On optimal trajectory tracking for mechanical systems with unilateral constraints

This paper addresses the optimal trajectory tracking problem for hybrid systems with discontinuous state trajectories. We consider state-triggered jumps in the state evolution. The tracking of a given reference trajectory is complicated by the fact that the time instants at which the closed-loop dynamics actually jumps differ from those at which the reference trajectory jumps. A recently introduced hybrid feedback tracking controller is used to cope with these differences. A method to design the corresponding time-varying feedback gains such that they minimize a cost criterion in a linear quadratic regulator (LQR) like idea is described in this work. The effectiveness of the proposed control law is investigated numerically by comparison with classic proportional feedback and constant gain hybrid feedback control for a mechanical system with a unilateral position constraint, namely a controlled mass interacting with a rigid obstacle.