Research Article Sliding Mode Reference Coordination of ...downloads.hindawi.com/journals/mpe/2013/764348.pdfmodifying each system feasible reference. e set (x,) is de ned as x, =

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 764348 11 pageshttpdxdoiorg1011552013764348

Research ArticleSliding Mode Reference Coordination ofConstrained Feedback Systems

Alejandro Vignoni1 Fabricio Garelli2 and Jesuacutes Picoacute1

1 Institut drsquoAutomatica i Informatica Industrial Universitat Politecnica de Valencia Camı de Vera sn 46022 Valencia Spain2 CONICET and Facultad de Ingenierıa Universidad Nacional de La Plata (UNLP) Calle 48 esq 116 sn 1900 La Plata Argentina

Correspondence should be addressed to Alejandro Vignoni alvig2upves

Received 4 September 2013 Revised 20 November 2013 Accepted 21 November 2013

Academic Editor Rongni Yang

Copyright copy 2013 Alejandro Vignoni et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper addresses the problem of coordinating dynamical systems with possibly different dynamics (eg linear and nonlineardifferent orders constraints etc) to achieve some desired collective behavior under the constraints and capabilities of each systemTo this end we develop a newmethodology based on reference conditioning techniques using geometric set invariance and slidingmode control the sliding mode reference coordination (SMRCoord) The main idea is to coordinate the systems referencesStarting from a general framework we propose two approaches a local one through direct interactions between the differentsystems by sharing and conditioning their own references and a global centralized one where a central node makes decisionsusing information coming from the systems references In particular in this work we focus in implementation on multivariablesystems like unmanned aerial vehicles (UAVs) and robustness to external perturbations To show the applicability of the approachthe problem of coordinating UAVs with input constraints is addressed as a particular case of multivariable reference coordinationwith both global and local configuration

1 Introduction

Coordination of multiagents and in particular formationcontrol ofmultiple unmanned aerial vehicles (UAVs) is a veryup-to-date topic and it supports many practical applicationssuch as surveillance weather forecasting damage assessmentand search and rescue [1ndash3]

Recently the consensus problem was addressed usingalgebraic graph theory and properties of the LaplacianMatrixfor single integrators (see [1 4] and references therein) Thisapproach was extended to a chain of integrators in [5] and tononlinear multiagent systems in [6]

Sliding mode control is an important topic in nonlinearsystems Research in nonlinear systems goes from stabilityanalysis (eg with fuzzy polynomial models and sum ofsquares tools [7]) to estimation (eg with second ordersliding mode observers for kinetic rates [8]) to control (egwith bounded L2 gain performance of Markovian jumpsingular time-delay systems [9])

The use of slidingmode (SM) techniques is generally usedfor control of swarms and multi-agent systems to achieveconsensus In those situations a master-slave or leader-follower configuration is implemented and the discontinu-ous action is a control signal In [10] higher-order slidingmode controllers are used in such configuration to maintainthe formation shape In [11] finite-time sliding mode estima-tors are used to achieve consensus in decentralized formationcontrol with virtual leader Also in [12] a discontinuouscontrol input is chosen to be proportional to the gradientof a positive semidefinite disagreement function definedby the graph Laplacian matrix leading to a sliding modeconsensus algorithm Recently in [13] consensus is achievedin connected and also in fully connected swarms of idealizedand identical first-order dynamic systems enforced by slidingmodes Also in [14] finite-time consensus algorithms fora swarm of self-propelling agents based on sliding modecontrol and graph algebraic theories are developed In [15]an LMI approach to multiagent systems control is performed

2 Mathematical Problems in Engineering

under time delay and uncertainties Finally in [16] exactformation control is achieved with binary information of theposition of the other agents

In a recent proposal from one of the coauthors slidingmode control has been used in a nontraditional way thesliding mode reference conditioning (SMRC) technique [17]This technique combines reference conditioning and slidingmode ideas and has been used in the beginning to boundcross-coupling interactions in multivariable linear systems[18 19] and for set-point seeking in nonlinear systems withstate dependent constraints [20] In [21] an SMRC auxiliaryloop has been implemented to reduce hypoglycemia in aclosed-loop glucose control for DM type 1 patients Also in[22] a geometric invariance and slidingmode ideas have beenproposed for redundancy resolution in robotic systems Andin [23] an integrated solution based on the same ideas hasbeen proposed for robotic trajectory tracking path planningand speed autoregulation

In the previous work [24ndash27] the authors use slidingmode reference conditioning to enforce coordination inmulti-agent systems In [24] sliding mode reference coordi-nation in SISO systems is imposed with a global supervisoryapproach and two layers of SMRC in a hierarchical structureIn [25] the coordination arises from the local interactionbetween the systems and the information flows in only onelevel as the swarm is assumed to have no leader In [26] theauthorsmake amultivariable reference coordination for idealunconstrained systems A preliminary unifying work is donein [27] with focus on SISO systems The present work is aunified approach of the sliding mode reference coordination(SMRCoord) integrating both local and global approacheswith focus on implementation of multivariable systems androbustness of perturbation to the coordination goals

The rest of the paper is organized as follows Next sectionpresents the problem of coordination in a general formThenin Section 3 some previous results in set invariance and ref-erence conditioning used thereafter to propose coordinationstrategy are described In Section 4 a decentralized versionis proposed to solve the coordination problem with onlyone hierarchical level and no leader Meanwhile in Section 5the supervised global coordination method is proposed InSection 6 the proposed strategies are used to coordinateUAVs showing the result obtained by simulations Finally inSection 7 some conclusions are presented and open issues forfuture study are considered

2 Problem Statement

In this section a general form of the coordination problem ispresented together with definitions and assumptions regard-ing constrained systems and systems coordination

Consider 119873 stable closed-loop systems with differentdynamic behaviour constraints and performance In thecontext of constrained systems the following feasible refer-ence concept arises [28] the fastest reference a system canfollow without violating its constraints while beiing in closedloop In the case of a system with actuator saturation as aconstraint a feasible reference will never lead the actuators

out of their operation range otherwise it may open the loopleaving the control system without feedback and leading to awindup effect

Hereafter coordination will be understood as the actionneeded in order to obtain a collective behaviour in a set ofconsidered systems In this work the coordination problemis approached by acting on the systems references

Among the possible collective behaviours one can count(i) keeping a function of the local references (120594) close to

the global reference(ii) maintaining a fixed distance between the systems

local references one to one or between flock cen-troids

(iii) obtaining generalized synchronization as a limit caseof the previous cases

Note the function 120594may be some function of the local refer-ences like themeanmode max ormin As a consequence ofthis the resulting definition of coordination is rather generaland broad depending on the selected 120594 function

21 Information Exchange Information exchange is one ofthe key elementswhen coordinating a set of systemsDepend-ing on the level on which this information is exchanged as inthe following architectures may arise Figure 1

Local Topology when information flows in the samelevel among neighbours systemsGlobal Topology when information exchange occursin a higher hierarchical level to a supervisory node

Information exchange still is one of the main bottlenecksof a centralized topology the data collection time dependson the number of systems involved unlike the decentralizedonewhich depends only on the network radiusMoreover theglobal topology presents the common issues of vulnerabilityand fragility If the central node has a communication prob-lem or if it is under attack all the network gets compromised

22 Constrained Systems When working with constrainedsystems if the constrained variables take different value fromthe desired ones inadequate values of the state variables 119909can destroy control performance In order to restore the stateadequacy auxiliary inputs r119891(x) called feasible references areused [28]Definition 1 The feasible reference r119891(x) of a closed-loopsystem is the closest input to the original reference 119903 suchthat if r119891(x) had been applied to the controller instead of 119903the system constraints would have not been violated that isthe trajectories of the systemand in particular the constrainedvariables would remain inside (or in the boundary of) a givenset Φ defined by the system constraints

23 Assumptions and Definitions Using the followingassumptionAssumption 2 Each system has a stabilizing control loop andcan follow a feasible reference

The coordination is defined as follows

Mathematical Problems in Engineering 3

(a) (b)

Figure 1 Network topology of the different architectures implemented in quadrotors (a) Local distributed topology (b) Supervised globaltopology

Definition 3 The systems are said to be coordinated as longas their references belong to the invariant set Φ119888(x 120588) bymodifying each system feasible reference The set Φ119888(x 120588) isdefined as

Φ119888 (x 120588) = x isin X 120601119888 (x 120588) =1003817100381710038171003817r (x) minus 120588

1003817100381710038171003817 minus 120575 le 0 (1)

where x isin X isin R119899 are the system states r(x) is a feasiblereference function of the states x 120588 is a function dependingon the information coming from the other systems and 120575 isa predefined value The norm sdot can be any norm definedinR119899 However hereafter we will use the Euclidean norm forsimplicity

3 Geometric Set Invariance and Sliding ModeReference Conditioning

In this section themethodology used in this paper to achievethe coordination of feasible references is described It is basedon concepts of geometric set invariance and sliding modereference conditioning

31 Geometric Set Invariance Consider the following dynam-ical system

Σ x = f (x) + g (x) uy = h (x) (2)

where x isin X sub R119899 is the state vector u isin U sub R119898 is acontrol input (possibly discontinuous) f R119899 rarr R119899 andg R119899 rarr R119899 are vector fields and h R119899 rarr R119887 is scalarfields all of them are defined in X

The variable y denotes the system output vector whichhas to be bounded so as to fulfill 119895 = 1 119873 user-specifiedsystem constraints 120601119894 The corresponding bounds on y aregiven by the set

Φ = x isin X | 120601119894 (y) le 0 119894 = 1 119873 (3)

From a geometrical point of view the goal is to find acontrol input u such that the region Φ becomes invariant

(ie trajectories originating in Φ remain in Φ for all times119905) while y is driven as close as possible to its desired value r

To ensure the invariance of Φ the control input u mustguarantee that the right hand side of the first equation in (2)points to the interior ofΦ at all points on the boundary layerof Φ denoted by 120597Φ defined as

120597Φ =

119873

⋃

119894=1

120597Φ119894 120597Φ119894 = x isin Φ | 120601119894 (y) = 0 (4)

The following assumption will be needed for later develop-ment and will allow us to compute the gradient vector nabla120601119895 ofthe functions 120601119894

Assumption 4 All the 120601119894 functions are assumed to be differ-entiable in the boundary 120597Φ119894

Mathematically the invariance of Φ may be ensured byan input u such that for all 119894 120601119894 le 0 when 120601119894(y) = 0 Thiscondition can be expressed as

120601119894 (x d u) = nabla120601⊤119894 x =

1003817100381710038171003817nabla1206011198941003817100381710038171003817

1003817100381710038171003817f + gu1003817100381710038171003817 cos 120579

= nabla120601⊤f + nabla120601⊤gu = 119871119891120601119894 + Lg120601119894u

forallx isin 120597Φ119894 119895 = 1 119873

(5)

which constitute in standard form the implicit invariancecondition [29 30]

infu 120601119894 (x du) le 0 forallx isin 120597Φ119894 119895 = 1 119873 (6)

Solving (6) foru results in the explicit invariance conditionfor system (2) and a particular constraint 120601119894 The set U119894 offeasible solutions is obtainedU119894 (x d)

=

u isin U | 119871119891120601119894 + Lg120601119894u le 0 x isin 120597Φ119894 and Lg120601119894 = 0⊤1198981015840empty x isin 120597Φ119894 and Lg120601119894 = 0⊤1198981015840 and 119871119891120601119894 gt 0

u = free x isin 120597Φ119894 and Lg120601119894 = 0⊤1198981015840 and 119871119891120601119894 le 0

u = free x isin Φ 120597Φ

(7)


rfi Σi

uiwi

rFi

yi

120601i

Figure 2 SM reference conditioning general scheme

where 0⊤119898 denotes the119898-dimensional null column vector andthe first set corresponding to Lg120601119894 = 0⊤119898 is always nonempty

Note that the control u in the interior of Φ can be freelyassigned Particularly u = 0⊤119898 could be taken so that thesystem evolves autonomously throughout the interior of ΦThen the control action becomes active only when someconstraint becomes active that is when the state trajectoryreaches the boundary 120597Φ trying to leave the set Φ Theinvariance condition will hold if the intersection⋂119894U119894(x) forall constraints of the solution setsU119894(x) is not empty

32 Sliding Mode Reference Conditioning Now in order tofind the necessary u to achieve the invariance of some setΦ(119910) (3) with the system Σ (2) consider the followingimplementation (Figure 2)

A discontinuous decision block will drive the search tofind u isin U119894(x) so as to fulfill the constraintΦ(119910) andmake r119891remain as close as possible to the external signal r Also a filter119865 is incorporated Its purpose is to filter out the conditionedsignal r119891 in order to feed the system Σ with a smooth signal

The filter 119865 is implemented as the first-order filter

r119891 = minusΛ (r119891 + u minus r) (8)

whereu isin U sub R119898 is the discontinuous control action r r119891 isinR119898 are the input and the conditioned input signals and Λ isin

R119898times119898 is a diagonal matrix a design parameter of the filterThe discontinuous decision block is implemented by

means of the variable structure control law

119906 = 0 if max119894 120601119894 (y) le 0

uSM otherwise(9)

where 120601119894(y) are the constraints defined previously that is theboundaries of the set Φ and uSM is such that 119906 isin ⋂119894U119894(x)

Notice that the block Σ in Figure 2 represents the entiredynamics from the constrained variables (y) to the inputsignal u Then system (2) becomes

x = f (x d) + g (x) r119891

r119891 = minusΛ (r119891 + u minus r)

y = h (x)

(10)

In the case of a control system (2) is the plant dynamictogether with a control loop in which case r119891 is the reference

and x in (2) is the extended state comprising the plant andcontroller

The choice of uSM depends on whether there is onlyone active constraint or more than one For a single activeconstraint the analysis is very similar to that of an SMRCin an SISO system [17 24 25] and is the approach we willuse hereafter The case of several active constraints (see [22])is not analyzed in this work since we are defining only oneconstraint per system in the formation control problem

4 Reference Coordination underLocal Topology

In this section the local topology scheme for systems coor-dination is presentedThe coordination problem approachedis the second one described in Section 2 To this endDefinition 3 is rewritten in the following way

Definition 5 The local objective between two connectedsystems is to bound the distance between their references bya predefined value 120575119894119895 The set Φ119894119895 is defined by

Φ119894119895 (x 119903119891119896) = x isin X 119903119891119896 isin R 119896 = 1 2

120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 le 0

(11)

where x isin X isin R119899 are the systems states and 119903119891119894 isin R 119903119891119895 isin R

are the conditioned references of systems 119894 and 119895 respectively

Moreover the systems are connected by the followingassumption

Assumption 6 The topology of the connection network isfixed in the sense of which system can communicate witheach other This network can be represented by a directedgraph with adjacency matrix 119860 = [119886119894119895] with 119886119894119895 = 1 when119894th system can communicate with 119895th one Otherwise 119886119894119895 = 0This graph is assumed to be connected [4]

Then the set Φ is defined as the union of the Φ119894119895 whenthey are connected (119886119894119895 = 1) for all the systems in the group

Φ =

119873

⋃

119894=1119895=1119894 = 119895

119886119894119895Φ119894119895 (12)

The coordination objective is to make the set Φ a controlledinvariant set To this end the following local coordinationscheme is proposed

41 Proposed Coordination Scheme Consider a swarm of 119873systems with 119873 ge 2 satisfying Assumption 2 connected ina network topology as in Assumption 6 with a sliding modereference conditioning auxiliary loop like in Section 32Thisloop allows us to handle the local constraints of each systemby commanding a feasible reference to each system closedloop Also the feasible reference provides information tothe other systems The systems will have a goal for everyconnected neighbor systems as in Definition 5 Then it is


possible to enforce coordination as in Section 2 among thesystem references using the following scheme incorporatingthe local goal into each system SMRC loop

In Figure 3 an implementation of the proposed scheme isdepicted with two systems for the sake of demonstrationThe119894th systemΣ119894 incorporates the plant and a biproper controller

Σ119894

119894 = 119891119894 (119909119894) + 119892119894 (119909119894) V119894119910119894 = ℎ119894 (119909119894)

119888119894 = 119860119888119894119909119888119894 + 119887119888119894119890119894

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894

(13)

where 119909119888119894 and 119890119894 are the state of the controller and the errorsignal of the control loop defined as 119890119894 = 119903119891119894 minus 119910119894 119860119888119894 119887119888119894 119888119888119894and 119889119888119894 are the parameters of the 119894th system controller

The local goal incorporated in the SMRC together withthe actuator saturation constraint leads to the followingstructure for the filter 119865119894

119865119894 119903119891119894 = minus120572119894 (119903119891119894 minus 119903 + 119908119894) (14)

with the combination function 119908119894 This will combine thediscontinuous actions from the physical constraints (119908119894) andfrom the virtual coordination constraints (119908119894119895) defined asfollows

119908119894 = 119908119894 +

119873

sum

119895=1119895 = 119894

119886119894119895119908119894119895 (15)

The discontinuous action 119908119894 is defined as in [18]

119908119894 =

119872119894 if 120601+119894 = V119894 minus V+119894119901 gt 0

minus119872119894 if 120601minus119894 = V119894 minus Vminus119894119901 lt 0

0 otherwise(16)

with 119872119894 being the amplitude of the discontinuous actionAnd the discontinuous action 119908119894119895 is defined according to thevirtual constraint 120601119894119895

119908119894119895 = 119872119894119895 sign (119903119891119894 minus 119903119891119895) if 120601119894119895 (119903119891119894 119903119891119895) ge 0

0 if 120601119894119895 (119903119891119894 119903119891119895) lt 0(17)

with 119872119894119895 being the amplitude of the discontinuous actionFinally the physical and vitrual coordination constraints are

120601plusmn119894 = V119894 minus Vplusmn119894119901 (18)

120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 (19)

with V+119894119901 and Vminus119894119901 being the upper and lower limits of theactuator saturation and 120575119894119895 being a preestablished value forthe references difference and (19) being defined according to(11)

In Appendix B the corresponding analysis is done toprove the invariance of the set Φ119894119895 obtaining the followingbound

119872119894119895 gt minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (20)

119872119894 gt sum

119894 = 119895

119872119894119895 minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894 (21)

5 Reference Coordination underGlobal Topology

To address the coordination problem under the global topol-ogy Definition 3 is rewritten as follows

Definition 7 The coordination objective can be defined interms of a set Φ120594 which by changing 119903 becomes an invariantset The set Φ120594 is defined by

Φ120594 (x 119903119891119894) = x isin X 119903119891119894 isin R119873

120601120594 (119903119891119894) =10038161003816100381610038161003816119903 minus 120594 (119903119891119894)

10038161003816100381610038161003816minus Δ le 0

(22)

where x isin X isin R119899 are the systems states 119903 isin R is theglobal conditioned references and 120594(119903119891119894) is a function of eachsystemrsquos reference Finally Δ is a fix predetermined value thewidth of the allowed band around 119903

Thus the coordination objective is tomakeΦ120594 become aninvariant set To this end the following coordination schemeis proposed

51 Proposed Coordination Scheme Consider a set of 119873dynamical systems Assumption 2 holds for each systemDefining the coordination objective as in Definition 7 itfollows that coordination understood as in Section 2 isenforced using the following scheme and incorporating thecoordination objective (Definition 7) in a global referenceconditioning loop

The proposed coordination scheme is shown in Figure 4Each agent 119894 has a local reference conditioning loop (120601119894 119908119894and 119865119894 like in Figure 2) to generate the feasible reference 119903119891119894from the global reference 119903

In a higher hierarchical level there is another referenceconditioning loop in this case to generate the global reference119903This loop comprises the switching function120601120594 the function120594 of the local references and the discontinuous action (119908119892)to create a smooth global reference 119903 from the global targetsignal 119888119892

511 Coordination Filter The coordination filter (119865119892)smooths the global discontinuous action (119908119892) with thefollowing dynamics

119903 = minus120582 (119903 minus 119888119892 minus 119896119908119892) (23)

where 119908119892 is the discontinuous action and 119896 is a weight todefine the strength of the coordination

512 Definition of the Switching Surface 120601120594 First we definethe boundaries of the set Φ120594 as constraints in the followingway

120601+120594 = 119903 minus 120594 (119903119891119894) minus Δ

120601minus120594 = 119903 minus 120594 (119903119891119894) + Δ

(24)


r

w21

12060121

rf1

rf2

Agent 1

Agent 1

Figure 3 Local SMRCoord local goal incorporation into the SMRCscheme

Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg

Figure 4 Coordination scheme for the global topology

The discontinuous action to force the system remain in theset Φ120594 when the system dynamics make the trajectories gooutside of the set is

119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)

To ensure the invariance of the set Φ120594 (7) we must choose119908minus119892 le minus119908

⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908

⋆119892 is the bound obtained in

Appendix A

6 Simulation

In this section the main features of the sliding modereference coordination (SMRCoord) and formation controldeveloped in Sections 5 and 4 are illustrated for quadrotorswith cartesian control through simulation results obtainedusingMATLAB and using non-linear identifiedmodels from[31] A simplified version of the Parrot AR-Drone quadrotor

models and software for development and implementationof control and navigation strategies can be found in [32]Note the strategy is straightforward to implement in MIMOsystems as soon as they are decoupled This is the case of thequadrotors in cartesian configuration

