Stability of discrete-time switching systems with constrained switching sequences

TL;DR

This paper presents a new algebraic framework using multinorms for analyzing the stability of discrete-time switching systems with constrained sequences, and provides accurate approximation algorithms for the joint spectral radius.

Contribution

Introduction of multinorms for stability analysis and development of the first arbitrarily accurate approximation schemes for constrained joint spectral radius.

Findings

Exact stability characterization via multinorms.

Algorithms achieve arbitrary accuracy in estimating the joint spectral radius.

Convex optimization approach with known complexity.

Abstract

We introduce a novel framework for the stability analysis of discrete-time linear switching systems with switching sequences constrained by an automaton. The key element of the framework is the algebraic concept of multinorm, which associates a different norm per node of the automaton, and allows to exactly characterize stability. Building upon this tool, we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius r, that is the exponential growth rate of a switching system with constrained switching sequences. More precisely, given a relative accuracy a > 0, the algorithms compute an estimate of r within the range [r; (1 + a)r]. These algorithms amount to solve a well defined convex optimization program with known time-complexity, and whose size depends on the desired relative accuracy a > 0.