Avoiding geometric intersection operations in reachability analysis of hybrid systems

Although a growing number of dynamical systems studied in various fields are hybrid in nature, the verification of prop-erties, such as stability, safety, etc., is still a challenging problem. Reachability analysis is one of the promising meth-ods for hybrid system verification, which together with all other verification techniques faces the challenge of making the analysis scale with respect to the number of continuous state variables. The bottleneck of many reachability analysis techniques for hybrid systems is the geometrically computed intersection with guard sets. In this work, we replace the in-tersection operation by a nonlinear mapping onto the guard, which is not only numerically stable, but also scalable, mak-ing it possible to verify systems which were previously out of reach. The approach can be applied to the fairly common class of hybrid systems with piecewise continuous solutions, guard sets modeled as halfspaces, and urgent semantics, i.e. discrete transitions are immediately taken when enabled by guard sets. We demonstrate the usefulness of the new ap-proach by a mechanical system with backlash which has 101 continuous state variables

Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution