Distance function design and Lyapunov techniques for the stability of hybrid trajectories

The comparison between time-varying hybrid trajectories is crucial for tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. In this paper, a generic distance function is designed that can be used for this purpose. The so-called “peaking phenomenon”, which occurs when using the Euclidean distance to compare two hybrid trajectories, is circumvented by taking the hybrid nature of the system explicitly into account. Based on the proposed distance function, we define the stability of a trajectory and present sufficient Lyapunov-type conditions for hybrid system with state-triggered jumps. A constructive Lyapunov function design is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. The stability conditions can then be verified using linear matrix conditions. Finally, for this class of systems, we present a tracking controller that asymptotically stabilises a given hybrid reference trajectory and we illustrate our results with an example.