On Stability Analysis of Switched Linear Time-Delay Systems under Arbitrary Switching

1. Introduction

A switched system is composed of an indexed family of subsystems described by continuous or discrete-time dynamics and a rule orchestrating the switching between them. As a special class of hybrid systems, switched systems have strong engineering background in various areas and are often used as a unified modeling tool for a great number of real-world systems such as power electronics chemical processes, mechanical systems, automotive industry, aircraft and air traffic control and many other fields (Morse, 1997; Liberzon & Morse, 1999; Liberzon, 2003; Sun & Ge; 2005, Sun & Ge, 2011; Shorten et al., 2003).

There has been an increasing research interest in stability analysis and control design for switched systems during the past two decades, Therefore, it is well known that stability of switched systems depends not only on the dynamics of every subsystem but also on the property of the switching signal. Thus, it is commonly recognized that there are mainly located in three basic types of problems considering the stability and stabilisation issues of switched systems (Lin & Antsaklis, 2009; Yang, Xiang & Lee, 2012; Hu et al., 2013):(i) asymptotical stability of the switched system with arbitrary switching, (ii) stability for certain useful classes of switching sequences, and (iii) construction of asymptotically stabilizing switching signals for a switched system. Specifically, stability analysis under arbitrary switching problem (i) which will be focused in this article is fundamental in the analysis and design of switched systems (Liberzon & Morse, 1999; Sun & Ge, 2011; Shorten et al., 2003; Zhang & Yu, 2008; Kim et al., 1999; Shorten et al., 2006; Liberzon & Tempo, 2004, Shorten et al., 2007). This problem deals with the case that all the subsystems are stable. Indeed, they exist many examples where all subsystems are stable but inappropriate switching rules can make the whole system unstable. In addition, if we know that a switched system is stable under arbitrary switching, then we can consider higher control specifications for the system. In this framework, it is well known that the existence of a common Lyapunov function for all subsystems is a sufficient condition for such systems to be asymptotically stable under arbitrary switching (Shorten et al., 2003; Shorten & Narendra, 2003; Narendra & Mason, 2003; Sun et al., 2006; Liberzon & Tempo, 2004). However, this method is usually very difficult to apply even for continuous-time switched linear systems (Liberzon, 2003; Shorten et al., 2007).