The Barabanov norm is generically unique, simple, and easily computed

TL;DR

This paper proves that for most discrete-time linear switching systems, the Barabanov norm is unique, simple, and can be efficiently computed, enabling precise analysis of system trajectories.

Contribution

It classifies all possible Barabanov norms for such systems, showing they are typically unique and either piecewise-linear or quadratic, with algorithms to verify assumptions.

Findings

Most systems have a unique, simple Barabanov norm.

Fastest growth trajectories are eventually periodic.

Numerical experiments confirm the theoretical classifications.

Abstract

Every irreducible discrete-time linear switching system possesses an invariant convex Lyapunov function (Barabanov norm), which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N.Guglielmi and M.Zennaro showed that many systems possess at least one simple Barabanov norm, which moreover, can be efficiently computed. In this paper we classify all possible Barabanov norms for discrete-time systems. We prove that, under mild assumptions, such norms are unique and are either piecewise-linear or piecewise quadratic. Those assumptions can be verified algorithmically and the numerical experiments show that a vast majority of systems satisfy them. For some narrow classes of systems, there are more complicated Barabanov norms but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. They turn out to be eventually periodic with special periods. Examples and numerical results are presented.