Stabilization and Synchronization of Complex Dynamical Networks ...
1786
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Stabilization and Synchronization of Complex Dynamical Networks With Different Dynamics of Nodes Via Decentralized Controllers Yinhe Wang, Yongqing Fan, Qingyun Wang, and Yun Zhang
Abstract—This paper investigates the stabilization and synchronization of complex dynamical networks with different dynamics of nodes by using decentralized linear control and linear matrix inequality. We propose a dynamical network model with similar nodes that the dimensions of node dynamics are different. For this kind of network model, decentralized linear controllers are designed for the stabilization and synchronization. In addition, the synchronization manifold is defined as an invariant manifold, which is regarded as the generalized case of the networks with same node dynamics. Some criteria for the synchronization of networks are derived in this paper. Finally, numerical examples are presented to verify the theoretical results. Index Terms—Decentralized control, dynamical network, similar nodes, stabilization, synchronization.
I. INTRODUCTION
C
OMPLEX dynamical networks have attracted increasing attention and become a hot topic in many fields of science and engineering in the past decade [1]–[3]. Dynamical processes of the complex dynamical networks such as stabilization and synchronization have been extensively investigated [1]–[13]. Stabilization is the recoverability of a dynamical network from arbitrary state to an equilibrium state [4]. Synchronization is one of the dynamical behaviors in a dynamical network where the states of many connected nodes are evolving in synchrony [3]. In the contexts of aforementioned papers, the emphasis is on some basic conditions and criteria for the network stabilizability and synchronizability via controllers such as decentralized and pining controllers, as well as some possible relationships between the network topology and the network synchronizability. For example, from the views of dynamics and control in the [12], [13], the authors studied the synchronization of complex delayed dynamical networks by introducing the impulsive effects into topological structure of the networks. Manuscript received July 22, 2011; revised October 09, 2011; accepted November 23, 2011. Date of publication January 26, 2012; date of current version July 24, 2012. This work was supported by the Guangdong Natural Science Foundation (8151009001000061) and the Natural Science Joint Research Program Foundation of Guangdong Province (8351009001000002) and the National Science Foundation of China under Project No. 11172017 and No. 10972001. This paper was recommended by Associate Editor Huijun Gao. Y. Wang, Y. Fan, and Y. Zhang are with the Faculty of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]; [email protected]; [email protected]). Q. Wang is with the Department of Dynamics and Control, Beihang University, Beijing 100191, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2011.2180439
However, the most existing works on the stabilization and synchronization are based on the dynamical networks with same node dynamics. In other words, from the viewpoint of mathematics, every node inside a dynamical network is an identical time-varying dynamical system. While synchronization in a network with different node dynamics is in an initial stage. In the [14], the authors investigated the stabilization for a class of dynamical networks with different dynamical nodes by impulsive control, where the dynamics of nodes in the same group are denoted by the same ODE (Ordinary Differential Equation) systems. Recently, synchronization of dynamical networks with non-identical nodes has been investigated in [15], but the same state dimension of every dynamical node is required. In addition, the generalized synchronization between two dynamical networks with different nodes is also investigated without controllers [16]. Synchronization for dynamical networks with different nodes, especially with different state dimensions of nodes has not yet been thoroughly explored. One reason is that a standardized mathematical description of synchronization has not been presented for universal dynamical networks. Some descriptions of synchronization in terms of invariant manifolds are provided in [17]–[21]. From mathematical point of view, the synchronization phenomena can be treated as the existence of an invariant manifold of a special type in the phase space of the coupled networks. We will adopt invariant manifold as the description of synchronization in this paper. The fundamental feature displayed in many real networks is that, in corresponding nodes, there are almost the same (or similar) qualities of nature or appearance (e.g., the power network in n plants with identical units [22], the feeding nodes in an electric power system consisting of connected synchronous machines [23], the symmetric and similar composite systems discussed in [24]–[26] also possess this property). These nodes are called similar nodes (a special case is with identical nodes). If a network is constructed by the similar nodes, especially by similar nodes with different state dimensions, then the network will exhibit different dynamical behaviors and the previous methods of stabilization and synchronization for networks with identical nodes or different nodes of the same state dimension will be invalid. Therefore, it is necessary to explore new stabilization and synchronization schemes for dynamical networks with similar nodes of different state dimensions in this paper. In this paper, we investigate the stabilization and synchronization of dynamical networks with different nodes (which
1549-8328/$31.00 © 2012 IEEE
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
maybe possess different state dimensions) by using decentralized control. Compared with the centralized control, decentralized control possesses many advantages for its lower dimensionality, easier implementation, lower cost, etc. This control method has become popular in large-scale systems theory [4], [26]. This paper first proposes a model of dynamical networks. Then, under some conditions, the decentralized controllers are synthesized to ensure the network to be stabilized and synchronized. Finally, two simulation examples of a small world coupled network and a scale-free coupled network with network size are given to show the effectiveness of the proposed stable and synchronous criteria.
