Seminars. Global asymptotic stability of a PID control system with Coulomb friction.

Abstract:

A classical problem in the friction literature considers regulating through a feedback PID controller the position of a point mass moving on a plane under the action of friction.

A basic model for friction involves a static component for zero velocity (the so-called stiction) and a dynamic component consisting of Coulomb and viscous friction.

In this general setting, the friction nonlinearity undermines the controller performance causing, for instance, limit cycles and undesired vibrations.

In this talk we address the particular case when the maximum magnitude of the stiction is equal to the magnitude of the Coulomb friction.

We model the friction force as a set-valued mapping, obtaining a differential inclusion for the complete model of mass and PID controller.

This formulation allows to prove that the set of all equilibria is globally asymptotically stable.

The proof involves a discontinuous Lyapunov-like function, and a suitable LaSalle’s invariance principle.

The robustness properties guaranteed by the well-posedness of the model will also be discussed together with the future developments of this result.