Stability of Switched Affine Systems: Arbitrary and Dwell-Time Switching

TL;DR

This paper investigates the stability properties of switched affine systems, introducing new analysis techniques for invariant sets and stability notions under arbitrary and dwell-time switching, supported by numerical methods.

Contribution

It presents a novel proof technique for invariant sets and explores stability concepts specific to switched affine systems, extending understanding beyond linear systems.

Findings

Existence of attractive invariant sets under certain stability conditions

Dwell-time switching can prevent forward invariant sets in stable systems

Numerical methods using LMIs and sum-of-squares support the analysis

Abstract

The dynamical behavior of switched affine systems is known to be more intricate than that of the well-studied switched linear systems, essentially due to the existence of distinct equilibrium points for each subsystem. First, under arbitrary switching rules, the stability analysis must be generally carried out with respect to a compact set with non-empty interior rather than to a singleton. We provide a novel proof technique for existence and outer approximation of attractive invariant sets of a switched affine system, under the hypothesis of global uniform stability of its linearization. On the other hand, considering dwell-time switching signals, forward invariant sets need not exist for this class of switched systems, even for stable ones. Hence, more general notions of stability/boundedness are introduced and studied, highlighting the relations of these concepts to the uniform stability of the linear part of the system under the same class of dwell-time switching signals. These results reveal the main differences and specificities of switched affine systems with respect to linear ones, providing a first step for the analysis of switched systems composed by subsystems not sharing the same equilibrium. Numerical methods based on linear matrix inequalities and sum-of-squares programming are presented and illustrate the developed theory.