Pinning Synchronization of Directed Networks With ... - IEEE Xplore

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 12, DECEMBER 2015

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Pinning Synchronization of Directed Networks With Switching Topologies: A Multiple Lyapunov Functions Approach Guanghui Wen, Member, IEEE, Wenwu Yu, Member, IEEE, Guoqiang Hu, Member, IEEE, Jinde Cao, Senior Member, IEEE, and Xinghuo Yu, Fellow, IEEE

Abstract— This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from M-matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results. Index Terms— Complex network, directed spanning tree, M-matrix, neural network, pinning synchronization.

I. I NTRODUCTION

M

ANY systems in nature and human society can be described by models of complex networks, where nodes represent the elements of the systems and links mimic the interactions among them. Prototypical examples including protein interaction networks, World Wide Web, smart grids, biological neural networks, citation networks, and so on. Manuscript received October 12, 2014; revised February 22, 2015 and May 18, 2015; accepted May 31, 2015. Date of publication June 30, 2015; date of current version November 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61304168 and Grant 61322302, in part by the Natural Science Foundation of Jiangsu Province, China, under Grant BK20130595, in part by the Research Fund for the Doctoral Program of Higher Education of China under Grant 20110092120024, in part by the Six Talent Peaks of Jiangsu Province, China, under Grant 2014-DZXX-004, and in part by the Australian Research Council through the Discovery Scheme under Grant DP140100544. G. Wen and W. Yu are with the Department of Mathematics, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]). G. Hu is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). J. Cao is with the Department of Mathematics, Southeast University, Nanjing 210096, China, and also with the Department of Mathematics, Faculty of Science, King Abdulazez University, Jeddah 21589, Saudi Arabia (e-mail: [email protected]). X. Yu is with the School of Electrical and Computer Engineering, RMIT University, Melbourne, VIC 3001, Australia, and also with the School of Automation, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2443064

Ubiquity of complex networks leads to a fascinating set of scientific problems concerning how network topology facilitates or impedes the emergence of network behavior [1]–[5]. With the development of sensor and computing technology, extensive empirical studies have been performed on real complex networks to understand the evolution principles for nontrivial statistical properties exhibited by them, such as the small-world effect [6] and the scale-free feature [7]. Beyond exploring the underlying mechanisms for these statistical properties, researchers have made substantial progress on analysis and synthesis of the dynamical behavior taking place on complex networks. One critical issue within this context is to design some coupling laws such that states of all the nodes in the considered network can achieve synchronization [8]–[11]. As one of the basic motions in nature, the synchronization behavior has been observed for a quite long time. It has been demonstrated that the study of synchronization in complex networks is very important for controlling various practical networks. For example, the velocity synchronization is a prerequisite in solving the flocking control problem of multiagent dynamical systems [12]. Furthermore, the formation flying problem of multiple aircraft can be transformed into synchronization problem using a linear transformation technique [13]. In addition, synchronization is an important collective behavior for the neuronal networks [14]–[16]. Note that synchronization in complex networks can be generally categorized into local synchronization [17], [18] and global synchronization [19]. Compared with the local synchronization, the global synchronization means that the network synchronization can be achieved for any given initial conditions, which is more favorable in practical applications. In [19], a distance between the nodes’ states and the synchronization manifold was introduced, based on which a new methodology was proposed to investigate the global synchronization of coupled systems. Then, a general algebraic connectivity was proposed in [20] to study the global synchronization as well as consensus problems in strongly connected networks. Global synchronization for coupled linear systems via state or output feedback control was studied in [21]. In [22] and [23], global synchronization for a class of complex networks with sampling-data coupling was addressed. For the case that the considered networks are not strongly connected or even do not contain any directed spanning tree, the pinning

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synchronization problem arises [24]–[27]. Pinning synchronization in scale-free and small-world complex networks was addressed in [24] and [25], respectively. Then, local and global pinning synchronization in random and scale-free networks were studied in [26]. In [27], pinning synchronization of undirected complex networks was further addressed. Without assuming the network topology is undirected or strongly connected, it was proved in [28] that a single controller can pin a coupled complex network to its homogenous trajectory under some suitable conditions. Global pinning synchronization for a class of complex networks has been investigated in [29] under a V-stability framework. More recently, pinning cluster synchronization was addressed in [30] via an adaptive control approach. Note that research on global pinning synchronization in complex networks is still ongoing today, such as pinning controllability analysis [31], pinning synchronization in some special kinds of coupled chaotic systems [32], pinning synchronization over general complex networks [33], and switching complex networks [34], [35]. Obviously, a variety of complicated factors may lead to the switching phenomenon of network topology, with the following two being the foremost: 1) the limitations of sensor hardware and 2) the external interferences caused by fading-channel, stochastic disturbances, and so on. Understanding the pinning synchronization behavior over switched networks is, thus, practically important. However, it is previously assumed in the aforementioned literature that each possible network topology contains a directed spanning tree with the target system being the root node. This indicates that each node in the considered network can be influenced by the target system directly or indirectly. In some real cases, the abovementioned condition is hard or even impossible to verify. Motivated by the aforementioned works on pinning synchronization of complex networks, this paper aims to solve the global pinning synchronization problem for a class of switched complex networks where some possible network topologies may not contain any directed spanning tree. To the best of our knowledge, this challenging problem is still unsolved. Using a combined tool from M-matrix theory and stability analysis of switched systems, a new kind of topologydependent multiple Lyapunov functions for the switched networks is constructed. Theoretical analysis indicates that the global pinning synchronization in such a complex network can be achieved if some carefully selected nodes are pinned such that the network topology contains a directed spanning tree rooted at the target node frequently enough as the network evolves with time. The rest of this paper is structured as follows. Some preliminaries are provided in Section II. Problem is formulated in Section III. In Section IV, the main theoretical results are derived and discussed. Some numerical simulations are given in Section V. Finally, the conclusions are drawn in Section VI. Throughout this paper, let R and N be the sets of real and positive natural numbers, respectively. Let Rn×n be the sets of real matrices. The superscript T is the transpose for real matrices. In is the n × n identity matrix. Symbol diag{δ1 , δ2 , . . . , δn } is a diagonal matrix with diagonal

