On the effect and robustness of zero-crossing detection algorithms in simulation of hybrid systems jumping on surfaces

Motivated by the fragility to perturbations of hybrid systems jumping on surfaces and the robustifying capabilities of zero-crossing detection algorithms, we propose a hybrid simulator model with incorporated zero-crossing detection. First, we reveal the effect of measurement noise and of discretization to hybrid systems jumping on surfaces. We prove that, under mild regularity conditions, zero-crossing detection algorithms have a robustifying effect on the original system. Then, we argue that, rather than computing the solutions to the discretization of the fragile nominal model, integration schemes with zero-crossing detection actually compute the solutions of a robustified version of the original model. We propose a mathematical model for the hybrid system with incorporated zero-crossing detection as well as a hybrid simulator for it. 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. An example illustrates the ideas and results throughout the paper.