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Exponential Synchronization of Complex Networks of Linear Systems and Nonlinear Oscillators: A Unified Analysis Jiahu Qin, Member, IEEE, Huijun Gao, Fellow, IEEE, and Wei Xing Zheng, Fellow, IEEE

Abstract— A unified approach to the analysis of synchronization for complex dynamical networks, i.e., networks of partialstate coupled linear systems and networks of full-state coupled nonlinear oscillators, is introduced. It is shown that the developed analysis can be used to describe the difference between the state of each node and the weighted sum of the states of those nodes playing the role of leaders in the networks, thus making it feasible to consider the error dynamics for the whole network system. Different from the other various methods given in the existing literature, the analysis employed in this paper is demonstrated successfully in not only providing the consistent convergence analysis with much simpler form, but also explicitly specifying the convergence rate. Index Terms— Complex networks, convergence rate, linear systems, nonlinear oscillators, synchronization, unified analysis.

I. I NTRODUCTION GREAT DEAL of complex systems in real world, such as large-scale sensor networks, neural networks, ecosystems, social systems, WWW, genetic regulatory networks, electrical power grids, and so on., can be modeled by complex networks [1], [2], [11], [12], [14], [38]. In the past years, synchronization of networks of coupled linear systems [17], [27], [35], [37], [39], [40], [51] and nonlinear oscillators has attracted the researchers from various disciplines of engineering and science [9], [10], [19]–[21], [24], [36], [47], [53]. A relevant topic, which has been

A

Manuscript received January 26, 2013; accepted April 4, 2014. Date of publication May 7, 2014; date of current version February 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61333012, Grant 61273201, and Grant 61105038, in part by the Program for New Century Excellent Talents in University, China, under Grant NCET-130544, in part by the Key Laboratory of Integrated Automation for the Process Industry, Northeastern University, in part by the Basic Research Plan, Shenzhen, under Grant JC201105160523A, in part by the Anhui Provincial Natural Science Foundation under Grant 1408085QF105, in part by the Australian Research Council, and in part by the University of Western Sydney, Penrith, NSW, Australia. J. Qin is with the University of Science and Technology of China, Hefei 230027, China, and also with the Australian National University, Canberra, ACT 0200, Australia (e-mail: [email protected]). H. Gao is with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China, and also with the King Abdulaziz University, Jeddah 22254, Saudi Arabia (e-mail: [email protected]). W. X. Zheng is with the School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith, NSW 2751, Australia (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.2014.2316245

widely investigated, is the consensus of multiagent systems (MASs) in which agents usually takes the integrator dynamics [8], [16], [28], [31]–[34], [49], [50] and the synchronization of neural networks [22]. Different from the consensus of MASs where it is the exchange of information only that determines the evolution of the states of agents, the synchronization for the networks of linear systems and nonlinear oscillators (termed synchronization problems for simplicity) depends on not only the couplings among the nodes (i.e., linear systems and nonlinear oscillators) in the networks, but also the self-dynamics governing the evolution of each isolated node (corresponding to s˙(t) = As(t) for linear system and s˙ (t) = f (s(t), t) for nonlinear oscillator, respectively). Because of this, most of the consensus analysis methods used for MASs cannot be extended directly to the synchronization analysis. However, it is worth mentioning that some ideas/results, in particular those presented from the graphical/structural viewpoints, may be useful in studying the synchronization problems. There has been extensive and increasing literature addressing the synchronization problems by employing different methods from different viewpoints. The existing results are mainly focused on the fixed network topology. More specifically, synchronization of coupled linear systems has been investigated in [17], [25], and [40], where different model transformations are performed on the network system dynamics and then the synchronization is considered in the sense of the stability of the transformed systems. As can be observed from these works, such transformations and methods used do not work for the synchronization of coupled nonlinear oscillators [9], [21]–[23], [30], [53], mainly due to the existence of nonlinear self-dynamics. In fact, totally different methods have been used to deal with the synchronization analysis for coupled nonlinear oscillators [21]–[23], [30], [47], [53], for Lyapunov method-based analysis, and [3]–[5] for the connection graph stability method-based analysis. The Lyapunov method-based analysis is motivated mainly from the master stability approach [29], and it can be roughly summarized as follows. First, synchronization is considered for all the nodes in the closed strong component (comprising the nodes having directed paths to all the other nodes in the network), and such nodes in fact play the role of leaders in the network as they influence directly or indirectly all the other nodes. Then, the synchronization analysis can be completed by

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QIN et al.: EXPONENTIAL SYNCHRONIZATION OF COMPLEX NETWORKS OF LINEAR SYSTEMS AND NONLINEAR OSCILLATORS

considering the error subsystems (i.e., the error between the synchronized state of the closed strong component and the other strong components) step by step, which finally leads to the synchronization of the whole network systems. Note that the mathematical manipulations incurred with the Lyapunov method in such synchronization analysis are complicated and moreover, similarly to the analysis for network of linear systems, it is difficult to explicitly specify the convergence rate. On the other hand, the connection graph stability approach given in [3]–[5] mainly consists in symmetrizing the graph and associating a weight to each edge of the undirected graph and to each path between any two nodes. This method is also feasible in handling networks of nonlinear oscillators with general network topology. However, it may not be easy to extend, if not impossible, this method to deal with the synchronization problem for networks of linear systems. Furthermore, it should also be mentioned that the synchronization behaviors of complex networks have been considered in [41]–[45]. However, the works in [41]–[45] have focused on scale-free or smallworld neuronal networks or the Macaque cortical network, which is in contrast with our work that will concentrate on the networks with general topological structure. Besides, as will be seen later on, the methodologies and technical analysis to be used in this paper are also very different from those in [41]–[45]. The gaps in the analysis for the two different synchronization problems as well as the specification of the convergence rate are two of the main motivations of this paper, which aims to provide a unified convergence analysis and further specify the convergence rate for both types of networks under fixed and dynamic network topologies. Motivated by the results on the consensus analysis of MASs, it is reasonable to assume that the state trajectory of each node in the network is decided by the state trajectories of the nodes in the closed strong component. Finally, all the nodes synchronize with each other if the convergence caused by the couplings among these nodes are strong enough to dominate the unstable mode caused by each individual system. More specifically, different from all the above-mentioned literature, we consider the error system dynamics for the whole network systems rather than just the subsystems consisting of the nodes in the closed strong component. Based on the Lyapunov method together with the exploration of the algebraic graph theory and matrix theory techniques, it is shown that the developed method can not only provide a unified convergence analysis, but also explicitly specify the convergence rates for both of the different synchronization problems. Moreover, the manipulation for the nonlinear oscillators is much simpler than that in [21]–[23], [47], and [53], and it can also be extended to deal with the synchronization of neural networks [22]. It is worthwhile to mention that the pinning synchronization, which is aimed at synchronizing all the nodes in the complex network to the solution of the individual uncoupled system by adding controllers to a small fractions of the nodes, will be demonstrated as a very special case of our model under fixed topology. As a separate issue from the fixed topology, the complexities caused by considering both the self-dynamics and the diffusive

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couplings among nodes make the synchronization problems under dynamic topology rather difficult to tackle, and a very few results have been reported as opposed to that for the consensus in MASs. In [35] and [51], the synchronization of network of linear systems is studied using dynamic controller rather than the static feedback law as considered in this paper. Using the static feedback control law, the leader-following synchronization under undirected topology is investigated in [27] based on the presumption of the simultaneous hold of Riccati and Lyapunov inequalities on the individual system dynamics. In [39], the discrete-time case in the absence of leader is examined, but the analysis is made under undirected network topology and the system matrix of each individual system is required to be neutrally stable. Moreover, weighting factors of the couplings among nodes are chosen from a finite set as opposed to the general infinite case to be considered in this paper. Within the dynamic topology framework, this paper focuses on performing the convergence analysis under more general individual system dynamics, i.e., the matrix pair (A, B) associated with each linear system dynamics is only required to be stabilizable, where A may even have exponentially unstable mode; the dynamically changing weighting factors are allowed to be chosen from an infinite set and the directed dynamic topologies are allowed to switch arbitrarily. The convergence analysis is made, however, under somewhat restrictive topology structure, that is, the network topology is required to be balanced and weakly connected at each time instant. Finally, using similar analysis for the network of linear systems, we will also study the synchronization of nonlinear oscillators under dynamic topologies. The unified analysis made in this paper will show that under certain connectivity assumptions on the network topologies, an appropriate feedback matrix can be designed to make the network of linear systems synchronize exponentially, while the nonlinear oscillators can achieve synchronization exponentially fast if the overall coupling strength of the network system is sufficiently large to overcome the divergence caused by the individual nonlinear system dynamics. Moreover, convergence rates for both cases can be explicitly specified as well. It is worthy of emphasis that the unified analysis as well as the explicit specification of the convergence may be obtained at the cost of a rough estimation of bound for the coupling strength for networks of nonlinear oscillators. For excellent works on the estimation of a critical coupling strength that is necessary to guarantee the synchronization, the readers are referred to [3] and its companion paper [4] for more details. The remainder of this paper is organized as follows. A brief summary of some relevant results in graph and matrix theory are provided in Section II. The main result concerning the synchronization under fixed topology is given in Section III, while Section IV deals with the case with switching topology. Finally, the main results of this paper are summarized in Section V. Notation: Throughout this paper, denote by M > 0 (M < 0) that M is symmetric positive (negative) definite and by M ≥ 0 (M ≤ 0) that M is symmetric and positive (negative) semidefinite. If all the eigenvalues of M are real, then denote by

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λmax (M) and λmin (M) the maximum and minimum eigenvalues of M, respectively. Denote by diag{A1 , A2 , . . . , An } the block diagonal matrix with its i th main diagonal matrix being a square matrix Ai , i = 1, . . . , n. By abuse of notation, for any m × 1 vector α, denote by diag(α) ∈ Rm×m the diagonal matrix with the i th (i = 1, 2, . . . , m) diagonal element being the i th element of α. A matrix is called nonnegative (positive) whenever all its elements are nonnegative (positive). II. P RELIMINARY Let G be a weighted digraph of order N, and A = [ai j ] ∈ R N×N be the associated adjacency matrix in which ai j > 0 whenever there is a directed edge from node j to node i . Moreover, assume aii = 0, i = 1, . . . , N. Denote by L(G) = L = [i j ] the Laplacian matrix associated with G, where i j = N −ai j , i = j, and ii = k=1,k =i aik . Conversely, any given N ×N matrix L = [i j ] with nonpositive off-diagonal elements and satisfying L1 N = 0 will be called graph Laplacian in the sequel for convenience. L can be deemed as a Laplacian matrix of a weighted digraph, say G(L) which is defined as follows: G(L) is a digraph with node set {1, 2, . . . , N} and there is an edge in G(L) from j to i ( j = i ) if and only if i j > 0. T Digraph G is called balanced if and only = 0  if 1 N L(G) 1 [28]. If G is a balanced digraph, then 2 L(G) + LT (G) is ˆ ˆ the Laplacian matrix  of G, the mirror graph of G, i.e., L(G) = 1 T (G) [28, Theorem 7]. L(G) + L 2 A digraph is called strongly connected if any two distinct nodes of the graph can be connected by a directed path, while it is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph. A directed graph has a directed spanning tree if there exists at least one node, called the root node, having a directed path to all other nodes. A strongly connected component (also termed strong component) of G is a maximal subgraph H of G such that H is strongly connected; and a closed strong component of G is a strong component of G, which has no incoming edges from any nodes outside. Lemma 1: Let L be the Laplacian matrix of a strongly connected digraph, and β = [β1 , β2 , . . . , β N ]T ∈ R N be the unique eigenvector satisfying β T L = 0 and β T 1 N = 1 [26]. Further, let x ∈ Rn N×n N be any column vector satisfying  T  β ⊗ eiT x = 0 (1) where ⊗ is the Kronecker product, ei ∈ Rn stands for the column vector in which only the i th entry is 1 and all the other entries are 0, and ⊗ denotes the Kronecker product. Then, for any symmetric positive semidefinite matrix B ∈ Rn×n , we have x T ( Lˆ ⊗ B)x ≥ a(L)x T ( ⊗ B)x

(2)

where Lˆ = (diag(β)L + L T diag(β))/2,  = diag(β), and a(L), the algebraic connectivity, is defined as (see [47], [52]) a(L) =

min

x T β=0,x =0

Proof: See the Appendix.

x T Lˆ x > 0. x T diag(β)x

(3)

When L is the Laplacian matrix of an undirected and connected graph, it is easy to obtain that a(L) = λ2 (L), where λ2 (L) is given by λ2 (L) =

min

x =0,1T x=0

x T Lx x Tx

(4)

which denotes the second smallest eigenvalue of L and is also called the algebraic connectivity of G [13]. With this notation, we have the following corollary from Lemma 1. Corollary 1: Assume that L is a Laplacian matrix of a connected undirected graph. Let x = [x 1T , x 2T , . . . , x NT ]T ∈ n Rn N , where xNi ∈ R , i = 1, . . . , N, be any column vector satisfying i=1 x i = 0. Then, for any symmetric positive semidefinite matrix B ∈ Rn×n , we have x T (L ⊗ B)x ≥ λ2 (L)x T (I N ⊗ B)x. The following lemma will be needed in the subsequent sections. Lemma 2 (Schur complement, [6]): Given constant matrices S1 , S2 and S3 , where S1 = S1T and S2 = S2T > 0, then S1 − S3T S2−1 S3 > 0 if and only if     S2 S3 S1 S3T > 0 or > 0. S3 S2 S3T S1 III. S YNCHRONIZATION U NDER F IXED N ETWORK T OPOLOGY The complex network consisting of N identical partial-state coupled linear systems [35], [39], [40], [51] is described as follows: x˙i = Ax i + B K

N 

ai j (x j (t)−x i (t)), i = 1, 2, . . . , N

(5)

j =1, j  =i

where A = [ai j ] ∈ Rn×n , B ∈ Rn×m , and K ∈ Rm×n is the feedback matrix to be designed. The complex network of N coupled nonlinear oscillators evolves according to the following dynamics [9], [31], [47], [53]: x˙i (t) = f (x i (t)) + c

n 

ai j (x j (t) − x i (t)),

j =1, j  =i

i = 1, 2, . . . , N

(6)

Rn

f : → Rn where c is the overall coupling strength,  T f (x i ) = [ f 1 (x i ), f2 (x i ), . . . , fn (x i )] is a continuous vector function, and  is the inner coupling matrix that is a positive definite constant matrix (i.e.,  > 0). For both of these two networks, x i = [x i1 , . . . , x in ]T ∈ Rn is the state of node i , and ai j is defined as follows: ai j > 0 if there is a coupling (i.e., direct edge) from node j to node i and otherwise ai j = 0. Definition 1: The complex networks (5) and (6) are said to be globally exponentially synchronized with the least rate of  if there exist T > 0 and M > 0 such that x i (t) − x j (t) ≤ Me−t hold for any initial values x i (0) and t > T , i, j = 1, 2, . . . , N. Without loss of generality, in what follows we shall assume that the network topology G has q (1 ≤ q ≤ N) strong

QIN et al.: EXPONENTIAL SYNCHRONIZATION OF COMPLEX NETWORKS OF LINEAR SYSTEMS AND NONLINEAR OSCILLATORS

components, say G 1 , . . . , G q , where each strong component has n  ,  = 1, . . . , q nodes, which are indexed, respectively, n + 1, . . . , as V(G  ) = { −1 j j =0 n j } (n 0 = 0); and the j =0 Laplacian matrix L associated with G takes the following Frobenius normal form [7]: ⎡

⎤

L 11 ⎢ . . ⎥ ⎢ . ⎥ .. ⎣ . ⎦ L q1 · · · L qq

0

513

obtains the following error dynamical system: e˙i (t) = Ax i (t) + Bu i (t) − = Aei (t) + B K



n1 

β1k (Ax k (t) + Bu k (t))

k=1

ai j (x j (t) − x ∗ (t) + x ∗ (t) − x i (t))

j ∈Ni

(7)

Rni ×ni ,

where L ii ∈ i = 1, . . . , q. Note that if G has a directed spanning tree, then for each p = 2, . . . , q, there must exists k with 1 ≤ k < p such that L pk = 0 [47]. Then, following exactly the same proof as that in Lemma 4 in [30], we obtain the following result. Lemma 3: If G has a directed spanning tree, then for any i = 2, . . . , q, there exists a positive column vector βi = [βi1 , . . . , βini ]T such that diag{βi1 , . . . , βini }L ii + L Tii diag{βi1 , . . . , βini } > 0 and βiT 1ni = 1. In fact, βi ∈ Rni can  be chosen as the positive left eigenvector of L ii + i−1 =1 R(L i ) associated with eigenvalue 0 satisfying βiT 1ni = 1, i = 2, . . . , q, where R(L i ) denotes the diagonal matrix with the kth (k = 1, . . . , n i ) diagonal element being the i th row sum of L i . In the sequel, denote   i = diag(βi ) = diag βi1 , . . . , βini , i = 1, . . . , q.

A. Networks of Linear Systems: Fixed Topology Assume that the matrix pair (A, B) satisfies the following assumption [25], [35], [39], [40]. Assumption 1: The pair (A, B) is stabilizable. Theorem 1: Consider the complex dynamical network system (5). If Assumption 1 holds and the network topology G has a directed spanning tree, then a feedback matrix K can be designed such that the network system (5) can be synchronized exponentially fast. Proof: The proof is mainly composed of two parts: we first derive the error dynamics for the whole network and then construct a Lyapunov function to perform the stability analysis which, as will be shown below, is feasible in not only performing the synchronization analysis, but also specifying the convergence rate. This idea will also be extended in the sequel to deal with the other models. In fact, this is the main streamline for the synchronization analysis employed throughout this paper, although the specific model transformation as well as the construction of the Lyapunov function may be different. Let β1 = [β11 , . . . , β1n1 ]T ∈ Rn1 be the positive vector satisfying β1T L 11 = 0 and β1T 1n1 = 1 and 1 = diag(β1 ). Denote agent i and x ∗ (t) = n1 kthe state error between the ∗ (t), i = 1, . . . , N. In the β x (t) as e (t) = x (t) − x i i k=1 1 k sequel, t might be dropped for notational simplicity. Then, one

−B K

n1 

β1k

k=1

= Aei (t) + B K



akj (x j (t) − x k (t))

j ∈Nk



ai j (e j (t) − ei (t))

j ∈Ni

  ˜ i = 1, . . . , n 1 + β1T L 11 ⊗ B K x(t),

(8)

where x(t) ˜ = [x 1T(t), x 2T (t), . . . , x nT1 (t)]T . Let e(t) = [e1T (t), e2T (t), . . . , eTN (t)]T . Note that β1T L 11 = 0, then system (8) can be written in the following compact form: e(t) ˙ = [I N ⊗ A − L ⊗ B K ]e(t).

(9)

Consider the following Lyapunov function candidate: V (t) = eT (t)( ⊗ P)e(t)

(10)

where  = diag{ 1 1 , 2 2 , . . . , q q }, P is a positive definite matrix, and i , i = 1, . . . , q, are positive scalars to be chosen so that V (t) is a valid Lyapunov function. Let K = B T P. Differentiating V (t) along the trajectories of (9) gives    V˙ (t) = eT (t)  ⊗ AT P + P A    − L T  + L ⊗ P B B T P e(t). (11) In what follows, choose 1 = 1. Then    eT (t) L T  + L ⊗ P B B T P e(t)    = e1T (t) 1 L 11 + L T11 1 ⊗ P B B T P e1 (t)   +eT (t) L T  + L − diag{1 L 11 + L T11 1 , 0, . . . , 0}  ⊗P B B T P e(t)    ≥ 2a(L 11 )e1T (t) 1 ⊗ P B B T P e1 (t) + eT (t) L T  + L   −diag{1 L 11 + L T11 1 , 0, . . . , 0} ⊗ P B B T P e(t)   = eT (t) q ⊗ P B B T P e(t) where i , i = 2, . . . , q, are given by (12), shown at the top of the next page. To find an appropriate matrix P such that V (t) is a valid Lyapunov function, we first prove that matrix q is positive definite if the positives scalars i , i = 2, . . . , q, are appropriately chosen. It follows directly from Lemma 3 that all the matrices i L ii + L Tii i , i = 2, . . . , q, are positive definite. Note also that 2a(L 11)1 > 0. Suppose that

i > 0, 2 ≤ i < q −1, to complete the proof for the argument, it suffices to prove by induction that i+1 > 0. To this end, according to the Schur complement lemma [6] and by noting

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⎡

2a(L 11 )1

⎢ ⎢ 2 2 L 21 ⎢

i = ⎢ .. ⎢ . ⎣ i i L i1

2 L T21 2

···

i L Ti1 i

2 (2 L 22 + L T22 2 ) · · · .. .. . .

i L Ti2 i .. .

i i L i2

i − i+1 Ti+1 (i+1 L i+1,i+1 + L Ti+1,i+1 i+1 )−1 i+1 > 0 where i+1 = [ i+1 L i+1,2 i+1 L i+1,3 · · · i+1 L i+1,i ] which can be guaranteed by choosing appropriate i+1 so that i+1 is sufficiently smaller than j for any j ≤ i . Assume now that i , i = 2, . . . , q, are properly chosen constants such that q > 0. Let δ be any positive number satisfying δ ≤ λmin ( q )/λmax (). Since (A, B) is stabilizable, we can choose P as a solution to the following Riccati inequality: (13)

With the above notation, we can obtain that     V˙ (t) ≤ eT (t)  ⊗ AT P + P A − q ⊗ P B B T P e(t)     ≤ eT (t) ⊗ AT P + P A −λmin ( q )I N ⊗ P B B T P e(t)    T   T = 1/2 ⊗ In e(t) I ⊗ A P + P A −λmin ( q )−1     ⊗P B B T P 1/2 ⊗ In e(t)    ≤ e˜ T (t) I ⊗ AT P + P A − δ P B B T P e(t) ˜ =

N 

  e˜iT (t) AT P + P A − δ P B B T P e˜i (t)

i=1

≤

N 

⎥ ⎥ ⎥ ⎥ , i = 2, . . . , q. ⎥ ⎦

(12)

· · · i (i L ii + L Tii i )

the fact that i+1 L i+1,i+1 + L Ti+1,i+1 i+1 > 0, it suffices to prove that

P A + AT P − δ P B B T P + δ In < 0.

⎤

−δ e˜iT (t)e˜i (t) = −δeT (t)( ⊗ In )e(t)

i=1

where e(t) ˜ = (1/2 ⊗ In )e(t) = [e˜1T (t), e˜2T (t), . . . , e˜TN (t)]T ∈ Nn R , e˜i (t) ∈ Rn , i = 1, . . . , N. This, together with the inequality that V (t) ≤ λmax (P)eT (t)( ⊗ In )e(t), yields that V˙ (t) ≤ − λmaxδ( P) V (t). This implies that e(t) approaches 0 exponentially fast with the states of the least speed of  = δ/2λmax (P). That is,  n1 k all the nodes are synchronized to x ∗ (t) = k=1 β1 x k (t) exponentially fast with the least rate of  = δ/2λmax (P). According to the proof of Theorem 1, an algorithm for finding a valid feedback matrix K to ensure the synchronization is provided as follows. Step 1: Rewrite the Laplacian matrix L of G such that it takes the Frobenius normal form (7). Then, compute the vectors β1 , β2 , . . . , βq , where β1 is the vector, which satisfies β1T L 11 = 0 and β1T 1 = 1 while β2 , . . . , βq are those vectors, which can be computed by the specific method, as stated in Lemma 3. Matrices i ,

i = 1, . . . , q, can thus be obtained accordingly and a(L 11 ) can be computed via (3). Step 2: Find such positive numbers 2 , . . . , q that the matrix q [see (12)] is positive definite. In fact, according to the proof of Theorem 1, one only needs to choose appropriate i+1 such that i+1 is sufficiently smaller than j for any j ≤ i . Step 3: Let δq be the largest diagonal element of matrix = diag{1 , 2 2 , . . . , q q }, and δ = λmin ( q ) · δq−1 . Finally, given any stabilizable matrix pair (A, B), one can choose P as a solution to the following Riccati inequality: P A + A T P − δ P B B T P + δ In < 0 and then design K as K = B T P. Remark 1: The above design is based on the initially given network topology. One may wonder that this is not a distributed synchronization scheme as in general the network topology may not be known a priori. In fact, K can be chosen without knowing the structure of the network topology since one can just choose a sufficiently small δ and then compute K according to the above Riccati inequality.  1 β1k x k (t), Remark 2: Theorem 1 shows that x ∗ (t) = nk=1 the weighted average value of all the states of the agents in G 1 , is the trajectory to which all the N agents synchronize. That is, the consensus trajectory is those agents, which can be connected to all the other nodes by directed paths. Practically speaking, such agents influence directly or indirectly all the other agents and thus play the roles of leaders in the group of agents. It is not difficult to derive x ∗ (t) = γ T x(t) for any given Laplacian matrix L, which not necessarily takes the Frobenius form, where γ is the column vector satisfying γ T L = 0 and γ T 1 = 1. Remark 3: For the special case that all the eigenvalues of A belong to the closed left-half complex plan and B is of full row rank, we can specify the final consensus value. In fact, one can choose K = B T (B B T)−1 . Thus, the compact form of the system dynamics is x(t) ˙ = [I N ⊗ A − L ⊗ B K ]x(t) = [I N ⊗ A − L ⊗ In ]x(t). This in turn implies x(t) = [exp(−Lt) ⊗ exp(At)]x(0). Since G has a directed spanning tree, there exists a nonnegative column vector β such that β T L = 0 and β T 1 = 1. From the fact that exp(−Lt) approaches 1β T exponentially fast as t approaches infinity together with the condition that A has no eigenvalues with positive real part, it then follows that limt →∞ (exp(−Lt) − 1β T ) ⊗ exp(At) = 0. This finally

QIN et al.: EXPONENTIAL SYNCHRONIZATION OF COMPLEX NETWORKS OF LINEAR SYSTEMS AND NONLINEAR OSCILLATORS

Fig. 1.

Pursuit graph with N = 20 nodes.

Fig. 3.

Fig. 2.

515

Evolution of E 1 (t) and E 2 (t).

X, Y trajectories of all the nodes’ states xi , i = 1, 2, . . . , 20.

yields

    lim x(t) − 1β T ⊗ exp(At) x(0) = 0 t →∞   i.e., the group decision value is β T ⊗ exp(At) x(0). Example 1: Consider a group of N = 20 harmonic oscillators in R2 with the underlying network topology as shown in Fig. 1, where     0 1 0 A= , B= . −1 0 1 Evidently, the matrix pair (A, B) is stabilizable. It is easy to compute that a(L) = 0.0489. Since G in Fig. 1 is strongly connected, q is given by

q = 2a(L)diag{1/20, . . . , 1/20}.

Then, one can choose δ = 0.08 ≤ λmin ( q )/λmax (), which, together with inequality (13), yields a positive solution   1.5867 0.0499 P= 0.0499 1.5804 and K = B T P = [0.0499 1.5804]. Let

  20  20  1  1   1 E 1 (t) = x  (t) x i (t) −   20 i=1

and

=1

  20  20  1  2   2 x  (t) E 2 (t) = x i (t) −   20 i=1

Fig. 4. Example of an interaction topology having a directed spanning tree.

while Fig. 3 shows that average synchronization is achieved since the pursuit graph in Fig. 1 is strongly connected and balanced. To further illustrate our result, we consider the following example where the network topology takes more general structure Example 2: Let ⎤ ⎡ −1 −1 0 ⎢ ⎥ 2 1 ⎦ , B = [0 1 0]T . A = ⎣ −1 0 0 −3 Obviously (A, B) is stabilizable. Assume that the network topology Is, as shown in Fig. 4. Evidently, G has a directed spanning tree since there are three root nodes, i.e., nodes 1–3. By some manipulation, one obtains δ = 1. According to (13), P can be chosen as ⎡ ⎤ 1.3322 −1.5792 −0.2676 ⎢ ⎥ 5.0263 0.8785 ⎦ P = ⎣ −1.5792 −0.2676 0.8785 0.4975 and thus K = B T P = [−1.5792 5.0263 0.8785]. From Fig. 5, which specifies, respectively, the X, Y , and Z trajectories of the error states ei (t) = x i (t)− 13 (x 1 (t)+x 2 (t)+x 3 (t)) ∈ R3 , i = 1, . . . , 6, it can be observed that the states of all the nodes converge to the weighted average of the states of nodes 1–3.

=1

be the quantities describing the process of the states of the nodes to their average value. It is easily observed from Fig. 2 that all the oscillators asymptotically achieve synchronization,

B. Network of Nonlinear Oscillators: Fixed Topology In this section, we will employ the techniques developed in the proof of Theorem 1 to deal with the synchronization of

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and further the compact dynamics is given by    e˙(t) = I N − 1ηT ⊗ In f(x(t)) − c(L ⊗ )e(t)

(15)

where f(x(t)) = [ f (x 1 (t))T , . . . , f (x N (t))T ]T . Consider the Lyapunov function candidate V (t) = eT (t)( ⊗ In )e(t)

Fig. 5.

X, Y, Z trajectories of error states ei , i = 1, 2, . . . , 6.

coupled nonlinear oscillators (6). Unless otherwise explicitly specified, all the notations used in the proof of Theorem 1 still work here. Lemma 4: For any given vector x, y ∈ Rm , we have 2x T y ≤ x T x + y T y. Similar to that in [21] and [22], assume that the nonlinear function f satisfies the following Lipschitz condition. Assumption 2: There exists a constant ρ > 0 such that

where  is exactly the same as that chosen for (10). Differentiating V (t) yields    V˙ (t) = −ceT (t) L + L T  ⊗  e(t)    +2eT (t)( ⊗ In ) I N − 1ηT ⊗ In f(x(t))    = −ceT (t) L + L T  ⊗  e(t)     +2eT (t)  − 1ηT ⊗ In f(x(t))−1 N ⊗ f (x ∗ (t))     (16) +2eT (t)  − 1ηT ⊗ In 1 N ⊗ f (x ∗ (t))    T T T = −ce (t) L + L  ⊗  e(t) + 2e (t)     × −1ηT ⊗ In f (x(t))−1 N ⊗ f (x ∗ (t)) (17) where (17) is obtained from (16) by observing the third term therein together with the fact that ηT 1 N = 1 and thus      − 1ηT 1 N =  1 N − 1 N ηT 1 N = 0.

