UC Santa Cruz

Characterising the distance between hybrid trajectories is crucial for solving tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. When the Euclidean distance function is used, the socalled peaking phenomenon for hybrid systems arises, which forms a major obstacle as trajectories cannot be stable in the sense of Lyapunov using such a distance. Therefore, in this paper, a novel and systematic way of designing appropriate distance functions is proposed that overcomes this hurdle and enables the derivation of sufficient Lyapunov-type conditions, using minimal or maximal average dwell-time arguments, for the stability of a hybrid trajectory. A constructive design method for piecewise quadratic Lyapunov functions is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. Finally, we illustrate our results with an example.