On stabilizability of switched linear systems under restricted switching

TL;DR

This paper establishes conditions for the exponential stability of discrete-time switched linear systems with unstable subsystems, under specific switching restrictions, using matrix commutation relations and graph theory.

Contribution

It provides new sufficient conditions for stability considering restricted switching signals and unstable subsystems, advancing control theory for such systems.

Findings

Derived stability conditions based on matrix commutation relations.

Applied graph-theoretic methods to analyze switching restrictions.

Ensured global exponential stability under specified switching constraints.

Abstract

This paper deals with stability of discrete-time switched linear systems whose all subsystems are unstable and the set of admissible switching signals obeys pre-specified restrictions on switches between the subsystems and dwell times on the subsystems. We derive 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 obeys the given restrictions. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-theoretic arguments.