II. DESCRIPTION OF NETWORKS WITH SIMILAR NODES Suppose that every node in a dynamical network with nodes is a continuous-time linear system described by
One group of the similar parameters can be obtained by using the following steps: firstly we find satisfying by solving the linear algebraic equation (3) denotes the associated vector of matrix ; where denotes Kronecker product between two matrices. Then we find satisfying the matrix by solving the linear algebraic equation . Finally, the gain matrix can be obtained from by using a given matrix . Assumption 2: If Assumption 1 is true, then there exist masatisfying , . trices Remark 3: Note that Assumption 1 does not require the matrices to be stabilized. Besides, if a matrices satisfy , then Assumption 2 is true with . III. DESIGNING DECENTRALIZED STABILIZATION CONTROLLERS
(1) where
is the state variable of node , the control input , and the given matrices . If the nodes in (1) are linearly coupled, then the dynamical interconnected equation of the network reads as
1787
If Assumption 1 and 2 are satisfied, then our task in this section is to find conditions and design the decentralized controllers which guarantee the globally exponential stabilization of the states of the network (2). Consider the following decentralized linear controller (4)
(2) where the inner coupling matrices , , . Let is the outer coupling configuration matrix. If there is a connection between node and node , then , otherwise ; . Remark 1: The dynamical network (2) admits different state dimensions in nodes, which is different from the networks with identical nodes in the [1]–[14] or ones with different nodes of the same state dimension in the [15]. Especially, if , and in (2), then the network (2) coincides with the one in the [4]. Assumption 1: Consider nodes as given in (1). There exist matrices , and matrices satisfying , . Remark 2: The conditions in Assumption 1 ensure the matrices and possess some common eigenvalues, and thus Assumption 1 implies that nodes as given in (1) contain certain similarly inner dynamical behavior (these nodes can be called as similar nodes, their state dimensions are not needed to be the same). Therefore, the network (2) is composed of similar nodes via linear coupling. From a viewpoint of large-scale systems, if Assumption 1 is true, then the network (2) is defined as a similar composite system [24], and and are called as similar parameters of the similar composite system. Generally speaking, there exist lots of the similar parameters.
are similar parameters in Assumption 1. is dewhere termined by the matrix inequality (7) given below. Let , . From controller (4), Assumption 1 and 2, the dynamical network (2) can be transformed as
(5) Let
. Equation (5) can be rewritten as (6)
where identity matrix Let matrix satisfy
,
, and the , denotes Kronecker product. and symmetric matrix
(7a) (7b) Let denotes the minimum eigenvalue of the matrix in the left side of the inequality (7a). From (7a), it is seen that . Choosing the positive function , then the orbit derivative of along (6) is
1788
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
IV. DESIGNING DECENTRALIZED SYNCHRONIZATION CONTROLLERS
(8) Notice
, from (8), we obtain (9)
This means that the states of the network (5) are globally exponentially stabilized by decentralized controller (4). It is noted that if the similar parameter has full column rank, then there exists the left-inverse matrix , i.e. . Therefore, means . Theorem 1: Consider the coupling network (2). If Assumption 1 and 2 hold, the similar parameters have full column rank, and there exist positive matrix and matrix satisfying (7), then the decentralized linear controller (4) can guarantee the globally exponential stabilization of the states of the network (2). Remark 4: The matrix inequality (7a) is nonlinear with the unknown matrices , and it is difficult to obtain the solution. However, the inequality (7a) can be transformed into a linear matrix inequality (LMI) via matrix transformation. The left and right multiplying (7a) by , it becomes
(10) where following LMIs with unknown matrices
is used. Hence, the are obtained.
Our task in this section is to find conditions and design the decentralized controller under Assumption 1–2, which ensures the globally exponential synchronization of the states of the network (2) on the synchronization manifold defined below. Definition 1: If Assumption 1 holds, then the synchronization manifold of the network (2) is defined as, where . Consider the following decentralized controller (13) are similar parameters in Assumption 1. is dewhere termined by the matrix inequality (19) given below. , . From Assumption 1–2 and Let controller (13), the coupled network (2) can be transformed as (14) . where , Remark 5: Introducing the error . Then , where . By direct calculation for (14), we have , and thus one easily observes that there exist a matrix M such that . Hence, from [27], one can know that the synchronization manifold in Definition 1 is an invariant manifold for the network (2). Especially, if the network (2) has identical nodes, then with are the identity matrices. and matrix Let the error (a non-square matrix). Notice that , , and . From (14), one can obtain
(11a) (11b) where
,
. Furthermore, (11a) is given as (12a)
where
(12b)
(15) From (15), we have (16) and are as shown in (17) at the bottom of the next where page. , and (16) reads as It is easily seen that
(12c)
(18)
By using LMI technology to (12), , one can easily deterand , then mine the optimal basic feasible solution and are derived. Corollary 1: Consider the coupling network (2). If Assumption 1–2 are satisfied, the similar parameters have full column rank and the LMI (12) possesses an optimal basic feasible solution and , then the decentralized controller (4) can guarantee the global exponential stabilization of the states of the network (2).
It is noted that the matrix in (18) is maybe a non-square one, which is a significant difference to the case in [4]. , where Consider the positive definitive matrix and in (13) satisfy the following matrix inequality
(19a) (19b)
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1789
where trix
is an adjustable positive real number, the identity ma. Now, choose the positive function , where ‘tr’ denotes the maalong (18) reads as trix trace. Then the orbit derivative of
(22) From (21) and (22), one can find (23) Therefore, by using (23), (20) can be rewritten as (20) Through direct calculation, one can obtain (24) where
(21)
. From (24), the states of the network (2) are globally exponential synchronization on the manifold in Definition 1. Theorem 2: Consider the coupling network (2). If Assumption 1–2 are satisfied and there exist symmetric matrix and matrix satisfying (19), then the decentralized controller (13) can guarantee the globally exponential synchronization of the states of the network (2). Remark 6: The matrix inequality (19a) is also nonlinear . Similar to the method in with the unknown matrices , the Section III, the left and right multiplying (19a) by inequality (19a) can be transformed into the following LMIs.
(25a) (25b) where
,
. Furthermore, (25a) is given as (26a)
.. .
.. .
.. .
.. .
.. .
.. .
(17a)
(17b)
1790
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 1. A small world coupled network.
where
(26b) (26c) By using LMI technology to (26), , one can determine the optimal basic feasible solution and , and thus , . Corollary 2: Consider the coupling network (2). If Assumption 1–2 are satisfied and the LMI (26) possesses an optimal and , then the decentralized basic feasible solution controller (13) can guarantee the global exponential synchronization of the states of the network (2) on the manifold in Definition 1. V. NUMERICAL EXAMPLES In what follows, we give two examples to illustrate our obtained results, including a small world coupled network and another scale-free coupled network with different nodes. Example 1: We consider a small world coupled network with 10 nodes shown in Fig. 1. Suppose that the first node is stabilizable and the stability of the other nodes is not required. The first node is given as (27a)
Fig. 2. State response curves of in (27), with different dimensions and initial values.
for the isolated nodes
Fig. 2 shows that some isolated nodes in (27) possess the oscillating states. By using the method displayed in Remark 2, one easily finds the following similar parameters:
The other nodes are given as
(27b) It is noted that the first node (27a) is different from the other nodes (27b) in dimension. Now, the initial states of the network are chosen as , , , , , , , , , . The state response curves of nodes in (27) are shown in Figs. 2.
Assume
in Remark 3, where
where ‘ ’ denotes normally distributed random numbers; is an adjustable constant satisfying . ,
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1791
Fig. 4. The total synchronous error curve of states of the small world coupled network with nodes (27), and coupling matrix (28).
By using LMI toolbox in Matlab for LMI (12), via the decentralized controller (4) and , stabilization curves of states of the small world coupled network is shown in Fig. 3. As can be seen from Fig. 3, the small world coupled network with nodes (27) and coupling matrix (28) can be stabilized by using the decentralized controller (4). Now, we consider the synchronization of the small world coupled network with nodes (27) and coupling matrix (28). For the sake of simplicity, the total synchronous error is defined equivalently as (29) Similarly, by using LMI toolbox in Matlab for LMI (26) with , via the decentralized controller (13) and , the total synchronization error curve is in Fig. 4. Example 2: Consider the stabilization and synchronization for a scale-free network with 27 nodes shown as Fig. 5 [29]. The isolated nodes in Fig. 5 are described as (30) where the matrices Fig. 3. The stabilization curves of states of the small world coupled network with nodes (27), and outer coupling matrix (28).
then Assumption 2 is true with Assumption 1. From [28], the outer coupling matrix is
(28)
and
are represented respectively as
1792
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 5. A scale-free coupled network.
The initial values of the network are chosen as
The state response curves of nodes in (30) are shown in Figs. 6. The Fig. 6 shows that the states of isolated nodes in (30) are divergent. By using the method in Remark 2, the similar parameters are obtained: , , , , , , , , , , , , , , , , , , , The inner coupling matrices satisfy in Remark 3. The following matrices are considered.
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1793
for the isolated nodes
Fig. 7. The stabilization curves of states of the scale-free coupled network with nodes (30) and outer coupling matrix in Fig. 5.
is an adjustable constant satisfying . Let denote the outer coupling matrix of the scalefree network in Fig. 5 satisfying the conditions in (2).
Stabilization curves of states of the coupled network with and via the decentralized controller (4) are in Fig. 7.
Fig. 6. State response curves of in (30) with different dimensions and initial values.
where
1794
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 8. The total synchronous error curves of the scale-free network with nodes (30) and outer coupling matrix in Fig. 5.
Similar to Example 1, the total synchronous error is defined equivalently to (31) . Fig. 8 shows By using LMI toolbox to LMI (26) with the total synchronous error curves of the scale-free network in Fig. 5 with nodes (30) and outer coupling matrix . It can be seen from the simulation figures in above two examples, the stabilization and synchronization of the complex dynamical networks with similar nodes can be obtained by using decentralized controllers and similar parameters. VI. CONCLUSION In this paper, the stabilization and synchronization problems have been investigated for some complex dynamical networks with different similar node dynamics. The decentralized controllers are constructed by the similar parameters. Some novel stabilization and synchronization criteria are derived by a set of linear matrix inequalities. The effectiveness of the proposed decentralized control scheme has been demonstrated by two numerical simulation examples. In addition, pinning control is also a good alternative method for synchronization of complex networks. In the future, we will try best to use pinning control to synthesize controllers for synchronization of complex dynamic networks with similar nodes. REFERENCES [1] T. Liu, J. Zhao, and D. J. Hill, “Exponential synchronization of complex delayed dynamical networks with switching topology,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 11, pp. 2967–2980, Nov. 2010. [2] G. Chen, X. Wang, X. Li, and J. Lü, “Some recent advances in complex networks synchronization,” Stud. Comput. Intell., vol. 254, pp. 3–16, 2009. [3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: Structure and dynamics,” Phys. Rep., vol. 424, no. 4–5, pp. 175–308, Feb. 2006. [4] Z. Duan, J. Wang, G. Chen, and L. Huang, “Stability analysis and decentralized control of a class of complex dynamical networks,” Automatica, vol. 44, no. 4, pp. 1028–1035, Apr. 2008. [5] J. Zhou, J. Lu, and J. H. Lü, “Adaptive synchronization of an uncertain complex dynamical network,” IEEE Trans. Autom. Control, vol. 51, no. 4, pp. 652–656, 2006, App.. [6] J. Zhao, D. J. Hill, and T. Liu, “Synchronization of complex dynamical networks with switching topology: A switched system point of view,” Automatica, vol. 45, no. 9, pp. 2502–2511, Sep. 2009.