elements δ1 , δ2 , . . . , δn . Symbols 0n and ∅ represent the n-dimensional column vector with each element being 0 and the empty set, respectively. For real symmetric matrices A and B, A > B means that A − B is positive definite. A column vector δ = (δ1 , δ2 , . . . , δn )T ∈ Rn is said to be positive if every entry δi > 0, 1 ≤ i ≤ n. Notations ⊗ and · denote the Kronecker product and Euclidean norm, respectively. Matrices are assumed to have compatible dimensions if not explicitly stated. II. P RELIMINARIES In this section, some useful preliminaries on algebraic graph theory and matrix theory are provided. A. Algebraic Graph Theory Let G(V, E, A) be a digraph with the set of nodes V = {υ1 , υ2 , . . . , υ N }, the set of directed links E ⊆ V × V, and an adjacency matrix A = [ai j ] N×N with ai j ≥ 0 for all i, j = 1, 2, . . . , N. A link ei j in G(V, E, A) is denoted by the ordered pair of nodes (υ j , υi ), where υ j and υi are called the parent and child nodes, respectively, and ei j ∈ E if and only if ai j > 0. Furthermore, only simple graph is considered in this paper, i.e., self-loops and multiple links are not allowed in G(V, E, A). For the sake of brevity, denote G(V, E, A) by G(A) if no confusion will occur. A path from node υi to υ j (υi = υ j ) is a sequence of directed links, (υi , υk1 ), (υk1 , υk2 ), . . . , (υkl , υ j ), with distinct nodes υkm , m = 1, 2, . . . , l. A digraph is strongly connected if it contains at least two nodes and for each distinct node pair (υi , υ j ), there exist a directed path from υi to υ j and another directed path from υ j to υi . The Laplacian matrix L = [li j ] N×N of G(A) is defined as li j = −ai j , i = j , N and lii = k=1, k =i aik , for i = 1, 2, . . . , N. For notational brevity, denote a time-varying digraph G(V(t), E(t), A(t)) by G(A(t)). Lemma 1 [36]: Suppose that the digraph G(A) contains a directed spanning tree. Then, 0 is a simple eigenvalue of its Laplacian matrix L, and all the other eigenvalues of L have positive real parts. B. Matrix Theory In this section, the following definitions and lemmas are introduced. Definition 1 [37]: Let Z N = {A = [ai j ] N×N ∈ R N×N : ai j ≤ 0 if i = j, i, j = 1, 2, . . . , N} denotes the set of real matrices whose off-diagonal elements are all nonpositive. Definition 2 [37]: A matrix A ∈ R N×N is called a nonsingular M-matrix if A ∈ Z N and all the leading principal minors of A are positive. Lemma 2 [37]: Suppose that matrix A = [ai j ] N×N ∈ R N×N has ai j ≤ 0, for all i = j, i, j = 1, . . . , N. Then, the following statements are equivalent. 1) A is a nonsingular M-matrix. 2) There exists a positive definite diagonal matrix = diag{φ1 , . . . , φn } ∈ R N×N such that A T +A > 0. 3) All the eigenvalues of A have positive real parts. Remark 1: Note that the Laplacian matrix of an arbitrarily given digraph G(A) with order N is a singular matrix, belonging to Z N .

WEN et al.: PINNING SYNCHRONIZATION OF DIRECTED NETWORKS WITH SWITCHING TOPOLOGIES

Lemma 3 [38]: Suppose that P ∈ R N×N is a positive definite matrix and M ∈ R N×N is symmetric. Then, for any vector x ∈ R N , the following inequality holds: λmin (P −1 M)x T Px ≤ x T M x ≤ λmax (P −1 M)x T Px where λmin (P −1 M) and λmax (P −1 M) are the minimum and maximum eigenvalues of P −1 M, respectively. III. P ROBLEM F ORMULATION In this section, the global pinning synchronization problem of switched complex networks is formulated. Suppose that the considered complex network consists of N nodes, the dynamics of the i th node are given by x˙i (t) = f (x i (t), t) + α

N

ai j (t)(x j (t) − x i (t))

(1)

j =1

where x i (t) = (x i1 (t), x i2 (t), . . . , x in (t))T ∈ Rn for i = 1, 2, . . . , N represent the states of the i th node, α > 0 is the coupling strength, and A(t) = [ai j (t)] N×N is the adjacency matrix of digraph G(A(t)) at time t. Throughout this paper, the derivatives of all functions at switching time points should be considered as their right-hand derivatives. According to the definition of Laplacian matrix for a graph, it follows from (1) that: x˙i (t) = f (x i (t), t) − α

N

li j (t)x j (t)

(2)

j =1

where L(t) = [li j (t)] N×N is the Laplacian matrix of digraph G(A(t)). The control goal in this paper is to design pinned controllers for some appropriately selected nodes in (2) such that the states of each node in the considered network will approach s(t) when t approaches +∞, i.e., limt →∞ x i (t)−s(t) = 0, for all i = 1, 2, . . . , N and arbitrarily given initial conditions where s˙ (t) = f (s(t), t).

(3)

Here, s(t) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Motivated in [26], [27], and [34], pinning network (2) using linear controllers −αci (t)(x i (t) − s(t)) to its i th node leads to x˙ i (t) = f (x i (t), t) − α

N

li j (t)x j (t) − αci (t)(x i (t) − s(t))

j =1

(4) where ci (t) ∈ {0, 1} and ci (t) = 1 if and only if the i th node of (2) is pinned at time t. Let ei (t) = x i (t) − s(t), i = 1, 2, . . . , N. It, thus, follows from (4) that: e˙i (t) = f (x i (t), t) − f (s(t), t) −α

N

li j (t)e j (t)−αci (t)ei (t).

j =1

(5) By taking the target system (3) as a virtual leader node of the complex network under consideration, one may get the ˜ augmented network topology G(A(t)) consisting of N + 1

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nodes. Labeling the index of the virtual node as N + 1, the ˜ Laplacian matrix L(t) of the augmented network topology ˜ G(A(t)) can be partitioned as ˆ ˜L(t) = L(t) c(t) ∈ R(N+1)×(N+1) (6) 0TN 0 ˆ where c(t) = (c1 (t), c2 (t), . . . , c N (t))T , L(t) = L(t) + diag{c1 (t), c2 (t), . . . , c N (t)}, and L(t) is the Laplacian matrix of graph G(A(t)). In this paper, the augmented network ˜ G(A(t)) does not need to contain a directed spanning tree all the time. Note that the initial network topologies of practical complex networks always meet some connectivity conditions. For example, in the flocking control problem, as studied in [39], the initial network topology is assumed to be connected. In the context of pinning synchronization, the ˜ condition that the augmented network topology G(A(0)) contains a directed spanning tree can be ensured by appropriately pinning some nodes in network G(A(0)) selected using the following linear time complexity algorithm. Algorithm 1: Find the strongly connected components and the node with zero in-degree in G(A(0)) by using Tarjan’s algorithm [40]. Suppose that there are ω˜ (ω˜ ≥ 0) strongly connected components, labeled as G1 , G2 , . . ., Gω˜ , and ωˆ (ωˆ ≥ 0) nodes with zero in-degree, labeled as ν1 , ν2 , . . ., νωˆ , in G(A(0)). Set κ0 = 0 and m = 1. 1) Check whether the following condition holds: ωˆ > 0. If it does not hold, go to step 2); else, pin all the ωˆ nodes with zero in-degree and update the value ˆ of κ0 by κ0 = κ0 + ω; 2) Check whether the following condition holds: ω˜ > 0. If it holds, go to step 3); else stop. 3) Check whether there exists at least one node belonging to Gm which is reachable from at least one node belong˜ j = m. If it holds, go to ing to G j , j = 1, 2, · · · , ω, step 4); if it does not hold, go to step 5). 4) Check whether the following condition holds: m < ω. ˜ If it holds, let m = m + 1 and re-perform step 3); else stop. 5) Arbitrarily select one node in Gm to be pinned. Let κ0 = κ0 + 1. Check whether the following condition holds: m < ω. ˜ If it holds, let m = m + 1 and re-perform step 3); else stop. Remark 2: Note that the time complexity of Tarjan’s algorithm is O(N + |E(0)|), where N and |E(0)| are the order and size of G(A(0)), respectively, and O(·) is a Landau symbol representing the complexity function. It it not ˜ hard to verify that G(A(0)) will contain a directed spanning tree if the κ0 nodes selected by Algorithm 1 are pinned. It can also be concluded that κ0 is the minimal number of pinned vertices such that the augmented network topology contains a directed spanning tree. However, some links may be lost as the networked systems evolve with time. To solve such a pinning synchronization problem, it is previously assumed in some existing works that each possible network topology contains at least a directed spanning tree or the nodes could synchronously discard the incoming links when the network topology does not contain

