To stick or to slip: A reset PID control perspective on positioning systems with friction

Abstract

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 a Coulomb-only effect, PID feedback produces a 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 inducing desirable exponential convergence of a relevant subset of the solutions. 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 clarify here the main steps of the Lyapunov-based proofs associated to our reset-enhanced PID controllers. These proofs involve building semiglobal hybrid representations of the solutions in the form of hybrid automata whose logical variables enable transforming the aforementioned discontinuous function into smooth or at least Lipschitz ones. Our theoretical results are illustrated by extensive simulations and experimental validation on an industrial nano-positioning system.

Stacking Integrators Without Sacrificing the Overshoot in Reset Control Systems

“…The open-loop DF and HOSDF analysis of control loop presented in Fig. 3 for n = 1, 2, tuned according to (10), (11) and ( 12) is depicted in Fig. 4.…”

“…Reset control has also recently been used to approximate the complex-order filters [9], [10]. Advantage of using reset control over linear control has been shown in many studies especially in precision motion control [8], [11]- [15], [15]- [19]. However, these studies are mostly focused on solving one problem.…”

Fractional-order single state reset element

“…This category of controllers, thereafter called reset controllers, has been further developed and more sophisticated elements such as first-order reset element (FORE) in [4,5], generalised FORE (GFORE) in [6] and second-order reset element (SORE) in [7]. These reset elements were used in different capacities such as phase lag reduction, decreasing sensitivity peak, narrowband and broadband phase compensation, control of positioning systems with friction see [8][9][10][11][12][13][14][15]. Reset elements stability was also investigated in the literature [16,17].…”

Feedback control method to suppress stick-slip in drill-strings featuring delay and actuation constraints

“…Consequently, this method lacks robustness in the face of uncertain values of WOB. PID and PD control has been proposed by [31][32][33][34] as a way with which to avoid stick-slip. Soft-Torque control (patented by Shell in [35]) and Z-torque control [36] are effectively PI controllers and have great sensitivity to actuator delays and measurement delays which also belong to the PID family.…”