Synchronization of Complex Dynamical Networks With ... - IEEE Xplore

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 8, AUGUST 2010

Synchronization of Complex Dynamical Networks With Time-Varying Delays Via Impulsive Distributed Control Zhi-Hong Guan, Zhi-Wei Liu, Gang Feng, Fellow, IEEE, and Yan-Wu Wang

Abstract—In this paper, the synchronization of complex dynamical networks (CDNs) with system delay and multiple coupling delays is studied via impulsive distributed control. The concept of control topology is introduced to describe the whole controller structure, which consists of some directed connections between nodes. The control topology can be designed either to be the same as the non-delayed coupling topology of the network, or to be independent of the intrinsic network topology. Based on the concept of control topology, an impulsive controller is designed to achieve the exponential synchronization of CDNs, and moreover, the exponential convergence rate can be specified. Illustrated examples have been given to show the effectiveness of the proposed impulsive distributed control strategy. Index Terms—Complex dynamical networks, control topology, global exponential synchronization, impulsive distributed control, time delay.

I. INTRODUCTION complex dynamical network (CDN) is a large set of interconnected nodes, in which each node represents an individual element in the network and edges represent the relations between them. It is well known that many systems in science, engineering, and nature can be described by models of CDNs, such as electrical power grids, world wide web, food webs, social networks, and so on [1]–[3]. Synchronization of complex networks is an important topic that has drawn a great deal of attentions [4]–[32], [35]–[38]. Network synchronization phenomena has been found in different forms both in nature and in man-made systems, such as fireflies in the forest, applause, description of hearts, distributed computing systems, routing messages in the internet, chaos-based communication network, and so on. In recent years, many synchronization criteria have been derived for CDNs with different special features, such as time varying coupling [14]–[16], time delay [7]–[13], impulsive

A

Manuscript received March 02, 2009; revised August 02, 2009 and October 24, 2009; accepted November 04, 2009. Date of publication February 17, 2010; date of current version August 11, 2010. This work was supported by the Hong Kong Research Grant Council under GRF Grant CityU 113708, and by the National Natural Science Foundation of China under Grants 60834002, 60873021, 60973039, and 60704035. This paper was recommended by Associate Editor O. De Feo. Z.-H. Guan, Z.-W. Liu, and Y.-W. Wang are with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]). G. Feng is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. Digital Object Identifier 10.1109/TCSI.2009.2037848

effects [23], and switching phenomena [21], [22], etc. Various control schemes, such as adaptive control [7], [13], [24], [35]–[38] impulsive control [25]–[29], and pinning control [30]–[34], were reported to achieve network synchronization. Some of control schemes [7], [24], [29]–[34] are based on a special solution of an isolated node, which may be difficult to obtain in some practical applications. The other control strategies [10], [28] are based on the weighted average of the states of all nodes. This may lead to the difficulty, since it is not easy for a node to obtain information of all other nodes in a large scale network. Consequently, it is more reasonable to realize the network synchronization by utilizing the information from neighboring nodes, or by utilizing the information from a small number of leading nodes. The decentralized adaptive coupling schemes were considered in [35]–[38], where the coupling gains self-adjust only based on the information from neighboring nodes. Zhang et al. [28] proposed an impulsive control scheme, in which each local controller shares information from partial nodes by taking the weights of great majority nodes as zero. In this paper, in order to avoid the implementation difficulty of the controller, the concept of control topology is introduced. By designing different control topologies according to different practical situations, the proposed synchronization controller will be easy to be implemented in practice. It is noted that in the practical cases time delays are often encountered. Ignoring them may lead to design flaws and incorrect analysis conclusions. Hence, time delays in couplings and in dynamical nodes have received considerable attention [7]–[13]. There are numerous examples in real-world which are characterized by network models with both coupling delays and node delays. For example, a distributed computer network, in which time delays in dynamical nodes represent computing time and coupling delays represent communication delays. It is noted that the synchronization of CDNs with time delays both in nodes and in couplings is still relatively unexplored. Yu et al. [12] studied the synchronization of CDNs with both node delay and coupling delay, but it was assumed that they are equal to each other. In many networks, nodes may influence each other not only by non-delayed state information but also by delayed state information. For example, in a financial system, decision-making of dealer is always influenced by both immediate transaction information and historical transaction information of some other dealers. Therefore, both non-delayed coupling and multiple delayed couplings should be considered. Motivated by the aforementioned discussion, this paper investigates the problem of distributed impulsive synchroniza-

