Generalized Outer Bounds on the Finite Geometric Sum of Ellipsoids

TL;DR

This paper derives a direct, closed-form formula for the boundary of the Minkowski sum of multiple ellipsoids in high-dimensional space, enabling precise and efficient computation of reachable sets in dynamical systems.

Contribution

It introduces a novel one-shot formula for summing multiple ellipsoids without iterative approximations, revealing new degrees of freedom in ellipsoidal bounds.

Findings

Exact closed-form boundary formula for Minkowski sum of ellipsoids

Efficient computation of reachable sets in discrete-time systems

New insights into ellipsoidal bounding techniques

Abstract

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done through iterative algorithms in which each new ellipsoid was added to an ellipsoid approximation of the sum of the previous ellipsoids. Here we provide one shot formulas to add $k$ ellipsoids directly with no intermediate approximations required. This allows us to observe a new degree of freedom in the family of ellipsoidal bounds on the geometric sum. We demonstrate an application of these tools to compute the reachable set of a discrete-time dynamical system.