Data-Driven Reachability Analysis Using Matrix Perturbation Theory

TL;DR

This paper introduces a matrix perturbation framework using matrix zonotopes for efficient, less conservative, real-time reachability analysis of dynamical systems affected by noise.

Contribution

It develops a new matrix zonotope perturbation framework with Cai-Zhang bounds and a scalable approximation method for improved reachability analysis.

Findings

The proposed method is significantly faster than traditional CMZ approaches.

It produces less conservative reachable sets compared to existing MZ-based methods.

Experimental results validate the efficiency and accuracy of the approach.

Abstract

We propose a matrix zonotope perturbation framework that leverages matrix perturbation theory to characterize how noise-induced distortions alter the dynamics within sets of models. The framework derives interpretable Cai-Zhang bounds for matrix zonotopes (MZs) and extends them to constrained matrix zonotopes (CMZs). Motivated by this analysis and the computational burden of CMZ-based reachable-set propagation, we introduce a coefficient-space approximation in which the constrained coefficient space of the CMZ is over-approximated by an unconstrained zonotope. Replacing CMZ-constrained-zonotope (CZ) products with unconstrained MZ-zonotope multiplication yields a simpler and more scalable reachable-set update. Experimental results demonstrate that the proposed method is substantially faster than the standard CMZ approach while producing reachable sets that are less conservative than those obtained with existing MZ-based methods, advancing practical, accurate, and real-time data-driven reachability analysis.