Constructing distance functions and piecewise quadratic Lyapunov functions for stability of hybrid trajectories

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.