Efficient Method for Computing Lower Bounds on the $p$-radius of Switched Linear Systems

TL;DR

This paper introduces a new, efficient method for computing lower bounds on the $p$-radius of a set of matrices, which is crucial for stability analysis in switched linear systems, extending and improving existing bounds.

Contribution

It proposes a novel spectral radius-based lower bound method for the $p$-radius that requires no restrictions on the matrix set and includes a Markovian extension.

Findings

The method extends existing lower bounds theoretically.

The bounds can be practically improved over previous methods.

A Markovian extension enhances applicability.

Abstract

This paper proposes lower bounds on a quantity called $L^p$-norm joint spectral radius, or in short, $p$-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the $p$-radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.