Converse Lyapunov theorems for discrete-time linear switching systems with regular switching sequences

TL;DR

This paper develops a stability analysis framework for discrete-time linear switching systems with regular switching sequences, introducing multinorms and converse Lyapunov theorems that enable finite, constructive stability certificates.

Contribution

It introduces the concept of multinorms and finite converse Lyapunov theorems for systems with regular switching sequences, providing a new algebraic and constructive approach to stability analysis.

Findings

Finite converse Lyapunov theorems are established for systems with regular switching sequences.

The framework allows for the computation of Lyapunov functions with memory or automaton-based structures.

An approximation scheme for the asymptotic growth rate of such systems is formulated.

Abstract

We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent results of X. Dai on the asymptotic growth rate of such systems, we introduce the concept of multinorm as an algebraic tool for stability analysis. We conjugate this tool with two families of multiple quadratic Lyapunov functions, parameterized by an integer T >= 1, and obtain converse Lyapunov Theorems for each. Lyapunov functions of the first family associate one quadratic form per state of the automaton defining the switching sequences. They are made to decrease after every T successive time steps. The second family is made of the path-dependent Lyapunov functions of Lee and Dullerud. They are parameterized by an amount of memory (T-1) >= 0. Our converse Lyapunov theorems are finite. More precisely, we give sufficient conditions on the asymptotic growth rate of a stable system under which one can compute an integer parameter T >= 1 for which both types of Lyapunov functions exist. As a corollary of our results, we formulate an arbitrary accurate approximation scheme for estimating the asymptotic growth rate of switching systems with constrained switching sequences.