61 Quadrotors and Controllers In this example we con-sidered three quadrotors The kind of motion consideredis cartesian (planar motion) Each one has its own localreference conditioning scheme as in Section 32 Each agent 119894sends and receives information from the other agents underthe local topology and to the supervisor under the globaltopology and incorporates them as constraints (Definition 5)in the SMRC as proposed in Section 4

62 Information Exchange in SMRCoord The way the infor-mation exchange is considered in this work is a key contri-bution The main idea is that each system sends key infor-mation regarding its local constraints to the other systems bysending its feasible reference In any of the two topologiesthe individual systems hide their states and output to onlyshare the feasible reference with the other systems and thusminimizing the information exchange The feasible referencecontains information of the system and its state regarding thelocal constraints

63 Simulation Results

631 UAV Formation with SMRCoord under Local TopologyFirst we implement a triangle-shape formation with threedrones using the SMRCoord under a local topologyThe timeevolution of the three agents references (119903119891119894) together with theresulting discontinuous actions (119908) is shown in Figure 5

It can be seen that the references are coordinated andfollow the target trajectory taking into account the saturationin the drones actuators

In the resulting discontinuous action (plots 3 to 5 inFigure 5) it is possible to see that the amplitude in eachcomponent is varying with time as the constraints 120601 alsochange depending on the relative positions of the agentbetween them Also the amplitude of the discontinuousaction is enough to make (21) hold so that the SM isestablished in the boundary 120597120601

In Figure 6 it is possible to see how despite the dynamicaldifference among the systems and their local controllersparameters coordination is also achieved in the output of thesystems

632 Formation Perturbation Rejection with SMRCoordunder Local Topology Simulation results in order to testthe formation perturbation rejection properties of theSMRCoorc under local topology can be seen in Figure 7Theperturbation consisted in manually stopping the movementof Drone 1 (red line) in the coordinate 119909 and has beenincorporated after 20 sec and after 50 sec It appears as aflattop in the reference (surrounded by a black ellipse in theplot) The strategy rejects the perturbation as the other twodrones wait that is they stop moving as well in order to keep


0 10 20 30 40 50

0

10

20C

ompo

nent

xConditioned references rf (m)

t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty

Conditioned references rf (m)

t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)

Figure 5 Local topology conditioned references and discontinuous actions of the individual systems

0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)

Drone 2Drone 1 Drone 3

0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)

Figure 6 Local topology drones trajectories in time and cartesian projectionThe trajectories are winding along the conditioned references


0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2

Drone 3Target trajectory

Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)

Figure 7 Target and conditioned references for 119909 and 119910 components under perturbations in the formation The black ellipses show theintervals where the perturbation was introduced to Drone 1 (red) and it is possible to see how the other two conditioned references wait forthe reference of Drone 1 until the perturbation vanishes and then the normal course continues

the formation As a drawback comparing with the first plotof Figure 5 the time it takes to finish the desired trajectory islonger because the drones had to wait until the perturbationwas gone

7 Conclusion

A novel strategy using ideas of set invariance and slidingmode reference conditioning is developed to deal with thecoordination of multi-agents and formation control problemThe proposed methodology has an interesting potential to beexpanded in order to overcome more general coordinationproblems This is inherent to its definition that is thecoordination goals are reflected in the design of the slidingmanifolds

The fact that the individual systems dynamics are hiddento the coordination system while only the necessary infor-mation about the subsystems constraints is communicated toit makes the proposed methodology transparent and allowsdealing with a broad kind of systems to be coordinated assoon as they can be reference conditioned

Additionally the features of the SMRC and the SM itselfare inherited by the proposal such as robustness properties ofSM control but not the usual problems of SM like chatteringbecause the technique is implemented as a part of a numericalgorithm in a digital environment

Practical applications of the proposed algorithm forexample in AR-Drone quadrotors flying in a controlledformation can be implemented as auxiliary supervisoryloops to the trajectory planning algorithm for the virtualleader and stabilizing controllers of the individual agentsTheresearch group is working towards this implementation

In the theoretical side an interesting future research lineon which we are working is on the extensions of the proposedmethodology to deal with multiple constraints per system inwhich case the problem is how to decide the direction of thecontrol action Alsowe areworking on the case of constrainedsystems and how to incorporate these constraints into thecoordination in order to have a formation control which alsotakes care of the individual systems constraint resulting in a

robust formation control against disturbances coming fromindividual limitations in the multi-agent systems

Appendices

A Global Topology Analysis

Consider the case of having 119873 = 2 two individual systemsfor the sake of clarity The state space representation of theclosed-loop system (when the reference conditioning loop isactive) is

x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)

where 119891 is the drift vector field 119892 is the control vector fieldp acts as a perturbation and ℎ comprises the discontinuousactions of the internal conditioning loops which are notconsidered as perturbations because these two signals areresponsible for the difference between 119903 and 119903119891119894 Calculatingthe gradient of the constraint 120601120594 we get

120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)

Note the difference between 120601+120594 and 120601

minus120594 is not reflected in

their gradients thus 120597120601120594120597119909will be used to refer to either 120601+120594 rsquos

gradient or 120601minus120594 rsquos gradientThe Lie derivative of 120601120594 in the direction of 119891 is

119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)


and in the direction of 119892 is

119871119892120601120594 = minus120582 (A5)

Then to ensure the invariance of the set Φ120594 the explicitinvariance condition (7) must hold When the filter 119865119892 isstable then 120582 gt 0 and

119871119892120601120594 = minus120582 lt 0 (A6)

which ensures (7) with

119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)

Consider a fixed constant |119908+119892 | |119908minus119892 | lt infin and 119908

⋆119892 gt

0 If 119865119892 is a filter BIBO stable with bounded input |119908119892| le

max|119908+119892 | |119908minus119892 | = 119908

⋆119892 and 119888119892 (assuming a bounded global

target) note that |119903| lt 119870119903 Since the filters 119865119894 also havebounded inputs (119903 and 119908119894) then |119903119891119894| lt 119870119903

119891119894

Also the bound on 120597120594120597119903119891119894 lt 119870120594

119894

is necessary but onlydepends on the selection of function 120594

Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)

and it is possible to choose 119908minus119892 le minus119908⋆119892 and 119908

+119892 ge 119908

⋆119892 from the

previous inequality to ensure the invariance of the setΦ120594

B Local Topology Analysis

The extended dynamics of the system will be analyzed inorder to design the SMRC parameters of the local topologyAssuming the systems start from inside the invariant set thatis all constraints are met their trajectories evolve up so theyreach the boundary then the corresponding constraint isactivated forcing the system to slide on the boundary surfaceuntil the system by itself comes back inside the invariantset This analysis is done with the 119894th systems consideringinformation coming from the 119895th one

For the analysis let us first rewrite the 119894th systemcontroller and filter dynamics as follows From (13) and (14)and using 119890119894 = 119889

minus1119888119894 (V119894 minus 119888119888119894119909119888119894) it follows that

119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894

V119894 = 119888119888119894 (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894 minus 120572119894) 119909119888119894