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GUAN et al.: SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS

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tion of CDNs with system delay and multiple coupling delays. Sufficient conditions for global exponential synchronization are derived. Different from existing control schemes in literature, each distributed controller in the proposed scheme is allowed to use information from different nodes. The concept of control topology is introduced to describe the control information flow which is transmitted between nodes. By appropriately designing the control topology, the synchronization control scheme can be easily implemented in practice. Two particular control topologies are discussed. A design procedure of impulsive controllers for network synchronization is presented. Numerical simulations are given to demonstrate the effectiveness of the proposed method. An outline of this paper is as follows. In Section II, the model of CDNs with delays is introduced and some necessary mathematical preliminaries are given. In Section III, an impulsive distributed control strategy, which can be expressed by control topology, are presented and some synchronization criteria are derived. Two kinds of particular control topologies are introduced in Section IV. In Section V, a design procedure of impulsive controllers is presented and some numerical simulations are given. Concluding remarks are finally stated in Section VI.

II. MODEL DESCRIPTION AND MATHEMATIC PRELIMINARIES First of all, some mathematical notations used throughout this paper are presented. denotes the set of real numbers, . For , denotes its transpose. The vector norm is defined as . are the set of real matrices. For , and are the maximum and the minimum eigenvalues of , the spectral norm used will be , . is the identity matrix of order , and . For , denote

A. Model Description linearly coupled identical

Consider a CDN consisting of nodes, which is described by

(2)

where , is matrix, the state vector of the th node, is a constant time delays and may be unknown (constant or time-varying) but is bounded by a known constant, , , the i.e., and denote the nonconstant delayed and the delayed coupling strength respectively, and represent the nondelayed and the delayed inner coupling matrices of the network respectively, is a control input, is a continuous map satisfying (3) ,

,

and are the non-delayed and the delayed coupling matrix respectively, where and are defined as follows: if there is a ( delayed) connection from , then, the ( delayed) coupling strength node to node ; otherwise, , for , is defined by (4)

For product of

and

, is defined as .. .

, the Kronecker

..

.

.. .

(1)

denotes the set of matrices such that the sum of the element is defined in each row is equal to the real number . The set , each row of contains as follow: if , and all other eleexactly one element 1 and one element ments are zeros. denotes the column indexes of the first is defined (second) nonzero element in the th row. The set . The set is by and if , defined as follow: for any pair of the column indexes and , there exist indexes with and such that for .

and , for , Remark 1: When CDN (2) becomes the network discussed in [17]–[20], [30], and , for , CDN [31], and when (2) becomes the network discussed in [8], [10], [11]. It is assumed that the initial conditions of network (2) is given by (5) , is the set of continuous functions from

where and to

.

B. Mathematic Preliminaries In this subsection, a definition and some lemmas are given as follows. Definition 1: The CDN (2) with the initial condition (5) is globally exponentially synchronized, if there exist ,

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and


such that

,

,

Inspired by [12], [18], the following lemma is given. Lemma 1: Let be a matrix and be a matrix. Then there exists a matrix such , where . Moreover, let be a that matrix and , then , constant where and . Furthermore, if is a matrix, then . , i.e., the function Denote the function that maps to by is defined as

III. DISTRIBUTED IMPULSIVE SYNCHRONIZATION OF CDNS WITH DELAYS In this section, based on the stability results for impulsive delayed systems [39], a new impulsive distributed control scheme is proposed and several novel criteria are derived for global exponential synchronization of CDN (2). A. Control Topology In order to achieve synchronization of CDN (2), the distributed impulsive controller, for the th node, is designed as

(6) Similar to [39], the following lemma holds. Lemma 2: Let

for each fixed and satisfy

,

be nondecreasing in , be nondecreasing in . Suppose that ,

then

, for , implies , for . Lemma 3: [40] For any matrix , , and , the Kronecker product has the following properties: , where is a constant; 1) ; 2) 3) . and , Lemma 4: Let , where , , then if , , for all . , for any pair Proof: According to the definition of of and , there exist indexes , where , and , for . Without loss of generality, assume that the indexes have no duplicate elements. Consequently,

,

,

. Then, from .

where , the impulsive instant sequence satisfies , let , are the control gains, is the Dirac denotes the influence weight of the impulsive function. state of the th node on that of the th node, and for all . is designed as follow: if the th node can obtain the non-delayed information of th node, then ; otherwise, . Denote the matrix . Remark 2: The distributed impulsive control (6) utilizes the local state information and the weighted average of available state of the other nodes. This is quite different from those impulsive controllers in [28], [29] and is practically much easier to implement because the controller utilizes only the available information. The control information flow of the controller (6) can be regarded as a weighted directed graph. In this paper, the weighted directed graph is defined as the control topology of the controller (6). The Laplacian matrix of the control topology is

.. .

.. .

..

(7)

.. .

.

Thus, (6) can be equivalently expressed as follows: (8) . where Hence, the impulsively controlled CDNs with delays (2) can then be described by the following impulsive differential equations shown in (9) at the bottom of the page, where

,

,

, (9)

,


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, , . Asis left-hand continuous at , . For sume that is continuous at . simplicity, it is assume that Rewrite the network (9) by Kronecker product, see equation , (10) at the bottom of the page, where , , , for , , , ,

For any trajectories of (10), let

, taking derivative of

along the

. B. Global Exponential Synchronization To simplify the presentation, some notations are given as follows:

From Lemma 3, , Therefore

, and according to Lemma 1, .

(11) where

,

, (12)

, , , . Based on the technique of Lyapunov function and comparison principle ([26], [27], [39]), the following conclusions can be derived. matrix , such that Theorem 1: If there exist a and (13) then the CDNs with delays (9) is globally exponentially synchronized in the following sense:

According to (3)

(14) where

,

is an unique solution of , in which ,

, , , and are defined by (3), (11) and (12) respectively. Proof: For , let , , , , . Note that denotes the column indexes of the first (second) nonzero ele, ment in the ith row. Then, . Construct a Lyapunov function in the form of

,

,

, (10)

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Then, for

Let

. Accordingly,

On the other hand, when

For any , let be a unique solution of the impulsive delayed system in (15), shown at the bottom of the page. Ac, for any . By the cording to lemma 2, satisfies the inteformula for the variation of parameters, gral equation

Denote , , and

. From (13), , , . Consequently,

, hence

has an unique solution , Since , ,

.

(16) where is the Cauchy matrix of the linear impulsive system shown in (17) at the bottom of the page. , (for And from (12), ), (for ). Therefore, when

Then, the following inequality will be proved by apagoge: (18)

If (18) is not true, there exists a

satisfying (19)

(20)

. (15)

(17) .


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Thus, see equation at the bottom of the page which leads , then to a contradiction. Therefore, (18) holds. Let , . Therefore, . From Lemma 4, , , . This completes the proof. Remark 3: It is worth noting that the synchronization of the controlled CDN with delays (2) is dependent on the control topology . In the proposed control scheme, to design an appropriate control topology for the network is one of the keys to achieve synchronization. , where is Corollary 1: Assume that matrix , such that a constant, if there exist a and

then, the CDNs with delays (9) is globally exponentially synchronized in the following sense:

(21)

(25)

then, the CDNs with delays (9) is globally exponentially synchronized in the following sense:

then, the CDN with delays (9) is globally exponentially synchronized in the following sense:

(24) for

, where

is an unique solution of , in which

. From the Theorem 1 and Theorem 2, it is easy to get the following corollary for the CDN without the node delay and the coupling delays. , , if there exist a Corollary 2: Assume that matrix , such that

(22) for

, where

is an unique solution of , in which

(26) ,

where , otherwise.

. Similarly, the following conclusion can be obtained for the . To avoid the repetition, the proof is omitted here. case matrix , such that Theorem 2: If there exist a and (23)

IV. TWO PARTICULAR CONTROL TOPOLOGIES According to the discussion in the Section III, control topology is very important to achieve the network synchronization. In this section, two graphes will be taken as examples of the control topologies to show how to design a suitable control topology for CDNs.

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Fig. 2. Control topology of pinning control.

Fig. 1. (a) Unidirectional star graph. (b) Unidirectional star graph.

A. Unidirectional Star-Like Graph In many real cases, if all members of a team have a common goal, it may be very easy to reach an agreement. Based on this idea, a special control topology can be designed, in which all controllers share the same state information from common satisfies , where nodes. That means the weight is a constant, for all , i.e,

.. . In this case, Accordingly,

.. . , for any

..

.

(27)

.. . matrix

. . Thus (28)

It is easy to find a suitable control gain such that . By taking most of as zero, the synchronization control can be designed using the information from a few number of nodes. and , the If control topology is a unidirectional star graph (see Fig. 1(a)) in which the center-node is the th node. Remark 4: The pinning control [30]–[34] is to place local controllers on a small fraction of nodes in the network, where each controller shares a common reference input and the ref. In erence input is a special solution of an isolated node some practical applications, it may be very hard to obtain for all controllers. In the proposed control strategy, this is not required, and only available state information from some nodes is used by each distributed controller. Moreover, the pinning control can be expressed by the control topology framework. If the common reference input is looked as information from a virtual node, all controllers should obtain control information from it. Thus, the control topology of pinning control is a disconnected graph (as shown in Fig. 2), which contains some disconnected nodes and a unidirectional star graph, where the center-node of the star is the common reference input. The local controller is placed on the nodes which connected to the center-node. Remark 5: Some other existing control schemes on network synchronization can also be expressed by the control topology framework. For example, the controller in [7], [24], [29] can be , . All expressed as local controllers should obtain as common reference input.

Similarly, if the common reference input is looked as information from a virtual node, the control topology can be expressed as ..

.

.. .

(29)

where the last node is the virtual node. This control topology is a unidirectional star graph. The controller In [28] can be ex, , pressed as then the control topology can be described as (27). B. Directed Spanning Tree From the engineering’s point of view, when designing a synchronization controller, one of the most important is the communication cost. Obviously, the less the communication cost is, the better the controller is. Therefore, it is natural to design a control topology following the intrinsic network topology. In many cases, the non-delayed coupling topology may contain a ). In other words, directed spanning tree (if there exists a node which directly or indirectly influences all other nodes. One can design the controller without breaking its intrinsic existing network topology by choosing the directed spanning tree (here, the weights of the tree should be adjusted, ) as control topology. The directed spansuch that ning tree is a unidirectional tree. A unidirectional tree (as shown in Fig. 1(b)) is a directed tree with unidirectional connection from its father node to itself, which means that each distributed controller will only use the information from its father node. V. DESIGN PROCEDURE AND ILLUSTRATIVE EXAMPLES In this section, a design procedure for the distributed impulsive controller is presented and examples are given to illustrative the effectiveness of the design procedure. A. Design Procedure The design procedure is summarized as follows. Step 1) According to the control demands and the network environment, design a suitable control topology , and a set of matrices such find an . Moreover, according to the analysis in that the Section IV, a directed spanning tree of the nondelayed coupling topology, or a unidirectional star-


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like graph can be chosen, as the control topology. In can be used as the control gains, addition, where is a constant diagonal matrix. Step 2) Calculate . Step 3) Determine a set of impulsive control instant according to the following rules: a) if , then take ; , then take , for b) if . all Furthermore, if the exponential convergence rate is required where is a positive constant, Step 3 is replaced to be by Step 3b. Step 3b) Determine a set of impulsive control instant according to the following rules: 1) if , then take ; , then take 2) if

B. Illustrative Examples In this subsection, some networks are considered to illustrate the effectiveness of the design procedure. Take the following 2-dimensional delayed system as the node system of the network, which is described by

Fig. 3. Chaotic orbits of the system (30).

, defined by

. The total error of the network (31) is

1) A Designed Network: Firstly, consider a designed network with eight nodes. The two coupling delays are and respectively. The tree , , and coupling matrices are

(30) ,

where

with ,

,

, and

It is well known that the system (30) with above parameters is chaotic [41] (see Fig. 3). The Lipschitz constants can be calcuand . lated: For simplicity, consider the network (2) with two terms of delayed coupling, described as follows:

(31)

,

, , . For CDN (31), the initial values are given as follows: , , , where and are chosen randomly from for where

,

,

Fig. 4 shows the evolution process of the total error and the . It is easy to states of the designated network (31) with see that the designed network (31) isn’t synchronized without impulsive control.

