Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis

Cent. Eur. J. Phys. • 11(6) • 2013 • 845-854DOI: 10.2478/s11534-013-0274-5

Central European Journal of Physics

Fractional-order linear time invariant swarm systems:asymptotic swarm stability and time response analysis

Research Article

Mojtaba Naderi Soorki1,∗, Mohammad Saleh Tavazoei1,†

1 Electrical Engineering Department, Sharif University of Technology,Azadi Avenue, Tehran, Iran

Received 7 February 2013; accepted 26 June 2013

Abstract: This paper deals with fractional-order linear time invariant swarm systems. Necessary and sufficient con-ditions for asymptotic swarm stability of these systems are presented. Also, based on a time responseanalysis the speed of convergence in an asymptotically swarm stable fractional-order linear time invariantswarm system is investigated and compared with that of its integer-order counterpart. Numerical simulationresults are presented to show the effectiveness of the paper results.

PACS (2008): 45.30.+s

Keywords: fractional-order linear time invariant swarm systems • asymptotic swarm stability • time response analysis© Versita sp. z o.o.

1. Introduction

In recent years, multi-agent autonomous systems imitat-ing natural swarms have attracted much interests [1, 2].Some real-world examples of swarms include mass move-ment of birds, schools of fish, and crowds of people [3].Swarm systems have many applications in various areas,such as physics, biology, robotic and engineering [4–6].Recently emerging applications such as formation controland distributed coordination of multi robot systems haveincreased the interest of engineers in swarms. The sta-bility analysis of swarm systems is more challenging dueto the difference between these systems and isolated sys-tems. Paper [7] has investigated the consensus for first-∗E-mail: [email protected]†E-mail: [email protected] (Corresponding author)

order continuous-time swarm systems. In paper [8], it hasbeen shown that the existence of an interaction graph in-cluding a spanning tree is necessary for asymptotic swarmstability. Besides these, there are also many studies re-lated to the stability of swarm systems [9, 10]. Paper [11]has presented sufficient conditions for consensus of high-order linear time-invariant swarm systems. Linear time in-variant swarm systems with specific structures have beenstudied in [12] and [13]. Recently, paper [14] has presentednecessary and sufficient conditions for swarm stability ofhigh-order linear time invariant swarm systems. The mostof these stability studies of swarm systems have been lim-ited to integer-order cases, even though fractional ordercontrol systems have attracted increasing attention in thepast few decades [15–18]. As a related work, the dis-tributed coordination of networked fractional-order sys-tems over a directed interaction graph has been studiedin [19]. In paper [19], sufficient conditions on the interac-845

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis

tion graph and the fractional order have been found suchthat the coordination is achieved.This paper presents necessary and sufficient conditionsfor asymptotic swarm stability of fractional-order lineartime invariant swarm systems. Also, the time responseof an asymptotically swarm stable fractional-order lineartime invariant swarm system is investigated and comparedwith that of its integer-order counterpart. To the best ofthe authors’ knowledge, this paper is the first researchwork that introduces fractional-order linear time invariantswarm systems and investigates the properties of thesesystems. In fact, the study of swarm stability and timeresponse analysis are new challenges in the field of swarmsystems which are addressed in this paper for fractional-order linear time invariant swarm systems.This paper is organized as follows. A model for afractional-order linear time invariant swarm system is pre-sented in Section 2. Moreover, the asymptotic swarm sta-bility of such a model is investigated in this section. Sec-tion 3 is devoted to the time response analysis of asymp-totically swarm stable fractional-order linear time invari-ant swarm systems. Also, numerical simulation results arepresented in Section 4. Finally, conclusions are given inSection 5.2. Fractional-order swarm systemsand their stability analysisFractional-order linear time invariant swarm systems andtheir stability analysis are addressed in this section.These systems are modelled as having N agents whosedynamics are described by pseudo state space modelshaving inner dimension d. It is assumed that the pseudostate of the ith agent is denoted by xi = [xi1, ..., xid]T ∈Rd. The communication among agents is specified by aweighted directed graph G of order N such that eachagent corresponds to a vertex. Also, the arc weight ofgraph G between ith and jth agents is specified by thevalue wij ≥ 0. This value can be considered as a measurefor the strength of the information link [14]. Graph G isspecified by its adjacency matrix which is denoted by W.

