Tracking control for hybrid systems with state-triggered jumps

This paper addresses the tracking problem in which the controller should stabilise time-varying reference trajectories of hybrid systems. Despite the fact that discrete events (or jumps) in hybrid systems can often not be controlled directly, as, e.g., is the case in impacting mechanical systems, the controller should still stabilise the desired trajectory. A major complication in the analysis of this hybrid tracking problem is that, in general, the jump times of the plant do not coincide with those of the reference trajectory. Consequently, the conventional Euclidean tracking error does not converge to zero, even if trajectories converge to the reference trajectory in between jumps, and the jump times converge to those of the reference trajectory. Hence, standard control approaches can not be applied. We propose a novel definition of the tracking error that overcomes this problem and formulate Lyapunov-based conditions for the global asymptotic stability of the hybrid reference trajectory. Using these conditions, we design hysteresis-based controllers that solve the hybrid tracking problem for two exemplary systems, including the well-known bouncing ball problem.