Stability analysis for a special class of dynamical networks

Abstract

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.