[7] J. Zhou, J. Lu, and J. Lü, “Pinning adaptive synchronization of a general complex dynamical network,” Automatica, vol. 44, no. 4, pp. 996–1003, Apr. 2008. [8] T. Liu, G. M. Dimirovski, and J. Zhao, “Exponential synchronization of complex delayed dynamical networks with general topology,” Phys. A, vol. 387, no. 2–3, pp. 643–652, Jan. 2008. [9] X. Liu, J. Z. Wang, and L. Huang, “Global synchronization for a class of dynamical complex networks,” Phys. A, vol. 386, no. 1, pp. 543–556, Dec. 2007. [10] Y. Wu, W. Wei, G. Li, and J. Xiang, “Pinning control of uncertain complex networks to a homogeneous orbit,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 3, pp. 235–239, Mar. 2009. [11] 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. [12] J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,” Phys. A, Stati. Mech. and its Appl., vol. 384, no. 2, pp. 684–692, Oct. 2007. [13] J. Zhou, Q. J. Wu, and L. Xiang, “Pinning complex delayed dynamical networks by a single impulsive controller,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 12, pp. 2882–2893, Dec. 2011. [14] Q. Zhang and J. Lu, “Impulsively control complex networks with different dynamical nodes to its trivial equilibrium,” Comput. Math. Appl., vol. 57, no. 7, pp. 1073–1079, Apr. 2009. [15] J. Zhao, D. J. Hill, and T. Liu, “Synchronization of dynamical networks with nonidentical nodes: Criteria and control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 3, pp. 584–594, Mar. 2011. [16] Y. Wu, C. Li, Y. Wu, and J. Kurths, “Generalized synchronization between two different complex networks,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 1, pp. 349–355, Jan. 2012. [17] K. Josic, “Invariant manifolds and synchronization of coupled dynamical systems,” Phys. Rev. Lett., vol. 80, no. 14, pp. 3053–3056, Apr. 1998. [18] K. Josic, “Synchronization of chaotic systems and invariant manifolds,” Nonlinearity, vol. 13, no. 4, pp. 1321–1336, Jul. 2000. [19] I. V. Belykh and V. N. Belykh, “Embedded invariant manifolds and ordering of chaotic synchronization of diffusivelly coupled systems,” Proc. 2th COC, vol. 2, pp. 346–349, Jul. 2000. [20] I. Chueshov, “Invariant manifolds and nonlinear master-slave synchronization in coupled systems,” Appl. Analys., vol. 86, no. 3, pp. 269–286, Apr. 2007. [21] P. Kloeden, “Synchronization of non-autonomous dynamical systems,” Elect. J. Differ. Equat., vol. 2003, no. 39, pp. 1–10, Apr. 2003. [22] C. S. Araujo and J. C. Castro, “Application of power system stabilizers in a plant with identical units,” IEE Proc. C, vol. 138, no. 1, pp. 11–18, Jan. 1991. [23] L. Bakule and J. Lunze, “Decentralized design of feedback control for large-scale systems,” Kybernetika, vol. 24, no. 8, pp. 3–96, 1988. [24] Y. Wang and S. Zhang, “Robust control for nonlinear similar composite systems with uncertain parameters,” IEE Proc., Contr. Theory Appl., vol. 147, no. 1, pp. 80–86, Jan. 2000. [25] X. Yan and G. Dai, “Decentralized output feedback robust control for nonlinear large-scale systems,” Automatica, vol. 34, no. 11, pp. 1469–1472, Nov. 1998. [26] L. Bakule, “Decentralized control: An overview,” Annu. Rev. Cont., vol. 32, no. 1, pp. 87–98, Apr. 2008. [27] M. Y. Li and J. S. Muldowney, “Dynamics of differential equations on invariant manifolds,” J. Differ. Equat., vol. 168, no. 2, pp. 295–320, Dec. 2000. [28] J. H. Lü, X. H. Yu, G. R. Chen, and D. Z. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 4, pp. 787–796, Apr. 2004. [29] A. Barabasi, E. Ravasz, and T. Vicsek, “Deterministic scale-free networks,” Phys. A, vol. 299, no. 3–4, pp. 559–564, Oct. 2001.
Yinhe Wang received the M.S. degree in mathematics from Sichuan Normal University, Chengdu, China, in 1990, and the Ph.D. degree in control theory and engineering from Northeastern University, Shenyang, China, in 1999. From 2000 to 2002, he was a Post-doctor in Department of Automatic Control, Northwestern Polytechnic University, Xi’an, China. From 2005 to 2006, he was a visiting scholar at the Department of Electrical Engineering, Lakehead University, Canada. He is currently a Professor with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include fuzzy adaptive robust control, analysis for nonlinear systems and complex dynamical networks.
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
Yongqing Fan received the M.S. degree in the Faculty of Applied Mathematics from Guangdong University of Technology, Guangzhou, China, in 2009. She is currently working toward the Ph.D. degree in the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. Her research interests include nonlinear systems and dynamical networks control.
Qingyun Wang received the M.Sc. degree in mathematics from Inner Mongolia University, Hohhot, China, in 2003 and the Ph.D. degree in general mechanics from Beihang University, Beijing, China, in 2006. Between 2008 and 2010, he was a Full Professor with Inner Mongolia Finance and Economics College, Hohhot, China. He is now a Full Professor with the Department of Dynamics and Control, Beihang University, Beijing, China, since June 2010. His research interests include neuronal dynamics modeling and analysis, dynamics and control of complex networks and the applications of nonlinear dynamics to mechanical and physical systems.