Furthermore, applying Assumption 2 to (17) gives      V˙ (t) ≤ −ceT (t) q ⊗  e(t) + eT (t)  − 1ηT ⊗ In  T  ×  − 1ηT ⊗ In e(t) 2   f (x 1 (t))− f (x 2 (t))2 ≤ ρx 1 (t)−x 2 (t)2 ∀x 1 , x 2 ∈ Rn ∀t ≥ 0. + f (x(t), t) − 1 N ⊗ f (x ∗ (t), t)

Remark 4: Note that if ∂ f i /∂ x j , i, j = 1, . . . , n, are bounded, then it is easy to see that the condition in the above assumption is satisfied automatically. As such, Assumption 2 holds for many well-known systems, such as Lorenz system, Chen system, Lü system, and various neural networks. Theorem 2: Consider the complex network (6) of nonlinear oscillators. Suppose that Assumption 2 holds. Then, the complex network (6) can be synchronized exponentially fast with  the least rate of δ + ρ − cλmin ( q ) · λmin () /2λmax () if the network topology G has a directed spanning tree and the coupling strength c satisfies c>

δ+ρ λmin ( q ) · λmin ()

where δ = λmax ( − 1ηT )( − 1ηT )T . Proof: Let η = [β1T , 0, . . . , 0]T ∈ R N and x(t) = T [x 1 (t), . . . , x NT (t)]T . Obviously ηT 1 N = 1. Similar to the derivation for (8), one obtains e˙i (t) = f (x i (t)) + c

N 

ai j (e j (t) − ei (t)) −

j =1

= f (x i (t)) −

n1 

n1 

β1k x˙k (t)

k=1

β1k f (x k (t), t)

+c

j =1

ai j (e j (t) − ei (t))

thus completing the proof. Remark 5: Theorem 2 shows that if the overall coupling strength is strong enough, then the synchronization generated by the couplings among nodes dominate over the unstable mode of the nonlinear self-dynamics and thus leads to the complete synchronization of the network systems. C. Application: Pinning Synchronization In general, the final synchronized state depends on the initial state and the dynamical rules governing the isolated nodes as well as the network topology. Thus, it is rather difficult to compute it. Pinning synchronization means to synchronize the states of the nodes in the network to the solution, say s(t), of the uncoupled system (i.e., s˙ (t) = As(t) for the linear system and s˙ (t) = f (s(t), t) for the nonlinear oscillator) by adding controllers to a small fraction of the nodes in the network. The pinning controlled network of N linear systems is described by  ai j (x j (t) − x i (t)) + di (s(t) x˙i (t) = Ax i (t) + B K j ∈Ni

k=1 N 

≤ −cλmin ( q ) · λmin ()e(t)2 + δe(t)2 + ρe(t)2   = − cλmin ( q ) · λmin () + δ + ρ e(t)2

(14)

 − x i (t)) ,

i = 1, 2, . . . , N

(18)

where di > 0 whenever node i is pinned and di = 0 otherwise.

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Index the pinning controller as virtual node 0, and let ai0 = bi , a0i = 0, i = 1, . . . , N. Then, the system dynamics (18) is rewritten as x˙i (t) = Ax i (t) + B K

N 

ai j (x j (t) − x i (t)), i = 0, 1, . . . , N.

n 

In this framework, the dynamics for node i, i 1, 2, . . . , N, in the complex network is described by

− i j ∈ {0} ∪ [α, α], ¯ i, j = 1, . . . , N, i = j ; 1T L = 0;  G(L) is weakly connected .

(20)

IV. S YNCHRONIZATION U NDER DYNAMIC N ETWORK T OPOLOGY This section aims to present some results concerning dynamic network topology with or without pinning controller, where both the network topological structure as well as the weighting factors ai j (t) are dynamically changing and further, differently from the finite weighting factors in [27], [28], [39], weighting factors can be chosen from an infinite set.

(21)

and σ (t) : [0, ∞) → IG be a switching signal whose image at time t is the index associated with a graph in G. We need the following result to help with the synchronization analysis. Lemma 5: Let ϒ be the set of all possible Laplacian matrices with which the associated digraph are balanced and weakly connected   ϒ = L = [i j ] L is a graph Laplacian;

j =1, j  =i

If G¯ has a directed spanning tree, then the network system (20) can be synchronized to s(t) exponentially fast for sufficiently large coupling strength c. Remark 6: A lower bound for c can be specified using exactly the same method as that for Theorem 2. Note that Corollary 3 is just the pinning synchronization considered in [9] and [30] in the context of, respectively, undirected and directed networks. Remark 7: It is worthwhile to note that the convergence rate may be investigated by looking at the second smallest eigenvalue (also termed as algebraic connectivity [13]) of the Laplacian matrix of the network topology, like that as done in [28]. However, this technique works only with undirected graphs or strongly connected and balanced digraphs. The algebraic connectivity for undirected graphs was later extended in [47] and [52] to digraphs, which are strongly connected but may not be balanced. Nevertheless, it is still difficult to explicitly specify the convergence rate for graphs with general topology in [47] and [52] mainly due to the technical analysis as summarized in Section I.

ai j (t)(x j (t) − x i (t)).

Assume that all the nonzero and hence positive weighting factors are chosen from [α, α], ¯ i.e., ai j (t) ∈ [α, α] ¯ if there exists a coupling from node j to node i , and ai j (t) = 0 otherwise. Let G be the set of all possible balanced network topologies of the N agents indexed by 1, 2, . . . , N    ¯ G = G G(V, , A = [ai j ]) is balanced and ai j ∈ 0 ∪ [α, α]

ai j (x j (t) − x i (t))

+ cdi (s(t) − x i (t)), i = 1, 2, . . . , N.

N 

=

j =1, j  =i

(19)

This implies that pinning synchronization of network systems with N nodes is transformed to the synchronization for the networks of N + 1 nodes. Let G¯ denote the virtual digraph consisting of graph G, node 0 (the pinning controller) and the directed edges from 0 to the nodes in G, which are pinned. Therefore, as a corollary of Theorem 1, we directly obtain the following result concerning the pinning synchronization. Corollary 2: Consider the pinning controlled system (18). Under Assumption 1, if G¯ has a directed spanning tree, then an appropriate feedback matrix K can be designed such that the network system (18) synchronizes to s(t) exponentially fast. Similarly, we have the following corollary of Theorem 2. Corollary 3: Consider the following pinning controlled dynamical nonlinear network system x˙i (t) = f (x i (t)) + c

A. Networks of Linear Systems: Dynamic Topology

x˙i (t) = Ax i (t) + B K

j =0, j  =i

517

2

Then, ϒ is a compact set in R N . Proof: Note that the set of all N × N matrices can 2 be viewed as the metric space R N . Each L = [i j ] in ϒ can be viewed as a vector [11 , . . . , 1N , 21 , . . . , 2 2N , . . . ,  N1 , . . . ,  N N ] in R N . To prove that ϒ is compact 2 in Euclidean Space R N , it is equivalent to proving that ϒ is a closed and bounded set. Let   ϒ1 = [i j ] − i j ∈ {0} ∪ [α, α], ¯ i, j = 1, . . . , N, i = j ;  ii ∈ [0, N α], ¯ i = 1, . . . , N     N i j = 0, i = 1, . . . , N ϒ2 = [i j ] j =1

 N    i j = 0, j = 1, . . . , N . ϒ3 = [i j ] 

i=1

It can be easily derived from the definition of Laplacian matrix that ϒ1 ∩ϒ2 ∩ϒ3 is the set consisting of all graph Laplacian L, which satisfies 1T L = 0 and whose the off-diagonal elements ¯ are chosen from the set 0 ∪ [α, α]. We first prove that ϒ1 ∩ ϒ2 ∩ ϒ3 is closed and bounded in 2 R N . In fact, ϒ1 is closed and bounded; and the sets ϒ2 , ϒ3 are closed but unbounded. The former argument holds since it is the product space of N 2 closed and bounded sets in R1 . For the latter argument, we only prove in the following that the set ϒ2 is closed as a similar proof can be derived for that of ϒ3 . Let   Si = [i,1 , . . . , i,N ][i,1 , . . . , i,N ] is the vector taken from  the i -th row of [i j ] ∈ ϒ2 , i = 1, 2, . . . , N.

