Stability analysis and decentralized control of a class of complex dynamical networks

doi:10.1016/j.automatica.2007.08.005

Automatica

Volume 44, Issue 4, April 2008, Pages 1028-1035

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Abstract

In this paper, stability analysis and decentralized control problems are addressed for linear and sector-nonlinear complex dynamical networks. Necessary and sufficient conditions for stability and stabilizability under a special decentralized control strategy are given for linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail. A dynamical network is viewed as a large-scale system composing of some subsystems, based on which the relationship between the stability of a network and the stability of its corresponding subsystems is investigated. It is pointed out that some subsystems must be unstable for the whole network to be stable in some special cases. Moreover, a controller design method based on a parameter-dependent Lyapunov function is provided. Furthermore, interconnected Lur’e systems and symmetrical networks of Lur’e systems are similarly studied. The test of absolute stability of a network of Lur’e systems is separated into the test of absolute stability of several independent Lur’e systems. Finally, several numerical examples are given to illustrate the theoretical results.

Introduction

Complex dynamical networks have attracted increasing attention from physicists, biologists, social scientists and control scientists in recent years (see Wang, 2002, Watts and Strogatz, 1998 and references therein). From a system-theoretic point of view, a complex dynamical network can be considered as a large-scale system with special interconnections among its dynamical nodes. The large-scale system theory has been extensively studied in the last three decades, and many interesting results have been established, on such basic issues as decentrally fixed modes, decentralized controllers design, diagonal Lyapunov function method, M-matrix method, etc. (Huang and Zhang, 1998, Siljak, 1978, Siljak, 1991). Compared with centralized control methods, decentralized control has many advantages for its lower dimensionality, easier implementation, lower cost, etc., where interchange of state information among subsystems is generally not needed. Therefore, this control methodology has become quite popular in large-scale systems theory since the 1970s. However, due to the structural constraints in decentralized control, it is very difficult to develop a unified and effective design strategy. As a result, many theoretical and practical problems remain unsolved in this field. On the other hand, stability analysis and synthesis are important problems in the study of dynamical systems. Without requiring interchange of information among different nodes, the decentralized controller design methodology is obviously noneffective, thereby bringing an interesting research problem to the field of complex dynamical networks. In addition, due to the dissipative coupling characteristic of many complex dynamical networks (Li, Wang, & Chen, 2004), subsystems of networks usually vary with the coupling structures. This is different from the typical large-scale systems in which subsystems remain unchanged under different interconnection relations. The concept of subsystems is actually ignored in the recent study of complex networks; instead, the roles of nodes receive more attention. This paper attempts to explore the connections between subsystems and nodes, and is devoted to stability analysis and decentralized controller design for linear and sector-nonlinear dynamical networks.

The rest of this paper is organized as follows. In Section 2, using a simple similarity transformation, some criteria for network stability and stabilizability under a special decentralized control strategy are established. An LMI controller design method based on a parameter-dependent Lyapunov function (PDLF) is given to reduce the design conservativeness. In Section 3, the absolute stability of Lur’e networks is studied. The test of absolute stability of a network is separated to the tests of absolute stability of some independent Lur’e systems, which significantly reduces the dimensionality of the problem. Finally, numerical examples are given for illustration in Section 4. The last section concludes the paper.

Section snippets

Stability of a linear dynamical network and stability of its subsystems

Suppose that every node in a dynamical network is a continuous-time linear system described by ${\dot{x}}_{i} = A_{1} x_{i} + B_{1} u_{i},$ where $x_{i}$ is the state of node $i$ , $u_{i}$ is the control input, and $A_{1} \in R^{n \times n}, B_{1} \in R^{n \times m}$ are given matrices. With a linear coupling, the $N$ nodes constitute a network (Li et al., 2004, Wang, 2002) as follows: ${\dot{x}}_{i} = A_{1} x_{i} + \sum_{j = 1, j \neq i}^{N} c_{ij} A_{12} (x_{j} - x_{i}) + B_{1} u_{i}, i = 1, \dots, N,$ where $c_{ij} \in R$ , $A_{1}$ and $B_{1}$ are given as in (1), and $A_{12} \in R^{n \times n}$ is the inner coupling matrix describing the interconnections among components of $(x_{j} - x_{i}), i, j = 1$

