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The comparison between time-varying hybrid trajectories is crucial fortracking, observer design and synchronisation problems for hybrid systems withstate-triggered jumps. In this paper, a systematic way of designing anappropriate distance function is proposed that can be used for this purpose.The so-called "peaking phenomenon", which occurs when using the Euclideandistance to compare two hybrid trajectories, is circumvented by taking thehybrid nature of the system explicitly into account in the design of thedistance function. Based on the proposed distance function, we define thestability of a trajectory of a hybrid system with state-triggered jumps andpresent sufficient Lyapunov-type conditions for stability of a hybridtrajectory. A constructive design method for the distance function is presentedfor hybrid systems with affine flow and jump maps and a jump set that is ahyperplane. For this case, the mentioned Lyapunov-type stability conditions canbe verified using linear matrix conditions. Finally, for this class of systems,we present a tracking controller that asymptotically stabilises a given hybridreference trajectory, and we illustrate our results with examples.

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