Using Seminorms To Analyze Contraction of Switched Systems With Only Non-Contracting Modes

TL;DR

This paper extends contraction analysis of switched systems to cases where all modes are non-contracting by using seminorms and subspace analysis to ensure overall stability.

Contribution

It introduces a novel approach using seminorms to analyze contraction in switched systems with only non-contracting modes, broadening existing stability conditions.

Findings

Conditions for contraction within subspaces are derived.

Mode activation time conditions guarantee overall contraction.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

This paper investigates contraction properties of switched dynamical systems for the case that all modes are non-contracting, thereby extending existing results that require at least one mode to be contracting. Leveraging the property that unstable systems may still exhibit stable behavior within certain subspaces, conditions are provided which ensure contracting evolution within a given subspace of the state space of the switched system. These conditions are derived using the concepts of seminorms and semi-contracting systems. Then, by selecting a set of subspaces whose corresponding seminorms form a separating family of the state space, and by verifying whether a given mode is contracting in each subspace, conditions on the activation time of each mode are provided by which contraction on the complete state space is guaranteed. Numerical examples are presented for illustration.