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any directed spanning tree [35]. However, it is sometimes difficult or even impossible to implement such an information discarding scheme for complex networks especially for those with huge size. In this paper, the global pinning synchronization in switched complex networks will be addressed without assuming that each possible network topology has a directed spanning tree. Furthermore, the dynamic nodes in the considered switched networks do not need to discard the incoming links synchronously as the networks evolve with time. To proceed the analysis, it is assumed that the switching among the different topologies is triggered by communication links’ loss or recovery and there is no node that will be deleted from the network. In particular, it is assumed that the augmented network contains a directed spanning tree at the beginning and some links will be lost as the network evolves with time. The above statements indicate that there may exist some time ˜ intervals over which the augmented network G(A(t)) does not contain any directed spanning tree. However, the augmented network may contain a directed spanning tree again by some repairing efforts, although it may take a certain period of time in practice. For the convenience of analysis, suppose that there exists an infinite sequence of uniformly bounded nonoverlapping time intervals [tk , tk+1 ), k ∈ N, with t1 = 0, ι1 ≥ tk+1 − tk ≥ ι0 > 0, across which the interaction graph is time invariant. Here, the positive constant ι0 is called the dwell time. It can be verified that the Zeno behavior is excluded during the network’s evolution, i.e., tk → +∞ as k → +∞. The time sequence t1 , t2 , . . . , is called the switching sequence, at which the network topology changes. For the convenience of expression, one may introduce a switching signal σ (t) : [0,+∞) → {1, . . . , q}. ˜ Then, let G(A(t)) = G(A˜σ (t ) ) be the augmented interaction graph of network (4) at time t. Obviously, G(A˜σ (t ) ) ∈ Gˆ for all t ≥ 0, where Gˆ = {G(A˜1 ), . . . , G(A˜q )}, q > 1, denotes the set of all possible augmented directed interaction graphs. Furthermore, let G¯ = {G(A˜ ϑ1 ), . . . , G(A˜ϑ p )} be the set of all possible augmented interaction graphs containing at least one directed spanning tree with {ϑ1 , . . . , ϑ p } ⊆ {1, . . . , q} and {ϑ1 , . . . , ϑ p } = ∅. One then has that {ϑ1 , . . . , ϑ p } = {1, . . . , q} if and only if each possible augmented interaction graph contains at least one directed spanning tree rooted at the target node. According to the above analysis, assume that there exists an infinite sequence of uniformly bounded nonoverlapping time intervals [t¯ρ , t¯ρ+1 ), ρ ∈ N, with t¯1 = 0, ¯ι1 > t¯ρ+1 − t¯ρ > ¯ι0 , such that σ (t¯ρ ) ∈ G¯ for all ρ ∈ N. Note that, for each ρ ∈ N, there exists a k ∈ N such that tk = t¯ρ . Obviously, the augmented network G(A˜σ (t¯ρ ) ) contains at least one directed spanning tree with the node N + 1 being the ρ root. For the convenience of expression, let tmin = mins∈N ts subject to ts > t¯ρ , σ (ts ) = σ (t¯ρ ). Note that the time points ρ t¯ρ and tmin can be designed offline or determined online as the network evolves with time (see Fig. 1 for illustration where it is assumed that q = 4 and G¯ = {G(A˜ 2 ), G(A˜4 )}. It can be 1 thus obtained from Fig. 1 that t¯1 = 0, tmin = 1.5, t¯2 = 4, 2 3 tmin = 6, t¯3 = 7, and tmin = 8.4).

Fig. 1.

Time points tk , k ∈ N.

ρ For the case of tmin < t¯ρ+1 , it follows from the above ρ analysis that σ (t) ∈ P, for t ∈ [t¯ρ , tmin ) and σ (t) ∈ Q\P, ρ for t ∈ [tmin , t¯ρ+1 ), where P = {ϑ1 , ϑ2 , . . . , ϑ p }, Q = {1, 2, . . . , q}. It thus follows from (5) that: ρ e(t) ˙ = F(e(t), t)−α( Lˆ σ (t¯ρ ) ⊗ In )e(t), t ∈ t¯ρ , tmin ρ (7) e(t) ˙ = F(e(t), t)−α( Lˆ σ (t ) ⊗ In )e(t), t ∈ t , t¯ρ+1 min

for the case of

ρ tmin

< t¯ρ+1 , and

e(t) ˙ = F(e(t), t)−α( Lˆ σ (t¯ρ ) ⊗ In )e(t), t ∈ t¯ρ , t¯ρ+1 ρ tmin

(8)

for the case of = t¯ρ+1 , where e(t) = (e1 2 e N (t)T )T , F(e(t), t) = (( f (x 1 (t), t) − f (s(t), t))T , ( f (x 2 (t), t) − f (s(t), t))T , . . . , ( f (x N (t), t) − f (s(t), t))T )T , ˆ σ (t ) σ (t ) c L L˜ σ (t ) = is the Laplacian matrix of G(A˜σ (t ) ). 0TN 0 Obviously, e(t) = 0 Nn is a fixed point of switched systems (7). Furthermore, it can be verified that the global pinning synchronization in networks (2) with target trajectory generated by (3) will be achieved if and only if the zero equilibrium point of (5) is globally attractive. Thus, it is sufficient to show that global pinning synchronization in network (2) with target trajectory given in (3) could be ensured if the zero equilibrium point of (7) is globally asymptotically stable. It should be noted that the right hand of systems (8) is discontinuous due the time-dependent switching over different topologies. However, it can be verified that e(t) is differentiable on the right. Furthermore, for any given e(0), the switched systems (8) possess a unique and absolutely continuous solution e(t) in the sense of Carathéodory [41]. To derive the main results, the following assumption is needed. Assumption 1: There exists a positive semidefinite matrix ∈ Rn×n such that (t)T , e