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Then, ϒ2 = S1 × S2 × · · · × SN . It is clear that ϒ2 is a closed 2 set in R N if each Si , i = 1, 2, . . . , N, is closed in R N . To prove that Si is closed, we introduce the following continuous multivariate function: f : R N → R1 , f (x) :=

N 

x i ∀x = [x 1 , x 2 , . . . , x N ] ∈ R N .

i=1

Since f is continuous and {1} is a closed set in R1 , f −1 ({1}) is closed in R N , i.e., each set Si , i = 1, . . . , N, is closed in R N . Therefore, ϒ1 ∩ ϒ2 ∩ ϒ3 is closed and bounded. On the other hand, denote by  the set of all N × N nonnegative matrix with zero diagonal elements. It is clear that there are only finite different types1 in  and all the digraphs associated with the matrices having the same type in  are also with the same topological structure. Note that  can be partitioned by the equivalence relation ∼. Let [ A] := {B ∈ |B ∼ A} denote the equivalence to which A ( A ∈ ) belongs. Without loss of generality, denote by [ A1 ], [ A2 ], . . . , [ Am ] all the equivalence classes with which the associated digraph are weakly connected, and let ϒ4 =

m  k=1

ϒ4k =

m    [i j ] − i j ∈ {0} ∪ [α, α], ¯ k=1

¯ i, j = 1, . . . , N, i = j ; ii ∈ [0, N α],  and − L + diag{11 , . . . ,  N N } ∼ Ak It is clear from the definition of ϒ that ϒ = ϒ1 ∩ϒ2 ∩ϒ3 ∩ϒ4 . Similar to the discussion above, it can be easily obtained that ϒ4 is also closed and bounded since each ϒ4k is a product spaces of N 2 closed and bounded set in R1 . This, combined 2 with the above analysis, implies that ϒ is compact in R N . Theorem 3: Consider the complex network system (21) with dynamic underlying network topology G(t). Then, the states of all the nodes are synchronized to x ∗ (t) = N1 (x 1 (t) + · · ·+x N (t)) exponentially fast if G(t) is kept weakly connected and balanced. Proof: i and the  N Denote the state error between node N x k (t) as ei (t) = x i (t) − N1 k=1 x k (t), i = state N1 k=1 1, 2, . . . , N. Let e(t) = [e1T (t), e2T (t), . . . , eTN (t)]T . By observing that 1TN L σ (t ) = 0, where L σ (t ) is the Laplacian matrix of G σ (t )2 , i.e., the network topology at time t, then similar to the derivation of (9), one can obtain the following compact form of the error dynamical system: e˙(t) = [I N ⊗ A − L(t) ⊗ B K ]e(t), G(t) ∈ G.

(22)

On the other hand, since G σ is weakly connected and balanced, it follows that λ2 (L σ + L σT /2) > 0 for any σ (·). Define the following multivariate function:   L + LT , ∀L ∈ ϒ. g : ϒ → R1 ; g(L) := λ2 2 From the well-known fact that the eigenvalues of any matrix are continuous functions of the elements of the matrix, it is 1 Two nonnegative matrices P and P are said of the same type, P ∼ P , 1 2 1 2 if they have zero elements and positive elements in the same places [26]. 2 Variable t may be dropped out from σ (t) sometimes for notational simplicity

Fig. 6.

Switching mode and the three possible interaction topologies.

clear that g(·) is a continuous function. This, together with Lemma 5 that the set ϒ is compact, implies that there exists a positive number λ¯ 2 such that λ¯ 2 = min{λ2 (L + L T /2)|L ∈ ϒ} > 0. Then, from the fact that (A, B) is stabilizable it follows that there exists a solution P > 0 to the following Riccati inequality: P A + AT P − 2λ¯ 2 P B B T P + 2λ¯ 2 In < 0.

(23)

Let K = B T P and consider the following Lyapunov function candidate: V (t) = eT (t)[I N ⊗ P]e(t). Differentiating V (t) along the trajectories of (22) yields    V˙ (t) = eT (t) I N ⊗ AT P + P A    − L σ + L σT ⊗ P B B T P e(t) (24)     ≤ eT (t) I N ⊗ AT P + P A − 2λ¯ 2 I N ⊗ P B B T P e(t) =

N 

  eiT (t) AT P + P A − 2λ¯ 2 P B B T P ei (t)

(25)

i=1

≤ −2λ¯ 2

N 

eiT (t)ei (t) < 0

i=1

for  N any e(t) = 0, where (25) is obtained by observing i=1 ei (t) = 0 and then applying Corollary 1 to the second term in (24). Note that N N   V (t) = eiT (t)Pei (t) ≤ λmax (P) eiT (t)ei (t) i=1

i=1

which, together with the above inequality, implies that V˙ (t) ≤ −2λ¯ 2 /λmax (P)V (t) and thus V (e(t)) ≤ V (e(0)) exp(−2λ¯ 2 /λmax (P)). This means that the error state e(t) approaches 0 exponentially fast with a least speed of γ = λ¯ 2 /λmax (P). Example 3: Assume that the topology G σ switches every 0.1 s periodically, as shown in the left graph of Fig. 6, from weighted digraphs Ga to Gb, Gb to Gc, and then Gc to Ga. It is easy to check that G a , G b , and G c are all strongly connected and balanced digraphs. Assume that the state trajectories of the nodes evolve in R3 . Let ⎡ ⎤ −2 1 1 ⎢ ⎥ A = ⎣ 1 −1 0 ⎦ , B = [0 1 0]T . 0 1 −1 It can be easily computed that λ¯ 2 = 0.7192. Then, according to (23), P can be chosen as ⎡ ⎤ 0.4385 0.4116 0.2344 ⎢ ⎥ P = ⎣ 0.4116 0.9552 0.4498 ⎦ . 0.2344 0.4498 0.6832

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519

Consider the Lyapunov function candidate V (t) = eT (t)(I N ⊗ In )e(t). Noting that (I N − N1 1 N 1TN )1 N = 0, then differentiating V (t) yields    V˙ (t) = −ceT(t) L(t) + L T (t) ⊗  e(t)    1 +2eT (t) I N − 1 N 1TN ⊗ In f(x(t)) N    = −ceT(t) L(t) + L T (t) ⊗  e(t) + 2eT (t)     1 × I N − 1 N 1TN ⊗ In f(x(t)) − 1 N ⊗ f (x ∗ (t)) . N Following the proof of Theorem 3, one obtains Fig. 7.

X, Y, Z trajectories of error states ei , i = 1, . . . , 4.

Fig. 7 shows, respectively, the X, Y, Z trajectories of the error states between each node andthe average states of all the nodes, i.e., ei (t) = x i (t)−1/4 4j =1 x j (t) ∈ R3 . It is observed from Fig. 7 that consensus can be reached by assigning K = B T P. B. Network of Nonlinear Oscillators: Dynamic Topology In this case, each node evolves according to the following dynamics: n 

x˙ i (t) = f (x i (t)) + c

i = 1, 2, . . . , N (26) where ai j (t) is as that defined in system (21). Theorem 4: Consider complex network of nonlinear oscillators (26). Suppose that Assumption 2 holds. Then, the network system (26) can be synchronized exponentially fast to x ∗ (t) = N1 (x 1 (t) + · · · + x N (t)) with the least rate of 1 ¯ 2 (γ + ρ − c λ2 · λmin ()) if the induced network topology ¯ G(t) is kept weakly connected and balanced, and the coupling strength c satisfies γ +ρ λ¯ 2 λmin ()

where γ = λmax (I N − N1 11T )(I N − N1 11T )T . ¯ and e(t) Proof: Let ei (t) = x i (t) − x(t) [e1T (t), e2T (t), . . . , eTN (t)]T . Then, we have e˙i (t) = f (x i (t)) −

N 1  f (x k (t)) N

≤ −cλ¯ 2 · λmin ()e(t)2 + γ e(t)2 + ρe(t)2   = − cλ¯ 2 · λmin () + γ + ρ e(t)2 where γ = λmax (I N − is complete.