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0

Fig. 4. Designed network (31) without impulsive control. Evolution of (a) the states x and (b) the total error err (t).

Fig. 6. Designed network (31) with the control topology (32), t t = 0:0243 and t = 15. Evolution of (a) the states x and (b) the total error err (t).

0

Fig. 7. Designed network (31) with the control topology (32), t t = 1:3, and t = 15. Evolution of (a) the states x and (b) the total error err (t).

0

Fig. 5. Designed network (31) with the control topology (32), t t = 0:0999, and t = 15. Evolution of (a) the states x and (b) the total error err (t).

According to [18], it can be obtained Case I. Unidirectional star control topology: Now, to design , follow the above proposed design a impulsive controller procedure. Step 1) Choose the unidirectional star control topology in which the center-node is the first node, i.e.,

.. . Take

.. .

.. . .. .

(32)

.. . . Choose

(33)

Calculate . . Step 2) Calculate , for all . Step 3) Let Fig. 5 shows the evolution process of the total error and the states of the designed network (31) with the control topology . If it is required that the exponential (32) and , It follows from step 3b: , convergence rate , take . Fig. 6 shows the evolution process of the total error and the states of the designed network (31) with the control topology (32) and . If impulsive interval is too big, the network can’t achieve synchronization (as shown in Fig. 7). Fig. 8 is


= 25

1 ( 0

Fig. 8. Total error at t with t t t network (31) with the control topology (32) and t

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1t), in the designed = 15.

0

t Fig. 10. Designed network (31) with the control topology (34), t and (b) the total error err

0:2270 and t = 15. Evolution of (a) the states x

= (t).

Fig. 9. Nondelayed coupling topology of the designated network (31) and its directed spanning tree.

given to show the relationship between the impulsive interval and the total error. Case II. Directed spanning tree: In the above control topology, the first node’s state information is shared by all other nodes to realize network synchronization. In other words, all nodes need to communicate with the first node, which means that additional connections are added. In this part, another control topology will be proposed for the network, by using the non-delay neighbor’s information. The non-delayed coupling of the designated network (31) contains a directed spanning tree, so it is reasonable to design the control topology according to the directed spanning tree. Choose the directed spanning tree of the non-delayed coupling graph (as shown in Fig. 9) as control topology and adjust the weights of the tree such . The control topology can be expressed as that follows:

Fig. 11. Unidirectional ring (31) without impulsive control. Evolution of (a) the states x and (b) the total error err t .

()

2) Unidirectional Ring: Consider a network with 30 dynamical nodes and the corresponding non-delayed coupling is a uniand directional ring. The two coupling delays are respectively. The coupling matrices are

.. .

.. .

..

.

..

.

..

.

.. .

.. .

(34) .. .

Choose ingly,

as (33), and take , accord, . Take . Fig. 10 shows the evolution process of the total error and the states of the designed network (31) with the control topology (34).

.. .

..

.

..

.

.. .

It is very difficult for the unidirectional ring to achieve synchronization if the number of network nodes is large enough. Fig. 11 isn’t synchroshows the unidirectional ring (31) with nized without control.

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0

=

t Fig. 13. Unidirectional ring (31) with the control topology (37), t and t . Evolution of (a) the states x and (b) the total error

1:6810210 err (t). 0

Fig. 12. Unidirectional ring (31) with the control topology (35), t t : . Evolution of (a) the states x and (b) the total error err and t

0 0556

=3

=3

= (t).

Case I. Unidirectional Star Control Topology: Choose

.. .

.. . .. .

.. .

.. .

..

Fig. 14. Parameter with number of nodes N in the chain.

.. .

(35) The directed spanning tree of unidirectional ring is a unidirectional chain. The parameter in this kind of control topology is dependent on numbers of nodes. Take (36)

.

.. . . Thus, , . , then take . Fig. 12 shows the evolution process of the states and the total error of (31) with the control topology (35) under impulsive control. Case II. Directed spanning tree: According to the original non-delayed coupling graph, directed spanning tree of the nondelayed coupling graph is chosen as the control topology,

..

