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

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.

Figures (4)

• Figure B1: Mixed bounded-Gaussian truncation. (a) 3D density surface. (b) $m\sigma$ level sets: CCG (solid) vs. probabilistic zonotope (dashed). The CCG uses a $2$-norm ball for the Gaussian part, avoiding the box over-approximation.
• Figure E1: Parameter-set comparison for a scalar system ($n\!=\!1$, $T\!=\!30$). The CMZ (red, dash-dot) over-approximates Gaussian noise by a $5\sigma$ box in $q\!=\!30$ dimensions, yielding a large polytope. The CMCG (green, solid) uses the $\chi^2_d$ radius with $d\!=\!2$, coinciding exactly with the MLE ellipsoid (blue, dashed). OLS estimate $\hat{\Theta}$ (black $+$); true parameters $\Theta^\star$ (red $\times$).
• Figure E2: Reachable-set comparison over 5 propagation steps for a 5D system, shown in three 2D projections: (a) $(x_1, x_2)$, (b) $(x_3, x_4)$, (c) $(x_4, x_5)$. $\mathcal{R}_k$ denotes the model-based reachable set (blue, gray fill), $\tilde{\mathcal{R}}_k^{\mathrm{CMZ}}$ the CMZ over-approximation (red, outermost), and $\tilde{\mathcal{R}}_k^{\mathrm{CMCG}}$ our CMCG-based set (green). The CMCG sets are consistently tighter because the CCG propagation preserves the correct $2$-norm for Gaussian generators.
• Figure E3: Gaussian-mixture case study. (a) Bimodal scalar density with its $95\%$ HDR (shaded) and the MVEE surrogate (red dashed). (b) Five-step reachable sets: conservative single-Gaussian (red), MVEE-based CMCG (green), and model-based (blue dashed). The MVEE construction gives a tighter outer approximation by convexifying the non-convex HDR.

Theorems & Definitions (35)

• Definition 1: Zonotope girard2005reachability
• Definition 2: Matrix zonotope althoff2010reachability
• Definition 3: Constrained Convex Generators (CCG) kousik2023ellipsotopes
• Definition 4: Constrained Matrix Convex Generators (CMCG)
• Definition 5: Probabilistic zonotope althoff2010reachability
• Definition 6: Probabilistic matrix zonotope althoff2010reachability
• Proposition 1: Confidence truncation: from probabilistic zonotope to CCG
• proof
• Remark 1: Norm mismatch in prior probabilistic zonotope approaches
• Definition 7: Highest Density Region hyndman1996hdrgraph