Basic Problems in Stability and Design ofSwitched Systems

By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications. Many systems encountered in practice exhibit switching between several subsystems that is dependent on various environmental factors. Some examples of such systems are discussed in 1-3]. Another source of motivation for studying switched systems comes from the rapidly developing area of switching control. Control techniques based on switching between diierent controllers have been applied extensively in recent years, particularly in the adaptive context, where they have been shown to achieve stability and improve transient response (see, among many references, 4-6]). The importance of such control methods also stems in part from the existence of systems that cannot be asymptotically stabilized by a single continuous feedback control law 7]. Switched systems have numerous applications in control of mechanical systems, automotive industry, aircraft and air traac control, switching power converters, and many other elds. The book 8] contains reports on various developments in some of these areas. In the last few years, every major control conference has had several regular and invited sessions on switching systems and control. Moreover, workshops and symposia devoted speciically to these topics are regularly taking place. Almost every major technical control journal has had or is planning to have a special issue on switched and hybrid systems. These sources can be consulted for further references. Mathematically, a switched system can be described by a diierential equation of the form _ x = f (x); (1) where ff p : p 2 Pg is a family of suuciently regular functions from R n to R n that is parameterized by some index set P, and : 0; 1) ! P is a piecewise constant function of time, called a switching signal. In speciic situations, the value of at a given time t might just depend on t or x(t), or both, or may be generated using more sophisticated techniques such as hybrid feedback with memory in the loop. We …