752 1198753 sdot sdot sdot 119875119897 + (minus119895120596)120572119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

119868119899 otimes Ω 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(1)

+ 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2)

= 0(16)

(1) = 119868119899 otimes Ω 119906 = 1198681198991199061 otimes (Ω [1 0 0]119879) + 1198681198991199062otimes (Ω [0 1 0 0]119879) + 1198681198991199063otimes (Ω [0 0 1 0 0]119879) + sdot sdot sdot + 119868119899119906119897minus1otimes (Ω [0 0 sdot sdot sdot 0 1 0]119879) + 119868119899119906119897otimes (Ω [0 0 1]119879) = 1198681198991199061 otimes [(minus119895120596)1205721 0 0]119879+ 1198681198991199062 otimes [minus1 (minus119895120596)1205722 0 0 1198752]119879 + 1198681198991199063 otimes [0minus 1 (minus119895120596)1205723 0 0 1198753]119879 + sdot sdot sdot + 119868119899119906119897minus1 otimes [0 0 minus1 (minus119895120596)120572119897minus1 119875119897minus1]119879 + 119868119899119906119897 otimes [0 0 minus1(minus119895120596)120572119897 + 119875119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061 0 0]119879 + 119868119899otimes [minus1199062 (minus119895120596)1205722 1199062 0 0 11987521199062]119879 + 119868119899 otimes [0minus 1199063 (minus119895120596)1205723 1199063 0 0 11987531199063]119879 + sdot sdot sdot + 119868119899 otimes [0 0 minus119906119897minus1 (minus119895120596)120572119897minus1 119906119897minus1 119875119897minus1119906119897minus1]119879 + 119868119899 otimes [0

0 minus119906119897 ((minus119895120596)120572119897 + 119875119897) 119906119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061minus 1199062 (minus119895120596)1205722 1199062 minus 1199063 (minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897 11987521199062+ 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897]119879 = [119868119899 ((minus119895120596)1205721 1199061 minus 1199062) 119868119899 ((minus119895120596)1205722 1199062 minus 1199063) 119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) 119868119899 (11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897)]119879

(17)

(2) = 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906 = 119872sum119898=1

(119871119898 otimes 119861) 119906119890119895120596120591119898

= 119872sum119898=1

(1198711198981199061) otimes (119861 [1 0 0]119879) + (1198711198981199062)otimes (119861 [0 1 0 0]119879) + sdot sdot sdot + (119871119898119906119897minus1)otimes (119861 [0 0 0 1 0]119879) + (119871119898119906119897)otimes (119861 [0 0 1]119879) 119890119895120596120591119898 = 119872sum

119898=1

(1198711198981199061)otimes ([0 0 0 1]119879) + (1198711198981199062) otimes ([0 0 0 0]119879)+ sdot sdot sdot + (119871119898119906119897minus1) otimes ([0 0 0 0 0]119879) + (119871119898119906119897)otimes ([0 0 0]119879) 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897) [(1199061otimes [0 0 0 1]119879) + (1199062 otimes [0 0 0 0]119879) + sdot sdot sdot+ (119906119897minus1 otimes [0 0 0 0]119879) + (119906119897otimes [0 0 0 0]119879)] 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897)

sdot [0 0 0 1199061]119879 119890119895120596120591119898 = [0 0 0119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898]119879

(18)

Because of (1) + (2) = 0 we have119868119899 ((minus119895120596)1205721 1199061 minus 1199062) = 0119868119899 ((minus119895120596)1205722 1199062 minus 1199063) = 0

Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

6 Complexity

119868119899

otimes

[[[[[[[[[[[

(minus119895120596)1205721 minus1 0 sdot sdot sdot 00 (minus119895120596)1205722 minus1 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot minus10 1198752 1198753 sdot sdot sdot 119875119897 + (minus119895120596)120572119897

]]]]]]]]]]]

+ 119872sum119898=1

(119871119898 otimes 119861) 119890minus119904120591119898

119906 = 0

119868119899 otimes Ω 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(1)

+ 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2)

= 0(16)

(1) = 119868119899 otimes Ω 119906 = 1198681198991199061 otimes (Ω [1 0 0]119879) + 1198681198991199062otimes (Ω [0 1 0 0]119879) + 1198681198991199063otimes (Ω [0 0 1 0 0]119879) + sdot sdot sdot + 119868119899119906119897minus1otimes (Ω [0 0 sdot sdot sdot 0 1 0]119879) + 119868119899119906119897otimes (Ω [0 0 1]119879) = 1198681198991199061 otimes [(minus119895120596)1205721 0 0]119879+ 1198681198991199062 otimes [minus1 (minus119895120596)1205722 0 0 1198752]119879 + 1198681198991199063 otimes [0minus 1 (minus119895120596)1205723 0 0 1198753]119879 + sdot sdot sdot + 119868119899119906119897minus1 otimes [0 0 minus1 (minus119895120596)120572119897minus1 119875119897minus1]119879 + 119868119899119906119897 otimes [0 0 minus1(minus119895120596)120572119897 + 119875119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061 0 0]119879 + 119868119899otimes [minus1199062 (minus119895120596)1205722 1199062 0 0 11987521199062]119879 + 119868119899 otimes [0minus 1199063 (minus119895120596)1205723 1199063 0 0 11987531199063]119879 + sdot sdot sdot + 119868119899 otimes [0 0 minus119906119897minus1 (minus119895120596)120572119897minus1 119906119897minus1 119875119897minus1119906119897minus1]119879 + 119868119899 otimes [0

0 minus119906119897 ((minus119895120596)120572119897 + 119875119897) 119906119897]119879 = 119868119899 otimes [(minus119895120596)1205721 1199061minus 1199062 (minus119895120596)1205722 1199062 minus 1199063 (minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897 11987521199062+ 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897]119879 = [119868119899 ((minus119895120596)1205721 1199061 minus 1199062) 119868119899 ((minus119895120596)1205722 1199062 minus 1199063) 119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) 119868119899 (11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + 119875119897119906119897 + (minus119895120596)120572119897 119906119897)]119879

(17)

(2) = 119872sum119898=1

(119871119898 otimes 119861) 119890119895120596120591119898119906 = 119872sum119898=1

(119871119898 otimes 119861) 119906119890119895120596120591119898

= 119872sum119898=1

(1198711198981199061) otimes (119861 [1 0 0]119879) + (1198711198981199062)otimes (119861 [0 1 0 0]119879) + sdot sdot sdot + (119871119898119906119897minus1)otimes (119861 [0 0 0 1 0]119879) + (119871119898119906119897)otimes (119861 [0 0 1]119879) 119890119895120596120591119898 = 119872sum