Absolute stability of networks with Lur’e nodes

The absolute stability of Lur’e systems has been extensively studied (see Arcak et al., 2003, Park, 1997 and references therein). In this section, consider the absolute stability of a Lur’e network. First, consider the absolute stability and the local instability of an interconnected Lur’e system which can be viewed as a network with two nodes. Given two Lur’e systems: $\{\begin{matrix} {\dot{x}}_{i} = A_{i} x_{i} + B_{i} f_{i} (y), \\ y_{i} = C_{i} x_{i}, i = 1, 2, \end{matrix}$ where $x_{i} = (x_{i 1}, x_{i 2}, \dots, x_{{in}_{1}})^{T}$ , $i = 1, 2$ , are the states of two Lur’e systems, $y_{i} = (y_{i 1}, y_{i 2}, \dots, y_{{im}_{1}})^{T}$ , $i = 1,$

Examples

Example 1

Consider network (2) with $A_{1} = (\begin{matrix} 0 & 1 \\ - 3 & - 5 \end{matrix}), A_{12} = (\begin{matrix} 0 & 0 \\ 1 & 2 \end{matrix}), C = (\begin{matrix} 3 & - 3 \\ 1 & - 1 \end{matrix}) .$ In this case, the eigenvalues of $C$ are $λ_{1} = 2$ and $λ_{2} = 0$ . Obviously, this network with two nodes is stable and all of $A_{1} + α_{1} λ_{1} A_{12} + α_{2} λ_{2} A_{12}$ are stable if $0 ⩽ α_{i} \in R, i = 1, 2$ , and $α_{1} + α_{2} = 1$ . However, the first subsystem, i.e., $A_{1} + 3 A_{12}$ , is unstable.

Example 2

Consider the star-shaped coupled network (6) with $N = 6$ and $A_{1} = (\begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}), A_{12} = (\begin{matrix} - 1 & 2 \\ 0 & 0 \end{matrix}), B_{1} = (\begin{matrix} - 1 \\ 7 \end{matrix}) .$ In this case, the conditions of Theorem 4 are satisfied. Here, $A_{1} - A_{12}$ is also unstable. By the general Lyapunov method, one can

Conclusion

In this paper, stability analysis and decentralized controller design problems have been addressed for linear and sector-nonlinear dynamical networks. The relationship between the stability of the whole network and the stability of its corresponding subsystems has been discussed. Different from the traditional large-scale systems theory in which subsystems are always required to be stable, it is shown here that unstable subsystems may exist in a stable network and that in some special cases

Zhisheng Duan received the M.S. degree in mathematics from Inner Mongolia University, China and the Ph.D. degree in control theory from Peking University, China in 1997 and 2000, respectively. From 2000 to 2002, he worked as a post-doctor in Peking University. He received the 2001 Chinese Control Conference Guan-ZhaoZhi Award. Since 2003, he has been an Associate Professor with the Department of Mechanics and Aerospace Engineering, Peking University. His research interests include robust

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Jinzhi Wang received the M.S. degree in mathematics from Northeast Normal University, China in 1988 and Ph.D. degree in control theory from Peking University, China in 1998. From 1998 to 2000 she was a post-doctor at the Institute of Systems Science, the Chinese Academy of Sciences, Beijing. In year 2000 she was a research associate in the University of Hong Kong. She is currently an Associate Professor at the Department of Mechanics and Engineering Science, Peking University. Her research interests include robust control, nonlinear control and control of systems with saturating actuators.

Guanrong Chen received his M.Sc. degree in computer science from Zhongshan (Sun Yat-sen) University, China in 1981 and Ph.D. degree in applied mathematics from Texas A&M University, USA in 1987. He is currently a Chair Professor and the founding Director of the Centre for Chaos and Complex Networks at the City University of Hong Kong (since January 2000), prior to which he was a tenured Full Professor at the University of Houston, Texas. He is a Fellow of the IEEE (since January 1997), with research interests in chaotic dynamics, complex networks and nonlinear controls.

Lin Huang received the B.S. and M.S. degrees in mathematics and mechanics from Peking University, China in 1957 and 1961, respectively. In 1961, he joined the Department of Mechanics, Peking University where he currently is a Professor of Control Theory. His research interests include stability of dynamical systems, robust control, and nonlinear systems. He has authored three books and (co)authored more than 150 papers in the fields of stability theory and control systems. He is a member of the Chinese Academy of Science.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Masayuki Fujita under the direction of Editor Ian Petersen. This work was supported by the National Science Foundation of China under Grants 60674093, 60334030, 10472001.

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Brief paperStability analysis and decentralized control of a class of complex dynamical networks☆

Abstract

Introduction

Section snippets

Stability of a linear dynamical network and stability of its subsystems

Absolute stability of networks with Lur’e nodes

Examples

Conclusion

Automatica

Systems and Control Letters

Automatica

Systems and Control Letters

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Systems Control Letters

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IEEE Transactions on Automatic Control

Brief paper
Stability analysis and decentralized control of a class of complex dynamical networks☆