A Characterization of Lyapunov Inequalities for Stability of Switched Systems

TL;DR

This paper characterizes all Lyapunov-based LMI conditions for the stability of discrete-time switched systems, providing a comprehensive understanding and complexity analysis of stability criteria.

Contribution

It introduces a meta-theorem that encapsulates all canonical Lyapunov inequalities for switched systems and establishes the PSPACE-completeness of stability verification.

Findings

All Lyapunov inequalities of a certain type are characterized by a single family.

Stability verification via these LMIs is PSPACE-complete.

A geometric interpretation of the stability conditions is provided.

Abstract

We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past fifteen years. We prove in this note that a family of languagetheoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.