Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies

Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

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Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies Haibo Jiang a,b,⇑, Qinsheng Bi a, Song Zheng a a b

Faculty of Science, Jiangsu University, Zhenjiang 212013, China School of Mathematics, Yancheng Teachers University, Yancheng 224002, China

a r t i c l e

i n f o

Article history: Received 22 February 2010 Received in revised form 27 April 2011 Accepted 30 April 2011 Available online 10 May 2011 Keywords: Multi-agent systems Networked nonlinear oscillators Directed networks Impulsive consensus Impulsive control protocols

a b s t r a c t In this paper, we investigate the problem of impulsive consensus of multi-agent systems, where each agent can be modeled as an identical nonlinear oscillator. Firstly, an impulsive control protocol is designed for directed networks with switching topologies based on the local information of agents. Then sufficient conditions are given to guarantee the consensus of the networked nonlinear oscillators. How to select the discrete instants and impulsive constants is also discussed. Numerical simulations show the effectiveness of our theoretical results. Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction Recently, multi-agent systems have been intensively studied in various disciplines, such as mathematical, physical, biological, and social sciences [1,2]. The goal of multi-agent systems is to generate a desired collective behavior by local interaction among the agents, such as group consensus, group coordination, oscillator synchronization and so on [3–12]. In [3,4], a systematical framework of consensus problem in networks of agents was investigated. Based on the frequency-domain analysis the authors studied the consensus problem for multi-agent systems with input and communication delays in [5]. In [6], the authors considered the consensus problem for multi-agent systems, in which all agents have an identical linear dynamic mode that can be of any order. Consensusability of linear multi-agent systems was studied in [7]. In [8], the authors investigated the synchronization of a network of identical linear systems under a possibly time-varying and directed interconnection structure. Synchronization of chaotic systems by the intermittent feedback method was investigated in [9,10]. In [11,12], global synchronization of two periodically or stochastically coupled oscillators in master-slave configuration was analytically and experimentally studied. In the real word the communication topologies of the multi-agent systems are dynamically changing over time. Very recently some consensus, synchronization and coordination problems of multi-agent systems have received much attention [13–19]. The problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies was considered in [13]. In [14], the authors addressed a coordination problem of a multi-agent system with jointly connected interconnection topologies by a proposed Lyapunov-based approach. The consensus and synchronization problems over random weighted networks were investigated in [15,16], respectively. The average-consensus problem in directed networks of agents with both switching topology and time-delay ⇑ Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang 212013, China. Tel.: +86 511 88791110. E-mail address: [email protected] (H. Jiang). 1007-5704/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.030

H. Jiang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

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was studied in [17,18]. More recently, some consensus, synchronization, and pinning control problems of complex networks over switching networks have received particular attention and many results have been reported in the literature [20–23]. Meanwhile, stability analysis and controller synthesis of systems with impulse have sparked the interest of many researchers [24–26]. Several criteria related to the eigenvalues and eigenvectors of coupling matrix for synchronizing a kind of impulsively coupled complex dynamical systems were established in [25]. In [26], contraction analysis of impulsively coupled oscillators was provided and very simple but very general results for synchronization of impulsively coupled oscillators were derived. On the other hand, impulsive control provides a new viewpoint when the plant has at least one changeable state variable or when the plant has impulsive effects. In many cases impulsive control can give an efficient way to deal with plants which is difficult to provide continuous control or cannot endure continuous control inputs [27]. It has been shown that impulsive control or synchronization approach is effective and robust in control or synchronization of chaotic systems [27–31] and complex dynamical networks [32–35]. In [36], the authors introduced impulsive control protocols for multiagent linear continuous dynamic systems. The convergence analysis of the impulsive control protocol for the networks with fixed or switching topologies was presented. In [37], the authors investigated the problem of impulsive synchronization of networked multi-agent systems, where each agent can be modeled as an identical nonlinear dynamical system. In [38], the authors studied synchronization problems of complex dynamical networks (CDNs) via distributed impulsive control. It is worth pointing out that impulsive control protocol for multi-agent systems has received relatively little attention. In this paper, we investigate the problem of impulsive consensus of multi-agent systems, where each agent can be modeled as an identical nonlinear oscillator. Sufficient conditions are given to guarantee the consensus of the networked nonlinear oscillators with switching topologies. This paper is organized as follows. In Section 2, we provide some results in algebraic graph theory and matrix theory. In Section 3, we formulate the impulsive consensus problem for networked nonlinear oscillators and introduce an impulsive control protocol. The impulsive consensus problem for directed networks with switching topologies is discussed in Section 4. In Section 5, numerical simulations are included to show the effectiveness of our theoretical results. Some conclusions are drawn in Section 6. Notation. Throughout this paper, the superscripts ‘1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; Rn denotes the n-dimensional Euclidean space; Let Rþ ¼ ½0; 1Þ; N ¼ f0; 1; 2; . . .g; Nþ ¼ f1; 2; . . .g; Rnm is the set of all n  m real matrices; In 2 Rnn is an identity matrix; For a vector x 2 Rn , let kxk denote the Euclidean vector norm, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi kxk ¼ xT x, and for A 2 Rnn , let kAk indicate the norm of A induced by the Euclidean vector norm, i.e., kAk ¼ kmax ðAT AÞ. The Kronecker product of two matrices A ¼ ½aij  2 Rmn and B ¼ ½bij  2 Rpq is denoted by A  B. For more properties of the Kronecker product the reader is referred to [39]. 2. Preliminaries In this section, we provide some results in algebraic graph theory [41,4] and matrix theory [40]. A directed graph G of order N consists of a vertex set V ¼ f1; 2; . . . ; Ng and an ordered edge set E ¼ fði; jÞ : i; j 2 Vg  V  V. The set of neighbors of vertex i is denoted by Ni ¼ fj 2 V : ði; jÞ 2 E; j – ig. A directed path is a sequence of ordered edges of the form ðsi1 ; si2 Þ; ðsi2 ; si3 Þ, . . ., where sij 2 V in a directed graph. A directed graph is said to be strongly connected, if there is a directed path from every node to every other node. A weighted adjacency matrix A ¼ ½aij  2 RNN , where aii = 0 and aij P 0, i – j. aij > 0 if and only if there is an ordered edge (i, j) P in G. The in-degree of vertex i is defined as follows deg in ðiÞ ¼ nj¼1 aji . The out-degree of vertex i is defined as follows Pn degout ðiÞ ¼ j¼1 aij . The vertex i is said to be balanced if and only if its in-degree and out-degree are equal, i.e. degin(i) = degout(i). Let D be the diagonal matrix with the out-degree of each vertex along the diagonal and call it the degree matrix of G. The Laplacian matrix of the weighted graph is defined as LG ¼ D  A. An important property of L is that all the row sums of L are zero and thus 1 ¼ ð1; 1; . . . ; 1ÞT 2 RN is an eigenvector of L associated with the zero eigenvalue. The graph is said to be balanced if and only P P if every vertex’s in-degree and out-degree are equal, i.e. nj¼1 aji ¼ nj¼1 aij ; i ¼ 1; 2; . . . ; N. If the graph is balanced, then 1TL = 0. Nr Given C ¼ ½cij  2 R , it is said that C P 0 (C is nonnegative) if all its elements cij are nonnegative, and it is said that C > 0 (C is positive) if all its elements cij are positive. Further, C P D if C  D P 0, and C > D if C  D > 0. If a nonnegative matrix C 2 Rnn satisfies C1 = 1, then it is said to be stochastic. A square matrix C 2 RNN is said to be doubly stochastic if both C and CT are stochastic. Let L be the graph Laplacian of the network. We refer to P = I  eL as Perron matrix of a graph G with parameter e. P Lemma 1 [4]. Let G be a directed graph with n nodes and maximum degree d ¼ maxi ð j–i aij Þ. Then, the perron matrix P with parameter e 2 (0, 1/d] satisfies the following properties. (1) P is a row stochastic nonnegative matrix with a trivial eigenvalue of 1; (2) All eigenvalues of P are in a unit circle; (3) If G is a balanced graph, then P is a doubly stochastic matrix. 3. Problem formulation Here we consider a system consisting of N agents indexed by i = 1, 2, . . . , N. The dynamics of each agent is

x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ þ ui ðtÞ;

xi ðtÞ 2 Rn ;

t P t0 P 0;

i ¼ 1; 2; . . . ; N;

ð1Þ

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H. Jiang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; . . . ; xin ðtÞÞT 2 Rn and ui ðtÞ 2 Rn are the state and the control input of agent i at time t, respectively; f ðxi ðtÞ; tÞ : Rn  R ! Rn is the nonlinear vector field function of agent i at time t. The control input of agent i is designed as

ui ðtÞ ¼

þ1 X

X

dðt  tk Þbk

aij ðtÞðxj ðtÞ  xi ðtÞÞ;

k 2 Nþ ;

i ¼ 1; 2; . . . N;

ð2Þ

j2Ni ðtÞ

k¼1

where the discrete instants tk satisfy 0 6 t0 < t1 < t2 <    < tk1 < tk <   , and limk?+1 tk = +1, d(t) is the Dirac delta function, R1 R aþe i.e., d(t) = 0 for t – 0, and 1 dðtÞdt ¼ 1. The Dirac delta function has the fundamental property that ae gðtÞdðt  aÞdt ¼ gðaÞ for e – 0 and all continuous compactly supported functions g(t). In many applications, the Dirac delta function is usually used to model a tall narrow spike function (an impulse). bk 2 R; k 2 Nþ are impulsive constants to be designed later, Ni ðtÞ is the set of neighbors of agent i at time t. Without loss of generality, we assume that limt!tþ xi ðtÞ ¼ xi ðt k Þ, which means k that the solution xi(t) is right continuous at time tk. Adopting a similar approach to that used in [31,34], from (1) and (2) we have

xi ðtk þ eÞ  xi ðt k  eÞ ¼

Z

t k þe

ðf ðxi ðsÞ; sÞ þ ui ðsÞÞds;

t k e

P i  i where e > 0 is sufficiently small. As e ? 0+, this becomes to Dxi ðtk Þ ¼ bk j2Ni ðxj ðt k Þ  x ðt k ÞÞ, where Dx ðt k Þ ¼  i þ i  i þ i i i x ðt k Þ  x ðt k Þ; x ðtk Þ ¼ limt!tþ x ðtÞ and x ðtk Þ ¼ limt!tk x ðtÞ. This implies that the agent i will suddenly update its state variable k according to the state variables of itself and its neighbors at the instants tk. Thus the control input ui(t) is called an impulsive control protocol. Under the impulsive control protocol (2), the dynamics of agent i satisfies the following equations

8 < x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ; t – t k ; P i i þ i  aij ðtk Þðxj ðt k Þ  xi ðtk ÞÞ; : Dx ðt k Þ ¼ x ðt k Þ  x ðtk Þ ¼ bk

i ¼ 1; 2; . . . N;

k 2 Nþ :

ð3Þ

j2Ni ðtÞ

Before proceeding, we recall some preliminaries and assumptions which will be used throughout the proofs of our main results. Definition 1. For the system (1) with initial conditions given by xi ðt0 Þ ¼ xi0 and yi ðt0 Þ ¼ yi0 ; i ¼ 1; 2; . . . ; N, the consensus is said to be achieved under the impulsive control protocol (2) if

lim kei;j ðtÞk ¼ 0;

i; j ¼ 1; 2; . . . N;

t!þ1

i,j

i

ð4Þ

j

where e (t) = x (t)  x (t). Definition 2 [27]. Let V : Rþ  Rn ! Rþ , then V is said to belong to class m0 if (i) V is continuous in each of the sets ½tk1 tk Þ  Rn , and for each x 2 Rn ; k 2 Nþ ; limðt;yÞ!ðtk xÞ Vðt; yÞ ¼ Vðt  k ; xÞ exits, (ii) V is Lipschitzian in x 2 Rn . _ Let xðtÞ ¼ f ðxðtÞ; tÞ, then we have the following upper right derivative of a Lyapunov function V(x(t), t) 2 m0. Definition 3 [27]. For ðxðtÞ; tÞ 2 Rn  ½t k1 tk Þ, we define

1 D Dþ VðxðtÞ; tÞ ¼ limþ sup ½VðxðtÞ þ hf ðxðtÞ; tÞ; t þ hÞ  VðxðtÞ; tÞ: h h!0 We make the following assumption for the system (1). Assumption 1. For any x; y 2 Rn , there exists a constant h, such that

kf ðx; tÞ  f ðy; tÞÞk 6 hkx  yk: Remark 1. The above condition is called the uniform Lipschitz condition. Many well-known nonlinear oscillators satisfy this condition, such as van der Pol-Duffing oscillators, Duffing system, Lorenz system, Chen system, Chua’s circuit.

4. Networks with switching topologies In this section, we provide the analysis of the impulsive consensus problem for networks with switching topologies.

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381

Here we consider m graphs indexed by G1 ; G2 ; . . . ; Gm . We define a switching signal r : [t0, +1) ? {1, 2, . . . , m}. The switch  ing signal is a piecewise constant right continuous function. Suppose that rðt  k Þ ¼ sk and its graph is Gsk with the Laplacian Lsk , where s k 2 f1; 2; . . . ; mg; k 2 Nþ . Assumption 2. The graphs Gi ; i ¼ 1; 2; . . . ; m of the networks are strongly connected and balanced. Let x(t) = (x1(t), x2(t), . . . , xN(t))T, then the system (3) can be described as

(

_ xðtÞ ¼ FðxðtÞ; tÞ;

t – tk ;

Dxðt k Þ ¼ ðbk Lsk  In Þxðt k Þ;

k 2 Nþ ;

ð5Þ

where F(x(t), t) = (f(x1(t), t), f(x2(t), t), . . . , f(xN(t), t))T. Then, we get

(

_ xðtÞ ¼ FðxðtÞ; tÞ;

t – tk ;

xðt þk Þ ¼ ðPtk  In Þxðtk Þ;

k 2 Nþ ;

ð6Þ

where Ptk ¼ IN  bk Lsk . By Assumption 2, P tk is a doubly stochastic matrix, so PTt P tk is also a doubly stochastic matrix. Then we can obtain the k following decomposition[42,43] N X i1 X

PTt Ptk ¼ IN  k

i¼1

wij ðtk Þðei  ej Þðei  ej ÞT ;

j¼1

T n  where wij ðt  k Þ be the (i, j) th entry of P t  P t k ; ei 2 R ; ei ðiÞ ¼ 1; ei ðjÞ ¼ 0; j – i. k

Define

xðtÞ ¼

N 1 X 1 xi ðtÞ ¼ ð1T  In ÞxðtÞ N i¼1 N

then by Assumption 2 and Lemma 1 we have

xðt þk Þ ¼

1 T 1 1 1 ð1  In Þxðt þk Þ ¼ ð1T  In ÞðPtk  In Þxðtk Þ ¼ ð1T Ptk  In Þxðt k Þ ¼ ð1T  In Þxðt k Þ ¼ xðt k Þ: N N N N

Therefore the dynamics of  x satisfies the following equations

8 > < x_ ðtÞ ¼ > :

1 N

N P

f ðxi ðtÞ; tÞ;

t – tk ;

i¼1

xðtþk Þ ¼ xðtk Þ;

ð7Þ

k 2 Nþ :

Theorem 1. Consider the system (1) with initial conditions given by xi ðt0 Þ ¼ xi0 and yi ðt0 Þ ¼ yi0 ; i ¼ 1; 2; . . . ; N. Assume that Assumptions 1 and 2 are satisfied. If there exist discrete instants tk and impulsive constants bk such that the conditions (i)–(iii) hold, then the consensus is achieved under the impulsive control protocol (2). (i) There exist two constants b1 and b2 such that 0 < b1 6 tk  tk1 6 b2 < þ1; k 2 Nþ ; (ii) There exist some constants bk > 0, dk > 0 such that Ptk ¼ IN  bk Lsk are nonnegative matrices with positive diagonal entries and every nonzero entry of P tk is no smaller than dk; (iii) There exists a constant l > 0 such that ð1  d2k =NÞe4hðtk tk1 Þ 6 l < 1; k 2 Nþ . Proof Consider the Lyapunov function candidate

VðxðtÞÞ ¼

N X

ðxi ðtÞ  xðtÞÞT ðxi ðtÞ  xðtÞÞ ¼ ðxðtÞ  1  xðtÞÞT ðxðtÞ  1  xðtÞÞ:

i¼1

Taking of V(x(t)) for t 2 ½t k1 t k Þ; k 2 Nþ , by Assumption 1 and the fact ða1 þ a2 þ    þ aN Þ2  2 the2 Dini derivative  2 6 N a1 þ a2 þ    þ aN , we obtain

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H. Jiang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

Dþ VðxðtÞÞ ¼ 2

    N N N X X 1 X   ðxi ðtÞ  xðtÞÞT ðx_ i ðtÞ  x_ ðtÞÞ 6 2 kxi ðtÞ  xðtÞkf ðxi ðtÞ; tÞ  f ðxj ðtÞ; tÞ   N i¼1 i¼1 j¼1

N N N N X X 2h X 2h X kxi ðtÞ  xðtÞk kxi ðtÞ  xj ðtÞk 6 kxi ðtÞ  xðtÞk ðkxi ðtÞ  xðtÞk þ kxj ðtÞÞ  xðtÞkÞ N i¼1 N j¼1 i¼1 j¼1 !2 N N N N X X X 2h X 2 i i 6 2h kx ðtÞ  xðtÞk þ kx ðtÞ  xðtÞk 6 2h kxi ðtÞ  xðtÞk2 þ 2h kxi ðtÞ  xðtÞk2 6 4hVðxðtÞÞ: N i¼1 i¼1 i¼1 i¼1

6

Then

VðxðtÞÞ 6 e4hðttk1 Þ Vðxðtþk1 ÞÞ;

t 2 ½tk1 t k Þ;

k 2 Nþ :

ð8Þ

On the other hand, when k 2 Nþ ,

Vðxðt þk ÞÞ ¼ ðxðt þk Þ  1  xðt þk ÞÞT ðxðt þk Þ  1  xðtþk ÞÞ ¼ ððP tk  In Þxðtk Þ  1  xðt k ÞÞT ððPtk  In Þxðt k Þ  1  xðt k ÞÞ ¼ ððPtk  In Þxðt k Þ  ðPtk  In Þð1  xðt k ÞÞÞT ððPtk  In Þxðt k Þ  ðPtk  In Þð1  xðt k ÞÞÞ ¼ ðxðt k Þ  1  xðt k ÞÞT ððP Tt Ptk Þ  In Þðxðt k Þ  1  xðt k ÞÞ k

¼ Vðxðt k ÞÞ 

N X i1 X i¼1

wij ðt k Þðxi ðt k Þ  xj ðtk ÞÞT ðxi ðt k Þ  xj ðt k ÞÞ:

j¼1

By condition (ii), every positive entry of PTt P tk is at least d2k because every positive entry of Ptk is at least dk. Furthermore, if k PN         i 2 Nj or j 2 Ni , then wij ðt l¼1 P t k ðliÞP t k ðljÞ P P t k ðiiÞP t k ðijÞ þ P t k ðjiÞP t k ðjjÞ > 0 and wji ðt k Þ ¼ wij ðt k Þ > 0. Then kÞ ¼ N X i1 X       T   V xðtþk Þ 6 V xðtk Þ  d2k sgn wij ðt k Þ xi ðtk Þ  xj ðt k Þ xi ðtk Þ  xj ðt k Þ : i¼1

ð9Þ

j¼1

Since the graph is strongly connected, it follows that for 6 p, q 6 N, p – q, there exists (p0, p1),  every 1 2 a path Ps  p  PN Pi1 q  2 c ðt Þ  xpcþ1 ðt  Þ 6 (p1, p2), . . . , (ps, ps+1), where 1 6 s 6 N  2, p0 = p, ps+1 = q. So xp ðt Þ  x ðt Þ 6 x k k k k c¼0 i¼1 j¼1 sgn i  j  T i  j  ðwij ðt k ÞÞðx ðt k Þ  x ðt k ÞÞ ðx ðt k Þ  x ðt k ÞÞ. Thus N  N X N  X X   xi ðt Þ  xðt Þ2 6 1 xi ðt  Þ  xj ðt Þ2 k k k k N i¼1 j¼1 i¼1

Vðxðt k ÞÞ ¼

N X N X N X i1 1X sgnðwij ðtk ÞÞðxi ðtk Þ  xj ðt k ÞÞT ðxi ðt k Þ  xj ðtk ÞÞ N i¼1 j¼1 i¼1 j¼1

6

¼N

N X i1 X i¼1

sgnðwij ðt k ÞÞðxi ðt k Þ  xj ðt k ÞÞT ðxi ðtk Þ  xj ðt k ÞÞ:

ð10Þ

j¼1

By (9) and (10), we obtain

Vðxðt þk ÞÞ 6 Vðxðt k ÞÞ 

d2k Vðxðt k ÞÞ ¼ N

1

! d2k Vðxðtk ÞÞ: N

ð11Þ

From (8) and (11), by mathematical induction we have 4hðtt k1 Þ

VðxðtÞÞ 6 e

k1 j¼1

P

! d2j 4hðt t Þ 1 e j j1 Vðxðt 0 ÞÞ; N

t 2 ½t k1 t k Þ;

k 2 Nþ ;

k P 2:

ð12Þ

From conditions (i) and (iii), we get

VðxðtÞÞ 6 e4hb2 lk1 Vðxðt 0 ÞÞ;

t 2 ½tk1 t k Þ;

k 2 Nþ ;

k P 2:

Thus V(x(t)) ? 0 as t ? 1. Therefore the consensus is achieved under the impulsive control protocol (2). This completes the proof. h

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Remark 2. Theorem 1 gives the sufficient conditions to guarantee the consensus of networked nonlinear oscillators (1) under the impulsive control protocol (2). However, for any multi-agent system in the form of (1), if the networks are switching and the graphs Gi ; i ¼ 1; 2; . . . ; m, are strongly connected and balanced, we can always choose proper discrete instants tk and impulsive constants bk such that the conditions (i)–(iii) in Theorem 1 hold. For simplicity, we choose the equidistant  be the maximpulsive interval Dtk ¼ tk  t k1 ¼ c; k 2 Nþ . The impulsive constants bk ; k 2 Nþ , are chosen as p; p 2 R. Let d  imum degree of the graphs Gi ; i ¼ 1; 2; . . . ; m. If 0 < p < 1=d then P tk ¼ IN  bk Lsk ; k 2 Nþ , are nonnegative matrices with positive diagonal entries. Let  d be the minimum nonzero entry of Ptk ; k 2 Nþ , and

0
ln 1d12 =N 4h

;

then ð1  d2k =NÞe4hðtk tk1 Þ 6 ð1   d2 =NÞe4hc ¼ l < 1; k 2 Nþ . So the conditions (i)–(iii) in Theorem 1 are satisfied. Remark 3. From condition (iii) of Theorem 1, we know that if N is very large then tk  tk1 should be very small. This is the weakness of our method based on algebraic graph theory and stochastic matrix theory. Remark 4. From the proof of Theorem 1, we know that when the consensus is achieved under the impulsive control protoPN i x ðtÞ . col, the agents will converge to their center  xðtÞ ¼ i¼1 N Remark 5. Theorem 1 is also applicable to the case that the topology of the network is fixed, i.e. GðtÞ ¼ G for time t. 5. Simulations In this section, two numerical examples are provided to show the effectiveness of our theoretical results. Example 1. Consider the following networked nonlinear oscillators, which consists of four Duffing systems,

x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ þ ui ðtÞ; i

i ¼ 1; 2; 3; 4;

ðxi1 ; xi2 ÞT ; f ðxi ðtÞ; tÞ

ð13Þ ðxi1 ðtÞÞ3

ðxi2 ðtÞ; xi1 ðtÞ

T

#xi2 ðtÞ

i

where x ¼ ¼   þ m cosðxtÞÞ . When # = 0.25, m = 0.4, x = 1 and u (t) = 0, the system (13) is chaotic. From Fig. 1, the Laplacian matrices of graphs G1 ; G2 ; G3 are obtained as follows

0

1

B 0 B L1 ¼ B @ 0 1

1

0

1

1

0

1

0

0

0

1

0 C C C; 1 A 1

0

2

1 1

B 0 B L2 ¼ B @ 1

1

1

0

2

1

0

0

0

1

0 C C C; 1 A 1

0

2

B 0 B L3 ¼ B @ 1

1 1 2 0

1 1

0

1

1 1 C C C: 2 1 A 0

2

We define a switching signal r : [t0, +1) ? {1, 2, 3}, r(t) = ((k  1) mod 3)+ 1, t 2 [tk1, tk). For simplicity, choose the impulsive constants 0 < bk ¼ p ¼ 0:33 < 1; k 2 Nþ , and the equidistant impulsive interval Dtk = tk  tk1 = c = 0.0006. It is easy to check that dk ¼ 0:33; ð1  ðdk Þ2 =NÞe4hðtk tk1 Þ ¼ ð1  ð dÞ2 =NÞe4hc ¼ l ¼ 0:9982 < 1, where  d ¼ 0:3; h ¼ 10:7694. Thus the conditions (i)–(iii) in Theorem 1 are satisfied. From Theorem 1, we know that the consensus is achieved. The initial values are chosen as x1(0) = (1.8 1.5)T, x2(0) = (1.8 1.5)T, x3(0) = (1.6 1)T, x4(0) = (0.5 0.5)T. Simulation results are shown in Figs. 2–4.

Fig. 1. Schematic representation of Gi ; i ¼ 1; 2; 3.

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H. Jiang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

4

e2,1(t) 1

3,1

e1 (t)

2

e4,1(t) 1

0

−2

0

0.005

0.01

Time t Fig. 2. The time histories of ei;1 1 ðtÞ; i ¼ 2; 3; 4.

1

0 2,1

e2 (t)

−1

3,1 (t) 2

e −2

e4,1(t) 2

−3

0

0.005

0.01

Time t Fig. 3. The time histories of ei;1 2 ðtÞ; i ¼ 2; 3; 4.

2

x1(t) 2

x (t)

1.5

3

x (t) x4(t)

i x 2(t)

1

x−(t)

0.5 0

−0.5 −1 −1.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x i (t) 1 Fig. 4. The phase graph of x1(t), x2(t), x3(t), x4(t) and  xðtÞ.

The simulation results show that the consensus is achieved under the proposed impulsive control protocol. Let  xðtÞ ¼ ðx1 ðtÞ þ x2 ðtÞÞ=2, from Fig. 4 the agents all converge to  xðtÞ.

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385

Fig. 5. Schematic representation of Gi ; i ¼ 1; 2; 3; 4.

0

i,1 1

e (t),i=2,...,10

5

−5

−10

0

0.5

Time t

1 −3

x 10

Fig. 6. The time histories of ei;1 1 ðtÞ; i ¼ 2; 3; . . . ; 10.

2

ei,1(t),i=2,...,10

2

0

−2

−4

0

0.5

Time t

−3

1

x 10

Fig. 7. The time histories of ei;1 2 ðtÞ; i ¼ 2; 3; . . . ; 10.

Example 2. The chaotic Chua’s circuit is used as agents of the networked nonlinear oscillators. The state equation of agent i is

8 i  i  i i i > < x_ 1 ðtÞ ¼ g x1 ðtÞ þ x2 ðtÞ  lðx1 ðtÞÞ þ u1 ðtÞ; i i i i x_ 2 ðtÞ ¼ x1 ðtÞ  x2 ðtÞ þ x3 ðtÞ þ u2 ðtÞ; > : i x_ 3 ðtÞ ¼ bxi2 ðtÞ þ ui3 ðtÞ;

ð14Þ

  i where lðxi1 ðtÞÞ ¼ bx1 ðtÞ þ 0:5ða  bÞ jxi1 ðtÞ þ 1j  jxi1 ðtÞ  1j ; i ¼ 1; 2; . . . ; 10. When g = 10, b = 18, a = 4/3, b = 3/4 and i u (t) = 0, the system (14) is chaotic. The graphs Gi ; i ¼ 1; 2; 3; 4, are shown in Fig. 5 [4]. We define a switching signal r : [t0, +1) ? {1, 2, 3, 4}, r(t) = ((k  1) mod 4) + 1, t 2 [tk1, tk). For simplicity, choose the impulsive constants bk ¼ p ¼ 0:2; k 2 Nþ , and the equidistant impulsive interval Dt k ¼ t k  t k1 ¼ c ¼ 0:00003; k 2 Nþ . It is easy to check that ð1  ðdk Þ2 =NÞe4hðtk tk1 Þ 6 ð1  ð dÞ2 =NÞe4hc ¼ l ¼ 0:9999 < 1, where  d ¼ 0:2; h ¼ 32:4096. Thus the

386

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3

ei,1(t),i=2,...,10

10

5

0

−5

0

0.5

Time t

−3

x 10

1

Fig. 8. The time histories of ei;1 3 ðtÞ; i ¼ 2; 3; . . . ; 10.

conditions (i)–(iii) in Theorem 1 are satisfied. The initial values are randomly chosen in the interval [3, 3]. Simulation results are shown in Figs. 6–8. The simulation results show that the impulsive control protocol is efficient to solve the consensus problem for the networks with switching topologies.

6. Conclusions In this paper, we investigate the problem of impulsive consensus of multi-agent systems, where each agent can be modeled as an identical nonlinear oscillator. Based on the local information of agents, an impulsive control protocol is designed for directed networks with switching topologies. Sufficient conditions are given to guarantee the impulsive consensus of the networked nonlinear oscillators. Furthermore, how to select the discrete instants and impulsive constants is also presented. Two numerical examples are provided to show the effectiveness of our theoretical results. The impulsive control protocol is an effective method in some cases when the multi-agent systems cannot endure continuous control protocol or it is impossible to give continuous control protocol. However the proposed method is not very applicable for the networks whose size is very large and the method for large networks deserves further study. In our future, we will consider the impulsive consensus problem of multi-agent systems with stochastic topologies. Acknowledgements The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that have helped to improve the presentation of the paper. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10972091, 20976075, 10872080), the Natural Science Foundation of Jiangsu Province of China (Grant Nos. BK2010292, BK2010293), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant Nos. 09KJB510018, 10KJB510026) and the Qin Lan Project of the Jiangsu Higher Education Institutions of China. References [1] Vicsek T, Czirok A, Jacob EB, Cohen I, Schochet O. Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 1995;75:1226–9. [2] Jabdabaie A, Lin J, Morse AS. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Automat Control 2003;48:988–1001. [3] Olfati-Saber R, Murray R. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Automat Control 2004;49:1520–33. [4] Olfati-saber R, Fax J, Murray R. Consensus and cooperation in networked multi-agent systems. Proc IEEE 2007;95:1–17. [5] Tian YP, Liu CL. Consensus of multi-agent systems with diverse input and communication delays. IEEE Trans Automat Control 2008;53:2122–8. [6] Wang JH, Cheng DZ, Hu XM. Consensus of multi-agent linear dynamic systems. Asian J Control 2008;10:144–55. [7] Ma CQ, Zhang JF. Necessary and sufficient conditions for consensusability of linear multi-agent systems. IEEE Trans Automat Control 2010;55:1263–8. [8] Scardovi L, Sepulchre R. Synchronization in networks of identical linear systems. Automatica 2009;45:2557–62. [9] Huang T, Li C, Yu W, Chen G. Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback. Nonlinearity 2009;22:569–84. [10] Huang T, Li C. Chaotic synchronization by the intermittent feedback method. J Comput Appl Math 2010;234:1097–104. [11] Porfiri M, Fiorilli F. Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments. Chaos Soliton Fract 2009;41:245–62. [12] Porfiri M, Pigliacampo R. Master-slave global stochastic synchronization of chaotic oscillators. SIAM J Appl Dyn Syst 2008;7:825–42. [13] Ren W, Beard RW. Consensus seeking in multi-agent systems using dynamically changing interaction topologies. IEEE Trans Automat Control 2005;50:655–61. [14] Hong YG, Gao LX, Cheng DZ, Hu JP. Lyapunov-based approach to multiagent systems with switching jointly connected interconnection. IEEE Trans Automat Control 2007;52:943–8. [15] Porfiri M, Stilwell DJ. Consensus seeking over random weighted directed graphs. IEEE Trans Automat Control 2007;52:1767–73. [16] Porfiri M, Stilwell DJ, Bollt EM. Synchronization in random weighted directed networks. IEEE Trans Circuits Syst I 2008;55:3170–7. [17] Lin P, Jia Y. Average consensus in networks of multi-agents with both switching topology and coupling time-delay. Physica A 2008;387:303–13.

H. Jiang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 378–387

387

[18] Sun YG, Wang L, Xie G. Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Systems Control Lett 2008;57:175–83. [19] Zhang Y, Tian YP. Consentability and protocol design of multi-agent systems with stochastic switching topology. Automatica 2009;45:1195–201. [20] Belykh IV, Belykh VN, Hasler M. Blinking model and synchronization in small-world networks with a time-varying coupling. Phys D 2004;195:188–206. [21] Zhao J, Hill DJ, Liu T. Synchronization of complex dynamical networks with switching topology: a switched system point of view. Automatica 2009;45:2502–11. [22] Porfiri M, Fiorilli F. Node-to-node pinning-control of complex networks. Chaos 2009;19:013122. [23] Stilwell DJ, Bollt EM, Roberson DG. Sufficient conditions for fast switching synchronization in time varying network topologies. SIAM J Appl Dyn Syst 2006;5:140–56. [24] Jiang HB, Yu JJ, Zhou CG. Robust fuzzy control of nonlinear fuzzy impulsive systems with time-varying delay. IET Control Theory Appl 2008;2:654–61. [25] Han XP, Lu JA, Wu XQ. Synchronization of impulsively coupled systems. Int J Bifur Chaos 2008;18:1539–49. [26] Jiang HB, Bi QS. Contraction theory based synchronization analysis of impulsively coupled oscillators. Nonlinear Dyn 2011, doi:10.1007/s11071-0110026-2. [27] Yang T. Impulsive control theory. Berlin: Springer; 2001. [28] Li C, Liao X, Yang X, Huang T. Impulsive stabilization and synchronization of a class of chaotic delay systems. Chaos 2005;15:043103. [29] Chen YS, Chang CC. Impulsive synchronization of Lipschitz chaotic systems. Chaos Soliton Fract 2009;40:1221–8. [30] Zhang YP, Sun JT. Impulsive robust fault-tolerant feedback control for chaotic Lur’e systems. Chaos Soliton Fract 2009;39:1440–6. [31] Zheng YA, Chen G. Fuzzy impulsive control of chaotic systems based on TS fuzzy model. Chaos Soliton Fract 2009;39:2002–11. [32] Liu B, Liu BX, Chen G. Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans Circuits Syst I 2005;52:1431–41. [33] Cai S, Zhou J, Xiang L, Liu Z. Robust impulsive synchronization of complex delayed dynamical networks. Phys Lett A 2008;372:4990–5. [34] Guan ZH, Zhang H. Stabilization of complex network with hybrid impulsive and switching control. Chaos Soliton Fract 2008;37:1372–82. [35] Jiang HB. Hybrid adaptive and impulsive synchronization of uncertain complex dynamical networks by the generalized Barbalat’s lemma. IET Control Theory Appl 2009;3:1330–40. [36] Jiang HB, Yu JJ, Zhou CG. Consensus of multi-agent linear dynamic systems via impulsive control protocols. Int J Systems Sci 2011;42:967–76. [37] Jiang HB, Bi QS. Impulsive synchronization of networked nonlinear dynamical systems. Phys Lett A 2010;374:2723–9. [38] Guan ZH, Liu ZW, Feng G, Wang YW. Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control. IEEE Trans Circuits Syst I 2010;57:2182–95. [39] Horn RA, Johnson CA. Topics in matrix analysis. Cambridge: Cambridge University Press; 1991. [40] Horn RA, Johnson CA. Matrix analysis. Cambridge: Cambridge University Press; 1985. [41] Godsil C, Royle G. Algebraic graph theory. New York: Springer-Verlag; 2001. [42] Xiao L, Boyd S, Kim SJ. Distributed average consensus with least-mean-square deviation. J Parallel Distrib Comput 2007;67:33–46. [43] Nedic A, Olshevsky A, Ozdaglar A, Tsitsiklis JN. On distributed averaging algorithms and quantization effects. IEEE Trans Automat Control 2009;54:2506–17.