Integrator Leakage for Limit Cycle Suppression in Servo Mechanisms With Stiction

This paper reports quantitative answers to two questions regarding the limit cycle behavior arising from the servo mechanisms with stiction under the positional PID (Proportional-Integral-Derivative) control. The first question is, how large the magnitude of the limit cycle is for a given value of stiction force. The second one is how we should modify the PID controller to avoid this limit cycle with minimal degradation in the serve performance. The main approach is based on understanding key algebraic properties of the state trajectory. As a result, a simple bisection algorithm has been devised to compute the exact magnitude of the periodic solution for a given value of stiction. A direct extension of this algorithm enables us to find the minimum value of the integrator leakage to avoid the limit cycle. A numerical example demonstrates the main results.