(t)T , . . . ,

(x − x) ˜ T ( f (x(t), t) − f (x(t), ˜ t)) ≤ (x − x) ˜ T (x − x) ˜

(9)

for all x, x˜ ∈ Rn , and t ≥ 0. Note that Assumption 1 is satisfied by many wellknown systems such as the Chua’s circuit systems and the Lorenz systems. Furthermore, it is not hard to verify that Assumption 1 holds if the nonlinear function f : Rn × [0, +∞) → Rn satisfies the global Lipschitz condition. IV. T HEORETICAL A NALYSIS In this section, the main results are provided and discussed. Based on the analysis in Section III, one has that, for each s ∈ P, G(A˜ s ) contains a directed spanning tree rooted at node N + 1. Denote the Laplacian matrix of G(A˜ s ) by L˜ s . Without loss of generality, let s Lˆ cs L˜ s = (10) ∈ R(N+1)×(N+1) 0TN 0


where cs = (c1s , c2s , . . . , csN )T , cis ∈ {0, 1} and cis = 1 if node i in graph G(As ) is pinned, i = 1, 2, . . . , N. According to the condition that, for each s ∈ P, G(A˜s ) contains a directed spanning tree, it follows from Lemmas 1 and 2 that Lˆ s is a nonsingular M-matrix. Then, in [37, Th. 2.3, p. 134] and [42, Lemma 4], the following lemma can be established. Lemma 4: For each s ∈ P, there exists a positive vector s ξ = (ξ1s , ξ2s , . . . , ξ Ns )T ∈ R N , such that ( Lˆ s )T ξ s = 1 N ¯ s Lˆ s + ( Lˆ s )T ¯ s = diag{ξ s , ξ s , . . . , ξ s }, ¯ s > 0, where and 1 2 N Lˆ s are defined in (10). For notational brevity, one may let σ (t¯ρ )

τ0

ρ τ1

σ (t¯ ) σ (t¯ )

= λminρ ξminρ

(11)

τ˜1h

(12)

= minρ

tmin h∈Qsub

σ (t¯ ) ¯ σ (t¯ρ ) Lˆ σ (t¯ρ ) + where λminρ is the smallest eigenvalue of ( ¯ σ ( t ) σ (t¯ ) ¯ σ (t¯ρ ) ), ξminρ = mini=1,2,...,N ξi ρ , ( Lˆ σ (t¯ρ ) )T σ (t¯ρ )

ξ σ (t¯ρ ) = (ξ1 ρ tmin

σ (t¯ρ )

, ξ2

Qsub = {σ (t) : eigenvalue of Furthermore, let

σ (t¯ρ ) T )

, . . . , ξN

is defined in Lemma 4,

ρ t ∈ [tmin , t¯ρ+1 )}, ¯ σ (t¯ρ ) Lˆ h ¯ σ (t¯ρ ) )−1 ( (

κ=

max

i = j,i, j ∈P

τ˜1h is the smallest ¯ σ (t¯ρ ) ). + ( Lˆ h )T

i ξmax

(13)

j

ξmin

where ξ s = (ξ1s , ξ2s , . . . , ξ Ns )T is defined in Lemma 4, s s ξmin = minr=1,2,...,N ξrs , ξmax = maxr=1,2,...,N ξrs , for each s ∈ P. Based on the above analysis, one may get the following theorem which summarizes the main results of this paper. Theorem 1: Under Assumption 1, the global pinning synchronization in network (2) with target trajectory given in (3) could be achieved if there exists a positive scalar ε0 such that, for each ρ ∈ N, the following conditions hold. () 1) α > 2λmax σ (t ρ ) . 2)

γρ >

τ0 ρ − γρ (t ρ+1 −tmin )+lnκ+ε0 . ρ (tmin −t ρ )

σ (t¯ )

In the above conditions, γˆρ = ατ0 ρ − 2λmax (), ρ γ˜ρ = ατ1 − 2λmax (), λmax () is the largest eigenvalue of , σ (t¯ )

ρ

τ0 ρ and τ1 are, respectively, defined in (11) and (12), and κ is defined in (13). Proof: Note that global pinning synchronization in networks (2) with target trajectory generated by (3) is achieved if and only if the zero equilibrium point of switched systems (5) is globally attractive. Since, for each ρ ∈ N, the digraph G(Aσ (t¯ρ ) ) contains at least one directed spanning tree rooted at node N +1, it follows ¯ σ (t¯ρ ) Lemma 4 that there exists a positive definite matrix ¯ ¯ ¯ ¯ ¯ σ (tρ ) Lˆ σ (tρ ) + ( Lˆ σ (tρ ) )T ¯ σ (tρ ) > 0. Then, one may such that construct the following multiple Lyapunov functions for the switched systems (7) and (8): σ (t¯ρ ) ¯ ⊗ In e(t), t ∈ [t¯ρ , t¯ρ+1 ) (14) V (t) = e(t)T ρ

for all ρ ∈ N. For t ∈ [t¯ρ , tmin ) and an arbitrarily given ρ ∈ N, taking the time derivative of V (t) along the trajectories of

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systems (7) gives σ (t¯ρ ) ¯ ⊗ In F(e(t), t) − αe(t)T V˙ (t) = 2e(t)T

σ (t¯ ) σ (t¯ ) σ (t¯ ) T σ (t¯ ) ¯ ρ ⊗ In e(t). (15) ¯ ρ Lˆ ρ + Lˆ ρ × Based on the above analysis and by Assumption 1, one gets that σ (t¯ρ ) ¯ ⊗ e(t) − αe(t)T V˙ (t) ≤ 2e(t)T

σ (t¯ρ ) σ (t¯ρ ) σ (t¯ρ ) T σ (t¯ρ ) ¯ ¯ Lˆ ⊗ In e(t), × + Lˆ ρ t ∈ t¯ρ , tmin . (16) Using the properties of Kronecker product, it follows from (16) that: σ (t¯ ) (17) V˙ (t) ≤ 2λmax () − α τ˜0 ρ V (t) where λmax () is the largest eigenvalue of , σ (t¯ ) ¯ σ (t¯ρ ) )−1 ( ¯ σ (t¯ρ ) Lˆ σ (t¯ρ ) + τ˜0 ρ is the smallest eigenvalue of ( ¯ σ (t¯ρ ) ). Note that, for each ρ ∈ N, the ( Lˆ σ (t¯ρ ) )T σ (t¯ρ )

σ (t¯ ) σ (t¯ )

