Stability Criteria for Switched and Hybrid Systems

Abstract

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

On linear co-positive Lyapunov functions for sets of linear positive systems

“…Such problems arise in the study of state dependent switching problems, see for example [21]. To be more precise, assume that there exist N closed pointed convex cones C j , such that the state space, R n + , the closed positive orthant of R n , can be written as R n + = ∪ N j=1 C j .…”

“…, A N } are the system matrices of the constituent systems, which are Metzler matrices. See [21,7] for more details on systems of this type.…”

Modal and transition dwell time computation in switching systems: A set-theoretic approach

“…Several notions of dwell time have been discussed [13], stability results being proposed mainly by means of multiple Lyapunov functions [7]. From the existing surveys [21][2] and monographs [16] [22] it is apparent that there are several unsolved problems in the computation of proper lower bounds for the dwell time to assure stability or, quite harder, the computation of its minimum value, typically possible only in special cases [9].…”

Controle de sistemas lineares com comutação

“…O objetivo consiste na determinação de uma regra de comutação σ(·) de tal forma a assegurar a estabilidade assintó-tica global. Em (Liberzon e Morse, 1999), (Liberzon, 2003), (DeCarlo et al, 2000), (Wirth, 2005), (Shorten et al, 2007) o leitor pode encontrar conjuntos de resultados bastante completos sobre sistemas lineares com comutação a tempo contí-nuo, em especial, para o caso de comutação envolvendo apenas dois subsistemas lineares. O artigo (Hespanha, 2005) utiliza extensões do Princípio da Invariância de LaSalle e fornece uma discussão interessante sobre resultados referentes à estabilidade uniforme de sistemas com comutação.…”