Data-Driven Reachability Analysis with Optimal Input Design

Abstract

This paper addresses the conservatism in data-driven reachability analysis for discrete-time linear systems subject to bounded process noise, where the system matrices are unknown and only input--state trajectory data are available. Building on the constrained matrix zonotope (CMZ) framework, two complementary strategies are proposed to reduce conservatism in reachable-set over-approximations. First, the standard Moore--Penrose pseudoinverse is replaced with a row-norm-minimizing right inverse computed via a second-order cone program (SOCP), which directly reduces the size of the resulting model set, yielding tighter generators and less conservative reachable sets. Second, an online A-optimal input design strategy is introduced to improve the informativeness of the collected data and the conditioning of the resulting model set, thereby reducing uncertainty. The proposed framework extends naturally to piecewise affine systems through mode-dependent data partitioning. Numerical results on a five-dimensional stable LTI system and a two-dimensional piecewise affine system demonstrate that combining designed inputs with the row-norm right inverse significantly reduces conservatism compared to a baseline using random inputs and the pseudoinverse, leading to tighter reachable sets for safety verification.

Figures (2)

• Figure D1: Reachable-set comparison on the five-dimensional LTI system, projected onto $(x_1,x_2)$, $(x_3,x_4)$, and $(x_4,x_5)$. $\mathcal{R}_{\mathrm{model}}$: model-based ground truth (light gray filled). Dashed lines show the random-input baselines from alanwar2022data: ($\hat{\mathcal{R}}_{\mathrm{MZ}}^{\mathrm{rand}}\!\mid\!\mathrm{pinv}$): MZ with pseudoinverse (red dashed); ($\hat{\mathcal{R}}_{\mathrm{CMZ}}^{\mathrm{rand}}\!\mid\!\mathrm{pinv}$): CMZ with pseudoinverse (blue dashed). Solid lines show the proposed designed-input variants: ($\hat{\mathcal{R}}_{\mathrm{MZ}}^{\mathrm{des}}\!\mid\!\mathrm{pinv}$): MZ with pseudoinverse (red solid); ($\hat{\mathcal{R}}_{\mathrm{MZ}}^{\mathrm{des}}\!\mid\!\mathrm{SOCP}(H)$): MZ with SOCP right inverse (green); ($\hat{\mathcal{R}}_{\mathrm{CMZ}}^{\mathrm{des}}\!\mid\!\mathrm{pinv}$): CMZ with pseudoinverse (blue solid); ($\hat{\mathcal{R}}_{\mathrm{CMZ}}^{\mathrm{des}}\!\mid\!\mathrm{SOCP}(H)$): CMZ with SOCP right inverse (cyan). All designed-input variants are visibly tighter than the corresponding random-input baselines, and the combination of CMZ with SOCP($H$) yields the tightest bound overall.
• Figure E1: PWA system reachable-set comparison over 10 steps. Black filled: initial set $\mathcal{X}_0$. $\hat{\mathcal{R}}_{\mathrm{rand}}\!\mid\!\mathrm{pinv}$: random-input data-driven over-approximation (unfilled contours). $\hat{\mathcal{R}}_{\mathrm{des}}\!\mid\!\mathrm{SOCP}(H)$: designed-input data-driven over-approximation (orange filled). $\mathcal{R}_{\mathrm{PWA}}$: model-based PWA reachable set (blue filled). The guard surface $x_1 = 0$ is shown as a dashed line. Both data-driven over-approximations soundly contain the model-based set, and the designed-input variant produces a visibly tighter bound.

Theorems & Definitions (19)

• Definition 1: Zonotope girard2005reachability
• Definition 2: Constrained Zonotope scott2016constrained
• Definition 3: Matrix Zonotope amr23reachable
• Definition 4: Constrained Matrix Zonotope alanwar2022data
• Proposition 1: CMZ--Zonotope Product Over-Approximation alanwar2022data
• Definition 5: Hybrid Zonotope bird2023hybrid
• Remark 1: Constraint dimensions and effectiveness
• Lemma 1: Proxy monotonicity under CMZ constraints
• proof
• Theorem 1: Tighter Over-Approximation via Input Design and Right-Inverse Optimization