Ellipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection

Abstract

Ellipsoids are a common representation for reachability analysis because they are closed under affine maps and allow conservative approximation of Minkowski sums; this enables one to incorporate uncertainty and linearization error in a dynamical system by exapnding the size of the reachable set. Zonotopes, a type of symmetric, convex polytope, are similarly frequently used due to efficient numerical implementation of affine maps and exact Minkowski sums. Both of these representations also enable efficient, convex collision detection for fault detection or formal verification tasks, wherein one checks if the reachable set of a system collides (i.e., intersects) with an unsafe set. However, both representations often result in conservative representations for reachable sets of arbitrary systems, and neither is closed under intersection. Recently, constrained zonotopes and constrained polynomial zonotopes have been shown to overcome some of these conservatism challenges, and are closed under intersection. However, constrained zonotopes can not represent shapes with smooth boundaries such as ellipsoids, and constrained polynomial zonotopes can require solving a non-convex program for collision checking (i.e., fault detection). This paper introduces ellipsotopes, a set representation that is closed under affine maps, Minkowski sums, and intersections. Ellipsotopes combine the advantages of ellipsoids and zonotopes, and enable convex collision checking at the expense of more conservative reachable sets than constrained polynomial zonotopes. The utility of this representation is demonstrated on several examples.

Stochastic Reachability-Based GPS Spoofing Detection with Chimera Signal Enhancement

“…We also provide a basic introduction to probabilistic zonotopes (p-zonotopes), which are used for representing stochastic sets. Other mathematical objects have been developed, including ellipsotopes (Kousik et al, 2021), which can provide tighter bounding stochastic sets than p-zonotopes. However, the primary objective of this work is to develop the overall framework and analysis for an SR-based spoofing detector and state estimator.…”