Stability analysis for a special class of dynamical networks

In this article, stability analysis and decentralised control problems are studied for a special class of linear dynamical networks. Necessary and sufficient conditions for stability and stabilisability under a decentralised control strategy are given for this type of linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail, respectively. A dynamical network can be viewed as a large-scale system composed of some subsystems with some coupling structures, based on this, the relationship between the stability of a network and the stability of its corresponding subsystems is studied. Different from the discussions that the subsystems in networks vary with different coupling structures (Duan, Z.S., Wang, J.Z., Chen, G.R., and Huang, L. (2008), ‘Stability Analysis and Decentralised Control of a Class of Complex Dynamical Networks’, Automatica, 44, 1028–1035), the subsystems in network discussed in this article remain unchanged with different interconnections which is the same as in general large-scale system. It is also pointed out that some subsystems must be made unstable for the whole network to be stable in some special cases. Moreover, the controller design method based on parameter-dependent Lyapunov function is provided.

[1]  B. F. Doolin,et al.  Large scale dynamic systems , 1975 .

[2]  M. Neumann,et al.  A conjecture concerning the Hadamard product of inverses of M-matrices , 1998 .

[3]  Mingzhou Ding,et al.  STABILITY OF SYNCHRONOUS CHAOS AND ON-OFF INTERMITTENCY IN COUPLED MAP LATTICES , 1997 .

[4]  Zou Yun.  Investigation on Mechanism and Extension of Feedback:An Extended Closed-loop Coordinate Control of Interconnection-regulation and Feedback , 2010 .

[5]  Nicholas R. Jennings,et al.  Decentralised Control of Complex Systems , 2007, Wirtschaftsinformatik.

[6]  Ji-Hong Li,et al.  Stable nonlinear adaptive controller for an autonomous underwater vehicle using neural networks , 2007, Int. J. Syst. Sci..

[7]  Lin Huang,et al.  Stability analysis and decentralized control of a class of complex dynamical networks , 2008, Autom..

[8]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[9]  Somdatta Sinha,et al.  GLOBAL AND LOCAL CONTROL OF SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICES , 1998 .

[10]  Dongbing Gu,et al.  Distributed network-based formation control , 2009, Int. J. Syst. Sci..

[11]  Shencan Chen,et al.  A lower bound for the minimum eigenvalue of the Hadamard product of matrices , 2004 .

[12]  R. K. Sharma,et al.  Collision avoidance between UAV clusters using swarm intelligence techniques , 2009, Int. J. Syst. Sci..

[13]  Guanrong Chen,et al.  Pinning a complex dynamical network to its equilibrium , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Xiao Fan Wang,et al.  Complex Networks: Topology, Dynamics and Synchronization , 2002, Int. J. Bifurc. Chaos.

[15]  John L. Junkins,et al.  Spacecraft Formation Flying Control using Mean Orbit Elements , 2000 .

[16]  Ting-Zhu Huang,et al.  Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse , 2007 .

[17]  Lin Huang,et al.  Some Special Decentralized Control Problems in Continuous-Time Interconnected Systems , 2006, Adv. Complex Syst..

[18]  Chai Wah Wu,et al.  Synchronization in Coupled Chaotic Circuits and Systems , 2002 .

[19]  M. E. Alexander,et al.  Modelling strategies for controlling SARS outbreaks , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[20]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[21]  L. Chua,et al.  Application of Kronecker products to the analysis of systems with uniform linear coupling , 1995 .

[22]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

[23]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[24]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[25]  Li Qiu,et al.  On the span of Hadamard products of vectors , 2007 .

[26]  Jinghua,et al.  Analytical study of spatiotemporal chaos control by applying local injections , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  J. K. Hedrick,et al.  Constant Spacing Strategies for Platooning in Automated Highway Systems , 1999 .

[28]  G. Visick.  A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product , 2000 .

[29]  Kai Sun,et al.  Splitting strategies for islanding operation of large-scale power systems using OBDD-based methods , 2003 .