Boundary and Taxonomy of Integrator Reach Sets

TL;DR

This paper investigates the precise geometry of integrator reach sets under input uncertainties, deriving exact formulas and boundary descriptions to aid in benchmarking and designing reach set over-approximation algorithms.

Contribution

It provides a closed-form support function, explicit boundary equations, and topological classification of the reach sets for integrators with box uncertainties.

Findings

Support functions derived explicitly

Exact boundary equations given by Hankel determinants

Reach sets are semialgebraic and translated zonoids

Abstract

Over-approximating the forward reach sets of controlled dynamical systems subject to set-valued uncertainties is a common practice in systems-control engineering for the purpose of performance verification. However, specific algebraic and topological results for the geometry of such sets are rather uncommon even for simple linear systems such as the integrators. This work explores the geometry of the forward reach set of the integrator dynamics subject to box-valued uncertainties in its control inputs. Our contribution includes derivation of a closed-form formula for the support functions of these sets. This result, then enables us to deduce the parametric as well as the implicit equations describing the exact boundaries of these reach sets. Specifically, the implicit equations for the bounding hypersurfaces are shown to be given by vanishing of certain Hankel determinants. Finally, it is established that these sets are semialgebraic as well as translated zonoids. Such results should be useful to benchmark existing reach set over-approximation algorithms, and to help design new algorithms for the same.