Scalable Zonotope-Ellipsoid Conversions using the Euclidean Zonotope Norm

Set-based computations become increasingly popular for safety-critical systems to ensure properties of controllers and observers. To efficiently compute various set operations, one often uses different set representations and conversions between them. Two popular set representations, for which scalable conversion algorithms do not yet exist, are zonotopes and ellipsoids. We provide computational approaches for all four conversion cases, i.e., overapproximations and underapproximations from zonotopes to ellipsoids and vice versa. By using upper bounds on the maximum and lower bounds on the minimum Euclidean norm of a given zonotope, our approaches have polynomial complexity and thus can be used for high-dimensional spaces. We show that the tightness of our approaches directly depends on the tightness of the Euclidean norm. Numerical experiments demonstrate the usefulness of our proposed methods.