Stability and L2 Gain Analysis of Switched Linear Discrete-Time Descriptor Systems

This article is focused on analyzing stability and L2 gain properties for switched systems composed of a family of linear discrete-time descriptor subsystems. Concerning descriptor systems, they are also known as singular systems or implicit systems and have high abilities in representing dynamical systems [1, 2]. Since they can preserve physical parameters in the coefficient matrices, and describe the dynamic part, static part, and even improper part of the system in the same form, descriptor systems are much superior to systems represented by state space models. There have been many works on descriptor systems, which studied feedback stabilization [1, 2], Lyapunov stability theory [2, 3], the matrix inequality approach for stabilization, H2 and/or H∞ control [4–6]. On the other hand, there has been increasing interest recently in stability analysis and design for switched systems; see the survey papers [7, 8], the recent books [9, 10] and the references cited therein. One motivation for studying switched systems is that many practical systems are inherently multi-modal in the sense that several dynamical subsystems are required to describe their behavior which may depend on various environmental factors. Another important motivation is that switching among a set of controllers for a specified system can be regarded as a switched system, and that switching has been used in adaptive control to assure stability in situations where stability can not be proved otherwise, or to improve transient response of adaptive control systems. Also, the methods of intelligent control design are based on the idea of switching among different controllers. We observe from the above that switched descriptor systems belong to an important class of systems that are interesting in both theoretic and practical sense. However, to the authors’ best knowledge, there has not been much works dealing with such systems. The difficulty falls into two aspects. First, descriptor systems are not easy to tackle and there are not rich results available up to now. Secondly, switching between several descriptor systems makes the problem more complicated and even not easy to make clear the well-posedness of the solutions in some cases. Next, let us review the classification of problems in switched systems. It is commonly recognized [9] that there are three basic problems in stability analysis and design of switched systems: (i) find conditions for stability under arbitrary switching; (ii) identify the limited but useful class of stabilizing switching laws; and (iii) construct a stabilizing switching law. Stability and L2 Gain Analysis of Switched Linear Discrete-Time Descriptor Systems 19