> λminρ ξminρ , σ (t¯ ) ¯ σ (t¯ρ ) Lˆ σ (t¯ρ ) + where λminρ is the smallest eigenvalue of ( ¯ ¯ σ ( t ) σ ( t ¯ σ (t¯ρ ) ), ξ ρ = mini=1,2,...,N ξ ρ ) , ξ σ (t¯ρ ) = ( Lˆ σ (t¯ρ ) )T min i

following inequality always holds: τ˜0

σ (t¯ )

σ (t¯ )

σ (t¯ρ ) T )

(ξ1 ρ , ξ2 ρ , . . . , ξ N then get that

is defined in Lemma 4. One may

ρ V˙ (t) ≤ −γˆρ V (t), t ∈ t¯ρ , tmin σ (t¯ )

σ (t¯ )

(18) σ (t¯ ) σ (t¯ )

where γˆρ = (ατ0 ρ − 2λmax ()), τ0 ρ = λminρ ξminρ , ρ ∈ N. It can be yielded from condition 1) that γˆρ > 0 for ρ ∈ N. ρ For the case of tmin = t¯ρ+1 , it can be directly obtained from (18) and condition 2) that − < e−ε0 V (t¯ρ ) (19) V (t¯ρ+1 ) ≤ κ V t¯ρ+1 where κ is defined in (13). ρ For the case of tmin < t¯ρ+1 , taking the time derivative of V (t) along the trajectories of systems (7) with ρ t ∈ [tmin , t¯ρ+1 ) yields σ (t¯ρ ) ¯ ⊗ In F(e(t), t) − αe(t)T V˙ (t) = 2e(t)T

σ (t¯ρ ) σ (t ) ¯ ¯ σ (t¯ρ ) ⊗ In e(t) Lˆ × + ( Lˆ σ (t ) )T ≤ 2λmax ()V (t)−αe(t)T

σ (t¯ ) σ (t ) ¯ ρ Lˆ ¯ σ (t¯ρ ) ⊗In e(t). × +( Lˆ σ (t ) )T ¯ σ (t¯ρ ) Lˆ σ (t ) +( Lˆ σ (t ))T ¯ σ (t¯ρ ) ) ⊗ In may Note that the matrix ( ρ not be positive definite for t ∈ [tmin , t¯ρ+1 ). However, it can be obtained from Lemma 3 that σ (t¯ρ ) σ (t ) σ (t ) T σ (t¯ρ )

¯ ¯ ⊗ In e(t) Lˆ + Lˆ −αe(t)T ≤ −α τ˜1σ (t ) V (t) ρ

σ (t )

for t ∈ [tmin , t¯ρ+1 ), where τ˜1 is the smallest eigenvalue of ρ ¯ σ (t¯ρ ) Lˆ σ (t ) + ( Lˆ σ (t ) )T ¯ σ (t¯ρ ) )−1 ( ¯ σ (t¯ρ ) ). Set Qtmin = {σ (t) : ( ρ

sub

ρ min is a proper subset of Q. t ∈ [tmin , t¯ρ+1 )}, one has that Qsub t

3244


ρ

Then, one may choose τ1 = min statements, one has

t

ρ

min i∈Qsub

τ˜1i . Based on the above

ρ V˙ (t) ≤ −γ˜ρ V (t), for t ∈ tmin , t¯ρ+1

(20)

ρ

where γ˜ρ = (ατ1 − 2λmax ()). Then, for the case of ρ tmin < t¯ρ+1 , it can be obtained from (20) and condition 2) that − < e−ε0 V (t¯ρ ). V (t¯ρ+1 ) ≤ κ V t¯ρ+1 (21) It can be concluded from the above analysis that V (t¯ρ+1 ) < e−ε0 V (t¯ρ ), for an arbitrarily given ρ ∈ N. For ρ = 1, one gets that V (t¯2 ) < V (0)e−ε0 . Furthermore, it can be obtained by recursion that V (t¯ρ+1 ) ≤ V (0)e−ρε0

(22)

for any given positive integer ρ. According to the fact that the dwell time ι0 > 0, one knows that no Zeno behavior will be emerged as the considered network evolves with time [43]. Thus, for an arbitrarily given t > 0, there exits a positive integer z such that t¯z < t ≤ t¯z+1 . 1 ), one gets When t ∈ (0, t¯min V (t) < V (0)e−γˆ1 t

(23)

¯

where γˆ1 = (ατ0σ (t1 ) − 2λmax ()) > 0. For the case of 1 = t¯ , it can be obtained from (23) that t¯min 2 V (t¯2 ) < κ V (0)e−γˆ1 t¯2

(24)

1 where κ is defined in (13). For the case of t¯min < t¯2 and γ˜1 > 0, some calculations give that

1 (25) V (t) < κ V (0)e−γ0 t , t ∈ tmin , t¯2 1 < t¯ and γ˜ ≤ 0, where γ0 = min{γˆ1 , γ˜1 }. For the case of t¯min 2 1 one has

V (t) < κ V (0)e−γ˜1 (t −tmin ) e−[(γˆ1 tmin )/t¯2 ]t 1

1

1 1 < κ V (0)e−γ˜1 ˆι1 e−[(γˆ1 tmin )/t¯2 ]t , t ∈ tmin , t¯2 .

(26)

z When t ∈ (t¯z , t¯min ), z ≥ 2, one has

V (t) ≤ V (tz )e−γˆz (t −t¯z ) < V (0)e−[((z−1)ε0)/(zˆι1 )]t < V (0)e−[ε0 /(2ˆι1 )]t

(27)

where the last inequality is obtained since z ≥ 2. For the case z of t¯min = t¯z+1 , it can be directly obtained from (27) that V (t¯z+1 ) < κ V (0)e−[ε0 /(2ˆι1 )]t¯z+1 .

(28)

z t¯min

< t¯z+1 and γ˜z > 0, it can be obtained that

z V (t) < κ V (0)e−0 t , t ∈ t¯min (29) , t¯z+1

For the case of

z where 0 = min{ε0 /(2ˆι1 ), γ˜z }. For the case of t¯min < t¯z+1 and γ˜z ≤ 0, some calculations give that

V (t) ≤ κ V (tz )e−γˆz (t¯min −t¯z ) e−γ˜z (t −t¯min ) < κ V (0)e−(z−1)ε0 e−γ˜z ˆι1 z

z

z < κ V (0)e−γ˜z ˆι1 e−[ε0 /(2ˆι1 )]t , t ∈ t¯min , t¯z+1 .

