Analysis of relay-based feedback compensation of Coulomb friction

TL;DR

This paper investigates relay-based feedback control for systems with Coulomb friction, analyzing stability, optimal gain, and residual oscillations, supported by numerical and experimental results.

Contribution

It introduces a time-optimal gain for relay feedback under Coulomb friction and analyzes residual oscillations with physical friction models.

Findings

Relay feedback stabilizes Coulomb friction systems.

Optimal gain minimizes Lyapunov function derivative.

Residual oscillations include limit cycles and chattering.

Abstract

Standard problem of one-degree-of-freedom mechanical systems with Coulomb friction is revised for a relay-based feedback stabilization. It is recalled that such a system with Coulomb friction is asymptotically stabilizable via a relay-based output feedback, as formerly shown in [1]. Assuming an upper bounded Coulomb friction disturbance, a time-optimal gain of the relay-based feedback control is found by minimizing the derivative of the Lyapunov function proposed in [2] for the twisting algorithm. Furthermore, changing from the discontinuous Coulomb friction to a more physical discontinuity-free one, which implies a transient presliding phase at motion reversals, we analyze the residual steady-state oscillations. This is in the sense of stable limit cycles, in addition to chattering caused by the actuator dynamics. The numerical examples and an experimental case study accompany the provided analysis.