To stick or to slip: Lyapunov-based reset PID for positioning systems with Coulomb and Stribeck friction
Résumé
Reset control systems for continuous-time plants were introduced in th 1950's by J.C. Clegg, then extended by Horowitz twenty years later and revisited using hybrid Lyapunov theory a few decades ago, to rigorously deal with the continuous-discrete interplay stemming from the reset laws. In this talk we overview a recent research activity where suitable reset actions induce stability and performance of PID-controlled positioning systems suffering from nonlinear frictional effects. With Coulomb-only effect, PID feedback produces a nontrivial set of equilibria whose asymptotic (but not exponential) stability can be certified by using a discontinuous Lyapunov-like function. With velocity weakening effects (the so-called Stribeck friction), the set of equilibria becomes unstable with PID feedback and the so-called ``hunting phenomenon'' (persistent oscillations) is experienced. Resetting laws can be used in both scenarios. With Coulomb friction only, the discontinuous Lyapunov-like function immediately suggests a reset action providing extreme performance improvement, preserving stability and increasing the convergence speed. With Stribeck, a more sophisticated set of logic-based reset rules recovers global asymptotic stability of the set of equilibria, providing an effective solution to the hunting instability. We will discuss the Lyapunov-based proofs of these hybrid laws, requiring nontrivial derivations, such as building semiglobal hybrid bisimulation models. The theoretical results will be illustrated by experiments carried out on an industrial nano-positioning system, showing the experimental advantages arising from our novel reset PID controllers.