Data-Driven Reachability Analysis for Piecewise Affine Systems

TL;DR

This work tackles safety verification for hybrid systems by presenting a data-driven reachability framework based on hybrid zonotopes to over-approximate reachable sets of Piecewise Affine (PWA) systems directly from noisy measurements. It introduces a complete pipeline: (i) a representation of hybrid zonotopes with dedicated set operations to handle region boundaries, (ii) a data-driven construction of a family of submodel representations that cover all modes, and (iii) online set-based estimation using three equivalent measurement-update schemes (Reverse Mapping, Implicit Intersection, Generalized Intersection) to fuse input-output data with noise. The authors prove the mathematical equivalence of the RM, IN, and GI methods under key conditions and validate the approach through numerical examples showing containment of true trajectories and analyzing computational performance, including the exponential scaling with horizon length. The results advance safety verification for multi-mode cyber-physical systems where explicit models are unavailable, with practical implications for autonomous and robotic systems operating across mode boundaries. Future work aims to refine boundary coupling and extend the framework to broader classes of hybrid models.

Abstract

Hybrid systems play a crucial role in modeling real-world applications where discrete and continuous dynamics interact, including autonomous vehicles, power systems, and traffic networks. Safety verification for these systems requires determining whether system states can enter unsafe regions under given initial conditions and uncertainties, a question directly addressed by reachability analysis. However, hybrid systems present unique difficulties because their state space is divided into multiple regions with distinct dynamic models, causing traditional data-driven methods to produce inadequate over-approximations of reachable sets at region boundaries where dynamics change abruptly. This paper introduces a novel approach using hybrid zonotopes for data-driven reachability analysis of piecewise affine systems. Our method addresses the boundary transition problem by developing computational algorithms that calculate the family of set models guaranteed to contain the true system trajectories. Additionally, we extend and evaluate three methods for set-based estimation that account for input-output data with measurement noise.

Figures (6)

• Figure 1: State space partition and reachable set evolution of a PWA system. The figure illustrates the state space divided into four regions ($\mathcal{C}_1$, $\mathcal{C}_2$, $\mathcal{C}_3$, and $\mathcal{C}_4$) by diagonal dashed lines. The red curve encloses $\tilde{\mathcal{R}}_{k,1}$ which is the reachable set at time $k$, intersecting with region $\mathcal{C}_1$. The solid dark red boundary encompasses the one-step reachable set $\tilde{\mathcal{R}}_{k+1}^1$ computed from $\tilde{\mathcal{R}}_{k,1}$.
• Figure 2: Reachable sets for the benchmark PWA system computed using Algorithm \ref{['alg:pwa-reachability']}. The green dashed line indicates the guard condition ($x_1 = 0$) separating the two subsystems. Blue regions show analytically derived reachable sets $\mathcal{R}_k$, while light purple regions represent the data-driven approximations $\tilde{\mathcal{R}}_k$.
• Figure 3: The experimental results show exponential growth of computation time with increasing number of steps.
• Figure 4: State-space representation of different estimation methods at time step $k$. The true state $x(k)$ is shown with a cross marker, while different estimation approaches are represented: two zonotope approximations (Zonotope-RM, Zonotope-IN), our three proposed methods (RM, IN, GI), and traditional Kalman Filter $3\sigma$ bounds.
• Figure 5: Set-based state estimation results for the PWA system showing the true state trajectory $x(k)$ (black dashed line) and the estimated bounds from three equivalent approaches: RM, IN, and GI.

Theorems & Definitions (12)

• Definition 1: Hybrid Zonotope bird2023hybrid
• Proposition 1
• Theorem 1
• proof
• Proposition 2: Reverse-Mapping
• proof
• Proposition 3: Implicit Intersection
• proof
• Proposition 4: Generalized Intersection
• proof