1 T N 11 )(I N

−

1 T T N 11 ) .

Hence, the proof

V. C ONCLUSION

ai j (t)(x j (t) − x i (t)),

j =1, j  =i

c>

V˙ (t) ≤ −cλ¯ 2 eT (t)[I N ⊗ ]e(t)    1 + eT (t) I N − 11T ⊗ In N   T 1 T × I N − 11 ⊗ In e(t) N  2 +f(x(t)) − 1 N ⊗ f (x ∗ (t))

=

We have unified, in this paper, the convergence analysis for synchronization of the network of linear systems and the network of nonlinear oscillators based on the exploration of the techniques from the algebraic graph theory, matrix analysis, and Lyapunov stability theory. With the unified analysis, we have systematically considered the synchronization problems under various different settings, including the cases under fixed and dynamic topologies, the cases with and without pinning controllers. Finally, we have shown that under certain connectivity assumptions on the network topologies, an appropriate feedback matrix can be designed to make the network of linear systems synchronize exponentially, while the nonlinear oscillators can achieve synchronization exponentially fast if the overall coupling strength of the network system is sufficiently large to overcome the divergence caused by the individual nonlinear system dynamics. In addition, we have explicitly specified the convergence rates for both of the different synchronization problems. In future work, we will consider the synchronization problem for more general cases of nonlinear oscillators under some relaxed assumption on nonlinear functions.

k=1

+c

N 

ai j (t)(e j (t) − ei (t)).

A PPENDIX P ROOF OF L EMMA 1

j =1

Furthermore, the compact error dynamics is described by    1 e(t) ˙ = I N − 11T ⊗ In f(x(t)) − c(L(t) ⊗ )e(t) N

Proof of Lemma 1: First, we prove the following inequality:

where f(x(t)) = [ f (x 1

holds for any x satisfying (1).

(t))T , . . . ,

f (x N

(t))T ]T .

x T ( Lˆ ⊗ In )x ≥ a(L)x T( ⊗ In )x

(27)

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For illustration convenience, denote x as x = [x 1T , x 2T , . . . , x NT ]T , where each x i = [x i1, . . . , x in ]T , i = 1, 2, . . . , N, is an n × 1 column vector. With this N βi x i = 0. Then notation, (1) holds if and only if i=1 x T ( Lˆ ⊗ In )x =

1  ˆ i j (x i − x j )T (x i − x j ) 2 N

N

i=1 j =1

2 1  ˆ  k i j x i − x kj 2 N

n

=

N

(28)

k=1 i=1 j =1

where Lˆ = [ˆi j ] N×N . Let x˜ k = [x 1k , x 2k (t), . . . , x Nk ]T , k = 1, 2, . . . , n. It then follows from (1) that β T x˜ k = 0, k = 1, 2, . . . , n, which together with (28) and the fact that 2  T 1  ˆ  k i j x i − x kj = x˜ k Lˆ x˜ k 2 N

N

i=1 j =1

 yields x T ( Lˆ ⊗ In )x = nk=1 (x˜ k )T Lˆ x˜ k . Note that β T x˜ k = 0, k = 1, . . . , n,. Then, it follows from (3) that: n     T a(L) x˜ k x˜ k x T Lˆ ⊗ In x ≥ k=1

= a(L)x T ( ⊗ In )x. Now, we proceed to prove the lemma. Since matrix B is symmetric positive semidefinite, there exists a matrix E ∈ Rm×n such that B = E T E. Thus    x T ( Lˆ ⊗ B)x = x T Lˆ ⊗ E T Im E x = [(I N ⊗ E)x]T ( Lˆ ⊗ Im )[(I N ⊗ E)x]. Let y = (I N ⊗ E)x and ςi be the m × 1 column vector with the i th entry being 1 and 0 elsewhere. By observing that N   T    β ⊗ ςiT y = β T ⊗ ςiT E x = ςiT E βjxj = 0 j =1

and inequality (27), we have y T ( Lˆ ⊗ Im )y ≥ a(L)y T ( ⊗ Im )y = a(L)x T ( ⊗ B)x thereby completing the proof. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the five anonymous reviewers for their valuable comments and suggestions that have helped them in improving this paper considerably. R EFERENCES [1] R. Albert and A. L. Barabási, “Statistical mechanics of complex networks,” Rev. Modern Phys., vol. 74, no. 1, pp. 47–97, 2002. [2] A. L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999. [3] V. N. Belykh, I. V. Belykh, and M. Hasler, “Connection graph stability method for synchronized coupled chaotic systems,” Phys. D, vol. 195, nos. 1–2, pp. 159–187, 2004.

[4] I. V. Belykh, V. N. Belykh, and M. Hasler, “Blinking model and synchronization in small-world networks with a time-varying coupling,” Phys. D, vol. 195, nos. 1–2, pp. 188–206, 2004. [5] I. V. Belykh, V. N. Belykh, and M. Hasler, “Generalized connection graph method for synchronization in asymmetrical networks,” Phys. D, vol. 224, nos. 1–2, pp. 42–51, 2006. [6] B. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities Systems Control Theory. Philadelphia, PA, USA: SIAM, 1994. [7] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory. Cambridge, U.K.: Cambridge Univ. Press, 1991. [8] M. Cao, A. S. Morse, and B. D. O. Anderson, “Reaching a consensus in a dynamically changing environment: Convergence rates, measurement delays, and asynchronous events,” SIAM J. Control Optim., vol. 47, no. 2, pp. 601–623, 2008. [9] T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 6, pp. 1317–1326, Jun. 2007. [10] D. Ding, Z. Wang, B. Shen, and H. Shu, “H∞ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 5, pp. 725–736, May 2012. [11] H. Dong, Z. Wang, and H. Gao, “Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 3164–3173, Jun. 2012. [12] H. Dong, Z. Wang, and H. Gao, “Distributed H∞ filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks,” IEEE Trans. Ind. Electron., vol. 60, no. 10, pp. 4665–4672, Oct. 2013. [13] C. Godsil and G. Doyle, Algebraic Graph Theory. New York, NY, USA: Springer-Verlag, 2001. [14] D. Guo, Q. Wang, and M. Perc, “Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses,” Phys. Rev. E, vol. 85, no. 6, pp. 061905-1–061905-8, 2012. [15] A. Jadbabaie, J. Lin, and S. A. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [16] Z. Ji, Z. Wang, H. Lin, and Z. Wang, “Interconnection topologies for multi-agent coordination under leader-follower framework,” Automatica, vol. 45, no. 12, pp. 2857–2863, 2009. [17] Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 1, pp. 213–224, Jan. 2010. [18] J. Liang, Z. Wang, and X. Liu, “Distributed state estimation for discretetime sensor networks with randomly varying nonlinearities and missing measurements,” IEEE Trans. Neural Netw., vol. 22, no. 3, pp. 486–496, Feb. 2011. [19] J. Liang, Z. Wang, X. Liu, and P. Louvieris, “Robust synchronization for 2-D discrete-time coupled dynamical networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 6, pp. 942–953, Jun. 2012. [20] Y. Liu, Z. Wang, J. Liang, and X. Liu, “Synchronization and state estimation for discrete-time complex networks with distributed delays,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 5, pp. 1314–1325, Oct. 2008. [21] J. Lu and D. W. C. Ho, “Globally exponential synchronization and synchronizability for general dynamical networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 2, pp. 2350–2361, Apr. 2010. [22] J. Lu, D. Ho, J. Cao, and J. Kurths, “Exponential synchronization of linearly coupled neural networks with impulsive disturbances,” IEEE Trans. Neural Netw., vol. 22, no. 2, pp. 329–335, Feb. 2011. [23] W. Lu and T. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Phys. D, vol. 213, no. 2, pp. 214–230, 2006. [24] J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 841–846, Jun. 2005. [25] C. Ma and J. Zhang, “Necessary and sufficient conditions for consensusability of linear multi-agent systems,” IEEE Trans. Autom. Control, vol. 55, no. 5, pp. 1263–1268, May 2010. [26] H. Minc, Nonnegative Matrices. New York, NY, USA: Wiley, 1988. [27] W. Ni and D. Cheng, “Leader-following consensus of multi-agent systems under fixed and switching topologies,” Syst. Control Lett., vol. 59, nos. 3–4, pp. 209–217, 2010.

