On convergence analysis of feedback control with integral action and discontinuous relay perturbation

We consider third-order dynamic systems which have an integral feedback action and discontinuous relay disturbance. More specifically for the applications, the focus is on the integral plus state-feedback control of the motion systems with discontinuous Coulomb-type friction. We recall the stiction region is globally attractive where the resulting hybrid system has also solutions in Filippov sense, while the motion trajectories remain in that idle state (called in tribology as stiction) until the formulated sliding-mode condition is violated by the growing integral feedback quantity. We analyze the conditions for occurrence of the slowly converging stick-slip cycles. We also show that the hybrid system is globally but only asymptotically stable, and almost always not exponentially. A particular case of the exponential convergence can appear for some initial values, assuming the characteristic equation of the linear subsystem has dominant real roots. Illustrative numerical examples are provided alongside with the developed analysis. In addition, a laboratory example is shown with experimental evidence to support the convergence analysis provided.