Computing Distances between Reach Flowpipes

TL;DR

This paper extends the Skorokhod distance to reachpipes, enabling the quantification of differences between hybrid dynamical systems' trace sets using geometric algorithms for evolving polyhedral sets.

Contribution

It introduces algorithms to compute bounds on distances between trace sets via reachpipes, extending Skorokhod distance computation to sets of trajectories.

Findings

Algorithms for upper and lower bounds on trace set distances

Extension of Skorokhod distance to reachpipes

Geometric methods for evolving polyhedral sets

Abstract

We investigate quantifying the difference between two hybrid dynamical systems under noise and initial-state uncertainty. While the set of traces for these systems is infinite, it is possible to symbolically approximate trace sets using \emph{reachpipes} that compute upper and lower bounds on the evolution of the reachable sets with time. We estimate distances between corresponding sets of trajectories of two systems in terms of distances between the reachpipes. In case of two individual traces, the Skorokhod distance has been proposed as a robust and efficient notion of distance which captures both value and timing distortions. In this paper, we extend the computation of the Skorokhod distance to reachpipes, and provide algorithms to compute upper and lower bounds on the distance between two sets of traces. Our algorithms use new geometric insights that are used to compute the worst-case and best-case distances between two polyhedral sets evolving with time.