A zero-crossing detection algorithm for robust simulation of hybrid systems jumping on surfaces

Hybrid systems are inherently fragile with respect to perturbations when their state experiences jumps on surfaces. Zero-crossing detection algorithms are capable of robustly detecting the crossing of such surfaces, but, up to now, the effects of adding such algorithms to the system being simulated are unknown. In this paper, we propose a mathematical model for hybrid systems that incorporates zero-crossing detection as well as a hybrid simulator for it. First, we discuss adverse effects that measurement noise and discretization can have on hybrid systems jumping on surfaces and prove that, under mild regularity conditions, zerocrossing detection algorithms can robustify the original system. Then, we show that integration schemes with zero-crossing detection actually compute a robustified version of the fragile nominal model. In this way, we rigorously characterize their effect on solutions to the simulated system. Finally, we show that both the model and simulator are not only robust, but also that the hybrid simulator preserves asymptotic stability properties, semiglobally and practically (on the step size), of the original system. Several examples throughout the paper illustrate these ideas and results.