.

Take

.. .

.. .

..

.

..

.

..

.

.. .

.. .

(37)

Take as (36) and . Thus, . The parameter , , then, . Fig. 13 shows the evolution take process of the states and the total error in the unidirectional ring (31) with the control topology (37).

and . As shown in Fig. 14, with inrapidly approach to 1. This is because with the creasing N, increase of the number of nodes, the distance between the root and the last node increased correspondingly. The influence of the root on the last node is reduced. Thus, for any connected non-delayed coupling topology, a minimal depth directed spanning tree can be chosen as the control topology to get better performance. 3) SW Network: Third, consider an SW network with 100 , , and dynamical nodes. Here, the parameters , and then the coupling matrix of the network can be randomly generated based on NW algorithm [42]. The two and respectively. The coupling delays are and is given as follows. delayed coupling

.. .

.. .

..

.

..

.

.. .


Fig. 15. SW network (31) without impulsive control. Evolution of (a) the states x and (b) the total error err(t).

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Fig. 17. SW network (31) with the directed spanning tree as control topology, t t = 0:0018 and t = 2:3. Evolution of (a) the states x and (b) the total error err (t).

0

VI. CONCLUSION

Fig. 16. SW network (31) with unidirectional star control topology t t = 0:0101, and t = 2:3. Evolution of (a) the states x and (b) the total error err (t).

0

Fig. 15 shows that the SW network without control isn’t synchronized. Case I. Choose a unidirectional star control topology in which the center-node is the first node. Take

then

. The parameters , . Fig. 16 shows the evolution process of the states and the total error in SW network (31) with unidirectional . star control topology and Case II. A minimal depth directed spanning tree of non-delayed coupling topology can also be found as the control topology (based on Dijkstra shortest path algorithm). Take . A can be found such that and . . Fig. 17 shows the evolution process of the total error and the states in SW network (31) with the directed spanning tree as . the control topology and

In this paper, the distributed impulsive synchronization of CDNs with system delay and multiple coupling delays is studied. To describe the directed connections introduced by the control scheme, the concept of control topology is introduced. Based on the concept of control topology, impulsive distributed control scheme is designed to achieve the network synchronization and some synchronization criteria have been derived by using the stability theory of impulsive delayed systems. Since the control topology can be designed by only utilizing the available state information, the control strategy presented is much easier to be implemented than the existing works on the topic. A design procedure for the impulsive controller is presented and the effectiveness of the impulsive control strategy have been demonstrated by applying two particular control topologies to three typical networks, i.e., a designated network, a unidirectional ring and a small-world network. It would be interesting to further develop the concept of control topology in future designs of various schemes to realize network synchronization. REFERENCES [1] S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, pp. 268–276, 2001. [2] M. E. J. Newman, “The structure and function of complex networks,” SIAM Rev., vol. 45, pp. 167–256, 2003. [3] R. Albert and A. L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys., vol. 74, pp. 47–97, 2002. [4] G. Osipov, J. Kurths, and C. Zhou, Synchronization in Oscillatory Networks. Berlin, Germany: Springer, 2007. [5] C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems. Singapore: World Scientific, 2007. [6] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep., vol. 469, pp. 95–153, 2008. [7] Q. Zhang, J. Lu, J. Lü, and C. K. Tse, “Adaptive feedback synchronization of a general complex dynamical network with delayed nodes,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 2, pp. 183–187, Feb. 2008. [8] J. Zhou and T. Chen, “Synchronization in general complex delayed dynamical networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 3, pp. 733–744, Mar. 2006. [9] W. Lu, T. Chen, and G. Chen, “Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay,” Phys. D, vol. 221, pp. 118–134, 2006.