119898=1

(1198711198981199061)otimes ([0 0 0 1]119879) + (1198711198981199062) otimes ([0 0 0 0]119879)+ sdot sdot sdot + (119871119898119906119897minus1) otimes ([0 0 0 0 0]119879) + (119871119898119906119897)otimes ([0 0 0]119879) 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897) [(1199061otimes [0 0 0 1]119879) + (1199062 otimes [0 0 0 0]119879) + sdot sdot sdot+ (119906119897minus1 otimes [0 0 0 0]119879) + (119906119897otimes [0 0 0 0]119879)] 119890119895120596120591119898 = 119872sum

119898=1

(119871119898 otimes 119868119897)

sdot [0 0 0 1199061]119879 119890119895120596120591119898 = [0 0 0119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898]119879

(18)

Because of (1) + (2) = 0 we have119868119899 ((minus119895120596)1205721 1199061 minus 1199062) = 0119868119899 ((minus119895120596)1205722 1199062 minus 1199063) = 0

Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Complexity 7

119868119899 ((minus119895120596)120572119897minus1 119906119897minus1 minus 119906119897) = 0119868119899 [11987521199062 + 11987531199063 + sdot sdot sdot + 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

+ 119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = 0(19)

and it yields that

1199062 = (minus119895120596)1205721 11990611199063 = (minus119895120596)1205722 1199062

119906119897 = (minus119895120596)120572119897minus1 119906119897minus1119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [11987521199062 + 11987531199063 + sdot sdot sdot+ 119875119897minus1119906119897minus1 + (119875119897 + (minus119895120596)120572119897) 119906119897]

(20)

According to (20) we have

119872sum119898=1

(119871119898 otimes 119868119897) 1199061119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721] 1199061

(21)

then we can multiply both sides of (21) by 119906119867 (the conjugatetranspose of 119906) the following equation can be obtained

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = minus [1198752 (minus119895120596)1205721

+ 1198753 (minus119895120596)1205722+1205721 + sdot sdot sdot + 119875119897minus1 (minus119895120596)120572119897minus2+120572119897minus3+sdotsdotsdot+1205721+ 119875119897 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 + (minus119895120596)120572119897+120572119897minus1+sdotsdotsdot+1205721]= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

(22)

where 119897 ⩾ 1 119875119897+1 = 1Due to

119906 = [1199061 1199062 1199063 119906119897]119879 = [1199061 (minus119895120596)1205721 1199061 (minus119895120596)1205721+1205722

sdot 1199061 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721 1199061]119879 1199061 = 119906

[1 (minus119895120596)1205721 (minus119895120596)1205721+1205722 (minus119895120596)120572119897minus1+120572119897minus2+sdotsdotsdot+1205721]119879 (23)

(22) can be simplified to

119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 11990611199061198671199061 119890119895120596120591119898 = 119872sum119898=1

119906119867 (119871119898 otimes 119868119897) 119906119906119867119906 119890119895120596120591119898

= minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 ≜ 119865119897 (120596) (24)

where 119897 ⩾ 1 119875119897+1 = 1Let 119906119867(119871119898 otimes 119868119897)119906(119906119867119906) = 119886119898 we take modulus of both

sides of (24) According to Lemma 1 we can get the followinginequality

119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981198901198951205961205911198981003816100381610038161003816100381610038161003816100381610038161003816 le

1003816100381610038161003816100381610038161003816100381610038161003816119872sum119898=1

1198861198981003816100381610038161003816100381610038161003816100381610038161003816

= 119906119867 (119871 otimes 119868119897) 119906119906119867119906 le 119872119897 (120596) = 1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899(25)

It is obvious that 119872119897(120596) is an increasing function for 120596 gt 0and if 120596 le 120596 we can get 119872119897(120596) le 119872119897(120596) = |119865119897(120596)| = 120582119899 thatis inequality (25) is true

According to (24) we can get

arg [119865119897 (120596)] = arg( 119872sum119898=1

119886119898119890119895120596120591119898) ≜ 120579119897 (120596) = arctan119877119897 (120596)

(26)

where 120579119897(120596) is the principal value of the argument of 119865119897(120596)119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] = tan[120579119897(120596)] and Re[119865119897(120596)]and Im[119865119897(120596)] respectively denote the real part and theimaginary part of 119865119897(120596)

According to (26) it is easy to obtain that

120579119897 (120596) = arg( 119872sum119898=1

119886119898119890119895120596120591119898) le max 120596120591119898 (27)

Consider a FOMAS with single integral we have 1198651(120596) =minus(minus119895120596)12057211198752 = 1205961205721119890119895(120587(2minus1205721)2)1198752 It is apparent that 1205791(120596) =120587(2 minus 1205721)2 and 1198721(120596) = 12059612057211198752(1198752 = 1) According to(25) we should only consider 120596 le 120596 = 120582(11205721)119899 and if all120591119898 lt 120591 = 1198791(120596) = 1198791[120582(11205721)119899 ] = (1120596)1205791(120596) = 120587(2 minus1205721)[2120582(11205721)119899 ] then 120596120591119898 lt 120582(11205721)119899 120591 = 120582(11205721)119899 1198791[120582(11205721)119899 ] =120587(2 minus 1205721)2 = 1205791(120596) which contradicts to (27) Thereforewhen all 120591119898 lt 120591 the characteristic eigenvalues of det[119866120591119898(119904)]

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

8 Complexity

of the MIFOMAS (10) with symmetric time-delays are allsituated in the LHP and the FOMAS with single integral willremain stable and can achieve consensus On the contrarythe FOMAS with single integral will not remain stableand can not achieve consensus Theorem 4 is proven for119897 = 1

In the following the FOMAS with multiple integral(double integral to sextuple integral) shall be analyzed stepby step For convenience of analysis we first need to definesome symbolic parameters

1199111 = 1205962120572111987522 1199112 = 1205962(1205721+1205722)11987523 1199113 = 2120596(21205721+1205722)cos12058712057222 119875211987531199114 = 2120596(21205721+1205722+1205723)cos120587 (1205722 + 1205723)2 119875211987541199115 = 2120596(21205721+1205722+1205723+1205724)cos120587 (1205722 + 1205723 + 1205724)2 119875211987551199116 = 2120596(21205721+1205722+1205723+1205724+1205725)cos120587 (1205722 + 1205723 + 1205724 + 1205725)2

sdot 119875211987561199117 = 2120596(21205721+1205722+1205723+1205724+1205725+1205726)

sdot cos 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 119875211987571199118 = 2120596(21205721+21205722+1205723)cos12058712057232 119875311987541199119 = 2120596(21205721+21205722+1205723+1205724)cos120587 (1205723 + 1205724)2 1198753119875511991110 = 2120596(21205721+21205722+1205723+1205724+1205725)cos120587 (1205723 + 1205724 + 1205725)2 1198753119875611991111 = 2120596(21205721+21205722+1205723+1205724+1205725+1205726)cos120587 (1205723 + 1205724 + 1205725 + 1205726)2

