Switched Systems With Multiple Equilibria Under Disturbances: Boundedness and Practical Stability

Abstract

This paper addresses robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. First, we prove that if each subsystem of the switched system is input-to-state stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. The size of this set varies monotonically with the supremum norm of the disturbance signal. These results generalize existing ones in the common equilibrium case to accommodate multiple equilibria. Then, we relax the (global) ISS conditions to consider equilibria that are locally exponentially stable (LES), and we establish practical stability for such switched systems under disturbances. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive movements, each corresponding to an equilibrium point of a dynamical system, must be composed to obtain more complex motions. As a concrete example, we consider the problem of realizing safe adaptive locomotion of a three-dimensional biped under persistent external force by switching among motion primitives characterized by LES limit cycles. The results of this paper, however, are relevant to a much broader class of applications, in which composition of different modes of behavior is required.