(30)

By (23)–(30), one gets that the zero equilibrium point of switched systems (7) and (8) is globally exponentially stable. One can thus conclude that the global pinning synchronization in switched networks (2) with target trajectory given in (3) could be achieved. Suppose that Assumption 1 holds, it can be obtained from Theorem 1 that global pinning synchronization in the considered complex networks can be ensured if the conditions 1) and 2) given in Theorem 1 are simultaneously satisfied. Here, condition 1) means that the coupling strength among the neighboring nodes is larger than a threshold value. With this condition, condition 2) can be equivalently rewritten ρ ρ as (tmin − t¯ρ ) > [−γ˜ρ (t¯ρ+1 − tmin ) + lnκ + ε0 ]/γˆρ . Intuitively speaking, condition 2) implies that, over each time interval [t¯ρ , t¯ρ+1 ), ρ ∈ N, the total activation time for the network topologies with a directed spanning tree is larger than a threshold quantity. Remark 3: Using tools from M-matrix theory and algebraic graph theory, a class of multiple quadratic Lyapunov functions in the form of (14) has been successfully constructed. It can be observed from the proof of Theorem 1 that the construction of the topology-dependent quadratic Lyapunov functions provides an efficient tool for analyzing synchronization behavior of the switched complex networks (2) with target system (3). Note that how to construct a common quadratic Lyapunov function for analyzing the synchronization behavior of switched networks under consideration is still a challenging today. It is also worth noting that there may even have no common quadratic Lyapunov function for switched linear complex networks [44]. Remark 4: When applying Theorem 1 to solve the pinning synchronization problem of practical complex networks, one σ (t¯ ) ρ needs to calculate λmax (), τ0 ρ , and τ1 . The stability and precision of the numerical computation method adopted here should be fully considered, since most of the typical complex networks are large scale. One may use the traditional Jacobi eigenvalue algorithm or Givens eigenvalue σ (t¯ ) algorithm [45] to calculate λmax () and τ0 ρ , since both ¯ σ (t¯ρ ) are real and symmetric. ¯ σ (t¯ρ ) Lˆ σ (t¯ρ ) + ( Lˆ σ (t¯ρ ) )T and Note that both the Jacobi eigenvalue algorithm and Givens eigenvalue algorithm have good numerical stability. Furtherρ more, to obtain τ1 , one needs to calculate the eigenvalues ¯ σ (t¯ρ ) Lˆ h + ( Lˆ h )T ¯ σ (t¯ρ ) )−1 ( ¯ σ (t¯ρ ) ) which is similar to of ( ¯ σ (t¯ρ ) )−1 . Obviously, ( ¯ σ (t¯ρ ) Lˆ h + ¯ σ (t¯ρ ) Lˆ h + ( Lˆ h )T ¯ σ (t¯ρ ) )( ( ¯ σ (t¯ρ ) )( ¯ σ (t¯ρ ) )−1 shares the same eigenvalues with ( Lˆ h )T symmetric matrix σ (t¯ ) −1 σ (t¯ρ ) h h T σ (t¯ρ ) σ (t¯ ) −1 ρ ˆ ˆ ¯ ¯ ¯ ¯ ρ Msy = L + L (31) where σ (t¯ )

¯ σ (t¯ρ )

−1

=

σ (t¯ρ ) 1/2 σ (t¯ ) ) , (ξ2 ρ )1/2 , . . . ,

diag{(ξ1

ρ

(ξ N ρ )1/2 }. The above analysis indicates that, to calculate τ1 , one may just need to use the traditional Jacobi eigenvalue algorithm or Givens eigenvalue algorithm to calculate the eigenvalues of Msy given in (31).


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Based on Theorem 1, one can obtain the following corollaries where the detailed proofs are omitted for brevity. Corollary 1: Under Assumption 1, the global pinning synchronization in network (2) with target trajectory given in (3) could be achieved if there exists a positive scalar ερ for each ρ ∈ N, such that the following conditions hold. () 1) α > 2λmax σ (t ρ ) . 2)

γρ >

τ0 ρ − γρ (t ρ+1 −tmin )+lnκρ +ερ . ρ −t ρ ) (tmin

σ (t¯ )

In the above conditions, γˆρ = ατ0 ρ − 2λmax (), ρ γ˜ρ = ατ1 − 2λmax (), λmax () is the largest eigenvalue of , σ (t¯ρ )

τ0

ρ

and τ1 are, respectively, defined in (11) and (12), and σ (t¯

κρ = where

σ (t¯ )

ξminρ

=

σ (t¯ ) maxr=1,2,...,N ξr ρ+1 ,

ξmaxρ+1

)

σ (t¯ )

ξminρ

σ (t¯ρ )

minr=1,2,...,N ξr

,

σ (t¯

ξmaxρ+1

(32)

Fig. 2. Communication topology G(A) which does not contain any directed spanning tree.

=

network (1) is available for pinning feedback. However, it is challenging yet important to further study the global pinning synchronization problem for directed switching complex networks with partial-state coupling where only the relative output information between the target system and each pinned node is available for pinning feedback. On the other hand, the present theoretical results are derived based on Assumption 1 which may be restricted in some applications. Therefore, it is still an unsolved problem about how to ensure global pinning synchronization in directed switching complex networks without Assumption 1.

)

σ (t¯ ) σ (t¯ ) σ (t¯ ) = (ξ1 ρ , ξ2 ρ , . . . , ξ N ρ )T = σ (t¯ ) σ (t¯ ) ( Lˆ σ (t¯ρ ) )−T 1 N , and ξ σ (t¯ρ+1 ) = (ξ1 ρ+1 , ξ2 ρ+1 , . . . , σ (t¯ ) ξ N ρ+1 )T = ( Lˆ σ (t¯ρ+1 ) )−T 1 N .

ξ σ (t¯ρ )

Corollary 2: Suppose that Assumption 1 holds and σ (t¯ ) ρ ρ ρ τ0 ρ (tmin − t¯ρ )+τ1 (t¯ρ+1 −tmin ) > 0, for ρ ∈ N. Then, global pinning synchronization in network (2) with target trajectory given in (3) could be achieved if there exists a positive constant ε0 such that α > αth with αth = supρ∈N αρ , αρ = σ (t¯ )

ε0 +lnκ+2λmax ()(t ρ+1 −t ρ )

σ (t ρ )

τ0

ρ ρ ρ −t ρ )+τ1 (t ρ+1 −tmin ) (tmin

,

ρ

where τ0 ρ and τ1 are, respectively, defined in (11) and (12), κ is defined in (13). Corollary 3: Suppose that Assumption 1 holds and G¯ = Gˆ = {1}. Then, global pinning synchronization in network (2) with target trajectory given in (3) could be achieved if α > λth

(33)

where λth = 2λmax ()λ0 , λ0 is the largest eigenvalue of ¯ 1 Lˆ 1 + ( Lˆ 1 )T ¯ 1 is defined in Lemma 4. ¯ 1 )−1 ¯ 1 , and ( Remark 5: Compared with those given in Theorem 1, the conditions given in Corollary 1 are less conservative. However, the consensus conditions given in Theorem 1 are more convenient to use in practical applications, since one does not need to calculate κρ for all time intervals. Remark 6: Corollary 3 indicates that global pinning synchronization can be ensured if the fixed network topology contains a directed spanning tree and the coupling strength is appropriately selected. We point out the results provided in Corollary 3 confirm those given in some existing literature on pinning synchronization over fixed complex networks, such as [26] and [46]–[48]. Remark 7: The coupling pattern among the nodes in the considered network (1) is full-state coupling. In addition, it is required in this paper that the relative full-state information between the target system (3) and each pinned node in

V. N UMERICAL E XAMPLES In this section, some numerical simulations on linearly coupled neural networks are provided to verify the presented analytical results. Note that neural network is one of the most important nonlinear systems, which has been widely studied in the context of stability analysis and bifurcation control [49]–[51]. Recently, synchronization of coupled neural networks has received much attention [52]–[56]. Example 1: Global pinning synchronization of a fixed complex network is numerically studied in this example. Suppose that there are 40 nodes, labeled from 1 to 40, in the considered complex networks (see Fig. 2 for illustration). By Algorithm 1, it can be obtained that at least five nodes should be pinned such that each node in G(A) can be directly or indirectly affected by the target node. In Fig. 3, the dashed lines originating from the target node (labeled as node 41) ending at the nodes of G(A) are the pinning links. The strongly connected components and the nodes with zero in-degree are highlighted in Fig. 3 (pale blue shading). Here, the weight on each link is assumed to be one. Consider the following linearly coupled neural networks with topology given in Fig. 3: x˙ i (t) = −Ax i (t)+g(W x i (t))+J (t) 41 ai j (t)(x j (t)−x i (t)) +α

(34)

j =1

where x i (t) = (x i1 (t), x i2 (t), . . . , x in (t))T ∈ R n is the neuron state, i = 1, 2, . . . , 40, W = (Wˆ 1T , Wˆ 2T , . . . , Wˆ nT ) stands for the synaptic connection weights, J = (J1 (t),

3246


Fig. 6. Synchronization error Err(t) in Example 1, which indicates that the global pinning synchronization in the considered complex network is achieved.

˜ which contains a directed spanning Fig. 3. Communication topology G(A) tree rooted at node 41.

Fig. 4.

Fig. 5.

Fig. 7.

Initial configuration of the scale-free network in Example 2.

Fig. 8.

State trajectories xi1 (t), i = 1, . . . , 401, in Example 2.

Fig. 9.




J2 (t), . . . , Jn (t))T represents the external inputs, g(W x i (t)) = (g1 (Wˆ 1 x i (t)), g2 (Wˆ 2 x i (t)), . . . , gn (Wˆ n x i (t)))T is the activation of neurons. In particular, choose A = diag{0.125, 0.125}; −0.15 0.45 W = ; g(y(t)) = (tanh(y1 (t)), tanh(y2 (t)))T , 0.6 0.65 for all y(t) = (y1 (t), y2 (t))T ∈ R2 ; J (t) = (30sin(30t), 30cos(30t))T . In view of Assumption 1, it can be obtained that = 0.7234I2 . In simulations, the target system is given as x˙41 (t) = −Ax 41(t) + g(W x 41 (t)) + J (t)

(35)

where x 41 (0) = (0.2, −0.15)T and the other parameters are defined the same as those in (34). It can be obtained form Corollary 3 that global pinning synchronization in network (34) with topology given in Fig. 3 can be achieved if α > αth = 34.2529. The state trajectories of the complex networks with α = 34.2530 are, respectively, shown in Figs. 4 and 5. Use Err(t) = (1/40) 2 1/2 to denote the synchronization x ( 40 j (t) − x 41 (t) ) j =1 error of the considered complex networks. Fig. 6 indicates that the global pinning synchronization problem is indeed solved. Here, the horizontal axes of Figs. 4–6 show the time in seconds.

Example 2: In this example, global pinning synchronization in an undirected scale-free network with 400 nodes is numerically studied. The network is generated using the well-known Barabási–Albert growth rule with parameters m 0 = 7, m = 4, and N = 400 [7]. The initial configuration of the network is assumed to consist of a fully connected component with four nodes and three isolated nodes (shown in Fig. 7). By Algorithm 1, one has that there are at least four nodes should be pinned such that the augmented graph contains a directed spanning tree. In simulations, nodes 5, 6, 7, and 120 are selected and pinned. The node dynamics are chosen the same as those given in Example 1. According to Corollary 3, one gets that the global pinning synchronization in the considered scale-free network can be achieved if the coupling strength α > 57.10. Choose α = 57.2, profiles of the evolution trajectories of the considered network are shown in Figs. 8 and 9, respectively. 2 1/2 denote the x Let Err(t) = (1/400)( 400 j (t) − x 401 (t) ) j =1 synchronization error of the considered networks. It can be seen from Fig. 10 that the global pinning synchronization problem is indeed solved. Note that the horizontal axes of Figs. 8–10 show the time in seconds. These figures demonstrate the effectiveness of the proposed theoretical results. Example 3: In this example, global pinning synchronization for complex network with switching directed topologies is


3247


Fig. 13.

Switching signal σ (t) in Example 3.

Fig. 14.


Fig. 15.


Fig. 11. Communication topologies G(A˜ 1 ) and G(A˜ 2 ), where only G(A˜ 1 ) contains a directed spanning tree rooted at node 11.

Fig. 12. Communication topologies G(A˜ 3 ) and G(A˜ 4 ), where only G(A˜ 3 ) contains a directed spanning tree rooted at node 11.

numerically studied. Consider the following linearly coupled neural networks: x˙i (t) = −Ax i (t) + g(W x i (t)) + J (t) 11 ai j (t)(x j (t)−x i (t)) +α

(36)

j =1

where the parameters are chosen the same as those in Example 1, i = 1, 2, . . . , 10. Let Gˆ = 1 2 3 4 {G(A˜ ), G(A˜ ), G(A˜ ), G(A˜ )} and G¯ = {G(A˜1 ), G(A˜3 )}. Topologies G(A˜i ), i = 1, . . . , 4, are, respectively, given in Figs. 11 and 12 where the weight on each edge is assumed to be one. In Figs. 11 and 12, the neighboring relationships between the nodes in network (36) and the single target node (labeled as node 11) are depicted by dashed lines with arrows. Furthermore, the strongly connected components and the nodes with zero in-degree are highlighted in Figs. 11 and 12 (pale blue shading). Let κ0i be the minimum number of the nodes that should be pinned such that the augmented network topology G(A˜i ) contains at least one directed spanning tree, i = 1, . . . , 4. According to Algorithm 1, one gets that κ01 = κ02 = κ03 = 4, κ04 = 6. Choose ⎧ 1, t ∈ [k, k + 0.4) ⎪ ⎪ ⎪ ⎨2, t ∈ [k + 0.4, k + 0.5) σ (t) = (37) ⎪ 3, t ∈ [k + 0.5, k + 0.95) ⎪ ⎪ ⎩ 4, t ∈ [k + 0.95, k + 1)

where k = 0, 1, 2, . . . According to (11), one can has that τ01 = τ03 = 0.41. Furthermore, it can be yielded from (37) that 2k−1 2k = k − 0.6, t¯2k = k − 0.5, tmin = k − 0.05, t¯2k−1 = k − 1, tmin where k ∈ N. It can be thus obtained from (12) that τ12k−1 = −0.2247 and τ12k = −0.4495. The switching signal is shown in Fig. 13. In addition, one may get from (13) that κ = 3. In the simulations, the target system is given as x˙11 (t) = −Ax 11(t) + g(W x 11 (t)) + J (t)

(38)

where x 11 (0) = (0.1, −0.1)T , the other parameters are defined the same as those in (36). Choose α = 12.88, it can be obtained from Theorem 1 that the global pinning synchronization can be ensured in the switched complex networks (36) with target trajectory given in (38). The state trajectories of the complex networks are, respectively, shown in Figs. 14 and 15. 2 1/2 to denote Use Err(t) = (1/10)( 10 j =1 x j (t) − x 11 (t) ) the synchronization error of the considered complex networks. Fig. 16 indicates that the global pinning synchronization problem is indeed solved. The profiles of Lyapunov function V (t) [defined in (14)] versus time are shown in Fig. 17. Furthermore, the detailed profiles of V (t)

3248


R EFERENCES


Fig. 17.

Profiles of V (t) [defined by (14)] in Example 3.

around the switching points t = 1 and t = 1.5 are shown in the subgraphs of Fig. 17. It can be seen from Fig. 17 that the value of V (t) converges to zero asymptotically though there are some chattering behaviors in the switching points. Here, the horizontal axes of Figs. 13–17 show the time in seconds. VI. C ONCLUSION Global pinning synchronization in a class of switched complex networks has been addressed in this paper. Unlike most existing related works, the underlying network topology of the complex network in this paper may be time varying. In particular, some possible augmented network topologies of the considered networks may not contain any directed spanning tree. Using tools from M-matrix theory and stability theory for switched systems, a class of multiple Lyapunov functions has been constructed and utilized for analyzing the convergence behavior of the synchronization errors. Some simple yet efficient conditions have been obtained for ensuring global pinning synchronization in such switched complex networks. Future work will focus on solving the global pinning synchronization in switched complex networks where only the sampled-data information of the target system is available for pinning feedback. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and all the anonymous reviewers for their careful reading of this paper and their constructive comments, which have led to the improvement of the presentation of this paper.

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Guanghui Wen (S’11–M’13) received the Ph.D. degree in mechanical systems and control from Peking University, Beijing, China, in 2012. He was a Research Associate and Post-Doctoral Fellow with the University of New South Wales at Canberra, Campbell, ACT, Australia, from September 2012 to January 2013. In 2014, he was a Visiting Research Fellow with the School of Electrical and Computer Engineering, RMIT University, Melbourne, VIC, Australia. He is currently a Lecturer with the Department of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, China. His current research interests include cooperative control of multiagent systems, networked control systems, and robust control. Dr. Wen was a recipient of the Best Student Paper Award in the 6th Chinese Conference on Complex Networks in 2010.

Wenwu Yu (S’07–M’12) received the B.Sc. degree in information and computing science and the M.Sc. degree in applied mathematics from the Department of Mathematics, Southeast University, Nanjing, China, in 2004 and 2007, respectively, and the Ph.D. degree from the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, in 2010. He held several visiting positions in Australia, China, Germany, Italy, The Netherlands, and USA. He is currently the Founding Director of the Laboratory of Cooperative Control of Complex Systems and a Full Professor with the Research Center for Complex Systems and Network Sciences, Department of Mathematics, Southeast University. He has authored or co-authored about 80 referred international journal papers, and a Reviewer of several journals. His current research interests include multiagent systems, nonlinear dynamics and control, complex networks and systems, smart grids, neural networks, cryptography, and communications. Dr. Yu was a recipient of the Best Master’s Degree Theses Award from Jiangsu Province, China, in 2008, the DAAD Scholarship from Germany in 2008, the Top 100 Most Cited Chinese Papers Published in International Journals in 2008, and the Best Student Paper Award in the 5th Chinese Conference on Complex Networks in 2009.

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Guoqiang Hu (M’05) received the B.Eng. degree from the University of Science and Technology of China, Hefei, China, in 2002, the M.Phil. degree from the Chinese University of Hong Kong, Hong Kong, in 2004, and the Ph.D. degree from the University of Florida, Gainesville, FL, USA, in 2007. He was a Post-Doctoral Research Associate with the University of Florida in 2008, and an Assistant Professor with Kansas State University, Manhattan, KS, USA, from 2008 to 2011. He is currently with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His current research interests include analysis, control and design of distributed intelligent systems, and its applications to smart grids, smart buildings, and networked robots. Dr. Hu serves as a Subject Editor of the International Journal of Robust and Nonlinear Control, and an Associate Editor of Unmanned Systems and the Asian Journal of Control. He serves on the Conference Editorial Board of the IEEE Control Systems Society.

Jinde Cao (M’07–SM’07) received the B.S. degree from Anhui Normal University, Wuhu, China, in 1986, the M.S. degree from Yunnan University, Kunming, China, in 1989, and the Ph.D. degree from Sichuan University, Chengdu, China, in 1998, all in mathematics/applied mathematics. He was with Yunnan University from 1989 to 2000. In 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. He was a Professor with Yunnan University from 1996 to 2000. From 2001 to 2002, he was a Post-Doctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He is currently a TePin Professor and Doctoral Advisor with Southeast University. He has authored or co-authored over 160 journal papers and five edited books, and is a Reviewer of Mathematical Reviews and Zentralblatt Math. His current research interests include nonlinear systems, neural networks, complex systems and complex networks, stability theory, and applied mathematics. Prof. Cao is an Associate Editor of the IEEE T RANSACTIONS ON C YBERNETICS , Differential Equations and Dynamical Systems, Mathematics and Computers in Simulation, and Neural Networks.

Xinghuo Yu (M’92–SM’98–F’08) received the B.Eng. and M.Eng. degrees from the University of Science and Technology of China, Hefei, China, in 1982 and 1984, respectively, and the Ph.D. degree from Southeast University, Nanjing, China, in 1988. He is currently with RMIT University, Melbourne, VIC, Australia, where he is the Founding Director of the RMIT’s Platform Technologies Research Institute. His current research interests include variable structure and nonlinear control, complex and intelligent systems and networks, and industrial applications. Prof. Yu received a number of awards for his achievements, including the Dr.Ing. Eugene Mittelmann Achievement Award of the IEEE Industrial Electronics Society in 2013, and the IEEE Industrial Electronics Magazine Best Paper Award in 2012. He served as the Vice President for Publications from 2012 to 2015. He is the IEEE Distinguished Lecturer of the IEEE Industrial Electronics Society. He holds a fellowship with the Institution of Engineering and Technology, the Engineers Australia, the Australian Computer Society, and the Australian Institute of Company Directors.