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Data-Driven Nonconvex Reachability Analysis using Exact Set Propagation

Zhen Zhang, M. Umar B. Niazi, Michelle S. Chong, Karl H. Johansson, Amr Alanwar

Abstract

This paper studies deterministic data-driven reachability analysis for dynamical systems with unknown dynamics and nonconvex reachable sets. Existing deterministic data-driven approaches typically employ zonotopic set representations, for which the multiplication between a zonotopic model set and a zonotopic state set cannot be represented algebraically exactly, thereby necessitating over-approximation steps in reachable-set propagation. To remove this structural source of conservatism, we introduce constrained polynomial matrix zonotopes (CPMZs) to represent data-consistent model sets, and show that the multiplication between a CPMZ model set and a constrained polynomial zonotope (CPZ) state set admits an algebraically exact CPZ representation. This property enables set propagation entirely within the CPZ representation, thereby avoiding propagation-induced over-approximation and even retaining the ability to represent nonconvex reachable sets. Moreover, we develop set-theoretic results that enable the intersection of data-consistent model sets as new data become available, yielding the proposed online refinement scheme that progressively tightens the data-consistent model set and, in turn, the resulting reachable set. Beyond linear systems, we extend the proposed framework to polynomial dynamics and develop additional set-theoretic results that enable both model-based and data-driven reachability analysis within the same algebraic representation. By deriving algebraically exact CPZ representations for monomials and their compositions, reachable-set propagation can be carried out directly at the set level without resorting to interval arithmetic or relaxation-based bounding techniques. Numerical examples for both linear and polynomial systems demonstrate a significant reduction in conservatism compared to state-of-the-art deterministic data-driven reachability methods.

Data-Driven Nonconvex Reachability Analysis using Exact Set Propagation

Abstract

This paper studies deterministic data-driven reachability analysis for dynamical systems with unknown dynamics and nonconvex reachable sets. Existing deterministic data-driven approaches typically employ zonotopic set representations, for which the multiplication between a zonotopic model set and a zonotopic state set cannot be represented algebraically exactly, thereby necessitating over-approximation steps in reachable-set propagation. To remove this structural source of conservatism, we introduce constrained polynomial matrix zonotopes (CPMZs) to represent data-consistent model sets, and show that the multiplication between a CPMZ model set and a constrained polynomial zonotope (CPZ) state set admits an algebraically exact CPZ representation. This property enables set propagation entirely within the CPZ representation, thereby avoiding propagation-induced over-approximation and even retaining the ability to represent nonconvex reachable sets. Moreover, we develop set-theoretic results that enable the intersection of data-consistent model sets as new data become available, yielding the proposed online refinement scheme that progressively tightens the data-consistent model set and, in turn, the resulting reachable set. Beyond linear systems, we extend the proposed framework to polynomial dynamics and develop additional set-theoretic results that enable both model-based and data-driven reachability analysis within the same algebraic representation. By deriving algebraically exact CPZ representations for monomials and their compositions, reachable-set propagation can be carried out directly at the set level without resorting to interval arithmetic or relaxation-based bounding techniques. Numerical examples for both linear and polynomial systems demonstrate a significant reduction in conservatism compared to state-of-the-art deterministic data-driven reachability methods.

Paper Structure

This paper contains 18 sections, 11 theorems, 78 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. Problem Statement
  5. Set Representations
  6. Assumptions
  7. Data-driven Reachability Analysis with Nonconvex Set Representations
  8. Set of Models
  9. Intersection Between Sets of models
  10. Iterative Refinement of Set of Models
  11. Exact Multiplication for Non-Convex Reachability
  12. Model Based and Data-Driven Reachability for Polynomial Systems
  13. Model-based reachability analysis
  14. Data-driven Reachability Analysis for Polynomial System
  15. Numerical simulations
  16. ...and 3 more sections

Key Result

Proposition 1

Given two CPZs the mergeID operator returns two adjusted CPZs that are equivalent to $\mathcal{P}_1$ and $\mathcal{P}_2$: with $\overline{\mathsf{id}}= $, $\mathcal{H} = \left\{ i~ |~ \mathsf{id}_{2}^{(i)} \not\in \mathsf{id}_{1} \right\}$, and where $i = 1, \dots, a$ with $a = |\mathcal{H}|+p_{1}$ for $\mathsf{id}_{1} \in \mathbb{N}^{1 \times p_{1} }$. $\lrcorner$$\blacktriangleleft$$\blacktr

Figures (5)

  • Figure C1: Data-driven reachability analysis with model refinement for LTI system. Legend: $\otimes$ denotes the exact multiplication operation and $\boxplus$ denotes the exact addition operation.
  • Figure C2: Illustration of the proof of Proposition \ref{['prop:multi']}, where the dashed lines indicate the boundary of PMZ and PZs, respectively. Let $\textrm{c}(\mathcal{Y})$ and $\textrm{c}(\mathcal{P})$ denote the constraints associated with CPMZ $\mathcal{Y}$ and CPZ $\mathcal{P}$, respectively. The exact multiplication of $\mathcal{Y}:=\{\textrm{PMZ}:\textrm{c}(\mathcal{Y}) \textrm{ hold}\}$ with $\mathcal{P}:=\{\textrm{PZ}:\textrm{c}(\mathcal{P}) \textrm{ hold}\}$ is a CPZ $\mathcal{Y}\otimes \mathcal{P}=\{\textrm{PZ}:\textrm{c}(\mathcal{Y}\otimes \mathcal{P}) \textrm{ hold}\}$, with constraints $\textrm{c}(\mathcal{Y}\otimes \mathcal{P}):= \textrm{Proj}_{\mathbb{R}^{n}} c(\mathcal{Y})$ and $c(\mathcal{P})$, where $\textrm{Proj}_{\mathbb{R}^{n}} c(\mathcal{Y})$ is the projection of $c(\mathcal{Y})$ from $\mathbb{R}^{n_x\times n}$ to $\mathbb{R}^{n}$.
  • Figure D1: Projection of Reachable Sets Computed from Input--State Data Using Algorithm \ref{['alg:offline-online']} with a Nonconvex Initial Set
  • Figure D2: Comparative Projection of Reachable Sets from Input--State Data Using Algorithm \ref{['alg:offline-online']} and Alanwar et al. Alanwar2023Datadriven
  • Figure D3: Projections of reachable sets for a polynomial system. (a) Model-based reachable sets computed using Algorithm \ref{['alg:PolyReachability']}, compared with the model-based reachability analysis results obtained using CORA althoff2015introduction, for both convex and nonconvex initial sets, together with Monte Carlo simulation trajectories. (b) Data-driven reachable sets computed using Algorithm \ref{['alg:DDPolyReachability_update_intersect']} for a convex initial set under a small disturbance magnitude, compared with the interval-based data-driven method of Alanwar et al. Alanwar2023Datadriven. (c) Data-driven reachable sets for a nonconvex initial set under a larger disturbance magnitude, illustrating the effect of online set refinement.

Theorems & Definitions (35)

  • Definition 1
  • Remark 1: Notion of exactness
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Definition 5
  • Proposition 1: mergeID kochdumper2020sparse
  • Example 2
  • Lemma 1
  • ...and 25 more