Stability analysis of switched systems

Abstract

Switched systems are a particular kind of hybrid systems described by a combination of continuous/discrete subsystems and a logic-based switching signal. Currently, switched systems are employed as useful mathematical models for many physical systems displaying different dynamic behavior in each mode. Among the challenging mathematical problems that have arisen in switched systems, stability is the main issue. It is well known that switching can introduce instability even when all the subsystems are stable while on the other hand proper switching between unstable subsystems can lead to the stability of the overall system. In the last few years, significant progress has been made in establishing stability conditions for switched systems. While major advances have been made, a number of interesting problems are left open, even in the case of switched linear systems. With respect to some of these problems, we present some new results in three chapters as follows:

In Chapter 2, we deal with the stability of switched systems under arbitrary switching. Compared to Lyapunov-function methods which have been widely used in the literature, a novel geometric approach is proposed to develop an easily verifiable, necessary and sufficient stability condition for a pair of second-order linear time invariant (LTI) systems under arbitrary switching. The condition is general since all the possible combinations of subsystem dynamics are analyzed.

In Chapter 3, we apply the geometric approach to the problem of stabilization by switching. Necessary and sufficient conditions for regional asymptotic stabilizability are derived, thereby providing an effective way to verify whether a switched system with two unstable second-order LTI subsystems can be stabilized by switching.

In Chapter 4, we investigate the stability of switched systems under restricted switching. We derive new frequency-domain conditions for the L2-stability of feedback systems with periodically switched, linear/nonlinear feedback gains. These conditions, which can be checked by a computational-graphic method, are applicable to higher-order switched systems.

We conclude the thesis with a summary of the main contributions and future direction of research in Chapter 5.