sdot 1198753119875711991112 = 1205962(1205721+1205722+1205723)11987524 11991113 = 2120596(21205721+21205722+21205723+1205724)cos12058712057242 1198754119875511991114 = 2120596(21205721+21205722+21205723+1205724+1205725)cos120587 (1205724 + 1205725)2 1198754119875611991115 = 2120596(21205721+21205722+21205723+1205724+1205725+1205726)cos120587 (1205724 + 1205725 + 1205726)2

sdot 11987541198757

11991116 = 1205962(1205721+1205722+1205723+1205724)11987525 11991117 = 2120596(21205721+21205722+21205723+21205724+1205725)cos12058712057252 1198755119875611991118 = 2120596(21205721+21205722+21205723+21205724+1205725+1205726)cos120587 (1205725 + 1205726)2 1198755119875711991119 = 1205962(1205721+1205722+1205723+1205724+1205725)11987526 11991120 = 2120596(21205721+21205722+21205723+21205724+21205725+1205726)cos12058712057262 1198756119875711991121 = 1205962(1205721+1205722+1205723+1205724+1205725+1205726)11987527

(28)

and

1198891 = 1205722120596(21205721+1205722minus1)11987521198753 sin 12058712057222 1198892 = (1205722 + 1205723) 120596(21205721+1205722+1205723minus1)11987521198754 sin 120587 (1205722 + 1205723)2 1198893 = (1205722 + 1205723 + 1205724) 120596(21205721+1205722+1205723+1205724minus1)11987521198755

sdot sin 120587 (1205722 + 1205723 + 1205724)2 1198894 = (1205722 + 1205723 + 1205724 + 1205725) 120596(21205721+1205722+1205723+1205724+1205725minus1) sdot 11987521198756

sdot sin 120587 (1205722 + 1205723 + 1205724 + 1205725)2 1198895 = (1205722 + 1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+1205722+1205723+1205724+1205725+1205726minus1)

sdot 11987521198757 sin 120587 (1205722 + 1205723 + 1205724 + 1205725 + 1205726)2 1198896 = 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin 12058712057232 1198897 = (1205723 + 1205724) 120596(21205721+21205722+1205723+1205724minus1)11987531198755 sin 120587 (1205723 + 1205724)2 1198898 = (1205723 + 1205724 + 1205725) 120596(21205721+21205722+1205723+1205724+1205725minus1) sdot 11987531198756

sdot sin 120587 (1205723 + 1205724 + 1205725)2 1198899 = (1205723 + 1205724 + 1205725 + 1205726) 120596(21205721+21205722+1205723+1205724+1205725+1205726minus1)

sdot 11987531198757 sin 120587 (1205723 + 1205724 + 1205725 + 1205726)2 11988910 = 1205724120596(21205721+21205722+21205723+1205724minus1)11987541198755 sin 12058712057242 11988911 = (1205724 + 1205725) 120596(21205721+21205722+21205723+1205724+1205725minus1)11987541198756

Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Complexity 9

sdot sin 120587 (1205724 + 1205725)2 11988912 = (1205724 + 1205725 + 1205726) 120596(21205721+21205722+21205723+1205724+1205725+1205726minus1) sdot 11987541198757

sdot sin 120587 (1205724 + 1205725 + 1205726)2 11988913 = 1205725120596(21205721+21205722+21205723+21205724+1205725minus1)11987551198756 sin 12058712057252 11988914 = (1205725 + 1205726) 120596(21205721+21205722+21205723+21205724+1205725+1205726minus1)11987551198757

sdot sin 120587 (1205725 + 1205726)2 11988915 = 1205726120596(21205721+21205722+21205723+21205724+21205725+1205726minus1)11987561198757 sin 12058712057262

(29)

For the FOMAS with double integral

1198652 (120596) = minus 2sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752+ (minus119895120596)1205721+1205722 1198753] = 1205961205721119890119895(120587(2minus1205721)2)1198752

+ 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753 = [12059612057211198752 cos 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2 ] = Re [1198652 (120596)]+ 119895 Im [1198652 (120596)]

(30)

[1198722 (120596)]2 = 10038161003816100381610038161198652 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722)cos12058712057222 11987521198753 = 1199111 + 1199112 + 1199113

(31)

Because of 1198772(120596) = Im[1198652(120596)]Re[1198652(120596)] we can get the firstderivative of 1198772(120596)

11987710158402 (120596) = minus [120572212059621205721+1205722minus111987521198753sin (12058712057222)][12059612057211198752cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753cos (120587 (2 minus 1205721 minus 1205722) 2)]2 = minus 1198891Re [1198652 (120596)]2 lt 0 (32)

For the FOMAS with triple integral

1198653 (120596) = minus 3sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 = minus [(minus119895120596)1205721 1198752

+ (minus119895120596)1205721+1205722 1198753 + (minus119895120596)1205721+1205722+1205723 1198754]= 1205961205721119890119895(120587(2minus1205721)2)1198752 + 1205961205721+1205722119890119895(120587(2minus1205721minus1205722)2)1198753+ 120596(1205721+1205722+1205723)119890119895(120587(2minus1205721minus1205722minus1205723)2)1198754= [12059612057211198752 cos 120587 (2 minus 1205721)2

+ 120596(1205721+1205722)1198753 cos 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 cos 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

+ 119895 [12059612057211198752 sin 120587 (2 minus 1205721)2+ 120596(1205721+1205722)1198753 sin 120587 (2 minus 1205721 minus 1205722)2+ 120596(1205721+1205722+1205723)1198754 sin 120587 (2 minus 1205721 minus 1205722 minus 1205723)2 ]

= Re [1198653 (120596)] + 119895 Im [1198653 (120596)] (33)

[1198723 (120596)]2 = 10038161003816100381610038161198653 (120596)10038161003816100381610038162 = 1205962120572111987522 + 1205962(1205721+1205722)11987523+ 2120596(21205721+1205722) cos 12058712057222 11987521198753 + 2120596(21205721+1205722+1205723)

sdot cos 120587 (1205722 + 1205723)2 11987521198754 + 2120596(21205721+21205722+1205723) cos 12058712057232 11987531198754+ 1205962(1205721+1205722+1205723)11987524 = 1199111 + 1199112 + 1199113 + 1199114 + 1199118 + 11991112

(34)

10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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10 Complexity

Because of 1198773(120596) = Im[1198653(120596)]Re[1198653(120596)] we can get thefirst derivative of 1198773(120596)

11987710158403 (120596)

= minus[1205722120596(21205721+1205722minus1)11987521198753 sin (12058712057222) + (1205722 + 1205723) 120596(21205721+1205722+1205723minus1) sdot 11987521198754 sin (120587 (1205722 + 1205723) 2) + 1205723120596(21205721+21205722+1205723minus1)11987531198754 sin (12058712057232)][12059612057211198752 sdot cos (120587 (2 minus 1205721) 2) + 120596(1205721+1205722)1198753 cos (120587 (2 minus 1205721 minus 1205722) 2) + 120596(1205721+1205722+1205723)1198754 sdot cos (120587 (2 minus 1205721 minus 1205722 minus 1205723) 2)]2

