It is well known that the positional proportional-integral-derivative (PID) control of a servo mechanism with stiction always leads to a limit cycle. Related to this fact, two basic questions have still remained unanswered. The first question is, if the limit cycle occurs, how large it becomes for a given value of stiction force. The second question, which is of more practical importance, is how we should modify the PID controller to avoid this limit cycle with minimal sacrifice of the servo performance. This paper presents a rigorous analysis to provide particular answers to these two questions, which turn out to be closely related to each other. More specifically, it is shown that, by exploiting algebraic properties of the state trajectory, a simple bisection algorithm can be devised to compute the exact magnitude of the periodic solution for a given value of stiction. Through the geometric analysis of the impact map, this result is then used to find the minimum value of the integrator leakage to avoid the limit cycle. The work in this paper will be useful as a specific reference in designing servo mechanisms with stiction free from limit cycle.

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