Unions and Complements of Hybrid Zonotopes

Hybrid zonotopes have been recently introduced as a mixed-integer set representation that is equivalent to the union of up to 2(N) constrained zonotopes through the use of N binary factors. This letter further develops this new set representation by deriving identities for union and complement set operations of hybrid zonotopes. The addition of these set operations proves the hybrid zonotope's closure under linear mappings, Minkowski sums, intersections, unions, and complements, thus increasing the usefulness of the set representation to a broad class of set-theoretic controls problems. Beyond set operations, a major contribution of this letter is a demonstration of how linear mixed-integer constraints may be embedded within the hybrid zonotope set definition. A numerical example using hybrid zonotopes as a nonconvex safety constraint in an obstacle avoidance problem is provided.