Reachability Analysis With Probabilistic Zonotopes Learning Realized Disturbances and Refining Aleatory Uncertainty

Jul 16, 2026

This paper develops a data-driven reachability framework for linear systems with disturbances modeled by probabilistic zonotopes, combining bounded deterministic and Gaussian stochastic components. Instead of requiring an exact disturbance model, it refines a conservative prior using trajectory-consistency constraints from data. It distinguishes realized disturbances (shaping the data-consistent model set) from aleatory disturbances (propagating future additive uncertainty). A constrained-PZ calculus absorbs stochastic constraints into an equivalent representation, removes infeasible latent directions, and identifies fusion rules for heterogeneous descriptions. Linear-program learning produces translated and scaled prior copies that satisfy nested confidence requirements, yielding high-probability reachable sets with reduced conservatism.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages16

Reading Time40 min

Introduction

Model-based reachability and limitations

Data-driven reachability and bounded disturbances

Stochastic/probabilistic reachability methods

Uncertainty refinement and motivation for PZ-based approach

IEEE TAC, VOL. XX, NO. XX, XXXX 2017 1 Reachability Analysis With Probabilistic Zonotopes: Learning Realized Disturbances and Refining Aleatory Uncertainty Amir Modares, Zhen Zhang, Themistoklis Charalambous, Amr Alanwar, Hamidreza Modares arXiv :2607 .01352v1 [ ee ss . SY] 1 Jul 2026 Abstract—This paper develops a data-driven reachability framework for linear systems whose disturbances are modeled by probabilistic zonotopes (PZs), combining bounded deterministic and Gaussian stochastic components. In contrast to methods that require a precisely known disturbance model (either purely deterministic or purely stochastic), we assume only a conservative prior PZand refine it from data. The framework separates two uncertainty sources: realized disturbances, which act along the collected trajectory and govern the size of the dataconsistent model set, and aleatory disturbances, which enter as future additive uncertainty during reachable-set propagation; both shape the reachable sets, but through different mechanisms. Refinement exploits prior system knowledge together with trajectory-consistency constraints induced by the data, which impose affine couplings between deterministic and Gaussian latent variables. We accordingly develop a constrained-PZ calculus that absorbs the stochastic part of these constraints into an equivalent representation, removes infeasible latent directions, and reduces stochastic covariance, together with identification-aware fusion rules for combining heterogeneous constrained-PZ descriptions. The refined realizeddisturbance proxies then serve as scenarios in a linear program that learns the smallest translated and scaled copy of the prior disturbance set that contains all proxy confidence sets while remaining nested in the prior. The resulting deterministic, high-probability reachable sets carry formal containment guarantees with substantially reduced conservatism, and numerical examples confirm that the pipeline tightens both the data-consistent model set and the propagated reachable sets. I. INTRODUCTION Reachability analysis is a fundamental tool in control theory for characterizing the set of states that a dynamical system can attain under uncertainty [1] . Accurate reachable-set overapproximations are essential for safety verification [2], robust control synthesis [3], and constraint satisfaction [4], particularly in safety-critical applications such as autonomous A. Modares and T. Charalambous are with the School of Engineering, University of Cyprus, 1678 Nicosia, Cyprus. E-mails:{modarres.amir, [charalambous.themistoklis](charalambous.themistoklis}@ucy.ac.cy)[}](charalambous.themistoklis}@ucy.ac.cy)[@ucy.ac.cy](charalambous.themistoklis}@ucy.ac.cy). T. Charalambous is also a Visiting Professor at the School of Electrical Engineering, Aalto University, 02150 Espoo, Finland. Z. Zhang and A. Alanwar are with Technical University of Munich, Germany. Emails: {zhenzhang.zhang, [alanwar](alanwar}@tum.de)[}](alanwar}@tum.de)[@tum.de](alanwar}@tum.de). Hamidreza Modares is with Michigan State University, USA. E-mail: [modaresh@msu.edu](modaresh@msu.edu). systems [5], robotics [6], and cyber–physical systems [7] . The practical usefulness of reachability methods hinges on their ability to balance computational tractability with tightness of the resulting sets [8] . Despite the extensive progress on reachability computation, the majority of existing approaches are inherently modelbased. This includes Hamilton–Jacobi reachability methods and their modern variants [9], [10], as well as setpropagation approaches based on structured set representationsand toolchains [11], [12] . In practice, however, constructing a sufficiently accurate model from first principles or identifying it reliably from imperfect measurements is often challenging [13], [14] . Recent years have seen growing interest in data-driven reachability methods, which leverage measured input–state trajectories to reduce modeli

FAQs

What disturbance model does the paper use for the linear systems?

Disturbances are modeled by probabilistic zonotopes that merge bounded deterministic parts with Gaussian stochastic components.

How does the framework use data to refine uncertainty?

It starts from a conservative prior probabilistic zonotope and refines it using trajectory-consistency constraints induced by observed input–state trajectories.

What is the difference between realized disturbances and aleatory disturbances in the approach?

Realized disturbances act along the collected trajectory and determine the size of the data-consistent model set, while aleatory disturbances persist as future additive uncertainty during reachable-set propagation.

Jul 16, 2026

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