1795
Yun Zhang received the B.S. and M.S. degrees in electrical engineering from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree in control theory and engineering from South China University of Technology, Guangzhou, China, in 1997. He is currently a Professor with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include robot control, analysis and design for complex network, and intelligent control.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Stabilization and Synchronization of Complex Dynamical Networks With Different Dynamics of Nodes Via Decentralized Controllers Yinhe Wang, Yongqing Fan, Qingyun Wang, and Yun Zhang
Abstract—This paper investigates the stabilization and synchronization of complex dynamical networks with different dynamics of nodes by using decentralized linear control and linear matrix inequality. We propose a dynamical network model with similar nodes that the dimensions of node dynamics are different. For this kind of network model, decentralized linear controllers are designed for the stabilization and synchronization. In addition, the synchronization manifold is defined as an invariant manifold, which is regarded as the generalized case of the networks with same node dynamics. Some criteria for the synchronization of networks are derived in this paper. Finally, numerical examples are presented to verify the theoretical results. Index Terms—Decentralized control, dynamical network, similar nodes, stabilization, synchronization.
I. INTRODUCTION
C
OMPLEX dynamical networks have attracted increasing attention and become a hot topic in many fields of science and engineering in the past decade [1]–[3]. Dynamical processes of the complex dynamical networks such as stabilization and synchronization have been extensively investigated [1]–[13]. Stabilization is the recoverability of a dynamical network from arbitrary state to an equilibrium state [4]. Synchronization is one of the dynamical behaviors in a dynamical network where the states of many connected nodes are evolving in synchrony [3]. In the contexts of aforementioned papers, the emphasis is on some basic conditions and criteria for the network stabilizability and synchronizability via controllers such as decentralized and pining controllers, as well as some possible relationships between the network topology and the network synchronizability. For example, from the views of dynamics and control in the [12], [13], the authors studied the synchronization of complex delayed dynamical networks by introducing the impulsive effects into topological structure of the networks. Manuscript received July 22, 2011; revised October 09, 2011; accepted November 23, 2011. Date of publication January 26, 2012; date of current version July 24, 2012. This work was supported by the Guangdong Natural Science Foundation (8151009001000061) and the Natural Science Joint Research Program Foundation of Guangdong Province (8351009001000002) and the National Science Foundation of China under Project No. 11172017 and No. 10972001. This paper was recommended by Associate Editor Huijun Gao. Y. Wang, Y. Fan, and Y. Zhang are with the Faculty of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]; [email protected]; [email protected]). Q. Wang is with the Department of Dynamics and Control, Beihang University, Beijing 100191, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2011.2180439
However, the most existing works on the stabilization and synchronization are based on the dynamical networks with same node dynamics. In other words, from the viewpoint of mathematics, every node inside a dynamical network is an identical time-varying dynamical system. While synchronization in a network with different node dynamics is in an initial stage. In the [14], the authors investigated the stabilization for a class of dynamical networks with different dynamical nodes by impulsive control, where the dynamics of nodes in the same group are denoted by the same ODE (Ordinary Differential Equation) systems. Recently, synchronization of dynamical networks with non-identical nodes has been investigated in [15], but the same state dimension of every dynamical node is required. In addition, the generalized synchronization between two dynamical networks with different nodes is also investigated without controllers [16]. Synchronization for dynamical networks with different nodes, especially with different state dimensions of nodes has not yet been thoroughly explored. One reason is that a standardized mathematical description of synchronization has not been presented for universal dynamical networks. Some descriptions of synchronization in terms of invariant manifolds are provided in [17]–[21]. From mathematical point of view, the synchronization phenomena can be treated as the existence of an invariant manifold of a special type in the phase space of the coupled networks. We will adopt invariant manifold as the description of synchronization in this paper. The fundamental feature displayed in many real networks is that, in corresponding nodes, there are almost the same (or similar) qualities of nature or appearance (e.g., the power network in n plants with identical units [22], the feeding nodes in an electric power system consisting of connected synchronous machines [23], the symmetric and similar composite systems discussed in [24]–[26] also possess this property). These nodes are called similar nodes (a special case is with identical nodes). If a network is constructed by the similar nodes, especially by similar nodes with different state dimensions, then the network will exhibit different dynamical behaviors and the previous methods of stabilization and synchronization for networks with identical nodes or different nodes of the same state dimension will be invalid. Therefore, it is necessary to explore new stabilization and synchronization schemes for dynamical networks with similar nodes of different state dimensions in this paper. In this paper, we investigate the stabilization and synchronization of dynamical networks with different nodes (which
1549-8328/$31.00 © 2012 IEEE
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
maybe possess different state dimensions) by using decentralized control. Compared with the centralized control, decentralized control possesses many advantages for its lower dimensionality, easier implementation, lower cost, etc. This control method has become popular in large-scale systems theory [4], [26]. This paper first proposes a model of dynamical networks. Then, under some conditions, the decentralized controllers are synthesized to ensure the network to be stabilized and synchronized. Finally, two simulation examples of a small world coupled network and a scale-free coupled network with network size are given to show the effectiveness of the proposed stable and synchronous criteria.
