Robustness of Exponential Stability of a Class of Switched Linear Systems with State Delays

TL;DR

This paper analyzes the robustness of exponential stability in switched linear systems with delays, introducing stability radius bounds and providing computable measures under certain positivity and delay conditions.

Contribution

It introduces the structured stability radius for switched systems with delays and derives bounds using common Lyapunov functions, enhancing robustness analysis methods.

Findings

Derived bounds for the stability radius of delayed switched systems.

Provided computable measures for systems with positive linear dynamics.

Illustrated results with examples demonstrating the method's applicability.

Abstract

This paper investigates the robustness of exponential stability of a class of switched systems described by linear functional differential equations under arbitrary switching. We will measure the stability robustness of such a system, subject to parameter affine perturbations of its constituent subsystems matrices, by introducing the notion of structured stability radius. The lower bounds and the upper bounds for this radius are established under the assumption that certain associated positive linear systems upper bounding the given subsystems have a common linear copositive Lyapunov function. In the case of switched positive linear systems with discrete multiple delays or distributed delay the obtained results yield some tractably computable bounds for the stability radius. Examples are given to illustrate the proposed method.