Second-Order Trajectory Sensitivity Analysis of Hybrid Systems

Hybrid dynamical systems are characterized by intrinsic coupling between continuous dynamics and discrete events. This paper has adopted a differential-algebraic impulsive switched (DAIS) model to capture such dynamic behavior. For such systems, trajectory sensitivity analysis provides a valuable approach for describing perturbations of system trajectories resulting from small variations in initial conditions and/or uncertain parameters. The first-order sensitivities have been fully described for hybrid system and used in a variety of applications. This paper formulates the differential-algebraic equations (DAE) that govern second-order sensitivities over regions where dynamics are smooth, i.e., away from events. It also establishes the jump conditions that describe the step change in second-order sensitivities at discrete (switching and state reset) events. These results together fully characterize second-order sensitivities for general hybrid dynamical system.