+ (119888119888119894119887119888119894119889minus1119888119894 minus 120572119894) V119894 minus 119889119888119894120572119894119908119894 + 119889119888119894120572119894 (119903 + 119910119894)

minus 119889119888119894 119910119894

(B1)

Therefore consider the extended state

119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)

The joint dynamics are given in

119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572

120588119894 = 119887minus1119894 [119889119888119894120572119894 (119903 minus 119910119894) minus 119889119888119894 119910119894]

(B4)

To make the invariance condition (7) hold for constraint120601+119894 we have from (18)

120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)

Now for the virtual constraints let us first incorporatethe 119895th systems and rewrite in a more convenient way theprevious joint dynamics using

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)

and considering the extended state

119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)

Remark 8 In this case in order to express the extended jointdynamics with respect to only one discontinuous action thefact that 119908119894119895 = minus119908119895119894 has been used


And to make the invariance condition (7) hold forconstraint 120601119894119895 we have from (19)

120597120601119894119895

120597119909119894119895119890

= [0 0 sign (119903119891119894 minus 119903119891119895) 0 0 minus sign (119903119891119894 minus 119903119891119895)]

(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)

Then the discontinuous signals amplitudes 119872119894 and 119872119894119895 aredesigned using the previous analysis

In first place the reason why 119908119894119895 is selected as in (17) isbecause 119871119892120601119894119895 = sign(119903119891119894 minus 119903119891119895)(120572119894 + 120572119895) changes its signdepending on 119903119891119894minus119903119891119895Then tomeet (7)119908119894119895 has also to changeits sign in the same way Then 119872119894119895 can be selected to make(B10) hold

119872119894119895 gt 119908120601119894119895

119894 (B11)

In second place 119872119894 has to be selected according to theusual procedure [17] according to (7) but large enough todominate the rest of the discontinuous terms 119872119894119895 as thesystems constraints are required to hold everywhere becausethey are more important than the virtual coordination onesThe previous statement allows us to give the systems afeasible reference that can be followed without violating theirconstraints

The worst case is when the terms (119903119891119894 minus 119903119891119895) have the samesign for all 119894 = 119895 and opposite to 119908119894 then

119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Research in this area is partially supported by Argen-tine government (ANPCyT PICT 2011-0888 and CONICETPIP 112-2011-00361) Spanish government (FEDER-CICYTDPI2011-28112-C04-01) andUniversitat Politecnica deValen-cia (Grant FPI2009-21)

References

[1] W Ren R W Beard and E M Atkins ldquoInformation consensusin multivehicle cooperative controlrdquo IEEE Control SystemsMagazine vol 27 no 2 pp 71ndash82 2007

[2] Y Cao W Yu W Ren and G Chen ldquoAn overview of recentprogress in the study of distributed multi-agent coordinationrdquoIEEE Transactions on Industrial Informatics vol 9 no 1 pp427ndash438 2013

[3] G Antonelli ldquoInterconnected dynamic systems an overview ondistributed controlrdquo IEEEControl SystemsMagazine vol 33 no1 pp 76ndash88 2013

[4] 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

[5] W He and J Cao ldquoConsensus control for high-order multi-agent systemsrdquo IET Control Theory and Applications vol 5 no1 pp 231ndash238 2011

[6] L Liu ldquoRobust cooperative output regulation problem fornonlinear multi-agent systemsrdquo IET Control Theory and Appli-cations vol 6 no 13 pp 2142ndash2148 2012

[7] J L Pitarch A Sala and C V Arino ldquoClosed-form estimatesof the domain of attraction for nonlinear systems via fuzzy-polynomial modelsrdquo IEEE Transactions on Systems Man andCybernetics B 2013

[8] S Nunez H D Battista F Garelli A Vignoni and J PicoldquoSecondorder sliding mode observer for multiple kinetic ratesestimation in bioprocessesrdquo Control Engineering Practice vol21 no 9 pp 1259ndash1265 2013

[9] L Wu X Su and P Shi ldquoSliding mode control with boundedl2 gain performance of markovian jump singular time-delaysystemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012

[10] D Galzi and Y Shtessel ldquoUAV formations control using highorder sliding modesrdquo in Proceedings of the American ControlConference (ACC rsquo06) pp 4249ndash4254 Minneapolis MinnUSA June 2006

[11] Y CaoW Ren and Z Meng ldquoDecentralized finite-time slidingmode estimators and their applications in decentralized finite-time formation trackingrdquo Systems and Control Letters vol 59no 9 pp 522ndash529 2010

[12] J Cortes ldquoFinite-time convergent gradient flows with applica-tions to network consensusrdquo Automatica vol 42 no 11 pp1993ndash2000 2006

[13] S Rao and D Ghose ldquoAchieving consensus amongst self-propelling agents by enforcing sliding modesrdquo in Proceedings ofthe 11th International Workshop on Variable Structure Systems(VSS rsquo10) pp 404ndash409 June 2010

[14] S Rao and D Ghose ldquoSliding mode control-based algorithmsfor consensus in connected swarmsrdquo International Journal ofControl vol 84 no 9 pp 1477ndash1490 2011

[15] P Guo J Zhang M Lyu and Y Bo ldquoSliding mode controlfor multiagent system with time-delay and uncertainties anlmi approachrdquoMathematical Problems in Engineering vol 2013Article ID 805492 p 12 2013

[16] M Jafarian and C de Persis ldquoExact formation control withvery coarse informationrdquo in Proceedings of the IEEE AmericanControl Conference (ACC rsquo13) pp 3026ndash3031 2013

[17] F Garelli R J Mantz and H de Battista Advanced Control forConstrained Processes and Systems The Institution of Engineer-ing and Technology 2011

[18] F Garelli R J Mantz and H de Battista ldquoLimiting interactionsin decentralized control of MIMO systemsrdquo Journal of ProcessControl vol 16 no 5 pp 473ndash483 2006

[19] F Garelli R J Mantz and H de Battista ldquoSliding mode com-pensation to preserve dynamic decoupling of stable systemsrdquoChemical Engineering Science vol 62 no 17 pp 4705ndash47162007

[20] J Pico F Garelli H de Battista and R J Mantz ldquoGeometricinvariance and reference conditioning ideas for control ofoverflow metabolismrdquo Journal of Process Control vol 19 no 10pp 1617ndash1626 2009


[21] A Revert F Garelli J Pico H de Battista P Rossetti andJ Vehi ldquoSafety auxiliary feedback element for the artificialpancreas in type 1 diabetesrdquo IEEE Transactions on BiomedicalEngineering vol 60 no 8 pp 2113ndash2122 2013

[22] L Gracia A Sala and F Garelli ldquoA supervisory loop approachto fulfill workspace constraints in redundant robotsrdquo Roboticsand Autonomous Systems vol 60 no 1 pp 1ndash15 2012

[23] L Gracia F Garelli and A Sala ldquoIntegrated sliding-mode algo-rithms in robot tracking applicationsrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 53ndash62 2013

[24] A Vignoni J Pico F Garelli and H de Battista ldquoDynamicalsystems coordination via sliding mode reference conditioningrdquoin Proceedings of the 18th IFAC World Congress vol 18 pp11086ndash11091 2011

