Inscribing and separating an ellipsoid and a constrained zonotope: Applications in stochastic control and centering

Constrained zonotopes are equivalent representations for convex polytopes that have recently enabled tractable implementations of some set-based control methods. We consider the problems of inscribing an ellipsoid within and separating an ellipsoid from a constrained zonotope. Such problems arise in several applications, including in stochastic optimal control problems when enforcing chance constraints involving constrained zonotopes. Given a parameterized ellipsoid, we propose a set of sufficient conditions that are convex in the parameters and guarantee that the ellipsoid is inscribed within a constrained zonotope. We use these conditions to solve a two-stage, return-guaranteed spacecraft rendezvous problem under uncertainty. We also apply these conditions to tractably approximate the Chebyshev center and the maximum volume inscribed ellipsoid of a constrained zonotope using linear and second-order cone programming. We also propose a set of necessary and sufficient conditions that separate an ellipsoid from a constrained zonotope, which has applications in enforcing probabilistic exclusion from a constrained zonotope.