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Data-Driven Tube-Based Zonotopic Predictive Control With Nonconvex Layered Terminal Sets

Zhen Zhang, Bogdan Gheorghe, Florin Stoican, Amr Alanwar

Abstract

This paper presents a data-driven tube-based zonotopic predictive control (DTZPC) framework with nonconvex layered terminal sets. Existing DTZPC schemes with closed-loop guarantees typically rely on a single ellipsoidal terminal set, which can be conservative and thereby limit feasibility. We propose a layered terminal-set design that decouples stability certification, feasibility enlargement, and motion-region screening into three components with distinct roles. First, an offline-designed feedback gain together with a contractive constrained zonotope provides a terminal ingredient for stability certification, while avoiding probabilistic feedback synthesis in high-dimensional DTZPC. Second, we derive a data-driven characterization of the inverse admissible closed-loop model set, avoiding the conservatism of interval-matrix relaxation and inversion. Combined with exact set multiplication, this yields inner and outer approximations of the maximal robust positively invariant (MRPI) set under fixed closed-loop dynamics. The inner approximation serves as a nonconvex terminal set to enlarge feasibility, whereas the outer approximation provides certified motion-region descriptions for fast screening and monitoring. Numerical examples demonstrate tighter inverse-set enclosures and improved feasibility over existing convex-terminal DTZPC schemes.

Data-Driven Tube-Based Zonotopic Predictive Control With Nonconvex Layered Terminal Sets

Abstract

This paper presents a data-driven tube-based zonotopic predictive control (DTZPC) framework with nonconvex layered terminal sets. Existing DTZPC schemes with closed-loop guarantees typically rely on a single ellipsoidal terminal set, which can be conservative and thereby limit feasibility. We propose a layered terminal-set design that decouples stability certification, feasibility enlargement, and motion-region screening into three components with distinct roles. First, an offline-designed feedback gain together with a contractive constrained zonotope provides a terminal ingredient for stability certification, while avoiding probabilistic feedback synthesis in high-dimensional DTZPC. Second, we derive a data-driven characterization of the inverse admissible closed-loop model set, avoiding the conservatism of interval-matrix relaxation and inversion. Combined with exact set multiplication, this yields inner and outer approximations of the maximal robust positively invariant (MRPI) set under fixed closed-loop dynamics. The inner approximation serves as a nonconvex terminal set to enlarge feasibility, whereas the outer approximation provides certified motion-region descriptions for fast screening and monitoring. Numerical examples demonstrate tighter inverse-set enclosures and improved feasibility over existing convex-terminal DTZPC schemes.

Paper Structure

This paper contains 14 sections, 9 theorems, 33 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. Set Representations and Operations
  5. Problem Statement
  6. Data-Driven Layered Terminal Set Construction
  7. Local Contractive Set and Feedback Synthesis
  8. Computation of an Over-Approximation of the MRPI Set
  9. Scenario-Based Computation of an Inner Approximation of the MRPI Set
  10. DTZPC Formulation and Closed-Loop Guarantees
  11. Numerical simulations
  12. Inverse-Set Tightness and Layered Terminal Sets
  13. Building Thermal Control: Case Study and Comparison
  14. Conclusion

Key Result

Lemma 1

Given a CPMZ ${\mathcal{Y}} = \langle C_{\mathcal{Y}}, G_{\mathcal{Y}}, E_{\mathcal{Y}}, A_{\mathcal{Y}}, B_{\mathcal{Y}}, R_{\mathcal{Y}}, \mathsf{id}_{\mathcal{Y}} \rangle_\text{CPMZ}\subset \mathbb{R}^{n_x \times n }$ and a CPZ $\mathcal{P} = \langle c_{\mathcal{P}}, G_{\mathcal{P}}, {E}_{\mathca where $\mathcal{Y}\otimes \mathcal{P} \subset \mathbb{R}^{n_x}$ and with $\overline{E}_\mathcal{Y}

Figures (2)

  • Figure D1: Inverse closed-loop model enclosures and layered terminal sets. (a)--(b) Projections of inverse closed-loop model enclosures on two different coordinate planes. (c) Phase portrait and projection of the layered terminal sets.
  • Figure D2: Building thermal control results: (a) terminal-set comparison; (b) closed-loop indoor temperature trajectory; (c) optimal control input trajectory. The sampling time is 4 min.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 1: Exact multiplication zhang2025data
  • Lemma 2
  • Definition 3
  • Definition 4: Controlled $\lambda$-contractive set blanchini2008set
  • Lemma 3
  • Lemma 4
  • Remark 1: Justification of Assumption \ref{['ass:invertibility']}
  • Lemma 5
  • ...and 12 more