QIN et al.: EXPONENTIAL SYNCHRONIZATION OF COMPLEX NETWORKS OF LINEAR SYSTEMS AND NONLINEAR OSCILLATORS

[28] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. [29] L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2109–2112, 1998. [30] J. Qin, W. X. Zheng, and H. Gao, “On pinning synchronisability of complex networks with arbitrary topological structure,” Int. J. Syst. Sci., vol. 42, no. 9, pp. 1559–1571, 2011. [31] J. Qin, H. Gao, and W. X. Zheng, “Second-order consensus for multiagent systems with switching topology and communication delay,” Syst. Control Lett., vol. 60, no. 6, pp. 390–397, 2011. [32] J. Qin, W. X. Zheng, and H. Gao, “Consensus of multiple second-order vehicles with a time-varying reference signal under directed topology,” Automatica, vol. 47, no. 9, pp. 1983–1991, 2011. [33] W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655–661, May 2005. [34] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control: Collective group behavior through local interaction,” IEEE Control. Syst. Mag., vol. 27, no. 2, pp. 71–82, Apr. 2007. [35] L. Scardovi and R. Sepulchre, “Synchronization in networks of identical linear systems,” Automatica, vol. 45, no. 11, pp. 2557–2562, 2009. [36] B. Shen, Z. Wang, and Y. S. Hung, “Distributed H∞ -consensus filtering in sensor networks with multiple missing measurements: The finite-horizon case,” Automatica, vol. 46, no. 10, pp. 1682–1688, 2010. [37] B. Shen, Z. Wang, and X. Liu, “Bounded H∞ synchronization and state estimation for discrete time-varying stochastic complex networks over a finite-horizon,” IEEE Trans. Neural Netw., vol. 22, no. 1, pp. 145–157, Jan. 2011. [38] S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, no. 6825, pp. 268–276, 2001. [39] Y. Su and J. Huang, “Two consensus problems for discrete-time multiagent systems with switching network topology,” Automatica, vol. 48, no. 9, pp. 1988–1997, 2012. [40] E. S. Tuna, “Synchronizing linear systems via partial-state coupling,” Automatica, vol. 44, no. 8, pp. 2179–2184, 2008. [41] Q. Wang, M. Aleksandrab, M. Perc, and Q. Lu, “Taming desynchronized bursting with delays in the Macaque cortical network,” Chin. Phys. B, vol. 20, no. 4, pp. 040504-1–040504-6, 2011. [42] Q. Wang, M. Perc, Z. Duan, and G. Chen, “Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling,” Phys. A, vol. 389, no. 16, pp. 3299–3306, 2010. [43] Q. Wang, G. Chen, and M. Perc, “Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling,” PLOS ONE, vol. 6, no. 1, p. e15851, 2011. [44] Q. Wang, M. Perc, Z. Duan, and G. Chen, “Synchronization transitions on scale-free neuronal networks due to finite information transmission delays,” Phys. Rev. E, vol. 80, no. 2, pp. 026206-1–026206-8, 2009. [45] Q. Wang, Z. Duan, M. Perc, and G. Chen, “Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability,” Europhys. Lett., vol. 83, no. 5, pp. 50008-1–50008-6, 2008. [46] Z. Wang, Y. Wang, and Y. Liu, “Global synchronization for discretetime stochastic complex networks with randomly occurred nonlinearities and mixed time-delays,” IEEE Trans. Neural Netw., vol. 21, no. 1, pp. 11–25, Jan. 2010. [47] C. W. Wu, “Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,” Nonlinearity, vol. 18, no. 3, pp. 1057–1064, 2005. [48] F. Xiao and L. Wang, “State consensus for multi-agent systems with switching topologies and time-varying delays,” Int. J. Control, vol. 79, no. 10, pp. 1277–1284, 2006. [49] F. Xiao and L. Wang, “Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays,” IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1804–1816, Sep. 2008. [50] W. Xiong, D. W. C. Ho, and Z. Wang, “Consensus analysis of multiagent networks via aggregate and pinning approaches,” IEEE Trans. Neural Netw., vol. 22, no. 8, pp. 1231–1240, Aug. 2011. [51] J. Xu, T. Li, L. Xie, and K. Y. Lum, “Dynamic consensus and formation: Fixed and switching topologies,” in Proc. 18th IFAC World Congr., Milan, Italy, 2011, pp. 9188–9193.

521

[52] W. Yu, G. Chen, M. Cao, and J. Kurths, “Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 881–891, Jun. 2010. [53] W. Yu, G. Chen, and M. Cao, “Consensus in directed networks of agents with nonlinear dynamics,” IEEE Trans. Autom. Control, vol. 56, no. 6, pp. 1436–1441, Jun. 2011.

Jiahu Qin (M’12) received the B.S., M.S., and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 2003, 2005, and 2012, respectively. He was a Visiting Fellow with the University of Western Sydney, Penrith, NSW, Australia, from July 2009 to December 2009. He joined the University of Science and Technology of China, Hefei, China, in 2013. He is currently a Research Fellow with the Australian National University. His current research interests include multiagent coordination and complex dynamical networks.

Huijun Gao (F’14) received the Ph.D. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2005. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from 2003 to 2004. From 2005 to 2007, he was a Post-Doctoral Researcher with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. Since 2004, he has been with the Harbin Institute of Technology, where he is currently a Professor and the Director of the Research Institute of Intelligent Control and Systems. His current research interests include network-based control, robust control/filter theory, time-delay systems, and their engineering applications. Dr. Gao is an Associate Editor of the journal Automatica, the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS , the IEEE T RANSACTIONS ON C YBERNETICS , the IEEE T RANSACTIONS ON F UZZY S YSTEMS , the IEEE/ASME T RANSACTIONS ON M ECHATRONICS , and the IEEE T RANS ACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY . He is an AdCom Member of the IEEE Industrial Electronics Society.

Wei Xing Zheng (M’93–SM’98–F’14) received the Ph.D. degree in electrical engineering from Southeast University, Nanjing, China, in 1989. He has held various faculty/research/visiting positions with Southeast University, the Imperial College of Science, Technology and Medicine, London, U.K., the University of Western Australia, Crawley, WA, Australia, the Curtin University of Technology, Bentley, WA, Australia, the Munich University of Technology, Munich, Germany, the University of Virginia, Charlottesville, VA, USA, and the University of California–Davis, Davis, CA, USA. He is currently a Full Professor with the University of Western Sydney, Penrith, NSW, Australia. Dr Zheng has served as an Associate Editor for a number of flagship journals, including the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS I: F UNDAMENTAL T HEORY AND A PPLICATIONS from 2002 to 2004, the IEEE S IGNAL P ROCESSING L ETTERS from 2007 to 2010, the IEEE T RANS ACTIONS ON C IRCUITS AND S YSTEMS -II: E XPRESS B RIEFS from 2008 to 2009, the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL from 2004 to 2007 and since 2013, Automatica since 2011, IET Control Theory and Applications since 2013, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS since 2014. He was a Guest Editor of the Special Issue on Blind Signal Processing and Its Applications for the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS -I: R EGULAR PAPERS from 2009 to 2010. He has also served as the Chair of the IEEE Circuits and Systems Society’s Technical Committee on Neural Systems and Applications, and the Chair of the IEEE Circuits and Systems Society’s Technical Committee on Blind Signal Processing.