Global Asymptotic Stability of a PID Control System With Coulomb Friction

• Published in: IEEE transactions on automatic control Vol. 63; no. 8; pp. 2654 - 2661
• Main Authors: Bisoffi, Andrea, Da Lio, Mauro, Teel, Andrew R., Zaccarian, Luca
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Asymptotic properties; Asymptotic stability; Automatic; Computer simulation; Control stability; Control systems; Coulomb friction; discontinuous Lyapunov function; Engineering Sciences; Force; Friction; Integral equations; Lyapunov methods; mechanical systems; nonlinear dynamical systems; PD control; PI control; PID control; position control; Proportional integral derivative; robust asymptotic stability; Robustness; Static friction; Well posed problems
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2017.2774443

For a point mass subject to Coulomb friction in feedback with a PID controller, we consider a model based on a differential inclusion comprising all the possible magnitudes of static friction during the stick phase and having unique solutions. 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. The well-posedness of the proposed model allows 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.