Stability of Uniformly Bounded Switched Systems and Observability

Abstract

This paper mainly deals with switched linear systems defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function. Its aim is to give sufficient conditions for such a system to be GUAS.We show that this property of being GUAS is equivalent to the uniform observability on [0, +∞) of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system. Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.The results are partially extended to nonlinear analytic systems.

Convergence rate of nonlinear switched systems

“…Then we turn our attention to switched systems defined by a set of analytic vector fields that share a common weak Lyapunov function (this is the framework of the previous papers [2,3,5,9,10]). In Section 4 we consider homogeneous vector fields and quadratic weak Lyapunov functions and we prove exponential stability, with a computable convergence rate, for all inputs with a fixed dwell-time.…”