II. DESCRIPTION OF NETWORKS WITH SIMILAR NODES Suppose that every node in a dynamical network with nodes is a continuous-time linear system described by
One group of the similar parameters can be obtained by using the following steps: firstly we find satisfying by solving the linear algebraic equation (3) denotes the associated vector of matrix ; where denotes Kronecker product between two matrices. Then we find satisfying the matrix by solving the linear algebraic equation . Finally, the gain matrix can be obtained from by using a given matrix . Assumption 2: If Assumption 1 is true, then there exist masatisfying , . trices Remark 3: Note that Assumption 1 does not require the matrices to be stabilized. Besides, if a matrices satisfy , then Assumption 2 is true with . III. DESIGNING DECENTRALIZED STABILIZATION CONTROLLERS
(1) where
is the state variable of node , the control input , and the given matrices . If the nodes in (1) are linearly coupled, then the dynamical interconnected equation of the network reads as
1787
If Assumption 1 and 2 are satisfied, then our task in this section is to find conditions and design the decentralized controllers which guarantee the globally exponential stabilization of the states of the network (2). Consider the following decentralized linear controller (4)
(2) where the inner coupling matrices , , . Let is the outer coupling configuration matrix. If there is a connection between node and node , then , otherwise ; . Remark 1: The dynamical network (2) admits different state dimensions in nodes, which is different from the networks with identical nodes in the [1]–[14] or ones with different nodes of the same state dimension in the [15]. Especially, if , and in (2), then the network (2) coincides with the one in the [4]. Assumption 1: Consider nodes as given in (1). There exist matrices , and matrices satisfying , . Remark 2: The conditions in Assumption 1 ensure the matrices and possess some common eigenvalues, and thus Assumption 1 implies that nodes as given in (1) contain certain similarly inner dynamical behavior (these nodes can be called as similar nodes, their state dimensions are not needed to be the same). Therefore, the network (2) is composed of similar nodes via linear coupling. From a viewpoint of large-scale systems, if Assumption 1 is true, then the network (2) is defined as a similar composite system [24], and and are called as similar parameters of the similar composite system. Generally speaking, there exist lots of the similar parameters.
are similar parameters in Assumption 1. is dewhere termined by the matrix inequality (7) given below. Let , . From controller (4), Assumption 1 and 2, the dynamical network (2) can be transformed as
(5) Let
. Equation (5) can be rewritten as (6)
where identity matrix Let matrix satisfy
,
, and the , denotes Kronecker product. and symmetric matrix
(7a) (7b) Let denotes the minimum eigenvalue of the matrix in the left side of the inequality (7a). From (7a), it is seen that . Choosing the positive function , then the orbit derivative of along (6) is
1788
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
IV. DESIGNING DECENTRALIZED SYNCHRONIZATION CONTROLLERS
(8) Notice
, from (8), we obtain (9)
This means that the states of the network (5) are globally exponentially stabilized by decentralized controller (4). It is noted that if the similar parameter has full column rank, then there exists the left-inverse matrix , i.e. . Therefore, means . Theorem 1: Consider the coupling network (2). If Assumption 1 and 2 hold, the similar parameters have full column rank, and there exist positive matrix and matrix satisfying (7), then the decentralized linear controller (4) can guarantee the globally exponential stabilization of the states of the network (2). Remark 4: The matrix inequality (7a) is nonlinear with the unknown matrices , and it is difficult to obtain the solution. However, the inequality (7a) can be transformed into a linear matrix inequality (LMI) via matrix transformation. The left and right multiplying (7a) by , it becomes
(10) where following LMIs with unknown matrices
is used. Hence, the are obtained.
Our task in this section is to find conditions and design the decentralized controller under Assumption 1–2, which ensures the globally exponential synchronization of the states of the network (2) on the synchronization manifold defined below. Definition 1: If Assumption 1 holds, then the synchronization manifold of the network (2) is defined as, where . Consider the following decentralized controller (13) are similar parameters in Assumption 1. is dewhere termined by the matrix inequality (19) given below. , . From Assumption 1–2 and Let controller (13), the coupled network (2) can be transformed as (14) . where , Remark 5: Introducing the error . Then , where . By direct calculation for (14), we have , and thus one easily observes that there exist a matrix M such that . Hence, from [27], one can know that the synchronization manifold in Definition 1 is an invariant manifold for the network (2). Especially, if the network (2) has identical nodes, then with are the identity matrices. and matrix Let the error (a non-square matrix). Notice that , , and . From (14), one can obtain
(11a) (11b) where
,
. Furthermore, (11a) is given as (12a)
where
(12b)
(15) From (15), we have (16) and are as shown in (17) at the bottom of the next where page. , and (16) reads as It is easily seen that
(12c)
(18)
By using LMI technology to (12), , one can easily deterand , then mine the optimal basic feasible solution and are derived. Corollary 1: Consider the coupling network (2). If Assumption 1–2 are satisfied, the similar parameters have full column rank and the LMI (12) possesses an optimal basic feasible solution and , then the decentralized controller (4) can guarantee the global exponential stabilization of the states of the network (2).
It is noted that the matrix in (18) is maybe a non-square one, which is a significant difference to the case in [4]. , where Consider the positive definitive matrix and in (13) satisfy the following matrix inequality
(19a) (19b)
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1789
where trix
is an adjustable positive real number, the identity ma. Now, choose the positive function , where ‘tr’ denotes the maalong (18) reads as trix trace. Then the orbit derivative of
(22) From (21) and (22), one can find (23) Therefore, by using (23), (20) can be rewritten as (20) Through direct calculation, one can obtain (24) where
(21)
. From (24), the states of the network (2) are globally exponential synchronization on the manifold in Definition 1. Theorem 2: Consider the coupling network (2). If Assumption 1–2 are satisfied and there exist symmetric matrix and matrix satisfying (19), then the decentralized controller (13) can guarantee the globally exponential synchronization of the states of the network (2). Remark 6: The matrix inequality (19a) is also nonlinear . Similar to the method in with the unknown matrices , the Section III, the left and right multiplying (19a) by inequality (19a) can be transformed into the following LMIs.
(25a) (25b) where
,
. Furthermore, (25a) is given as (26a)
.. .
.. .
.. .
.. .
.. .
.. .
