Hybrid systems with state-triggered jumps: sensitivity-based stability analysis with application to trajectory tracking

The definition of asymptotic stability for a trajectory of a hybrid system with state-triggered jumps is not straightforward. Nearby solutions jump at close but non-coincident times, making the standard notion of closeness, based on vector difference, unsuitable to compare trajectories point-wise in time. With tracking control as ultimate goal, we propose a notion of stability and a constructive stability proof based on sensitivity analysis applicable to single-jump-flow trajectories. A key role in the analysis is played by a time-triggered linear system, associated with the discontinuous trajectory of interest, whose uniform asymptotic stability suffices to guarantee the asymptotic stability of the original discontinuous trajectory. As an illustrative example, the stability analysis is applied to guarantee closed-loop stable tracking for a trajectory with velocity jumps of a 2 DoF mechanical system with unilateral constraint.