Bridging Data-Driven Reachability Analysis and Statistical Estimation via Constrained Matrix Convex Generators

TL;DR

This paper introduces a novel approach for data-driven reachability analysis using mixed-norm uncertainty sets and Constrained Convex Generators, resulting in tighter safety verification bounds especially for Gaussian noise.

Contribution

It develops a new geometric framework with matrix convex generators that better fit Gaussian and mixed noise distributions, reducing conservatism in reachability analysis.

Findings

Achieves tighter reachable sets than box-based methods for Gaussian disturbances.

CMC representations coincide with maximum-likelihood confidence ellipsoids for Gaussian noise.

Numerical examples show improved accuracy in safety verification.

Abstract

Data-driven reachability analysis enables safety verification when first-principles models are unavailable. This requires constructing sets of system models consistent with measured trajectories and noise assumptions. Existing approaches rely on zonotopic or box-based approximations, which do not fit the geometry of common noise distributions such as Gaussian disturbances and can lead to significant conservatism, especially in high-dimensional settings. This paper builds on ellipsotope-based representations to introduce mixed-norm uncertainty sets for data-driven reachability. The highest-density region defines the exact minimum-volume noise confidence set, while Constrained Convex Generators (CCG) and their matrix counterpart (CMCG) provide compatible geometric representations at the noise and parameter level. We show that the resulting CMCG coincides with the maximum-likelihood confidence ellipsoid for Gaussian disturbances, while remaining strictly tighter than constrained matrix zonotopes for mixed bounded-Gaussian noise. For non-convex noise distributions such as Gaussian mixtures, a minimum-volume enclosing ellipsoid provides a tractable convex surrogate. We further prove containment of the CMCG times CCG product and bound the conservatism of the Gaussian-Gaussian interaction. Numerical examples demonstrate substantially tighter reachable sets compared to box-based approximations of Gaussian disturbances. These results enable less conservative safety verification and improve the accuracy of uncertainty-aware control design.