Limit cycle behavior and convergence to zero error in learning control with stick-slip friction

Stick-slip friction exists in virtually all mechanical systems. Learning control was developed motivated by robots performing repetitive tasks, with the aim of learning to improve performance based on previous experience performing the task. The behavior of learning control applied to nonlinear systems has been studied, but usually under the assumption that the system equations satisfy a Lipschitz condition. Relatively little research has appeared concerning the behavior of learning control in systems having stick-slip friction that does not satisfy such a condition. Here we study the learning behavior for the simplest form of learning control based on integral control concepts applied in the repetition domain. Methods of predicting when it will converge to zero tracking error are developed, and guidelines to producing this situation are given. When the desired trajectory is not physically executable by the mechanical system, then it is shown that limit cycle like behavior can occur, and formulas are given that characterize certain parameters of this behavior