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[37] P. deLellis, M. diBernardo, and F. Garofalo, “Synchronization of complex networks through local adaptive coupling,” Chaos, vol. 18, p. 037110, 2008. [38] P. DeLellis, M. diBernardo, and F. Garofalo, “Novel decentralized adaptive strategies for the synchronization of complex networks,” Automatica, vol. 45, pp. 1312–1318, 2009. [39] Z. Yang and D. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Trans. Autom. Control, vol. 52, pp. 1448–1454, 2007. [40] R. Horn and C. Johnson, Martix Analysis. New York: SpringerVerlag, 2001. [41] H. Lu, “Chaotic attractors in delayed neural networks,” Phys. Lett. A, vol. 298, pp. 109–116, 2002. [42] M. J. E. Newman and D. J. Watts, “Renormalization group analysis of the small-world network model,” Phys. Lett. A, vol. 263, pp. 341–346, 1999. Zhi-Hong Guan graduated in mathematics from the Central China Normal University, Wuhan, China, in 1979, and received the Ph.D. degree in control theory and control engineering from the South China University of Technology, Guangzhou, China, in 1994. From 1994 to 1996, he was a Postdoctoral Fellow with the Faculty of Electrical Engineering and Communications, the South China University of Technology. From 1979 to 1994, he was with the Jianghan Petroleum Institute, Jingzhou, China, where he was Lecturer, Associate Professor, and then Full Professor of Mathematics and Automatic Control. Since December 1997, he has been Full Professor of the Department of Control Science and Engineering, Executive Associate Director of the Centre for Nonlinear and Complex Systems and Director of the Control and Information Technology in the Huazhong University of Science and Technology (HUST), Wuhan, China. From 1999 to 2003, he held the Vice Chairman in the Department of Control Science and Engineering at HUST. Since 1999, he has held visiting positions at the Central Queensland University, Australia, the National University of Singapore, the University of Hong Kong, the City University of Hong Kong, the Harvard University, USA, and the Loughborough University, UK. His research interests include nonlinear and complex systems, impulsive and hybrid control systems, networked control systems, chaos control and synchronization. Dr. Guan was an Associate Editor of the Journal of Dynamics of Continuous, Discrete and Impulsive Systems (DCDIS). Currently, he is the Associate Editor of the Journal of Control Theory and Applications, and severs as Member of the Committee of Control Theory of the Chinese Association of Automation, Executive Committee Member and also Director of the Control Theory Committee of the Hubei Province Association of Automation.

Zhi-Wei Liu was born in Yichun, China, 1982. He received the B.S. degree in Management Information System from Southwest Jiaotong University, Chengdu, China, 2004. Currently, he is working towards the Ph.D. degree at the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan. His research interests include complex dynamical networks and impulsive and hybrid control systems.

Gang Feng received the B.Eng. and M.Eng. degrees in automatic control from Nanjing Aeronautical Institute, China in 1982 and in 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is at present a Chair Professor and the Associate Provost. He is a ChangJiang Chair professor at Nanjing University of Science and Technology, awarded by Ministry of Education, China. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992-1999.


His current research interests include piecewise linear systems, robot networks, and intelligent systems and control. Prof. Feng was awarded an Alexander von Humboldt Fellowship in 1997-1998, and the IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award in 2007. He is an Associate Editor of IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON FUZZY SYSTEMS, and Mechatronics, and was an Associate Editor of IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: JOURNAL OF CONTROL THEORY AND APPLICATIONS, and the Conference Editorial Board of IEEE Control System Society.

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Yan-Wu Wang was born in Huixian, China, 1976. She received the B.S. degree in automatic control, the M.S. and Ph.D. degrees in control theory and control engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1997, 2000, and 2003, respectively. From 2003 to 2005, she worked as a Post-Doctoral Researcher in the Department of Electronics and Information Engineering, HUST, Wuhan, China. From February to April, 2005, she held the visiting position with the School of Electrical and Electronic Engineering in Nanyang Technological University. From September 2008 to September 2009, she held the position of visiting scholar with the Department of Mechanical Engineering in Boston University. Since November 2004, she has been an Associate Professor in the Department of Control Science and Engineering, HUST. Her current research interests include complex networks and hybrid systems. Dr. Wang is a recipient of the 2006 Natural Science Award of Ministry of Education Nominated State Science and Technology Award, China, the 2005 Natural Science Award of Hubei Province, China, and the 2004 Excellent Ph.D. Dissertation of Hubei Province, China. In 2008, she was awarded the title of “New Century Excellent Talents” by the Chinese Ministry of Education.