Qualitative analysis of discrete-time switched systems

We investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total, activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the /spl Lscr//sub 2/ gain of the switched system.

1 Introduction

By a switched system, we mean a hybrid dynamical system that is composed of a family of continuous-time or discrete-time subsystems and a rule orchestrating the switching between the subsystems. In the last two decades, there has been increasing interest in the stability analysis and control design for such switched systems; see, e.g., [1]–[18] and the references cited therein. The motivation for studying switched systems is from the fact that many practical systems are inherently multimodal in the sense that several dynamical subsystems are required to describe their behavior which may depend on various environmental factors [1], and that the methods of intelligent control design are based on the idea of switching between different controllers [2]–[4]. For recent progress and perspectives in the field of switched systems, see the survey papers [3] and [5].