[25] A Vignoni J Pico F Garelli and H de Battista ldquoSliding modereference conditioning for coordination in swarms of non-identical multi-agent systemsrdquo in Proceedings of the 12th IEEEInternational Workshop on Variable Structure Systems pp 231ndash236 2012

[26] A Vignoni F Garelli S Garcıa-Nieto and J Pico ldquoUavreference conditioning for formation control via set invarianceand sliding modesrdquo in Proceedings of the 3rd IFACWorkshop onDistributed Estimation and Control in Networked Systems vol 3pp 317ndash322 2012

[27] A Vignoni F Garelli and J Pico ldquoCoordinacion de sistemascon diferentes dinamicas utilizando conceptos de invarianzageometrica y modos deslizantesrdquo Revista Iberoamericana deAutomatica e Informatica Industral RIAI vol 10 no 4 pp 390ndash401 2013

[28] R Hanus M Kinnaert and J L Henrotte ldquoConditioning tech-nique a general anti-windup and bumpless transfer methodrdquoAutomatica vol 23 no 6 pp 729ndash739 1987

[29] H Amann Ordinary Differential Equations An Introduction toNonlinear Analysis Walter de Gruyter 1990

[30] J Mareczek M Buss and M Spong ldquoInvariance control fora class of cascade nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 47 no 4 pp 636ndash640 2002

[31] X Blasco S Garcıa-Nieto and G Reynoso-Meza ldquoControlautonomo del seguimiento de trayectorias de un vehıculocuatrirrotor simulacion y evaluacion de propuestasrdquo RevistaIberoamericana de Automatica e Informatica Industrial RIAIvol 9 no 2 pp 194ndash199 2012

[32] S Garcıa-Nieto X Blasco J Sanchıs J M Herrero GReynoso-Meza and M Martınez-Iranzo ldquoTrackdrone literdquo2012 httphdlhandlenet1025116427

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


under time delay and uncertainties Finally in [16] exactformation control is achieved with binary information of theposition of the other agents

In a recent proposal from one of the coauthors slidingmode control has been used in a nontraditional way thesliding mode reference conditioning (SMRC) technique [17]This technique combines reference conditioning and slidingmode ideas and has been used in the beginning to boundcross-coupling interactions in multivariable linear systems[18 19] and for set-point seeking in nonlinear systems withstate dependent constraints [20] In [21] an SMRC auxiliaryloop has been implemented to reduce hypoglycemia in aclosed-loop glucose control for DM type 1 patients Also in[22] a geometric invariance and slidingmode ideas have beenproposed for redundancy resolution in robotic systems Andin [23] an integrated solution based on the same ideas hasbeen proposed for robotic trajectory tracking path planningand speed autoregulation

In the previous work [24ndash27] the authors use slidingmode reference conditioning to enforce coordination inmulti-agent systems In [24] sliding mode reference coordi-nation in SISO systems is imposed with a global supervisoryapproach and two layers of SMRC in a hierarchical structureIn [25] the coordination arises from the local interactionbetween the systems and the information flows in only onelevel as the swarm is assumed to have no leader In [26] theauthorsmake amultivariable reference coordination for idealunconstrained systems A preliminary unifying work is donein [27] with focus on SISO systems The present work is aunified approach of the sliding mode reference coordination(SMRCoord) integrating both local and global approacheswith focus on implementation of multivariable systems androbustness of perturbation to the coordination goals

The rest of the paper is organized as follows Next sectionpresents the problem of coordination in a general formThenin Section 3 some previous results in set invariance and ref-erence conditioning used thereafter to propose coordinationstrategy are described In Section 4 a decentralized versionis proposed to solve the coordination problem with onlyone hierarchical level and no leader Meanwhile in Section 5the supervised global coordination method is proposed InSection 6 the proposed strategies are used to coordinateUAVs showing the result obtained by simulations Finally inSection 7 some conclusions are presented and open issues forfuture study are considered

2 Problem Statement

In this section a general form of the coordination problem ispresented together with definitions and assumptions regard-ing constrained systems and systems coordination

Consider 119873 stable closed-loop systems with differentdynamic behaviour constraints and performance In thecontext of constrained systems the following feasible refer-ence concept arises [28] the fastest reference a system canfollow without violating its constraints while beiing in closedloop In the case of a system with actuator saturation as aconstraint a feasible reference will never lead the actuators

out of their operation range otherwise it may open the loopleaving the control system without feedback and leading to awindup effect

Hereafter coordination will be understood as the actionneeded in order to obtain a collective behaviour in a set ofconsidered systems In this work the coordination problemis approached by acting on the systems references

Among the possible collective behaviours one can count(i) keeping a function of the local references (120594) close to

the global reference(ii) maintaining a fixed distance between the systems

local references one to one or between flock cen-troids

(iii) obtaining generalized synchronization as a limit caseof the previous cases

Note the function 120594may be some function of the local refer-ences like themeanmode max ormin As a consequence ofthis the resulting definition of coordination is rather generaland broad depending on the selected 120594 function

21 Information Exchange Information exchange is one ofthe key elementswhen coordinating a set of systemsDepend-ing on the level on which this information is exchanged as inthe following architectures may arise Figure 1

Local Topology when information flows in the samelevel among neighbours systemsGlobal Topology when information exchange occursin a higher hierarchical level to a supervisory node

Information exchange still is one of the main bottlenecksof a centralized topology the data collection time dependson the number of systems involved unlike the decentralizedonewhich depends only on the network radiusMoreover theglobal topology presents the common issues of vulnerabilityand fragility If the central node has a communication prob-lem or if it is under attack all the network gets compromised

22 Constrained Systems When working with constrainedsystems if the constrained variables take different value fromthe desired ones inadequate values of the state variables 119909can destroy control performance In order to restore the stateadequacy auxiliary inputs r119891(x) called feasible references areused [28]Definition 1 The feasible reference r119891(x) of a closed-loopsystem is the closest input to the original reference 119903 suchthat if r119891(x) had been applied to the controller instead of 119903the system constraints would have not been violated that isthe trajectories of the systemand in particular the constrainedvariables would remain inside (or in the boundary of) a givenset Φ defined by the system constraints

23 Assumptions and Definitions Using the followingassumptionAssumption 2 Each system has a stabilizing control loop andcan follow a feasible reference

The coordination is defined as follows


(a) (b)




1003817100381710038171003817 minus 120575 le 0 (1)





Σ x = f (x) + g (x) uy = h (x) (2)



Φ = x isin X | 120601119894 (y) le 0 119894 = 1 119873 (3)




120597Φ =

119873

⋃

119894=1

120597Φ119894 120597Φ119894 = x isin Φ | 120601119894 (y) = 0 (4)




120601119894 (x d u) = nabla120601⊤119894 x =

1003817100381710038171003817nabla1206011198941003817100381710038171003817

1003817100381710038171003817f + gu1003817100381710038171003817 cos 120579


forallx isin 120597Φ119894 119895 = 1 119873

(5)




=




(7)


rfi Σi

uiwi

rFi

yi

120601i











119906 = 0 if max119894 120601119894 (y) le 0

uSM otherwise(9)



x = f (x d) + g (x) r119891


y = h (x)

(10)







Φ119894119895 (x 119903119891119896) = x isin X 119903119891119896 isin R 119896 = 1 2

120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 le 0

(11)






Φ =

119873

⋃

119894=1119895=1119894 = 119895

119886119894119895Φ119894119895 (12)






Σ119894

119894 = 119891119894 (119909119894) + 119892119894 (119909119894) V119894119910119894 = ℎ119894 (119909119894)

119888119894 = 119860119888119894119909119888119894 + 119887119888119894119890119894

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894

(13)



119865119894 119903119891119894 = minus120572119894 (119903119891119894 minus 119903 + 119908119894) (14)


119908119894 = 119908119894 +

119873

sum

119895=1119895 = 119894

119886119894119895119908119894119895 (15)


119908119894 =

119872119894 if 120601+119894 = V119894 minus V+119894119901 gt 0


0 otherwise(16)


119908119894119895 = 119872119894119895 sign (119903119891119894 minus 119903119891119895) if 120601119894119895 (119903119891119894 119903119891119895) ge 0

0 if 120601119894119895 (119903119891119894 119903119891119895) lt 0(17)



120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 (19)



119872119894119895 gt minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (20)

119872119894 gt sum

119894 = 119895

119872119894119895 minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894 (21)




Φ120594 (x 119903119891119894) = x isin X 119903119891119894 isin R119873

120601120594 (119903119891119894) =10038161003816100381610038161003816119903 minus 120594 (119903119891119894)

10038161003816100381610038161003816minus Δ le 0

(22)










120601+120594 = 119903 minus 120594 (119903119891119894) minus Δ

120601minus120594 = 119903 minus 120594 (119903119891119894) + Δ

(24)


r

w21

12060121

rf1

rf2

Agent 1

Agent 1


Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg



119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)


⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908


Appendix A

6 Simulation












0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)




1003817100381710038171003817 minus 120575 le 0 (1)





Σ x = f (x) + g (x) uy = h (x) (2)



Φ = x isin X | 120601119894 (y) le 0 119894 = 1 119873 (3)




120597Φ =

119873

⋃

119894=1

120597Φ119894 120597Φ119894 = x isin Φ | 120601119894 (y) = 0 (4)




120601119894 (x d u) = nabla120601⊤119894 x =

1003817100381710038171003817nabla1206011198941003817100381710038171003817

1003817100381710038171003817f + gu1003817100381710038171003817 cos 120579


forallx isin 120597Φ119894 119895 = 1 119873

(5)




=




(7)


rfi Σi

uiwi

rFi

yi

120601i











119906 = 0 if max119894 120601119894 (y) le 0

uSM otherwise(9)



x = f (x d) + g (x) r119891


y = h (x)

(10)







Φ119894119895 (x 119903119891119896) = x isin X 119903119891119896 isin R 119896 = 1 2

120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 le 0

(11)






Φ =

119873

⋃

119894=1119895=1119894 = 119895

119886119894119895Φ119894119895 (12)






Σ119894

119894 = 119891119894 (119909119894) + 119892119894 (119909119894) V119894119910119894 = ℎ119894 (119909119894)

119888119894 = 119860119888119894119909119888119894 + 119887119888119894119890119894

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894

(13)



119865119894 119903119891119894 = minus120572119894 (119903119891119894 minus 119903 + 119908119894) (14)


119908119894 = 119908119894 +

119873

sum

119895=1119895 = 119894

119886119894119895119908119894119895 (15)


119908119894 =

119872119894 if 120601+119894 = V119894 minus V+119894119901 gt 0


0 otherwise(16)


119908119894119895 = 119872119894119895 sign (119903119891119894 minus 119903119891119895) if 120601119894119895 (119903119891119894 119903119891119895) ge 0

0 if 120601119894119895 (119903119891119894 119903119891119895) lt 0(17)



120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 (19)



119872119894119895 gt minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (20)

119872119894 gt sum

119894 = 119895

119872119894119895 minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894 (21)




Φ120594 (x 119903119891119894) = x isin X 119903119891119894 isin R119873

120601120594 (119903119891119894) =10038161003816100381610038161003816119903 minus 120594 (119903119891119894)

10038161003816100381610038161003816minus Δ le 0

(22)










120601+120594 = 119903 minus 120594 (119903119891119894) minus Δ

120601minus120594 = 119903 minus 120594 (119903119891119894) + Δ

(24)


r

w21

12060121

rf1

rf2

Agent 1

Agent 1


Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg



119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)


⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908


Appendix A

6 Simulation












0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










rfi Σi

uiwi

rFi

yi

120601i











119906 = 0 if max119894 120601119894 (y) le 0

uSM otherwise(9)



x = f (x d) + g (x) r119891


y = h (x)

(10)







Φ119894119895 (x 119903119891119896) = x isin X 119903119891119896 isin R 119896 = 1 2

120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 le 0

(11)






Φ =

119873

⋃

119894=1119895=1119894 = 119895

119886119894119895Φ119894119895 (12)






Σ119894

119894 = 119891119894 (119909119894) + 119892119894 (119909119894) V119894119910119894 = ℎ119894 (119909119894)

119888119894 = 119860119888119894119909119888119894 + 119887119888119894119890119894

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894

(13)



119865119894 119903119891119894 = minus120572119894 (119903119891119894 minus 119903 + 119908119894) (14)


119908119894 = 119908119894 +

119873

sum

119895=1119895 = 119894

119886119894119895119908119894119895 (15)


119908119894 =

119872119894 if 120601+119894 = V119894 minus V+119894119901 gt 0


0 otherwise(16)


119908119894119895 = 119872119894119895 sign (119903119891119894 minus 119903119891119895) if 120601119894119895 (119903119891119894 119903119891119895) ge 0

0 if 120601119894119895 (119903119891119894 119903119891119895) lt 0(17)



120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 (19)



119872119894119895 gt minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (20)

119872119894 gt sum

119894 = 119895

119872119894119895 minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894 (21)




Φ120594 (x 119903119891119894) = x isin X 119903119891119894 isin R119873

120601120594 (119903119891119894) =10038161003816100381610038161003816119903 minus 120594 (119903119891119894)

10038161003816100381610038161003816minus Δ le 0

(22)










120601+120594 = 119903 minus 120594 (119903119891119894) minus Δ

120601minus120594 = 119903 minus 120594 (119903119891119894) + Δ

(24)


r

w21

12060121

rf1

rf2

Agent 1

Agent 1


Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg



119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)


⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908


Appendix A

6 Simulation












0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Σ119894

119894 = 119891119894 (119909119894) + 119892119894 (119909119894) V119894119910119894 = ℎ119894 (119909119894)

119888119894 = 119860119888119894119909119888119894 + 119887119888119894119890119894

V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894

(13)



119865119894 119903119891119894 = minus120572119894 (119903119891119894 minus 119903 + 119908119894) (14)


119908119894 = 119908119894 +

119873

sum

119895=1119895 = 119894

119886119894119895119908119894119895 (15)


119908119894 =

119872119894 if 120601+119894 = V119894 minus V+119894119901 gt 0


0 otherwise(16)


119908119894119895 = 119872119894119895 sign (119903119891119894 minus 119903119891119895) if 120601119894119895 (119903119891119894 119903119891119895) ge 0

0 if 120601119894119895 (119903119891119894 119903119891119895) lt 0(17)



120601119894119895 =10038161003816100381610038161003816119903119891119894 minus 119903119891119895

10038161003816100381610038161003816minus 120575119894119895 (19)



119872119894119895 gt minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (20)

119872119894 gt sum

119894 = 119895

119872119894119895 minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894 (21)




Φ120594 (x 119903119891119894) = x isin X 119903119891119894 isin R119873

120601120594 (119903119891119894) =10038161003816100381610038161003816119903 minus 120594 (119903119891119894)

10038161003816100381610038161003816minus Δ le 0

(22)










120601+120594 = 119903 minus 120594 (119903119891119894) minus Δ

120601minus120594 = 119903 minus 120594 (119903119891119894) + Δ

(24)


r

w21

12060121

rf1

rf2

Agent 1

Agent 1


Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg



119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)


⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908


Appendix A

6 Simulation












0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r

w21

12060121

rf1

rf2

Agent 1

Agent 1


Agent 1rf1

rfn Agent n

r120590

wg

Fg

cg



119908119892 =

119908+119892 if 120601

+120594 gt 0

119908minus119892 if 120601

minus120594 lt 0

0 otherwise(25)


⋆119892 and 119908

+119892 ge 119908

⋆119892 where 119908


Appendix A

6 Simulation












0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

0

10

20C

ompo

nent


t (s)

(a)

0 10 20 30 40 50minus5

0

5

10

Com

pone

nty


t (s)

(b)

0 10 20 30 40 50minus5

0

5

wD

rone

1


t (s)

(c)


0 10 20 30 40 50minus5

0

5

wD

rone

2t (s)

(d)


0 10 20 30 40 50minus5

0

5

wD

rone

3

t (s)

(e)


0 5 10 15minus20246810

0

10

20

30

40

50

60

Trajectories

t(s

)

x (m)y (m)

(a)


0

5

10

15

minus20

24

68

10

x(m

)y (m)

(b)



0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70minus5

05

101520

Drone 1Drone 2


Com

pone

nty


t (s)

(a)

0 10 20 30 40 50 60minus5

0

5

10

Drone 1Drone 2


Com

pone

nty


t (s)

(b)



7 Conclusion







Appendices



x = 119891 (119909) + 119892 (119909)119908119892 + ℎ (119909) + p (A1)

with

119909 = [

[

1199031198911

1199031198912

119903

]

]

119891 (119909) =[[

[

minus1205721 (1199031198911 minus 119903)

minus1205722 (1199031198912 minus 119903)

minus120582119903

]]

]

119892 (119909) = [

[

0

0

minus120582

]

]

ℎ (119909) = [

[

minus12057211199081

minus12057221199082

0

]

]

p = [

[

0

0

120582119888119892

]

]

(A2)


120597120601120594

120597119909= [minus

120597120594

1205971199031198911

minus120597120594

1205971199031198912

1] (A3)





119871119891120601120594 =120597120594

1205971199031198911

1205721 (1199031198911 minus 119903) +120597120594

1205971199031198912

1205722 (1199031198912 minus 119903) minus 120582119903 (A4)



119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871119892120601120594 = minus120582 (A5)


119871119892120601120594 = minus120582 lt 0 (A6)


119908120601120594

119892 =1205721

120582

120597120594

1205971199031198911

1199031198911 +1205722

120582

120597120594

1205971199031198912

1199031198912

minus (1205721

120582

120597120594

1205971199031198911

+1205722

120582

120597120594

1205971199031198912

+ 1) 119903

(A7)


⋆119892 gt


max|119908+119892 | |119908minus119892 | = 119908



119891119894


119894


Then exist 119908120590120594

119892 such as

100381610038161003816100381610038161003816119908120601120594

119892

100381610038161003816100381610038161003816le1

120582[

119873

sum

119894=1

(119870119903119894

119870120594119894

) + 119870119903] le 119870 le 119908⋆119892 (A8)


+119892 ge 119908

⋆119892 from the






119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894) 119909119888119894 + 119887119888119894119889

minus1119888119894 V119894



minus 119889119888119894 119910119894

(B1)


119909119894119890 ≜ [119909119894 119909119888119894 V119894]119879isin R119899+119899119888+1 (B2)


119894119890 =[[

[

119891119894 (119909119894) + 119892119894 (119909119894) V119894(119888119888119894119887119888119894 + 120572119894) V119894 + 119888119888119894 (119860119888119894 minus 120572119894) 119909119888119894

119887119888119894V119894 + 119860119888119894119909119888119894 + 119887119894120588119894

]]

]

+ [

[

0

0

119887119894

]

]

119908119894 (B3)

with

119860119888119894 = (119860119888119894 minus 119887119888119894119889minus1119888119894 119888119888119894)

119887119888119894 = 119887119888119894119889minus1119888119894

119887119894 = minus119889119888119894120572


(B4)


120597120601plusmn119894

120597119909119894119890

= [0 0 1] (B5)

119908120601119894

119894 = minus

119887119888119894Vplusmn119894119901 + 119860119888119894119909119888119894

119887119894

minus 120588119894(B6)


V119894 = 119888119888119894119909119888119894 + 119889119888119894119890119894 = 119888119888119894119909119888119894 + 119889119888119894119903119891119894 minus 119889119888119894119910119894 (B7)


119909119894119895119890 ≜ [119909119894 119909119888119894 119903119891119894 119909119895 119909119888119895 119903119891119895]119879isin R119899119894+119899119888119894+1+119899119895+119899119888119895+1

119894119895119890 =

[[[[[[[[

[

119891119894 (119909119894) + 119892119894 (119909119894) (119888119888119894119909119888119894 + 119889119888119894119903119891119894) minus 119892119894 (119909119894) 119889119888119894119910119894

119860119888119894119909119888119894 + 119887119888119894119903119891119894 minus 119887119888119894119910119894

minus120572119894119903119891119894 + 120572119894119903

119891119895 (119909119895) + 119892119895 (119909119895) (119888119888119895119909119888119895 + 119889119888119895119903119891119895) minus 119892119895 (119909119895) 119889119888119895119910119895

119860119888119895119909119888119895 + 119887119888119895119903119891119895 minus 119887119888119895119910119895

minus120572119895119903119891119895 + 120572119895119903

]]]]]]]]

]

+

[[[[[[[

[

0

0

minus120572119894

0

0

120572119895

]]]]]]]

]

119908119894119895

(B8)




120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597120601119894119895

120597119909119894119895119890


(B9)

119908120601119894119895

119894 = minus

120572119894119903119891119894 minus 120572119895119903119891119895

120572119894 + 120572119895

minus

120572119894 minus 120572119895

120572119894 + 120572119895

119903 (B10)



119872119894119895 gt 119908120601119894119895

119894 (B11)



119872119894 gt sum

119894 = 119895

119872119894119895 + 119908120601119894 (B12)

with 119908120601119894 from (B6)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Sliding Mode Reference Coordination of ...downloads.hindawi.com/journals/mpe/2013/764348.pdfmodifying each system feasible reference. e set (x,) is de ned as x, =