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Reachability Analysis of Nonlinear Discrete-Time Systems Using Polyhedral Relaxations and Constrained Zonotopes

Brenner S. Rego, Guilherme V. Raffo, Marco H. Terra, Joseph K. Scott

TL;DR

This work addresses the challenge of computing guaranteed reachability enclosures for nonlinear discrete-time systems with bounded uncertainty. It introduces a novel propagation framework that combines constrained zonotopes with lifted polyhedral relaxations of factorable nonlinear representations, yielding tighter enclosures than traditional linearization-based CZ methods while maintaining linear growth in complexity. The method relies on factorable representations to build a lifted polyhedron in an augmented z-space, then projects back to obtain CZ enclosures for the nonlinear image, enabling recursive reachability analysis. Numerical experiments show improved accuracy over interval arithmetic and existing CZ approaches, with favorable scalability and practical performance for nonlinear dynamics.

Abstract

This paper presents a novel algorithm for reachability analysis of nonlinear discrete-time systems. The proposed method combines constrained zonotopes (CZs) with polyhedral relaxations of factorable representations of nonlinear functions to propagate CZs through nonlinear functions, which is normally done using conservative linearization techniques. The new propagation method provides better approximations than those resulting from linearization procedures, leading to significant improvements in the computation of reachable sets in comparison to other CZ methods from the literature. Numerical examples highlight the advantages of the proposed algorithm.

Reachability Analysis of Nonlinear Discrete-Time Systems Using Polyhedral Relaxations and Constrained Zonotopes

TL;DR

This work addresses the challenge of computing guaranteed reachability enclosures for nonlinear discrete-time systems with bounded uncertainty. It introduces a novel propagation framework that combines constrained zonotopes with lifted polyhedral relaxations of factorable nonlinear representations, yielding tighter enclosures than traditional linearization-based CZ methods while maintaining linear growth in complexity. The method relies on factorable representations to build a lifted polyhedron in an augmented z-space, then projects back to obtain CZ enclosures for the nonlinear image, enabling recursive reachability analysis. Numerical experiments show improved accuracy over interval arithmetic and existing CZ approaches, with favorable scalability and practical performance for nonlinear dynamics.

Abstract

This paper presents a novel algorithm for reachability analysis of nonlinear discrete-time systems. The proposed method combines constrained zonotopes (CZs) with polyhedral relaxations of factorable representations of nonlinear functions to propagate CZs through nonlinear functions, which is normally done using conservative linearization techniques. The new propagation method provides better approximations than those resulting from linearization procedures, leading to significant improvements in the computation of reachable sets in comparison to other CZ methods from the literature. Numerical examples highlight the advantages of the proposed algorithm.

Paper Structure

This paper contains 21 sections, 1 theorem, 17 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement and Preliminaries
  3. Interval analysis
  4. Convex polytopes and constrained zonotopes
  5. Lifted Halfspace Polyhedral Enclosure of a Nonlinear Function over an Interval
  6. Arithmetic operations
  7. Sum
  8. Subtraction
  9. Multiplication
  10. Division
  11. Univariate functions
  12. Exponential
  13. Logarithm
  14. Even integer power
  15. Odd integer power
  16. ...and 6 more sections

Key Result

Proposition 1

Let $Z = ({\mathbf{G}}_z, {\mathbf{c}}_z, {\mathbf{A}}_z, {\mathbf{b}}_z)_\text{CZ} \subset \mathbb{R}^n$ be a constrained zonotope and $P \triangleq ({\mathbf{H}}_p, {\mathbf{k}}_p, {\mathbf{A}}_p, {\mathbf{b}}_p)_\text{P}$ be a convex polytope in H-rep with $n_h$ halfspaces. Then, with ${\mathbf{G}}_q \triangleq \frac{1}{2} \text{diag} ({\mathbf{k}}_p-\bm{\sigma})$, and ${\mathbf{c}}_q \triangl

Figures (2)

  • Figure 1: Enclosures for Example 1 obtained by IA (dash-dotted lines), CZMV (solid blue), Alg\ref{['alg:czreachability']} with complexity reduction (solid yellow), and Alg\ref{['alg:czreachability']} without complexity reduction (dashed lines), along with uniform samples from $\bar{X}_0$ propagated through \ref{['eq:nonlinearDC']} (dots). Panels (a)--(c) show $\bar{X}_1$ for $\alpha = 0.1$, $0.5$, $1$, respectively, while panels (d)--(f) show $\bar{X}_2$ for $\alpha = 0.1$, $0.5$, $1$, respectively. This figure has been generated using the YALMIP function plot.
  • Figure 2: The 1-norm radii of the enclosures $\bar{X}_k$ obtained by IA (dashed purple), CZMV (dash-dotted blue), CZFO (dotted red), and Alg\ref{['alg:czreachability']} (solid yellow) for Example 2.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3