Stationary Set and Stability: a Case Study for Mechanical Systems with Discontinuities

This paper focuses on the position control of a class of mechanical systems with static friction force. A discontinuous mapping is introduced to represent the static friction such that the closed-loop systems are described by the differential equations with discontinuous right-hand side. Based on the Filippov´s framework, it´s proved the equilibrium set of the closed-loop systems under PD control corresponds to a stationary mode, and the size of the equilibrium set can be regulated as small as possible with feasible feedback gain. Furthermore, we will show that the positioning error can be rendered into such equilibrium set. To demonstrate the presented theoretical results, the experiments tested on a three-axis flight simulation table are carried out.