Geometry of the Limit Sets of Linear Switched Systems

TL;DR

This paper investigates the stability of linear switched systems, introducing non chaotic inputs and providing conditions under which these systems are asymptotically stable, even in the presence of potentially chaotic switching signals.

Contribution

It introduces the concept of non chaotic inputs and offers new stability criteria for systems with shared quadratic Lyapunov functions, extending understanding of switched system behavior.

Findings

Large class of switching signals ensuring stability

Definition of non chaotic inputs generalizing dwell time notions

Sufficient condition for stability with pairs of Hurwitz matrices

Abstract

The paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a non strict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call non chaotic inputs, which generalize the different notions of inputs with dwell time. Next we turn our attention to the behaviour for possibly chaotic inputs. To finish we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.