A characterization of switched linear control systems with finite L 2 -gain

TL;DR

This paper extends the concept of Barabanov norms to analyze the L2-gain of switched linear control systems, establishing a key equivalence with the spectral radius condition for a broad class of systems.

Contribution

It introduces a generalized approach to characterize the L2-gain of switched systems using extended extremal trajectories and spectral radius conditions.

Findings

Finiteness of L2-gain is equivalent to spectral radius being less than one.

Extension of Barabanov norms to non-closed switching classes.

Provides a new criterion for stability analysis of switched systems.

Abstract

Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.