Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered discontinuities in the state evolution, potentially accompanied by mode switching in the control vector field. For a given trajectory with state jumps, we show how to construct an approximation of the nearby perturbed trajectory corresponding to a given variation of the initial condition and input signal. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal to those of the nominal one. The main contribution of this work is the development of a notion of error to clarify in which sense the approximate trajectory is, at each instant of time, a first-order approximation of the perturbed trajectory. This notion of error naturally finds application in the (local) tracking problem of a time-varying reference trajectory of a hybrid system. To illustrate the possible use of this new error definition in the context of trajectory tracking, we outline how the standard linear trajectory tracking control for nonlinear systems could be generalized for hybrid systems.