Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

TL;DR

This paper analyzes how external disturbances affect a specific class of hybrid dynamical systems, establishing bounds on the impact of perturbations based on their integral, with applications to Max-Weight scheduling.

Contribution

It introduces bounds on the sensitivity of piecewise linear hybrid systems to perturbations, including systems modeling Max-Weight scheduling, which was not previously characterized.

Findings

The effect of perturbations is bounded by a constant times the integral of the disturbance.

The results apply to systems with finitely many linear pieces and include fluid-level Max-Weight dynamics.

Provides a framework for understanding robustness of hybrid systems to external influences.

Abstract

We consider a class of continuous-time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces, and which include the fluid-level dynamics of the Max-Weight scheduling policy as a special case. We study the effect of an external disturbance/perturbation on the state trajectory, and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation.