A note on the existence of stabilizing switching signals for switched linear systems

TL;DR

This paper establishes conditions under which unstable switched linear systems can be stabilized through specific switching signals, using matrix commutation and graph theory, with numerical validation.

Contribution

It introduces new sufficient conditions for stability of unstable subsystems in switched systems via time-dependent switching signals, employing commutation relations and graph theory.

Findings

Conditions for stability of unstable subsystems derived

Switching signals can stabilize systems with all unstable modes

Numerical experiments confirm theoretical results

Abstract

This paper deals with stability of discrete-time switched linear systems whose all subsystems are unstable. We present sufficient conditions on the subsystems matrices such that a switched system is globally exponentially stable under a set of purely time-dependent switching signals that are allowed to activate all subsystems. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-theoretic arguments. We present a numerical experiment to demonstrate our results.