Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

TLDR

Research on stability analysis and switching stabilization of switched systems has grown, yet the switching stabilizability problem remains one of the most elusive challenges. The paper surveys recent results and investigates conditions under which a switched system can be stabilized by designing appropriate switching control laws. The survey reviews stability analysis under arbitrary and restricted switching, highlights necessary and sufficient conditions for asymptotic stability, discusses multiple Lyapunov function theory, and outlines various switching stabilization methods. A necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described.

Abstract

During the past several years, there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems. This paper aims to briefly survey recent results in this field. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After a brief review of the stability analysis under restricted switching and the multiple Lyapunov function theory, the switching stabilization problem is studied, and a variety of switching stabilization methods found in the literature are outlined. Then the switching stabilizability problem is investigated, that is under what condition it is possible to stabilize a switched system by properly designing switching control laws. Note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature. A necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here.