Hybrid Zonotopes: A Mixed-Integer Set Representation for the Analysis of Hybrid Systems

Set-based methods have been leveraged in many engineering applications from robust control and global optimization, to probabilistic planning and estimation. While useful, these methods have most widely been applied to analysis over sets that are convex, due to their ease in both representation and calculation. The representation and analysis of nonconvex sets is inherently complex. When nonconvexity arises in design and control applications, the nonconvex set is often over-approximated by a convex set to provide conservative results. However, the level of conservatism may be large and difficult to quantify, often leading to trivial results and requiring repetitive analysis by the engineer. Nonconvexity is inherent and unavoidable in many applications, such as the analysis of hybrid systems and robust safety constraints. 

In this dissertation, I present a new nonconvex set representation named the hybrid zonotope. The hybrid zonotope builds upon a combination of recent advances in the compact representation of convex sets in the controls literature with methods leveraged in solving mixed-integer programming problems. It is shown that the hybrid zonotope is equivalent to the union of an exponential number of convex sets while using a linear number of continuous and binary variables in the set’s representation. I provide identities for, and derivations of, the set operations of hybrid zonotopes for linear mappings, Minkowski sums, generalized intersections, halfspace intersections, Cartesian products, unions, complements, point containment, set containment, support functions, and convex enclosures. I also provide methods for redundancy removal and order reduction to improve the compactness and computational efficiency of the represented sets. Therefore proving the hybrid zonotopes expressive power and applicability to many nonconvex set-theoretic methods. Beyond basic set operations, I specifically show how the exact forward and backward reachable sets of linear hybrid systems may be found using identities that are calculated algebraically and scale linearly. Numerical examples show the scalability of the proposed methods and how they may be used to verify the safety and performance of complex systems. These exact methods may also be used to evaluate the level of conservatism of the existing approximate methods provided in the literature.