Data-Driven Nonconvex Reachability Analysis using Exact Set Propagation

This paper studies deterministic data-driven reachability analysis for dynamical systems with unknown dynamics and nonconvex reachable sets. Existing deterministic data-driven approaches typically employ zonotopic set representations, for which the multiplication between a zonotopic model set and a zonotopic state set cannot be represented algebraically exactly, thereby necessitating over-approximation steps in reachable-set propagation. To remove this structural source of conservatism, we introduce constrained polynomial matrix zonotopes (CPMZs) to represent data-consistent model sets, and show that the multiplication between a CPMZ model set and a constrained polynomial zonotope (CPZ) state set admits an algebraically exact CPZ representation. This property enables set propagation entirely within the CPZ representation, thereby avoiding propagation-induced over-approximation and even retaining the ability to represent nonconvex reachable sets. Moreover, we develop set-theoretic results that enable the intersection of data-consistent model sets as new data become available, yielding the proposed online refinement scheme that progressively tightens the data-consistent model set and, in turn, the resulting reachable set. Beyond linear systems, we extend the proposed framework to polynomial dynamics and develop additional set-theoretic results that enable both model-based and data-driven reachability analysis within the same algebraic representation. By deriving algebraically exact CPZ representations for monomials and their compositions, reachable-set propagation can be carried out directly at the set level without resorting to interval arithmetic or relaxation-based bounding techniques. Numerical examples for both linear and polynomial systems demonstrate a significant reduction in conservatism compared to state-of-the-art deterministic data-driven reachability methods.