= minus(1198891 + 1198892 + 1198896)Re [1198653 (120596)]2 lt 0

(35)

In a similar way the [119872119897(120596)]2 and 1198771015840119897 (120596) of theFOMASs with quadruple integral to sextuple integral can

be respectively calculated under the appropriate parametersand they are as follows

[1198724 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199118 + 1199119 + 11991112 + 11991113 + 11991116

11987710158404 (120596) = minus(1198891 + 1198892 + 1198893 + 1198896 + 1198897 + 11988910)Re [1198654 (120596)]2 lt 0(36)

[1198725 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199118 + 1199119 + 11991110 + 11991112 + 11991113 + 11991114 + 11991116 + 11991117 + 11991119

11987710158405 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198896 + 1198897 + 1198898 + 11988910 + 11988911 + 11988913)Re [1198655 (120596)]2 lt 0(37)

[1198726 (120596)]2 = 1199111 + 1199112 + 1199113 + 1199114 + 1199115 + 1199116 + 1199117 + 1199118 + 1199119 + 11991110 + 11991111 + 11991112 + 11991113 + 11991114 + 11991115 + 11991116 + 11991117 + 11991118 + 11991119 + 11991120+ 11991121

11987710158406 (120596) = minus(1198891 + 1198892 + 1198893 + 1198894 + 1198895 + 1198896 + 1198897 + 1198898 + 1198899 + 11988910 + 11988911 + 11988912 + 11988913 + 11988914 + 11988915)Re [1198656 (120596)]2 lt 0(38)

In summary we have found that the first derivatives of119877119897(120596) (2 le 119897 le 6) listed above are negative values and1198771015840119897 (120596) lt 0 means that 119877119897(120596) = tan[120579119897(120596)] are monotonicallydecreasing with the growth of 120596 Then it can be deducedthat the arguments 120579119897(120596) also decrease monotonically andcontinuously about 120596 because the values of 119865119897(120596) varysmoothly Evidently we can analyze the features of 120579119897(120596) (2 le119897 le 6) together

If 0 lt 1205961 lt 1205962 we have 12059621205961 gt 1 and because thearguments 120579119897(120596) decrease monotonically and continuouslyabout120596 we have 120579119897(1205961) gt 120579119897(1205962) ie 120579119897(1205961)120579119897(1205962) gt 1Thus

119879119897 (1205961)119879119897 (1205962) = 120579119897 (1205961)1205961 sdot 1205962120579119897 (1205962) = 120579119897 (1205961)120579119897 (1205962) sdot 12059621205961 gt 1 (39)

so we can get 119879119897(1205961) gt 119879119897(1205962) which means 119879119897(120596) alsodecreasemonotonically and continuously about120596When120596 le120596 we have

119879119897 (120596) ge 119879119897 (120596) = 120591 (40)

It is worth noting that inequality (40) can be obtainedwhen the characteristic eigenvalue of det[119866120591119898(119904)] of theMIFOMAS (10) is 119904 = minus119895120596 = 0 If we let all 120591119898 lt 120591 thenwe can obtain the following inequality

119879119897 (120596) = 120579119897 (120596)120596 = arg (sum119872119898=1 119886119898119890119895120596120591119898)120596 le max 120596120591119898120596lt 120596120591120596 = 120591

(41)

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

Complexity 11

Inequality (41) is in contradiction with inequality (40)Therefore as long as all 120591119898 lt 120591 we can ensure all thecharacteristic eigenvalues of det[119866120591119898(119904)] of the MIFOMAS(10) with symmetric time-delays are situated in the LHPand the MIFOMAS (10) with symmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with symmetric time-delays will not bestable and can not achieve consensus Theorem 4 is provenfor 119897 isin 2 3 4 5 6Remark 5 Consensus of the MIFOMAS (10) without sym-metric time-delays is necessary for consensus of this systemwith symmetric time-delays

Remark 6 Although 119897 isin 1 2 3 4 5 6 in Theorem 4 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 7 If we suppose that a FOMAS with multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 1 and 1205721 = 1205722 = sdot sdot sdot = 120572119897 = 1then the MIFOMAS (10) with symmetric time-delays can betransformed into high-order MAS with symmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119865119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

120579119897 (120596) = arg [119865119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]= arctan119877119897 (120596)

119879119897 (120596) = 1120596120579119897 (120596)

(42)

where 119897 isin 1 2 3 4 5 6 119877119897(120596) ≜ Im[119865119897(120596)]Re[119865119897(120596)] =tan[120579119897(120596)] and Re[119865119897(120596)] and Im[119865119897(120596)] respectively denotethe real part and the imaginary part of 119865119897(120596)

For the high-order MAS with symmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = 119879119897(120596) = (1120596)120579119897(120596) then the controlprotocol (7) can resolve the consensus problem for the high-order MAS with symmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem for the high-order MAS with symmetric time-delayse value of 120596 corresponding to 119897 in 119879119897(120596) is determined by thefollowing equation

1003816100381610038161003816119865119897 (120596)1003816100381610038161003816 = 120582119899 119897 isin 1 2 3 4 5 6 (43)

where |119865119897(120596)| is the modulus of 119865119897(120596) and 120582119899 is the maximumeigenvalue of 119871Remark 8 The dynamic model and the control protocol inCorollary 7 were discussed in [21] and the conclusion inCorollary 7 is less conservative than that in [21] The proofabout Corollary 7 is the same as that of Theorem 4

Theorem 9 Suppose that a FOMAS with multiple integral isgiven byMIFOMAS (4) whose corresponding network topologyG satisfies Lemma 2 Define the following functions

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(44)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 is the 119894-theigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)]and Im[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part andreal part of 119861119897(120596) respectively

If all 120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) for theMIFOMAS (10) with asymmetric time-delays then the controlprotocol (7) can resolve the consensus problem of the MIFO-MAS (10) with asymmetric time-delays and on the contrarythen the control protocol (7) can not resolve the consensusproblem of the MIFOMAS (10) with asymmetric time-delayse value of 120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by thefollowing equation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (45)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)Proof By adopting the proof method similar to Theorem 4we suppose that 119904 = minus119895120596 = 0 is the characteristic eigenvalueof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays on the imaginary axis 119906 = 1199061 otimes [1 0 0 0]119879 +1199062 otimes [0 1 0 0]119879 + sdot sdot sdot + 119906119897minus1 otimes [0 0 1 0]119879 + 119906119897 otimes[0 0 0 1]119879 is the corresponding eigenvector and 119906 = 11199061 1199062 119906119897minus1 119906119897 isin C119899 According to Lemma 2 we can getthe following equation

