[Paper Review] Hybrid zonotopes: a new set representation for reachability analysis of mixed logical dynamical systems

Trevor J. Bird, Herschel C. Pangborn+2 more|arXiv (Cornell University)|Jun 28, 2021

Formal Methods in Verification4 citations

TL;DR

This paper introduces hybrid zonotopes—a new nonconvex set representation that compactly encodes the exact reachable set of discrete-time linear hybrid systems modeled as mixed logical dynamical (MLD) systems. By combining continuous and binary zonotope factors, it represents the union of $2^N$ constrained zonotopes using only linear growth in representation complexity, enabling exact reachability analysis without exponential blowup.

ABSTRACT

This article presents a new set representation named the hybrid zonotope that is equivalent to the union of $2^N$ constrained zonotopes -- convex polytopes -- through the addition of $N$ binary zonotope factors. The major contribution of this manuscript is a closed-form solution for exact forward reachable sets of discrete-time, linear hybrid systems modeled as mixed logical dynamical systems. The proposed approach captures the worst-case exponential growth in the number of convex sets required to represent the nonconvex reachable set while exhibiting only linear growth in the complexity of the hybrid zonotope set representation. Redundancy removal techniques are provided that leverage binary trees to store the combinations of binary factors of the hybrid zonotope that map to nonempty convex subsets. Numerical examples show the hybrid zonotope's ability to compactly represent nonconvex reachable sets with an exponential number of features. Furthermore, the hybrid zonotope is shown to be closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections.

Motivation & Objective

• Address the exponential growth in convex set representations during reachability analysis of hybrid systems.
• Develop a nonconvex set representation that captures the exact reachable set of linear MLD systems without over-approximation.
• Enable efficient computation of reachable sets by reducing redundancy in the representation of nonconvex unions of convex sets.
• Ensure closure under key set operations such as linear mappings, Minkowski sums, and intersections for use in set-theoretic analysis.
• Provide a scalable and exact alternative to over-approximation-based methods that suffer from conservatism and computational intractability.

Proposed method

• Propose hybrid zonotopes as a union of $2^N$ constrained zonotopes via $N$ binary zonotope factors, enabling exact representation of nonconvex reachable sets.
• Derive a closed-form solution for forward reachable sets of discrete-time MLD systems using hybrid zonotopes, embedding guard crossings, dynamics switching, and reset maps implicitly.
• Leverage binary trees and mixed-integer programming to identify and remove redundant combinations of binary factors that map to empty subsets.
• Use a mixed-integer linear program (MILP) formulation to compute support functions and detect halfspace intersections efficiently.
• Ensure closure under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections via algebraic properties of the hybrid zonotope structure.
• Implement the method using YALMIP and compare against two baseline methods: M1 (exponential growth via constrained zonotopes) and M2 (MILP-based compact representation).

Experimental results

Research questions

• RQ1 Can a nonconvex set representation be developed that captures the exact reachable set of linear hybrid systems with only linear growth in representation complexity?
• RQ2 How can the exponential branching of reachable sets due to guard crossings and mode switches be encoded without exploding the number of convex sets?
• RQ3 To what extent can redundancy in hybrid zonotope representations be removed while preserving exactness and computational efficiency?
• RQ4 How does the hybrid zonotope compare to existing methods in terms of computational time, representation size, and scalability on benchmark problems?
• RQ5 Can hybrid zonotopes support essential set operations like Minkowski sums and halfspace intersections in a computationally efficient and exact manner?

Key findings

• The hybrid zonotope representation achieves exact reachable set computation for MLD systems with only linear growth in representation complexity, in contrast to the worst-case exponential growth of traditional methods.
• For Case $(12,4)$, the M1 method failed to complete after 20 hours due to exponential blowup, while the hybrid zonotope method (HCG-r) completed in under 1 second with minimal overhead.
• The reduced hybrid zonotope (HCG-r) achieved the lowest average computation time for support function evaluation and halfspace intersection detection, outperforming both M1 and M2.
• The method using hybrid zonotopes with redundancy removal (HCG-r) required only 1312 continuous generators and 1604 binary variables for Case $(12,4)$, compared to over 1.7 million convex sets in M1.
• The hybrid zonotope is closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections, enabling its use in broader set-theoretic analysis pipelines.
• Numerical results show that hybrid zonotopes compactly represent nonconvex reachable sets with an exponential number of features using a linear number of parameters, significantly improving scalability and efficiency.