Stability of uniformly bounded switched systems and Observability

TL;DR

This paper investigates conditions under which switched linear systems with shared quadratic Lyapunov functions are globally uniformly asymptotically stable (GUAS), linking stability to observability of an associated bilinear system, with extensions to nonlinear systems.

Contribution

It establishes a novel equivalence between GUAS of switched systems and observability of a related bilinear system, providing new stability criteria and extending results to nonlinear analytic systems.

Findings

GUAS is equivalent to uniform observability of a bilinear system.

Provides sufficient conditions for uniform asymptotic stability.

Results are extended to nonlinear analytic systems.

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 $[0,+\infty)$ 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.