119861119897 (120596) ≜ 119872sum119898=1

119886119898119890119895120596120591119898 = minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1 (46)

Take modulus of both sides of (46) |119861119897(120596)| is an increas-ing function for 120596 gt 0 thus 120596(|119861119897(120596)|) is also an increasingfunction for |119861119897(120596)|

Calculate the principal value of the argument of (46) wehave

Θ119897 (120596) ≜ arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)sum119896119894=1 120572119894 119875119896+1]= arctan119877119897 (120596)

(47)

where 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] = tan[Θ119897(120596)] andIm[119861119897(120596)] and Re[119861119897(120596)] denote the imaginary part and realpart of 119861119897(120596) respectively

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Submit your manuscripts atwwwhindawicom

12 Complexity

According to the definition of 119861119897(120596) in (46) we have

Θ119897 (120596) le arg( 119872sum119898=1

119886119898) + max (120596120591119898) (48)

so there is

max (120596120591119898) ge Θ119897 (120596) minus arg( 119872sum119898=1

119886119898) (49)

Due tosum119872119898=1 119886119898 = 119906119867(119871 otimes 119868119897)119906(119906119867119906) the possible valuesofsum119872119898=1 119886119898 must be nonzero eigenvalues of 119871 of graphG iesum119872119898=1 119886119898 = 120582119894 (120582119894 = 0) which makes the delay margin 120591minimized So when |119861119897(120596)| le |120582119894| 120596(|119861119897(120596)|) le 120596(|120582119894|) = 120596119894If we let all 120591119898 lt 120591 there is

max (120596120591119898) lt 120596119894120591 = 120596119894min|120582119894| =0

Γ119897 (120596119894)= min|120582119894| =0

[Θ119897 (120596119894) minus arg (120582119894)]120596119894 120596119894le Θ119897 (120596) minus arg( 119872sum

119898=1

119886119898) (50)

Inequality (50) is in contradiction with inequality (49)Therefore as long as all 120591119898 lt 120591 the characteristic eigenvaluesof det[119866120591119898(119904)] of the MIFOMAS (10) with asymmetric time-delays can not reach or pass through the imaginary axisthen the MIFOMAS (10) with asymmetric time-delays willremain stable and can achieve consensus On the contrarythe MIFOMAS (10) with asymmetric time-delays will notbe stable and can not achieve consensus Theorem 9 isproven

Remark 10 Consensus of the MIFOMAS (10) without asym-metric time-delays is necessary for consensus of this systemwith asymmetric time-delays

Remark 11 Although 119897 isin 1 2 3 4 5 6 in Theorem 9 due tocomputational complexity the value of 119897 may be greater than6 under the appropriate parameters

Corollary 12 If we suppose that a FOMASwith multiple inte-gral is given by MIFOMAS (4) whose corresponding networktopology G satisfies Lemma 2 and 1205721 = 1205722 = = 120572119897 = 1then the MIFOMAS (10) with asymmetric time-delays can betransformed into high-orderMASwith asymmetric time-delayswhose dynamic model is an integer-order dynamic model andthe following functions can be obtained

119861119897 (120596) = minus 119897sum119896=1

(minus119895120596)119896 119875119896+1

Θ119897 (120596) = arg [119861119897 (120596)] = arg[minus 119897sum119896=1

(minus119895120596)119896 119875119896+1]

1 2

4 3

Figure 1 The undirected network topology

= arctan119877119897 (120596) Γ119897 (120596) = 1120596 [Θ119897 (120596) minus arg (120582119894)]

(51)

where 119897 isin 1 2 3 4 5 6 arg(120582119894) isin (minus1205872 1205872) 120582119894 isthe 119894-th eigenvalue of 119871 119877119897(120596) ≜ Im[119861119897(120596)]Re[119861119897(120596)] =tan[Θ119897(120596)] and Im[119861119897(120596)] and Re[119861119897(120596)] denote theimaginary part and real part of 119861119897(120596) respective-ly

For the high-order MAS with asymmetric time-delays if all120591119898 satisfy 120591119898 lt 120591 = min|120582119894| =0Γ119897(120596119894) then the control protocol(7) can resolve the consensus problem for the high-order MASwith asymmetric time-delays and on the contrary then thecontrol protocol (7) can not resolve the consensus problem forthe high-order MAS with asymmetric time-delayse value of120596119894 corresponding to 119897 in Γ119897(120596119894) is determined by the followingequation

1003816100381610038161003816119861119897 (120596119894)1003816100381610038161003816 = 10038161003816100381610038161205821198941003816100381610038161003816 119897 isin 1 2 3 4 5 6 (52)

where |119861119897(120596119894)| is the modulus of 119861119897(120596119894)6 Simulation Results

The correctness and validity of the theoretical results forTheorems 4 and 9 will be verified by some numerical sim-ulations in this section Under different network topologiesthe FOMAS with different multiple integral will be consid-ered

First of all to validate Theorem 4 we consider aFOMAS composed of 4 agents Figure 1 shows the net-work topology depicted with a connected and undirectedgraph G and Figure 1 has five different time-delays whichare symmetric time-delays and it shows full connectivityAll the delays are marked with 120591119894119895 where 119894 and 119895 arethe indexes which are used to represent the connectedagents 119894 and 119895 If we suppose the weight of each edge ofgraph G in Figure 1 is 1 then the adjacency matrix and

Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Complexity 13

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)Figure 2 The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 lt 120591 in Example 1

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

minus40

minus30

minus20

minus10

0

10

20

30

40

ＲＣ1

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

ＲＣ2

(t)

Figure 3The trajectories of all agentsrsquo states in the FOMAS with double integral and symmetric time-delays when all 120591119898 gt 120591 in Example 1

the corresponding Laplacian matrix of G are respective-ly

A = [[[[[[

2 1 0 11 3 1 10 1 2 11 1 1 3

]]]]]]

119871 = [[[[[[

2 minus1 0 minus1minus1 3 minus1 minus10 minus1 2 minus1minus1 minus1 minus1 3

]]]]]]

(53)

where 120582119899 = 4 is the maximum eigenvalue of 119871

Example 1 For a FOMAS with double integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 = 1 1198754 =1198755 = 1198756 = 1198757 = 0 thus the delay margin 120591 = 1015119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 101119904 12059114 = 12059141 = 100119904 12059123 = 12059132 =099119904 12059124 = 12059142 = 098119904 12059134 = 12059143 = 097119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 102119904 12059114 =12059141 = 103119904 12059123 = 12059132 = 104119904 12059124 = 12059142 = 105119904 12059134 =12059143 = 106119904 which exceed the delay margin 120591 The simulationresults about Example 1 are displayed in Figures 2 and 3the two subfigures in Figure 2 show the trajectories of allagentsrsquo states when all symmetric time-delays are less than thedelay margin 120591 which indicates that the FOMASwith doubleintegral and symmetric time-delays is stable and consensus

