On infinitesimal contraction analysis for hybrid systems

S. Burden, S. Coogan
IEEE Conference on Decision and Control, 2022


Infinitesimal contraction analysis, wherein global convergence results are obtained from properties of local dynamics, is a powerful analysis tool. In this paper, we generalize infinitesimal contraction analysis to hybrid systems in which state-dependent guards trigger transitions defined by reset maps between modes that may have different norms and need not be of the same dimension. In contrast to existing literature, we do not restrict mode sequence or dwell time. We work in settings where the hybrid system flow is differentiable almost everywhere and its derivative is the solution to a jump-linear-time-varying differential equation whose jumps are defined by a saltation matrix determined from the guard, reset map, and vector field. Our main result shows that if the vector field is infinitesimally contracting, and if the saltation matrix is non-expansive, then the intrinsic distance between any two trajectories decreases exponentially in time. When bounds on dwell time are available, our approach yields a bound on the intrinsic distance between trajectories regardless of whether the dynamics are expansive or contractive. We illustrate our results using wo examples: a constrained mechanical system and an electrical circuit with an ideal diode.