On the design of a position feedback control law for a simple mechanical system subject to impacts

The control of mechanical systems subject to smooth impacts is considered in this paper. As the dynamic model of the mechanical system varies in dependence on the condition of contact or non-contact, a family of dynamic compensators from position measurements is first derived, thus guaranteeing the asymptotic stability in the two different conditions. As the system can switch between the two conditions an arbitrary number of times, the stability of the overall system is not guaranteed by the stability in the two separate phases; hence, a Lyapunov approach is proposed in order to guarantee the asymptotic stability of the overall system independently of the number of transitions between the two phases. Second, it is shown how to select among the considered family of compensators the one that reduces as much as possible the effects of the impacts on the trajectory of the system, while considering a regulation problem, with zero steady-state regulation error. Both simulation and experimental results have been carried out showing the effectiveness of the proposed technique.