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zonoLAB: A MATLAB toolbox for set-based control systems analysis using hybrid zonotopes

Justin Koeln, Trevor J. Bird, Jacob Siefert, Justin Ruths, Herschel Pangborn, Neera Jain

TL;DR

zonoLAB addresses the need for scalable, expressive set-based analysis in dynamic systems by leveraging hybrid zonotopes to exact-represent reachability, MPC policies, and neural-network mappings within a MATLAB toolbox. The approach unifies zonotopes, constrained zonotopes, and hybrid zonotopes to enable convenient set operations, set-based mappings, and open-/closed-loop analysis for MLD, PWA, MPC, and nonlinear systems. Key contributions include formal definitions of the three representations, conversion utilities, plotting capabilities, and a set-mapping framework, plus demonstrations for MLD/PWA, MPC, neural networks, and nonlinear dynamics. This work facilitates rigorous safety verification and controller design with a practical, open-source tool that can be extended to more activation functions, nonlinear bounds, and complexity-reduction techniques.

Abstract

This paper introduces zonoLAB, a MATLAB-based toolbox for set-based control system analysis using the hybrid zonotope set representation. Hybrid zonotopes have proven to be an expressive set representation that can exactly represent the reachable sets of mixed-logical dynamical systems and tightly approximate the reachable sets of nonlinear dynamic systems. Moreover, hybrid zonotopes can exactly represent the continuous piecewise linear control laws associated with model predictive control and the input-output mappings of neural networks with piecewise linear activation functions. The hybrid zonotope set representation is also highly exploitable, where efficient methods developed for mixed-integer linear programming can be directly used for set operation and analysis. The zonoLAB toolbox is designed to make these capabilities accessible to the dynamic systems and controls community, with functionality spanning fundamental operations with hybrid zonotope, constrained zonotope, and zonotope set representations, powerful set analysis tools, and general-purpose algorithms for reachability analysis of open- and closed-loop systems.

zonoLAB: A MATLAB toolbox for set-based control systems analysis using hybrid zonotopes

TL;DR

zonoLAB addresses the need for scalable, expressive set-based analysis in dynamic systems by leveraging hybrid zonotopes to exact-represent reachability, MPC policies, and neural-network mappings within a MATLAB toolbox. The approach unifies zonotopes, constrained zonotopes, and hybrid zonotopes to enable convenient set operations, set-based mappings, and open-/closed-loop analysis for MLD, PWA, MPC, and nonlinear systems. Key contributions include formal definitions of the three representations, conversion utilities, plotting capabilities, and a set-mapping framework, plus demonstrations for MLD/PWA, MPC, neural networks, and nonlinear dynamics. This work facilitates rigorous safety verification and controller design with a practical, open-source tool that can be extended to more activation functions, nonlinear bounds, and complexity-reduction techniques.

Abstract

This paper introduces zonoLAB, a MATLAB-based toolbox for set-based control system analysis using the hybrid zonotope set representation. Hybrid zonotopes have proven to be an expressive set representation that can exactly represent the reachable sets of mixed-logical dynamical systems and tightly approximate the reachable sets of nonlinear dynamic systems. Moreover, hybrid zonotopes can exactly represent the continuous piecewise linear control laws associated with model predictive control and the input-output mappings of neural networks with piecewise linear activation functions. The hybrid zonotope set representation is also highly exploitable, where efficient methods developed for mixed-integer linear programming can be directly used for set operation and analysis. The zonoLAB toolbox is designed to make these capabilities accessible to the dynamic systems and controls community, with functionality spanning fundamental operations with hybrid zonotope, constrained zonotope, and zonotope set representations, powerful set analysis tools, and general-purpose algorithms for reachability analysis of open- and closed-loop systems.
Paper Structure (15 sections, 16 equations, 4 figures, 2 tables)

This paper contains 15 sections, 16 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Set Representations and Basic Operations
  3. Zonotopic Set Representations
  4. Zonotopic Set Operations
  5. The zonoLAB Toolbox
  6. Toolbox Structure
  7. Set Definitions and Conversions
  8. Plotting
  9. Set-based Mappings
  10. Reachability Analysis Examples
  11. MLD and PWA System Analysis
  12. Closed-loop MPC Analysis
  13. Neural Network Representation
  14. Nonlinear System Analysis
  15. Conclusions

Figures (4)

  • Figure 1: Examples of how the constrained factor spaces (a), (c), and (e) are projected to create a zonotope (b), constrained zonotope (d), and hybrid zonotope (f), respectively, generated by the file provided with zonoLAB.
  • Figure 2: Reachable sets for the two-equilibrium example system from Bird2023Automatica, generated by the file provided with zonoLAB.
  • Figure 3: Hybrid zonotope representation of a neural network example from Ortiz2023 generated by the file provided with zonoLAB.
  • Figure 4: Hybrid zonotope representations of a nonlinear function (a), a piecewise linear control law (b), and the resulting closed-loop reachable sets generated by the file provided with zonoLAB.

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3