(17a)
(17b)
1790
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 1. A small world coupled network.
where
(26b) (26c) By using LMI technology to (26), , one can determine the optimal basic feasible solution and , and thus , . Corollary 2: Consider the coupling network (2). If Assumption 1–2 are satisfied and the LMI (26) possesses an optimal and , then the decentralized basic feasible solution controller (13) can guarantee the global exponential synchronization of the states of the network (2) on the manifold in Definition 1. V. NUMERICAL EXAMPLES In what follows, we give two examples to illustrate our obtained results, including a small world coupled network and another scale-free coupled network with different nodes. Example 1: We consider a small world coupled network with 10 nodes shown in Fig. 1. Suppose that the first node is stabilizable and the stability of the other nodes is not required. The first node is given as (27a)
Fig. 2. State response curves of in (27), with different dimensions and initial values.
for the isolated nodes
Fig. 2 shows that some isolated nodes in (27) possess the oscillating states. By using the method displayed in Remark 2, one easily finds the following similar parameters:
The other nodes are given as
(27b) It is noted that the first node (27a) is different from the other nodes (27b) in dimension. Now, the initial states of the network are chosen as , , , , , , , , , . The state response curves of nodes in (27) are shown in Figs. 2.
Assume
in Remark 3, where
where ‘ ’ denotes normally distributed random numbers; is an adjustable constant satisfying . ,
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1791
Fig. 4. The total synchronous error curve of states of the small world coupled network with nodes (27), and coupling matrix (28).
By using LMI toolbox in Matlab for LMI (12), via the decentralized controller (4) and , stabilization curves of states of the small world coupled network is shown in Fig. 3. As can be seen from Fig. 3, the small world coupled network with nodes (27) and coupling matrix (28) can be stabilized by using the decentralized controller (4). Now, we consider the synchronization of the small world coupled network with nodes (27) and coupling matrix (28). For the sake of simplicity, the total synchronous error is defined equivalently as (29) Similarly, by using LMI toolbox in Matlab for LMI (26) with , via the decentralized controller (13) and , the total synchronization error curve is in Fig. 4. Example 2: Consider the stabilization and synchronization for a scale-free network with 27 nodes shown as Fig. 5 [29]. The isolated nodes in Fig. 5 are described as (30) where the matrices Fig. 3. The stabilization curves of states of the small world coupled network with nodes (27), and outer coupling matrix (28).
then Assumption 2 is true with Assumption 1. From [28], the outer coupling matrix is
(28)
and
are represented respectively as
1792
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 5. A scale-free coupled network.
The initial values of the network are chosen as
The state response curves of nodes in (30) are shown in Figs. 6. The Fig. 6 shows that the states of isolated nodes in (30) are divergent. By using the method in Remark 2, the similar parameters are obtained: , , , , , , , , , , , , , , , , , , , The inner coupling matrices satisfy in Remark 3. The following matrices are considered.
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
1793
for the isolated nodes
Fig. 7. The stabilization curves of states of the scale-free coupled network with nodes (30) and outer coupling matrix in Fig. 5.
is an adjustable constant satisfying . Let denote the outer coupling matrix of the scalefree network in Fig. 5 satisfying the conditions in (2).
Stabilization curves of states of the coupled network with and via the decentralized controller (4) are in Fig. 7.
Fig. 6. State response curves of in (30) with different dimensions and initial values.
where
1794
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 8, AUGUST 2012
Fig. 8. The total synchronous error curves of the scale-free network with nodes (30) and outer coupling matrix in Fig. 5.
Similar to Example 1, the total synchronous error is defined equivalently to (31) . Fig. 8 shows By using LMI toolbox to LMI (26) with the total synchronous error curves of the scale-free network in Fig. 5 with nodes (30) and outer coupling matrix . It can be seen from the simulation figures in above two examples, the stabilization and synchronization of the complex dynamical networks with similar nodes can be obtained by using decentralized controllers and similar parameters. VI. CONCLUSION In this paper, the stabilization and synchronization problems have been investigated for some complex dynamical networks with different similar node dynamics. The decentralized controllers are constructed by the similar parameters. Some novel stabilization and synchronization criteria are derived by a set of linear matrix inequalities. The effectiveness of the proposed decentralized control scheme has been demonstrated by two numerical simulation examples. In addition, pinning control is also a good alternative method for synchronization of complex networks. In the future, we will try best to use pinning control to synthesize controllers for synchronization of complex dynamic networks with similar nodes. REFERENCES [1] T. Liu, J. Zhao, and D. J. Hill, “Exponential synchronization of complex delayed dynamical networks with switching topology,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 11, pp. 2967–2980, Nov. 2010. [2] G. Chen, X. Wang, X. Li, and J. Lü, “Some recent advances in complex networks synchronization,” Stud. Comput. Intell., vol. 254, pp. 3–16, 2009. [3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: Structure and dynamics,” Phys. Rep., vol. 424, no. 4–5, pp. 175–308, Feb. 2006. [4] Z. Duan, J. Wang, G. Chen, and L. Huang, “Stability analysis and decentralized control of a class of complex dynamical networks,” Automatica, vol. 44, no. 4, pp. 1028–1035, Apr. 2008. [5] J. Zhou, J. Lu, and J. H. Lü, “Adaptive synchronization of an uncertain complex dynamical network,” IEEE Trans. Autom. Control, vol. 51, no. 4, pp. 652–656, 2006, App.. [6] J. Zhao, D. J. Hill, and T. Liu, “Synchronization of complex dynamical networks with switching topology: A switched system point of view,” Automatica, vol. 45, no. 9, pp. 2502–2511, Sep. 2009.
[7] J. Zhou, J. Lu, and J. Lü, “Pinning adaptive synchronization of a general complex dynamical network,” Automatica, vol. 44, no. 4, pp. 996–1003, Apr. 2008. [8] T. Liu, G. M. Dimirovski, and J. Zhao, “Exponential synchronization of complex delayed dynamical networks with general topology,” Phys. A, vol. 387, no. 2–3, pp. 643–652, Jan. 2008. [9] X. Liu, J. Z. Wang, and L. Huang, “Global synchronization for a class of dynamical complex networks,” Phys. A, vol. 386, no. 1, pp. 543–556, Dec. 2007. [10] Y. Wu, W. Wei, G. Li, and J. Xiang, “Pinning control of uncertain complex networks to a homogeneous orbit,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 3, pp. 235–239, Mar. 2009. [11] 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. [12] J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,” Phys. A, Stati. Mech. and its Appl., vol. 384, no. 2, pp. 684–692, Oct. 2007. [13] J. Zhou, Q. J. Wu, and L. Xiang, “Pinning complex delayed dynamical networks by a single impulsive controller,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 12, pp. 2882–2893, Dec. 2011. [14] Q. Zhang and J. Lu, “Impulsively control complex networks with different dynamical nodes to its trivial equilibrium,” Comput. Math. Appl., vol. 57, no. 7, pp. 1073–1079, Apr. 2009. [15] J. Zhao, D. J. Hill, and T. Liu, “Synchronization of dynamical networks with nonidentical nodes: Criteria and control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 3, pp. 584–594, Mar. 2011. [16] Y. Wu, C. Li, Y. Wu, and J. Kurths, “Generalized synchronization between two different complex networks,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 1, pp. 349–355, Jan. 2012. [17] K. Josic, “Invariant manifolds and synchronization of coupled dynamical systems,” Phys. Rev. Lett., vol. 80, no. 14, pp. 3053–3056, Apr. 1998. [18] K. Josic, “Synchronization of chaotic systems and invariant manifolds,” Nonlinearity, vol. 13, no. 4, pp. 1321–1336, Jul. 2000. [19] I. V. Belykh and V. N. Belykh, “Embedded invariant manifolds and ordering of chaotic synchronization of diffusivelly coupled systems,” Proc. 2th COC, vol. 2, pp. 346–349, Jul. 2000. [20] I. Chueshov, “Invariant manifolds and nonlinear master-slave synchronization in coupled systems,” Appl. Analys., vol. 86, no. 3, pp. 269–286, Apr. 2007. [21] P. Kloeden, “Synchronization of non-autonomous dynamical systems,” Elect. J. Differ. Equat., vol. 2003, no. 39, pp. 1–10, Apr. 2003. [22] C. S. Araujo and J. C. Castro, “Application of power system stabilizers in a plant with identical units,” IEE Proc. C, vol. 138, no. 1, pp. 11–18, Jan. 1991. [23] L. Bakule and J. Lunze, “Decentralized design of feedback control for large-scale systems,” Kybernetika, vol. 24, no. 8, pp. 3–96, 1988. [24] Y. Wang and S. Zhang, “Robust control for nonlinear similar composite systems with uncertain parameters,” IEE Proc., Contr. Theory Appl., vol. 147, no. 1, pp. 80–86, Jan. 2000. [25] X. Yan and G. Dai, “Decentralized output feedback robust control for nonlinear large-scale systems,” Automatica, vol. 34, no. 11, pp. 1469–1472, Nov. 1998. [26] L. Bakule, “Decentralized control: An overview,” Annu. Rev. Cont., vol. 32, no. 1, pp. 87–98, Apr. 2008. [27] M. Y. Li and J. S. Muldowney, “Dynamics of differential equations on invariant manifolds,” J. Differ. Equat., vol. 168, no. 2, pp. 295–320, Dec. 2000. [28] J. H. Lü, X. H. Yu, G. R. Chen, and D. Z. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 4, pp. 787–796, Apr. 2004. [29] A. Barabasi, E. Ravasz, and T. Vicsek, “Deterministic scale-free networks,” Phys. A, vol. 299, no. 3–4, pp. 559–564, Oct. 2001.
Yinhe Wang received the M.S. degree in mathematics from Sichuan Normal University, Chengdu, China, in 1990, and the Ph.D. degree in control theory and engineering from Northeastern University, Shenyang, China, in 1999. From 2000 to 2002, he was a Post-doctor in Department of Automatic Control, Northwestern Polytechnic University, Xi’an, China. From 2005 to 2006, he was a visiting scholar at the Department of Electrical Engineering, Lakehead University, Canada. He is currently a Professor with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include fuzzy adaptive robust control, analysis for nonlinear systems and complex dynamical networks.
WANG et al.: STABILIZATION AND SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS WITH DIFFERENT DYNAMICS OF NODES
Yongqing Fan received the M.S. degree in the Faculty of Applied Mathematics from Guangdong University of Technology, Guangzhou, China, in 2009. She is currently working toward the Ph.D. degree in the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. Her research interests include nonlinear systems and dynamical networks control.
Qingyun Wang received the M.Sc. degree in mathematics from Inner Mongolia University, Hohhot, China, in 2003 and the Ph.D. degree in general mechanics from Beihang University, Beijing, China, in 2006. Between 2008 and 2010, he was a Full Professor with Inner Mongolia Finance and Economics College, Hohhot, China. He is now a Full Professor with the Department of Dynamics and Control, Beihang University, Beijing, China, since June 2010. His research interests include neuronal dynamics modeling and analysis, dynamics and control of complex networks and the applications of nonlinear dynamics to mechanical and physical systems.
1795
Yun Zhang received the B.S. and M.S. degrees in electrical engineering from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree in control theory and engineering from South China University of Technology, Guangzhou, China, in 1997. He is currently a Professor with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include robot control, analysis and design for complex network, and intelligent control.