G : W =

w11 . . . w1N... . . . ...wN1 . . . wNN

Let the dynamics of each agent be described by pseudostate space modelsDαt xi = Axi + F

N∑j=1 wij (xj − xi), i ∈ {1, 2, . . . , N} (1)

where A ∈ Rd×d, F ∈ Rd×d, and Dαt denotes the Caputofractional derivative operator defined as follows [21].

Dαt f (t) = { 1Γ(n−α) ∫ t0 f (n)(τ)(t−τ)α−n+1 dτ, n− 1 < α < n ∈ N

f (n)(t), α = n ∈ N

In the above definition, n is the first integer larger thanor equal to α . It is worth mentioning that the consideredswarm system is the fractionalized version of that intro-duced in [14]. Now, consider the following definition.Definition 2.1.Asymptotic swarm stability [14]: A linear time invariantswarm system is said to be asymptotically swarm stableif it achieves the full state consensus, i.e. for each ε > 0there is t∗ > 0 such that xi(t) − xj (t) < ε for t > t∗(i, j ∈ {1, 2, ..., N}).If the pseudo state vector of agents is defined by x =[xT1 , . . . , xTN ]T , the system dynamics in (1) can be expressedas

Dαt x = (IN ⊗ A− L⊗ F )x (2)

where L = L(G) is the Laplacian matrix of the graph G [20].The following lemma reveals a property of the Laplacianmatrix of a graph.Lemma 2.1.The Laplacian matrix of any graph always has a zeroeigenvalue. Also, the Laplacian matrix of a directed graphhas exactly one zero eigenvalue λ1 = 0 with the corre-sponding eigenvector [1, 1, ..., 1]T if and only if this graphincludes a spanning tree. In such a case, all of the othereigenvalues λ2, ..., λN are placed in the open right halfplane [7, 8].

By considering the eigenvalues of L as λ1 = 0, λ2, ..., λN ∈C, the Jordan canonical form of L can be written as

J =

0 0 0 . . . 00 λ2 ∗

. . . 0... . . . . . . . . . ...0 0 . . . . . . ∗0 0 . . . 0 λN

where ′∗′ may either be 1 or 0. By assuming TLT−1 = Jand defining x = (T ⊗ Id)x , (2) is rewritten as

Dαt x = (IN ⊗ A− J ⊗ F )x (3)

846

Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei

where matrix (IN ⊗ A− J ⊗ F ) is in the form

(IN ⊗ A− J ⊗ F ) =

A 0 0 . . . 00 A− λ2F ×

. . . 0... . . . . . . . . . ...0 0 . . . . . . ×0 0 . . . 0 A− λNF

(4)In (4), each ′×′ represents a matrix in Rd×d that may ei-ther equal −F or 0 (For more details, see [14]). Accordingto (4), (3) can be decoupled into pseudo state space equa-tions having one of the following forms

Dαt xi = (A− λiF )xi (5)

orDαt xi = (A− λiF )xi − Fxi+1 (6)

Before deriving the main result, we need the following twolemmas that are similar to what was expressed in [19] (Infact, Lemmas 2.2 and 2.3 are vector forms of Lemmas 2 and3 of [19], which investigate the time response propertiesof systems (5) and (6) in the scalar case).Lemma 2.2.Let xi(t) be the solution of (5). This solution has thefollowing properties:

1. If α ∈ (0, 2θiπ ), where θi = min | arg(eig(A − λiF))|

then limt→∞ xi(t) = 0.

2. If α ∈ (0, 1] and λi = 0, then xi(t) = Eα,1(Atα )xi(0).13. If α ∈ (1, 2) and λi = 0, then xi(t) =Eα,1(Atα )xi(0) + tEα,2(Atα )˙xi(0).

Proof.By the stability theorem of fractional-order systems de-scribed by pseudo state space models [23], it can be easilyshown that if α ∈ (0, 2θiπ ), where θi = min | arg(eig(A −

λiF))|, then system (5) will be an asymptotically stablesystem, and consequently limt→∞ xi(t) = 0. Also, proper-ties 2 and 3 can be respectively concluded from [[24]: Eq.(3.47)] and [[25]: Eq. (16)]

1 Eα,β (.) denotes the two-parameter Mittag-Leffler func-tion [21].

Lemma 2.3.Let xi+1(t) be a continuous function satisfying condi-tion limt→∞ xi+1(t) = 0. In this case, (6) yields inlimt→∞ xi(t) = 0, provided that α ∈ (0, 2θi

π ), where θi =min | arg(eig(A − λiF))|.Proof.By taking the Laplace transform from the both sides of (5),it is deduced thatL{xi(t)} = [sα I−A+λiF ]−1(sα−1xi(0)−FL{xi+1}), α ∈ (0, 1](7)

L{xi(t)} =[sα I − A+ λiF ]−1(sα−1xi(0) + sα−1˙xi(0)− FL{xi+1}), α ∈ (1, 2) (8)

If α ∈ (0, 2θiπ ), where θi = min | arg(eig(A − λiF))|, theroots of equation det(sα I−A+ λiF ) = 0 are placed in theleft half plane. Therefore, in such a case the final valuetheorem can be used to obtain limt→∞ xi(t) from equalities(7) and (8). By applying this theorem, we have

limt→∞

xi(t) = lims→0 s[sα I − A+ λiF ]−1(sα−1xi(0)− FL{xi+1})(9)

limt→∞

xi(t) = lims→0 s[sα I − A+ λiF ]−1(sα−1xi(0) + sα−1˙xi(0)− FL{xi+1}) (10)

By considering (9), (10), and assumption limt→∞ xi+1(t) =0 which results in limt→∞ L{xi+1(t)} = 0, it is concludedthat limt→∞ xi(t) = 0.Now, consider the following theorem as the main result ofthis section.Theorem 2.2.Let α ∈ (0, 2θ

π ) where θ = mini=1,...,N θi, θi =min | arg(eig(A − λiF ))|, and λ1 = 0 ,and λ2, . . . , λN ∈ Care eigenvalue of the Laplacian matrix L with the Jordanform J = TLT−1. In this case, the swarm system (1) whichdoes not satisfy condition | arg(eigA)| > α π2 is asymptot-ically swarm stable if and only if the graph topology Ghas a spanning tree. In addition, in the steady state thepseudo states of the agents of the asymptotically swarmstable system are obtained as follows.

x(t) = (T ⊗ Id)−1x(t)limt→∞

x(t) = [Eα,1(Atα )x1(0), 0, . . . , 0]T , α ∈ (0, 1]limt→∞

x(t) = [Eα,1(Atα )x1(0) + tEα,2(Atα )˙x1(0), 0, . . . , 0]T ,α ∈ (1, 2θπ ) (11)

847


Proof. If graph G has a directed spanning tree, fromLemma 2.1 it deduced that L has a simple zero eigen-value, and other eigenvalues L of have positive real parts.Since λ1 = 0 is a simple zero eigenvalue for L, fromProperties 2 and 3 of Lemma 2.2 it is concluded thatx1(t) = Eα,1(Atα )x1(0) for the case α ∈ (0, 1] and x1(t) =Eα,1(Atα )x1(0) + tEα,2(Atα )˙x1(0) for the case α ∈ (1, 2).On the other hand, if condition α ∈ (0, 2θ

π ) is satis-fied, Property 1 of Lemma 2.2 and Lemma 2.3 yield inlim0→∞ xi(t) = 0 for i 6= 1. Considering these points, theasymptotic relations (11) are derived. Hence, if graph Ghas a directed spanning tree and α ∈ (0, 2θπ ), we have

limt→∞

x(t) = (T ⊗ Id)−1x(t) = (T−1 ⊗ Id)Eα,1(Atα )x1(0)0...0

=

t11IdEα,1(Atα )x1(0)...t11IdEα,1(Atα )x1(0)

(12)for the case α ∈ (0, 1], andlimt→∞

x(t) = (T ⊗ Id)−1x(t)= (T−1 ⊗ Id)

Eα,1(Atα )x1(0) + tEα,2(Atα )˙x1(0)0...0

=

t11IdEα,1(Atα )x1(0) + tEα,2(Atα )˙x1(0)...t11IdEα,1(Atα )x1(0) + tEα,2(Atα )˙x1(0)

(13)

for the case α ∈ (0, 2θπ ), where [t11, . . . , t11]T is the firstcolumn of matrix T−1 (Note that from Lemma 2.1 we knowthat this vector is a coefficient of [1, 1, . . . , 1]T ). Accordingto (12) and (13), x1(t) = . . . = xN (t) when t → ∞ whichresults in the asymptotic swarm stability of the consideredswarm system.If the interaction graph G does not include a spanningtree and condition | arg(eigA)| > α π2 is not satisfied, bythe same approach presented in the proof of Theorem 2.1in [14], it can be shown that the considered swarm systemcannot be asymptotically swarm stable.

For the case α = 1 (integer-order case), similar that pre-sented in Theorem 2 of [14], Theorem 2.1 reveals that theconsidered swarm system with non-Hurwitz matrix A isasymptotically swarm stable if and only if matrices A−λiFfor i = 2, . . . , N are Hurwitz and the graph topology Ghas a spanning tree.

3. Time response analysisTheorem 2.1 in the previous section has specified thepseudo states of agents in the steady state. Based onthis theorem, in this section some analyses are presentedon the time response of an asymptotically swarm stablefractional-order linear time invariant swarm system withan order in the range (0, 1). Also, the obtained resultsare compared with the results of the integer-order case,i.e. the case α = 1. Note that the Jordan canonical formof structure (IN ⊗ A − J ⊗ F ) in (3) can be expressed asfollows:

φ , B−1(IN⊗A−J⊗F )B =

γ1 0 0 . . . 00 γ2 × . . . 0... . . . . . . . . . ...0 0 . . . . . . ×0 0 . . . 0 γNd

(14)

In (14), γi(i = 1, . . . , Nd) denotes the eigenvalues of thematrices A − λiF where λ1 = 0 , and λ2, . . . , λN ∈ Care eigenvalues of the Laplacian matrix L and each ′×′may either be 0 or 1. For simplicity, in the rest of thissection we assume that the eigenvalues γi(i = 1, . . . , Nd)are distinct and nonzero. According to the structure of(IN ⊗ A− J ⊗ F ), matrix B has the following form

B =Sd×d . . . . . .

0d×d . . ....... . . ....0d×d . . . . . .

(15)where Sd×d is the similarity transformation matrix usedto diagonalize matrix A. According to Theorem 2.1, foran asymptotically swarm stable fractional-order lineartime invariant swarm system the eigenvalues of matricesA − λiF for i = 2, . . . , N are in the stable region, i.e.| arg(eig(A − λiF))| > α π2 . By transformation z = B−1x ,equation (3) can be rewritten as

Dαt z(t) = φz(t) (16)Since it is assumed that the eigenvalues γi(i = 1, . . . , Nd)are distinct, the solution of (16) is obtained as [22]

z(t) ={c1 I1Eα,1(γ1tα ) + c2 I2Eα,1(γ2tα ) + . . .+cNd ´INdEα,1(γNdtα )} (17)

where Ii(i = 1, . . . , Nd) are Nd×1 vectors in which the ithelement is 1 and the other elements are zero. Also, ci(i =848


1, . . . , Nd) are fixed values obtained based on the initialconditions. Finally, according to relations x = (T ⊗ Id)xand x = Bz(t) , agents responses are obtained as follows.x(t) = (T ⊗ Id)−1x = (T−1 ⊗ Id)Bz(t) (18)

By defining M = (T−1 ⊗ Id)B ∈ RNd×Nd and consideringthe structure of matrix B in (15),

M = (T−1 ⊗ Id)B =t11Id . . . . . .

t11Id . . ....... . . ....

t11Id . . . . . .

Sd×d . . . . . .

0d×d . . ....... . . ....0d×d . . . . . .

=md×d . . . . . .

md×d . . ....... . . ....

md×d . . . . . .

(19)

where md×d = t11IdSd×d. (19) can be expressed as follows

x(t) =x1(t)...xN (t)

=m11 . . . m1Nd... . . . ...mNd1 . . . mNdNd

c1Eα,1(γ1tα )...cNdEα,1(γNdtα )

=

m11c1Eα,1(γ1tα ) + . . .+m1,NdcNdEα,1(γNdtα )...md1c1Eα,1(γ1tα ) + . . .+md,NdcNdEα,1(γNdtα )

...

mNd−d+1,1c1Eα,1(γ1tα ) + . . .+mNd−d+1,NdcNdEα,1(γNdtα )...mNd,1c1Eα,1(γ1tα ) + . . .+mNd,NdcNdEα,1(γNdtα )

(20)

As equation (20) shows, the time response of each agentcan be stated in terms of Mittag-Leffler functions. Someparts of the solution related to the eigenvalues of matri-ces A − λiF (i = 2, . . . , N) tend to zero as t →∞ (Theseeigenvalues satisfy condition | arg(γi)| > α π2 ). Also, theterms related to the eigenvalues of matrix A not satis-fying condition | arg(γi)| > α π2 do not converge to zero.Now, using Theorem 3 of [19], in the following theoremthe speed of convergence in an asymptotically swarm sta-ble fractional-order swarm system is compared with thatof its integer-order counterpart for sufficiently small times.Theorem 3.1.There is T > 0 such that the elements of difference vector

between pseudo states of the ith agent and the jth agent(i.e. vector xi(t)− xj (t)) in an asymptotically swarm stablefractional-order swarm system with an order between 0and 1 (i.e. 0 < α < 1 ) decrease faster than those of itsinteger-order counterpart for t ∈ (0, T ).Proof. Suppose that γl for l = 1, . . . , d are the eigen-values of matrix A, and γl for l = d+1, . . . , Nd are eigen-values of matrices A − λiF (i = 2, . . . , N). In an asymp-totically swarm stable fractional-order swarm system, weknow that | arg(γl)| > α π2 for l = d+ 1, . . . , Nd (Theorem2.1). According to (20), the difference of vectors xi(t) andxj (t) can be expressed as follow

849


xi(t)− xj (t) =mid−d+1,1c1Eα,1(γ1tα ) + . . .+mid−d+1,NdcNdEα,1(γNdtα )...

mid,1c1Eα,1(γ1tα ) + . . .+mid,NdcNdEα,1(γNdtα )

−

mjd−d+1,1c1Eα,1(γ1tα ) + . . .+mjd−d+1,NdcNdEα,1(γNdtα )...

mjd,1c1Eα,1(γ1tα ) + . . .+mjd,NdcNdEα,1(γNdtα )

=mid−d+1,1 −mjd−d+1,1 . . . mid−d+1,d −mjd−d+1,d... . . . ...

mid,1 −mjd,1 . . . mid,d −mjd,d

c1Eα,1(γ1tα )...cdEα,1(γdtα )

+

mid−d+1,d+1 −mjd−d+1,d+1 . . . mid−d+1,Nd −mjd−d+1,Nd... . . . ...

mid,d+1 −mjd,d+1 . . . mid,Nd −mjd,Nd

cd+1Eα,1(γd+1tα )...cNdEα,1(γNdtα )

(21)

According to (19), mid−d+1,1 . . . mid−d+1,d... . . . ...mid,1 . . . mid,d

=mjd−d+1,1 . . . mjd−d+1,d... . . . ...mjd,1 . . . mjd,d

= md×d (22)Hence, (21) can be written as

xi(t)− xj (t) =mid−d+1,d+1 −mjd−d+1,d+1 . . . mid−d+1,Nd −mjd−d+1,Nd... . . . ...


cd+1Eα,1(γd+1tα )...cNdEα,1(γNdtα )

(23)If α = 1 (the integer-order case), (23) results in

xi(t)− xj (t) =mid−d+1,d+1 −mjd−d+1,d+1 . . . mid−d+1,Nd −mjd−d+1,Nd... . . . ...


cd+1eγd+1t...cNdeγNdt

(24)

By comparison the convergence rate of functions Eα,1(γitα )and eγit for sufficiently small t (See Theorem 3 of [19]),from (23) and (24) one can conclude that there exists T >0 such that the elements of difference vector xi(t) − xj (t)in an asymptotically swarm stable fractional-order swarmsystem with 0 < α < 1 decrease faster than those of itsinteger-order counterpart for t ∈ (0, T ).As an important result, Theorem 2 reveals that in asymp-totically swarm stable fractional-order swarm systems theconvergence rate of agents in the initial moments of mo-tion is higher than that of their integer-order counterparts.This means that in the fractional-order case the conver-

gence speed is fast in the beginning of motion, i.e. whenthe initial distance between agents may be large. Now,we want to study the speed of convergence for the suffi-ciently large times. To achieve this goal, first consider thefollowing lemma from [21].Lemma 3.1.If 0 < α < 2 and µ is an arbitrary real number such thatπα2 < µ < min{π, πα}, then for each γi ∈ C satisfyingcondition µ ≤ | arg(γitα )| ≤ π, we have

Eα,1(γitα ) ≈ (−t−α )(γiΓ(1− α)) (25)850


as t →∞.

The following theorem reveals the convergence rate in anasymptotically swarm stable fractional-order swarm sys-tem for sufficiently large times.Theorem 3.2.In an asymptotically swarm stable fractional-order swarmsystem in the form (1) with the order α ∈ (0, 1), theaverage distance between agents decreases like as t−αwhen t → ∞ if the eigenvalues of matrices A − λiF fori = 1, . . . , N are different and nonzero.

Proof. Let γ1, . . . , γd be the eigenvalues of matrix Aand γd+1, . . . , γNd be the eigenvalues of matrices A− λiFfor i = d+ 1, . . . , Nd. In an asymptotically swarm stable

fractional-order swarm system, | arg(γi)| > α π2 for i =d + 1, . . . , Nd. Hence, according to (21) and Lemma 3.1one can concluded that

x(t) ≈

m11c1Eα,1(γ1tα ) + . . .+m1,dcdEα,1(γdtα )...mNd,1c1Eα,1(γ1tα ) + . . .+mNd,dcdEα,1(γdtα )

+ −t−αΓ(1− α)

m1,d+1cd+1

γd+1 + . . .+ m1,NdcNdγNd...

mNd,d+1cd+1γd+1 + . . .+ mNd,NdcNd

γNd

(26)

when t →∞. Vector x(t) ∈ RNd in (26) can be expressedas

x(t) =x1(t)...xN (t)

=

x11 (t)...x1d

...xN1...xNd

≈

m11c1Eα,1(γ1tα ) + . . .+m1,dcdEα,1(γdtα )...md1c1Eα,1(γ1tα ) + . . .+md,dcdEα,1(γdtα )

...

mNd−d+1,1c1Eα,1(γ1tα ) + . . .+mNd−d+1,dcdEα,1(γdtα )...mNd,1c1Eα,1(γ1tα ) + . . .+mNd,dcdEα,1(γdtα )

+ −t−αΓ(1− α)

m1,d+1cd+1

γd+1 + . . .+ m1,NdcNdγNd...

md,d+1cd+1γd+1 + . . .+ md,NdcNd

γNd

...

mNd−d+1,d+1cd+1γd+1 + . . .+ mNd−d+1,NdcNd

γNd...mNd,d+1cd+1

γd+1 + . . .+ mNd,NdcNdγNd

(27)

Let the average distance between agents be defined byh(t) = 2(N × (N − 1)) × N−1∑

k=1N∑

i=k+1√(xk1 (t)− x i1(t))2 + . . .+ (xkd (t)− x id(t))2 (28)

From (19),md×d =

m11 . . . m1,d... . . . ...md,1 . . . md,d

= . . . =mNd−d+1,1 . . . mNd−d+1,d... . . . ...mNd,1 . . . mNd,d

(29)According to (27) – (29),

h(t) ≈ 2pt−αΓ(1− α)N × (N − 1) (30)851


where the constant p is given byp = N∑

k=1N∑

i=k+1

√√√√√ Nd∑j=d+1

cjγj

(mdk−(d−1),j −mdi−(d−1),j ))2 + . . .+ ( Nd∑j=d+1

cjγj

(mdk,j −mdi,j )2 (31)

Theorem 3.2 reveals that for sufficiently large times thespeed of convergence in an asymptotically swarm sta-ble fractional-order swarm system is less than that of itsinteger-order counterpart.4. Numerical simulationsIn this section, the effectiveness of the presented analy-sis is shown by a numerical example (Numerical simula-tions of this section have been carried out by using theAdams-type predictor-corrector method which has beenintroduced in [26] for solving fractional-order differentialequations). To this end, consider the following fractional-order linear time invariant swarm system

Dαt xi = Axi + F

5∑j=1 wij (xj − xi), i ∈ {1, . . . , 5} (32)

which has 5 agents and the pseudo state vector of eachagent is denoted by xi = [xi1, xi2]T ∈ R2. Graph Ga shownin Figure 1 is used to express the communication amongagents with

W =

0 0 0 0 0.50.75 0 0 0 0.450 0.3 0 0 00.6 0 0 0 00 0 0 0.3 0

The eigenvalues of the Laplacian matrix for this graph are

λ(Ga) = {0, 1.2, 0.3, 0.7± 0.3742i}.Assume that α = 0.8 and matrices A and F are given by

A = [ 0 0.3−5 2

], A = [7 −98 6

]

In this case, the eigenvalues of matrix A are 1 ± 0.71ithat are not in the stable region {s ∈ C|| arg(eig(s))| >

Figure 1. Graph Ga which includes a spanning tree

α π2 }, whereas the eigenvalues of A − λiF (i = 2, . . . , 5)are −0.95 ± 4.57i,−6.86 ± 5.88i,−0.24 ± 10.74i and−6.8 ± 12.63i are in the aforementioned stable re-gion. According to Theorem 2.1, the swarm sys-tem in question is asymptotically swarm stable. Fig-ure 2 shows the trajectories of the agents which con-firms the consensus. Also, Figure 3 shows the av-erage distance between agents, i.e. function h(t) =110 ∑4

k=1 ∑5i=k+1 √(xk1(t)− xi1(t))2 + (xk2(t)− xi2(t))2. Toverify the results of Theorem 3.2, the diagram log(h(t)) ver-sus log(t) has been plotted in Figure 4. For large enoughtimes, this diagram specifies a line with slope −α = −0.8which confirms the results of Theorem 3.2.

5. ConclusionIn this paper, fractional-order linear time invariant swarmsystems were introduced and studied in the aspect ofswarm stability and time response analysis. The maincontributions of the paper can be summarized as follows:

• Finding a closed-form expression for time responseof fractional-order linear time invariant swarm sys-tems (Equation (20))• Providing an asymptotic swarm stability analysisfor fractional-order linear time invariant swarm sys-tems (Theorem 2.1)

852


Figure 1. Graph �� which includes a spanning tree

Figure 2. Trajectories of agents in the swarm system considered in Section 4 (��[0,7])

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10

-80

-70

-60

-50

-40

-30

-20

-10

0

10

trajectory of agents

x

y

Figure 2. Trajectories of agents in the swarm system considered inSection 4 (t ∈ [0, 7]).

Figure 3. Average distance between agents.

• Comparing the speed of convergence in an asymp-totically swarm stable fractional-order linear timeinvariant swarm system with that of its integer-order counterpart for sufficiently small and largetimes (Theorems 3.1 and 3.2)The obtained results in this paper can be considered as abasis for further research works on fractional-order swarmsystems.

Figure 4. log(h) versus log(t).

AcknowledgmentsThe authors thank the Research Council of Sharif Univer-sity of Technology for supporting this work.References

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Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis