Hybrid Polynomial Zonotopes: A Set Representation for Reachability Analysis in Hybrid Nonaffine Systems

Reachability analysis for hybrid nonaffine systems remains computationally challenging, as existing set representations--including constrained, polynomial, and hybrid zonotopes--either lose tightness under high-order nonaffine maps or suffer exponential blow-up after discrete jumps. This paper introduces Hybrid Polynomial Zonotope (HPZ), a novel set representation that combines the mode-dependent generator structure of hybrid zonotopes with the algebraic expressiveness of polynomial zonotopes. HPZs compactly encode non-convex reachable states across modes by attaching polynomial exponents to each hybrid generator, enabling precise capture of high-order state-input couplings without vertex enumeration. We develop a comprehensive library of HPZ operations, including Minkowski sum, linear transformation, and intersection. Theoretical analysis and computational experiments demonstrate that HPZs achieve superior tightness preservation and computational efficiency compared to existing approaches for hybrid system reachability analysis.