14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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14 Complexity

Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

20 40 60 80 100 120 140 160 180 2000

minus08

minus06

minus04

minus02

0

02

04

06

ＲＣ3

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

minus4

minus3

minus2

minus1

0

ＲＣ1

(t)

1

2

3

4

Figure 4 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 2

of the FOMAS with double integral and symmetric time-delays can be reached the two subfigures in Figure 3 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that theFOMAS with double integral and symmetric time-delays isunstable and consensus of the FOMAS with double integraland symmetric time-delays can not be reached

Example 2 For a FOMAS with triple integral and symmetrictime-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1398119904 accordingto Theorem 4 Two groups of symmetric time-delays are set12059112 = 12059121 = 130119904 12059114 = 12059141 = 125119904 12059123 = 12059132 =120119904 12059124 = 12059142 = 115119904 12059134 = 12059143 = 110119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 140119904 12059114 =12059141 = 141119904 12059123 = 12059132 = 142119904 12059124 = 12059142 = 143119904

12059134 = 12059143 = 144119904 which exceed the delay margin 120591 Thesimulation results about Example 2 are displayed in Figures4 and 5 the three subfigures in Figure 4 show the trajectoriesof all agentsrsquo states when all symmetric time-delays are lessthan the delay margin 120591 which indicates that the FOMASwith triple integral and symmetric time-delays is stable andconsensus of the FOMAS with triple integral and symmetrictime-delays can be reached the three subfigures in Figure 5show the trajectories of all agentsrsquo states when all symmetrictime-delays exceed the delay margin 120591 which indicates thatthe FOMAS with triple integral and symmetric time-delaysis unstable and consensus of the FOMAS with triple integraland symmetric time-delays can not be reached

Example 3 For a FOMASwith sextuple integral and symmet-ric time-delays under the undirected graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04 and

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

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International Journal of

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Operations ResearchAdvances in

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 15

Time (s) Time (s)20 40 60 80 100 120 140 160 180 2000 20 40 60 80 100 120 140 160 180 2000

minus2

minus15

minus1

minus05

0

05

1

15

2

ＲＣ2

(t)

20 40 60 80 100 120 140 160 180 2000Time (s)

minus15

minus1

minus05

0

05

1

15

ＲＣ3

(t)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 5 The trajectories of all agentsrsquo states in the FOMAS with triple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 2

1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 = 1 thus 120591 =02515119904 according to Theorem 4 Two groups of symmetrictime-delays are set 12059112 = 12059121 = 024119904 12059114 = 12059141 = 022119904 12059123 =12059132 = 020119904 12059124 = 12059142 = 018119904 12059134 = 12059143 = 016119904 which arebounded by the delay margin 120591 12059112 = 12059121 = 026119904 12059114 =12059141 = 027119904 12059123 = 12059132 = 028119904 12059124 = 12059142 = 029119904 12059134 =12059143 = 030119904 which exceed the delay margin 120591 The simulationresults about Example 3 are displayed in Figures 6 and 7 thesix subfigures in Figure 6 show the trajectories of all agentsrsquostates when all symmetric time-delays are less than the delaymargin 120591 which indicates that the FOMAS with sextupleintegral and symmetric time-delays is stable and consensusof the FOMAS with sextuple integral and symmetric time-delays can be reached the six subfigures in Figure 7 showthe trajectories of all agentsrsquo states when all symmetric time-delays exceed the delay margin 120591 which indicates that the

FOMAS with sextuple integral and symmetric time-delays isunstable and consensus of the FOMAS with sextuple integraland symmetric time-delays can not be reached

Next to examine Theorem 9 we give a network topologydescribed in Figure 8 which is a directed graph G with aspanning tree It also contains five different time-delayswhichare asymmetric time-delays and displays full connectivity Ifwe suppose the weight of each edge of graph G in Figure 8 is1 then the adjacency matrix and the corresponding Laplacianmatrix are

A = [[[[[[

2 1 0 10 1 0 10 1 1 00 0 1 1

]]]]]]

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

16 Complexity

minus4

minus3

minus2

minus1

0

1

2

3

4

minus1minus08minus06minus04minus02

002040608

1

minus06

minus04

minus02

0

02

04

06

minus05minus04minus03minus02minus01

00102030405

minus06minus05minus04minus03minus02minus01

001020304

minus15

minus1

minus05

0

05

1

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 6The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 lt 120591 in Example 3

119871 =[[[[[[[

2 minus1 0 minus10 1 0 minus10 minus1 1 00 0 minus1 1

]]]]]]]

(54)

where 1205821 = 2 1205822 = 0 1205823 = 15 + 0866119894 and 1205824 = 15 minus 0866119894are all eigenvalues of 119871Example 4 For a FOMAS with double integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 120572119897 = 0 (3 le 119897 le 6) and 1198752 = 25 1198753 =1 1198754 = 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1649119904 according to

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 17

minus30

minus20

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

15

20

minus15

minus10

minus5

0

5

10

15

minus10minus8minus6minus4minus2

02468

10

minus8

minus6

minus4

minus2

0

2

4

6

8

minus5minus4minus3minus2minus1

012345

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

ＲＣ1

(t)ＲＣ3

(t)ＲＣ5

(t)

ＲＣ2

(t)ＲＣ4

(t)ＲＣ6

(t)

Figure 7The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and symmetric time-delays when all 120591119898 gt 120591 in Example 3

Theorem 9 Two groups of asymmetric time-delays are set12059112 = 164119904 12059114 = 163119904 12059124 = 162119904 12059132 = 161119904 12059143 =160119904 which are bounded by the delay margin 120591 12059112 =166119904 12059114 = 167119904 12059124 = 168119904 12059132 = 169119904 12059143 = 170119904which exceed the delay margin 120591 The simulation resultsabout Example 4 are displayed in Figures 9 and 10 the two

subfigures in Figure 9 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwithdouble integral and asymmetric time-delays can be reachedthe two subfigures in Figure 10 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed the

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom

18 Complexity

1 2

4 3

Figure 8 The directed network topology

minus4

minus3

minus2

minus1

0

1

2

3

4

ＲＣ1

(t)

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus1

minus05

0

05

1

15

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 9 The trajectories of all agentsrsquo states in the FOMAS with double integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 4

100 200 300 400 500 6000Time (s)

100 200 300 400 500 6000Time (s)

minus15

minus10

minus5

0

5

10

15

ＲＣ2

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ1

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Figure 10The trajectories of all agentsrsquo states in the FOMASwith double integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 4

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 19

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus04

minus03

minus02

minus01

0

01

02

03

04

05

06

ＲＣ2

(t)

minus02

minus01

0

01

02

03

04

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 11 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 5

delaymargin 120591 which indicates that consensus of the FOMASwith double integral and asymmetric time-delays can not bereached

Example 5 For a FOMASwith triple integral and asymmetrictime-delays under the directed graph let us set 1205721 = 09 1205722 =08 1205723 = 07 120572119897 = 0 (4 le 119897 le 6) and 1198752 = 25 1198753 =9 1198754 = 1 1198755 = 1198756 = 1198757 = 0 thus 120591 = 1327119904 accordingtoTheorem 9 Two groups of asymmetric time-delays are set12059112 = 132119904 12059114 = 129119904 12059124 = 126119904 12059132 = 123119904 12059143 =121119904 which are bounded by the delay margin 120591 12059112 =133119904 12059114 = 136119904 12059124 = 139119904 12059132 = 142119904 12059143 = 145119904which exceed the delay margin 120591 The simulation resultsabout Example 5 are displayed in Figures 11 and 12 the threesubfigures in Figure 11 show the trajectories of all agentsrsquostates when all asymmetric time-delays are less than the delaymargin 120591 which indicates that consensus of the FOMASwith

triple integral and asymmetric time-delays can be reachedthe three subfigures in Figure 12 show the trajectories of allagentsrsquo states when all asymmetric time-delays exceed thedelaymargin 120591 which indicates that consensus of the FOMASwith triple integral and asymmetric time-delays can not bereached

Example 6 For a FOMAS with sextuple integral and asym-metric time-delays under the directed graph let us set 1205721 =09 1205722 = 08 1205723 = 07 1205724 = 06 1205725 = 05 1205726 = 04and 1198752 = 25 1198753 = 9 1198754 = 4 1198755 = 8 1198756 = 12 1198757 =1 thus 120591 = 04335119904 according to Theorem 9 Two groupsof asymmetric time-delays are set 12059112 = 043119904 12059114 =040119904 12059124 = 037119904 12059132 = 034119904 12059143 = 031119904 whichare bounded by the delay margin 120591 12059112 = 044119904 12059114 =047119904 12059124 = 050119904 12059132 = 053119904 12059143 = 057119904 which exceed thedelay margin 120591 The simulation results about Example 6 are

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

20 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus60

minus40

minus20

0

20

40

60

ＲＣ2

(t)

minus150

minus100

minus50

0

50

100

150

200ＲＣ1

(t)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

ＲＣ3

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Figure 12 The trajectories of all agentsrsquo states in the FOMAS with triple integral and asymmetric time-delays when all 120591119898 gt 120591 in Example 5

displayed in Figures 13 and 14 the six subfigures in Figure 13show the trajectories of all agentsrsquo states when all asymmetrictime-delays are less than the delay margin 120591 which indicatesthat consensus of the FOMAS with sextuple integral andasymmetric time-delays can be reached the six subfiguresin Figure 14 show the trajectories of all agentsrsquo states whenall asymmetric time-delays exceed the delay margin 120591 whichindicates that consensus of the FOMASwith sextuple integraland asymmetric time-delays can not be reached

7 Conclusion

The consensus problems of a FOMAS with multiple integralunder nonuniform time-delays are studied in this paperTaking into account two kinds of nonuniform time-delaysthe sufficient conditions have been derived in the form ofinequalities for theMIFOMASwith nonuniform time-delaysNumerical simulations of the MIFOMAS with nonuniform

time-delays over undirected topology and directed topologyare performed to verify these theorems Finally the simula-tion results show that the selected examples have achieved thedesired results the MIFOMAS with nonuniform time-delaysunder given conditions can achieve the consensus With thehelp of the above research of this paper distributed formationcontrol of the MIFOMAS with nonuniform time-delays willbe one of the most significant topics which will be one of ourfuture research tasks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 21

0

minus4

minus3

minus2

minus1

0

1

2

3

4ＲＣ1

(t)

minus03

minus02

minus01

01

02

03

ＲＣ3

(t)

minus015

minus01

minus005

0

005

01

015

02

025

03

035

ＲＣ5

(t)

minus04

minus02

0

02

04

06

08

ＲＣ2

(t)

minus015

minus01

minus005

0

005

01

015

02

025ＲＣ4

(t)

minus04

minus02

0

02

04

06

08

1

ＲＣ6

(t)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

100 200 300 400 500 600 700 8000Time (s)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 13The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delays when all 120591119898 lt 120591 in Example 6

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

22 Complexity

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

0 800700600500400300200100Time (s)

minus20

minus15

minus10

minus5

0

5

10

15

20

ＲＣ5

(t)

minus15

minus10

minus5

0

5

10

15

ＲＣ6

(t)

minus40

minus30

minus20

minus10

0

10

20

30

40ＲＣ4

(t)

minus80

minus60

minus40

minus20

0

20

40

60

80

ＲＣ3

(t)

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500ＲＣ1

(t)

minus200

minus150

minus100

minus50

0

50

100

150

200

ＲＣ2

(t)

Ｒ11 (t)Ｒ21 (t)

Ｒ31 (t)Ｒ41 (t)

Ｒ12 (t)Ｒ22 (t)

Ｒ32 (t)Ｒ42 (t)

Ｒ14 (t)Ｒ24 (t)

Ｒ34 (t)Ｒ44 (t)

Ｒ13 (t)Ｒ23 (t)

Ｒ33 (t)Ｒ43 (t)

Ｒ16 (t)Ｒ26 (t)

Ｒ36 (t)Ｒ46 (t)

Ｒ15 (t)Ｒ25 (t)

Ｒ35 (t)Ｒ45 (t)

Figure 14The trajectories of all agentsrsquo states in the FOMASwith sextuple integral and asymmetric time-delayswhen all 120591119898 gt 120591 in Example 6

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 23

Acknowledgments

This work was supported by the Science and TechnologyDepartment Project of Sichuan Province of China (Grantsnos 2014FZ0041 2017GZ0305 and 2016GZ0220) the Edu-cation Department Key Project of Sichuan Province of China(Grant no 17ZA0077) and the National Natural ScienceFoundation of China (Grant nos U1733103 61802036)

References

[1] M Shi K Qin and J Liu ldquoCooperative multi-agent sweepcoverage control for unknown areas of irregular shaperdquo IetControl eory and Applications vol 18 no 2 pp 115ndash117 2008

[2] A E Turgut H Celikkanat F Gokce and E Sahin ldquoSelf-organized flocking inmobile robot swarmsrdquo Swarm Intelligencevol 2 no 2-4 pp 97ndash120 2008

[3] T R Krogstad and J T Gravdahl Coordinated Attitude Controlof Satellites in Formation Springer Berlin Heidelberg 2006

[4] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[5] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[6] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[7] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[8] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[9] P Li and K Qin ldquoDistributed robust hinfin consensus controlfor uncertain multiagent systems with state and input delaysrdquoMathematical Problems in Engineering vol 2017 Article ID5091756 15 pages 2017

[10] Y Wu B Hu and Z Guan ldquoExponential Consensus Analy-sis for Multiagent Networks Based on Time-Delay ImpulsiveSystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 99 pp 1ndash8 2017

[11] Y Wu B Hu and Z Guan ldquoConsensus Problems OverCooperation-Competition Random Switching Networks WithNoisy Channelsrdquo IEEE Transactions on Neural Networks andLearning Systems vol 99 pp 1ndash9 2018

[12] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[13] J Qin H Gao and W X Zheng ldquoSecond-order consensus formulti-agent systems with switching topology and communica-tion delayrdquo Systemsamp Control Letters vol 60 no 6 pp 390ndash3972011

[14] H Li X Liao X Lei T Huang and W Zhu ldquoSecond-order consensus seeking in multi-agent systems with nonlineardynamics over random switching directed networksrdquo IEEE

Transactions on Circuits and Systems I Regular Papers vol 60no 6 pp 1595ndash1607 2013

[15] P Li K Qin and M Shi ldquoDistributed robust 119867infin rotatingconsensus control for directed networks of second-order agentswith mixed uncertainties and time-delayrdquoNeurocomputing vol148 pp 332ndash339 2015

[16] M Shi and K Qin ldquoDistributed Control for Multiagent Con-sensus Motions with Nonuniform Time Delaysrdquo MathematicalProblems in Engineering vol 2016 Article ID 3567682 10 pages2016

[17] Y Liu and Y M Jia ldquoConsensus problem of high-order multi-agent systems with external disturbances an h8 analysisapproachrdquo International Journal of Robust amp Nonlinear Controlvol 20 no 14 pp 1579ndash1593 2010

[18] P Lin Z Li Y Jia and M Sun ldquoHigh-order multi-agentconsensus with dynamically changing topologies and time-delaysrdquo IETControleory ampApplications vol 5 no 8 pp 976ndash981 2011

[19] Y-P Tian and Y Zhang ldquoHigh-order consensus of hetero-geneous multi-agent systems with unknown communicationdelaysrdquo Automatica vol 48 no 6 pp 1205ndash1212 2012

[20] J Liu and B Zhang ldquoConsensus Problem of High-OrderMultiagent Systems with Time DelaysrdquoMathematical Problemsin Engineering vol 2017 Article ID 9104905 7 pages 2017

[21] M Shi K Qin P Li and J Liu ldquoConsensus Conditions forHigh-Order Multiagent Systems with Nonuniform DelaysrdquoMathematical Problems in Engineering vol 2017 Article ID7307834 12 pages 2017

[22] W Ren and Y C Cao Distributed Coordination of Multi-agentNetworks Springer London UK 2011

[23] R C Koeller ldquoToward an equation of state for solid materialswith memory by use of the half-order derivativerdquoActaMechan-ica vol 191 no 3-4 pp 125ndash133 2007

[24] J Sabatier O P Agrawal and J A Machado ldquoAdvance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineeringrdquo Biochemical Journal vol 361 no 1pp 97ndash103 2007

[25] J Bai G Wen A Rahmani X Chu and Y Yu ldquoConsensuswith a reference state for fractional-order multi-agent systemsrdquoInternational Journal of Systems Science vol 47 no 1 pp 222ndash234 2016

[26] G Ren and Y Yu ldquoConsensus of fractional multi-agent systemsusing distributed adaptive protocolsrdquo Asian Journal of Controlvol 19 no 6 pp 2076ndash2084 2017

[27] C Tricaud and Y Chen ldquoTime-Optimal Control of Systemswith Fractional Dynamicsrdquo International Journal of DifferentialEquations vol 2010 Article ID 461048 16 pages 2010

[28] Z Liao C Peng W Li and Y Wang ldquoRobust stability analysisfor a class of fractional order systems with uncertain parame-tersrdquo Journal of e Franklin Institute vol 348 no 6 pp 1101ndash1113 2011

[29] Y C Cao Y Li W Ren and Y Q Chen ldquoDistributed coordina-tion of networked fractional-order systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 40no 2 pp 362ndash370 2010

[30] H Y Yang X L Zhu and K C Cao ldquoDistributed coordinationof fractional order multi-agent systems with communicationdelaysrdquo Fractional Calculus Applied Analysis vol 17 no 1 pp23ndash37 2013

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

24 Complexity

[31] H Y Yang L Guo X L Zhu K C Cao and H L ZouldquoConsensus of compound-ordermulti-agent systems with com-munication delaysrdquo Central European Journal of Physics vol 11no 6 pp 806ndash812 2013

[32] W Zhu B Chen and Y Jie ldquoConsensus of fractional-ordermulti-agent systems with input time delayrdquo Fractional Calculusand Applied Analysis vol 20 no 1 pp 52ndash70 2017

[33] J Liu K Qin W Chen P Li and M Shi ldquoConsensus ofFractional-Order Multiagent Systems with Nonuniform TimeDelaysrdquoMathematical Problems in Engineering vol 2018 Arti-cle ID 2850757 9 pages 2018

[34] J Liu K Y Qin W Chen and P Li ldquoConsensus of delayedfractional-order multi-agent systems based on state-derivativefeedbackrdquo Complexity vol 2018 Article ID 8789632 12 pages2018

[35] C Yang W Li and W Zhu ldquoConsensus analysis of fractional-order multiagent systems with double-integratorrdquo DiscreteDynamics in Nature and Society 8 pages 2017

[36] S Shen W Li and W Zhu ldquoConsensus of fractional-ordermultiagent systems with double integrator under switchingtopologiesrdquo Discrete Dynamics in Nature and Society 7 pages2017

[37] J Bai G Wen A Rahmani and Y Yu ldquoConsensus for thefractional-order double-integrator multi-agent systems basedon the sliding mode estimatorrdquo IET Control eory amp Applica-tions vol 12 no 5 pp 621ndash628 2018

[38] J Liu W Chen K Qin and P Li ldquoConsensus of Fractional-Order Multiagent Systems with Double Integral and TimeDelayrdquoMathematical Problems in Engineering vol 2018 ArticleID 6059574 12 pages 2018

[39] J Liu KQin P Li andWChen ldquoDistributed consensus controlfor double-integrator fractional-ordermulti-agent systems withnonuniform time-delaysrdquo Neurocomputing vol 321 pp 369ndash380 2018

[40] I Podlubny ldquoFractional differential equationsrdquo Mathematics inScience amp Engineering vol 198 no 3 pp 553ndash556 1999

[41] C Godsil and G Royle Algebraic Graph eory World BookPublishing Company Beijing Inc 2014

[42] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo Proceedings of the Fractional DifferentialSystems Models Methods and Applications vol 5 pp 145ndash1581998

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom