Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeB18.6
Synchronization of complex dynamical networks via distributed impulsive control Zhi-Wei Liu, Zhi-Hong Guan, Yan-Wu Wang and Rui-Quan Liao Abstract— This paper studies synchronization problems of complex dynamical networks (CDNs) via distributed impulsive control. The concept of control topology is introduced to describe the whole controller structure, which is constituted by some directed connections between nodes. The control topology can be designed either to be the same with the coupling topology, or to be independent to the intrinsic network topology. Based on the concept of control topology, the impulsive controller is designed to ensure the synchronization of the CDNs. Illustrated examples are given to show the effectiveness of the distributed impulsive control strategy.
I. INTRODUCTION A complex dynamical network (CDN) is a large set of interconnected nodes, in which each node represents an individual element in the network and edges represent the relations between them. Models of CDNs have been widely used to describe many natural and man-made systems, such as electrical power grids, the internet, food webs, VLSI circuits, social networks, and so on [1]-[3]. In recent years, synchronization and control of complex dynamical networks has become an active area of research and attracted much attention [5]-[7], [24]-[26] from multidisciplinary researchers in a wide range including statistical physics, applied mathematics, system control theory and computer science. Synchronization problem have a long history, and the first study on synchronization phenomenon was made by Huygens [4] in the 17th century. Network synchronization phenomena has been found in different forms both in nature and man-made systems, such as fireflies in the forest, applause, description of the heart, distributed computing system, routing messages in the internet, chaos-based communication network, and so on [24]-[26]. A wide variety of synchronization criteria have been derived for CDNs with different special features, such as time varying [8], [9], time delay [10]-[14], impulsive effects [15], or switching phenomena [16], [17], etc. Various control schemes, such as adaptive control [10], [18], impulsive control [19], [20], and pinning control [21]-[23], were reported to realize the network synchronization. Some of control schemes [10], [18], [20] are based on a solution of the homogenous system, This work was supported by the National Natural Science Foundation of China under Grants 60834002, 60873021, 60704035, the International Science and Technology Cooperative Project under Grant 2008DFA12150, and the Program for New Century Excellent Talents in University. Zhi-Wei Liu, Zhi-Hong Guan and Yan-Wu Wang are with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China. Rui-Quan Liao is with the Petroleum Engineering College, Yangtze University, Jingzhou, 420400, P. R. China. Corresponding Author:
[email protected] (Z.H. Guan).
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
which leads to the difficulty for all nodes to obtain the state information of an isolated node. The other control strategies [14], [19] are based on the weight average of the state of all nodes. It may lead to the difficulty for every node to obtain the weight average, due to the large scale of the real CDNs. Consequently, it is more reasonable to realize the network synchronization by utilizing the information from neighborhood, or by utilizing the information from a small number of nodes. Lu [27] studied the network synchronization by designing an adaptive coupling for the undirected network without intrinsic coupling. Zhang et al. [19] proposed an impulsive control scheme, in which each distributed control share information from partial nodes by taking the weights of great majority nodes as zero. In this paper, in order to avoid the implement difficulty of the controller, the conception of control topology is introduced. By design different control topology according to different practical situation, the synchronization controller will be easy to implement in practice. Motivated by the aforementioned comments, this paper investigates the problem of distributed impulsive synchronization of CDNs. Based on the stability results for impulsive systems, sufficient conditions for global synchronization are derived. Different from previous relative literatures, in the presented control scheme, each distributed controller is allowed to use the state information from different nodes. The concept of control topology is introduced to describe of the control information flow, which are transmitted between nodes. By appropriately choosing the control topology, the synchronization control schema can be fairly easy to implement in practice. Two particular control topologies are discussed. Numerical simulations are given to demonstrate the effectiveness of the control scheme. An outline of this paper is as follows. In section II, the model of CDNs is introduced and some necessary mathematical preliminaries are given. In section III, an impulsive control strategy, which can be expressed by control topology, are presented and some synchronization criteria are derived. Two kinds of particular control topologies are introduced in section IV. In section V, some numerical simulations are given. Concluding remarks are finally stated in Section VI. II. P RELIMINARIES First of all, some mathematical notations used throughout this paper are presented. R denotes the set of real numbers, Rn = R × R × . . . × R. For x ∈ Rn , xT denotes its trans{z } | n p pose. The vector norm used will be kxk = (xT x). Rn×m
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WeB18.6 are the set of real matrices. For A ∈ Rn×n , λmax (A) and λmin (A) are the maximum and the minimum eigenvalues of ¡ ¢1/2 , A, the spectral norm used will be kAk = λmax (AT A) µ2 (A) = λmax (A+AT )/2. In is the identity matrix of order n, and N = {0, 1, 2, 3 . . . }. For v(t) : R+ → R, denote v(t + h) − v(t) D v(t) = lim+ sup . h h→0
ˆ ˆ Γ = Aˆ ⊗ Γ and M = M ⊗ In . Furthermore, AM, where A if M is a (N − 1) × N matrix, then M GM = IN −1 . Denote the function that maps A to Aˆ by SM , i.e., the function is defined as Aˆ = SM (A) = M AGM .
+
n×m
p×q
For A = (aij ) ∈ R , B = (bij ) ∈ R , product of A and B is defined as a11 B · · · a1m B .. .. A⊗B = . . an1 B
anm B
the Kronecker
.
T (ε) denotes the set of matrices with real entries such that the sum of the entries in each row is equal to the real number ε. The set M1 and M2 is defined as follow: M1 is composed of matrices with N columns; each row of M ∈ M1 contains zeros and exactly one 1 and one −1. M2 ⊆ M1 and if M ∈ M2 , for any pair of indices i and j there exist indices i1 , i2 , . . . il with i1 = i and il = j such that for all 1 ≤ q < l, Mp,iq 6= 0 and Mp,iq+1 6= 0 for some p. Denote Miki1 (Miki2 ) as the first (second) nonzero element in ith row, where M ∈ M2 . A. Model Description Consider a CDN consisting of N linearly coupled identical nodes, which is described by x˙ i (t) = Axi (t) + g (t, xi (t)) + c
N X
Lemma 2: [28] For any matrices A, B, C and D, the Kronecker product has the following properties: 1) (µA) ⊗ B = A ⊗ (µB), where µ is a constant 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C 3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) 2 Lemma 3: [12][26] Let d(x) = k(M ⊗ In )xk , M ∈ M2 , d(x) → 0 then if and only if kxi − xj k → 0 for all i, j = 1, 2, . . . N . III. D ISTRIBUTED I MPULSIVE S YNCHRONIZATION OF CDN S In this section, based on the stability results for impulsive systems, a new distributed impulsive control scheme is proposed and several novel criteria are derived for the synchronization of network (1). A. Control topology In order to achieve the synchronization of (1), the distributed impulsive controller, for ith node, is designed as: ui =
bij Γxj (t) + ui , (1)
j=1
where i = 1, 2, . . . N , xi = (xi1 , xi2 , . . . , xin )T ∈ Rn is the state vector of the ith node, A ∈ Rn×n is the constant matrix, the constant c > 0 denotes the coupling strength, Γ ∈ Rn×n represents the inner coupling matrices of network, ui is a control input, g : [0, +∞) × Rn → Rn is continuous map satisfying kg(t, x) − g(t, y)k ≤ K kx − yk , ∀x, y ∈ Rn ,
(2)
B = (bij ) ∈ RN ×N is the coupling matrix , where bij defined as follows: if there is a connection from node j to node i (i 6= j), then, the coupling strength bij ≥ 0; otherwise, bij = 0, for i = j, bij is defined by: XN bii = − bij . (3) j=1,j6=i
B. Mathematic Preliminaries Definition 1: The network system 1 is globally exponentially synchronized, if there exist ε > 0, T > t0 and M > 0 such that kxi − xj k ≤ M e−εt for any initial condition, t > T , and i, j = 1, 2, . . . N. Inspired by [12][26], the following lemma is given. Lemma 1: Let M ∈ M2 be a P ×N matrix and A ∈ T (ε) be a N × N matrix. Then there exists a N × P matrix GM ˆ , where Aˆ = M AGM . Moreover, let such that M A = AM Γ be a n×n constant matrix and AΓ = A⊗Γ, then MAΓ =
∞ X
[Ck (xi (t− k)−
N X
ξij xj (t− k ))]δ(t − tk ), k ∈ N,
j=1
k=1
(4) ) = lim where i = 1, 2, · · · N , xi (t− x (t), the impulsive i k − t→tk
∞
instant sequence {tk }k=1 satisfies 0 ≤ t1 < t2 < · · · < tk < · · · , lim tk = ∞ , Ck ∈ Rn×n are the control gains, k→∞
δ(•) is the Dirac impulsive function. ξij ≥ 0 denote the influence PNweight of the state information of jth node on ith node, j=1 ξij = 1 for all i = 1, 2, · · · , N , ξij is designed as follow: if ith node can obtain the information of jth node, then ξij ≥ 0 ; otherwise, ξij = 0. Denote the matrix ξ = (ξij ). Remark 1: The distributed impulsive control (4) utilizes the local state information and the weighted average of available state information of other nodes. This is quite different from those impulsive controllers in [19], [20] and is practically much easier to implement because the controller utilizes only the available information. The control information flow of the controller (4) can be regarded as a weighted directed graph. In this paper, the weighted directed graph is defined as the control topology of the controller (4). The Laplacian matrix of the control topology is 1 − ξ11 −ξ12 · · · −ξ1N −ξ21 1 − ξ22 · · · −ξ2N (5) ϑ= . .. .. .. .. . . . .
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−ξN 1
−ξN 2
···
−ξN N
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Thus, the (4) can be equivalently expressed by (5) as follows: ui =
∞ X
k=1
Ck
N X
ϑij xj (t− k )δ(t
− tk )), k ∈ N,
(6)
j=1
where i = 1, 2, 3 . . . N. Hence, the impulsively controlled CDNs (1) can then be described by the following impulsive differential equations: N P bij Γxj (t), x˙ i (t) = Axi (t) + g(t, xi (t)) + c j=1 t 6= tk , k ∈ N, t ≥ 0, (7) N P ϑij xj (t− ∆xi (tk ) = xi (t+ ) − xi (t− ) = Ck ), k k k j=1 t = tk ,
− where i = 1, 2, . . . , N , ∆xi (tk ) = xi (t+ k ) − xi (tk ) is the jump of the state variable at the time instant tk , where xi (t+ k ) = lim+ xi (t). Assume that x(t) is right-hand contint→tk
where x = (xT1 , xT2 , . . . xTN )T ∈ R , BΓ ˜ (t, x(t)) c(B ⊗ Γ), AN = (IN ⊗ A), G ¡ T ¢T T T g (t, x1 (t)) , g (t, x2 (t)) , . . . , g (t, xN (t)) , ϑn = ϑ ⊗ In , CN k = IN ⊗ Ck .
According to (2), PP 2 i=1 kxki1 (t) − xki2 (t)k kg(t, xki1 (t)) − g(t, xki2 (t))k PP 2 ≤ 2K i=1 kyi (t)k = 2KV (t) Then,
¯ ¯ V˙ (t)¯
≤ βk V (t− k)
Therefore, for any t ∈ (tk , tk+1 ], Q −λ(t−t0 ) V (t) ≤ V (t+ 0 )e
βk t0
