Global asymptotic stability of a PID control system with Coulomb friction

TL;DR

This paper analyzes the global asymptotic stability of a PID-controlled point mass system with Coulomb friction, introducing a differential inclusion model and employing Lyapunov methods to prove stability and robustness.

Contribution

It presents a novel differential inclusion model for Coulomb friction in PID control systems and proves their global stability using Lyapunov functions and invariance principles.

Findings

Global asymptotic stability of the system established

Robustness results including ISS property demonstrated

Simulation results validate theoretical findings

Abstract

We propose a model for representing a point mass subject to Coulomb friction in feedback with a PID controller, based on a differential inclusion comprising all the possible magnitudes of static friction during the stick phase. For this model we study the set of all equilibria and we establish its global asymptotic stability using a discontinuous Lyapunov-like function, and a suitable LaSalle's invariance principle. We finally use well-posedness of the proposed model to establish useful robustness results, including an ISS property from a suitable input in a perturbed context. Simulation results are also given to illustrate our statements.