Safe & Robust Reachability Analysis of Hybrid Systems

TL;DR

This paper introduces a safe reachability analysis method for hybrid systems that guarantees over-approximation of reachable states and discusses robustness to small perturbations, addressing limitations of traditional reachability definitions.

Contribution

It proposes a novel safe reachability approach that always computes a supersets of reachable states and links robustness to Scott continuous approximations in hybrid systems.

Findings

Safe reachability computes supersets of reachable states.

Best Scott continuous approximation aligns with robust analysis.

Illustrates the gap between actual reachable states and over-approximations.

Abstract

Hybrid systems - more precisely, their mathematical models - can exhibit behaviors, like Zeno behaviors, that are absent in purely discrete or purely continuous systems. First, we observe that, in this context, the usual definition of reachability - namely, the reflexive and transitive closure of a transition relation - can be unsafe, ie, it may compute a proper subset of the set of states reachable in finite time from a set of initial states. Therefore, we propose safe reachability, which always computes a superset of the set of reachable states. Second, in safety analysis of hybrid and continuous systems, it is important to ensure that a reachability analysis is also robust wrt small perturbations to the set of initial states and to the system itself, since discrepancies between a system and its mathematical models are unavoidable. We show that, under certain conditions, the best Scott continuous approximation of an analysis A is also its best robust approximation. Finally, we exemplify the gap between the set of reachable states and the supersets computed by safe reachability and its best robust approximation.