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Control Invariant Sets for Neural Network Dynamical Systems and Recursive Feasibility in Model Predictive Control

Xiao Li, Tianhao Wei, Changliu Liu, Anouck Girard, Ilya Kolmanovsky

TL;DR

The paper addresses safety guarantees for neural-network-based dynamical systems by offline synthesizing a control invariant set (CIS) and integrating it into model predictive control (MPC) to ensure forward invariance and recursive feasibility. It combines state-space quantization, reachability analysis, and mixed-integer linear constraints to produce a CIS offline and to encode online control within a CIS-centered MPC, demonstrated in a lane-keeping scenario. The key contributions are (i) a finite-step CIS synthesis framework with provable termination, (ii) an MILC/MIQP-based MPC that preserves safety and recursive feasibility, and (iii) practical validation showing real-time feasibility and robust safety during lane keeping. The results highlight the practical impact of combining CIS theory with NNDS modeling to enable safe, data-driven control in safety-critical applications.

Abstract

Neural networks are powerful tools for data-driven modeling of complex dynamical systems, enhancing predictive capability for control applications. However, their inherent nonlinearity and black-box nature challenge control designs that prioritize rigorous safety and recursive feasibility guarantees. This paper presents algorithmic methods for synthesizing control invariant sets specifically tailored to neural network based dynamical models. These algorithms employ set recursion, ensuring termination after a finite number of iterations and generating subsets in which closed-loop dynamics are forward invariant, thus guaranteeing perpetual operational safety. Additionally, we propose model predictive control designs that integrate these control invariant sets into mixed-integer optimization, with guaranteed adherence to safety constraints and recursive feasibility at the computational level. We also present a comprehensive theoretical analysis examining the properties and guarantees of the proposed methods. Numerical simulations in an autonomous driving scenario demonstrate the methods' effectiveness in synthesizing control-invariant sets offline and implementing model predictive control online, ensuring safety and recursive feasibility.

Control Invariant Sets for Neural Network Dynamical Systems and Recursive Feasibility in Model Predictive Control

TL;DR

The paper addresses safety guarantees for neural-network-based dynamical systems by offline synthesizing a control invariant set (CIS) and integrating it into model predictive control (MPC) to ensure forward invariance and recursive feasibility. It combines state-space quantization, reachability analysis, and mixed-integer linear constraints to produce a CIS offline and to encode online control within a CIS-centered MPC, demonstrated in a lane-keeping scenario. The key contributions are (i) a finite-step CIS synthesis framework with provable termination, (ii) an MILC/MIQP-based MPC that preserves safety and recursive feasibility, and (iii) practical validation showing real-time feasibility and robust safety during lane keeping. The results highlight the practical impact of combining CIS theory with NNDS modeling to enable safe, data-driven control in safety-critical applications.

Abstract

Neural networks are powerful tools for data-driven modeling of complex dynamical systems, enhancing predictive capability for control applications. However, their inherent nonlinearity and black-box nature challenge control designs that prioritize rigorous safety and recursive feasibility guarantees. This paper presents algorithmic methods for synthesizing control invariant sets specifically tailored to neural network based dynamical models. These algorithms employ set recursion, ensuring termination after a finite number of iterations and generating subsets in which closed-loop dynamics are forward invariant, thus guaranteeing perpetual operational safety. Additionally, we propose model predictive control designs that integrate these control invariant sets into mixed-integer optimization, with guaranteed adherence to safety constraints and recursive feasibility at the computational level. We also present a comprehensive theoretical analysis examining the properties and guarantees of the proposed methods. Numerical simulations in an autonomous driving scenario demonstrate the methods' effectiveness in synthesizing control-invariant sets offline and implementing model predictive control online, ensuring safety and recursive feasibility.
Paper Structure (20 sections, 19 theorems, 61 equations, 6 figures, 3 algorithms)

This paper contains 20 sections, 19 theorems, 61 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem
  3. Preliminaries
  4. Closed-Loop Neural Network Dynamical System
  5. Control Invariant Set Synthesis
  6. Algorithmic Setup and State Space Quantization
  7. Problem Reduction and Numerical Algorithms
  8. Sub-problems: Reachability Analysis and Decision-Variable-Dependent Set Propagation
  9. Theoretical Properties
  10. Reachability Analysis
  11. MILC-OP and Decision-Variable-Dependent Set Propagation
  12. CIS Synthesis Process and Finite Determination
  13. Model Predictive Control
  14. Model Predictive Control and Recursive Feasibility
  15. Computational Aspects: Receding-Horizon Scheme and Warm Start
  16. ...and 5 more sections

Key Result

Proposition III.1

Consider the SSQ process, for all $\mathcal{S}\in \mathcal{P}_{\cup}(\mathcal{X}_{\Delta})$, there exists a unique set of indices $\mathcal{J}=\Delta(\mathcal{S}) \subseteq [n_\Delta]$ such that $\mathcal{S} = \cup_{j\in\mathcal{J}} \mathcal{X}_{\Delta}^{(j)}$, namely, where $\Delta(\cdot): \mathcal{P}_{\cup}(\mathcal{X}_{\Delta}) \to \mathbb{N}$ extracts the indices of the basis hyperboxes that

Figures (6)

  • Figure 1: A schematic diagram of a closed-loop NNDS.
  • Figure 2: A schematic diagram of the proposed numerical algorithms for synthesizing a CIS: Algorithm \ref{['al:synthesisInvSet']} executes a set recursion that generates a sequence of $i$-Step Admissible Subsets and terminates when reaching a fixed point. The set recursion is advanced by Algorithm \ref{['al:synthesisUnderQ']}, which computes a One-Step Returnable Subset $\mathcal{Q}(\tilde{\mathcal{A}}_i(\mathcal{X}_s), \tilde{\mathcal{A}}_i(\mathcal{X}_s))$ of the $i$-Step Admissible Subset $\tilde{\mathcal{A}}_i(\mathcal{X}_s)$. Algorithm \ref{['al:isReturnable']} supports this process by verifying the one-step returnability of hyperboxes in the set $\tilde{\mathcal{A}}_i(\mathcal{X}_s)$ to itself.
  • Figure 3: A lane-keeping problem: a) A vehicle traveling at a constant speed is controlled through steering to ensure it remains within the lane boundaries. b) The safe set $\mathcal{X}_s$ is approximated from within using a hyperbox, denoted as $\mathcal{X}_s^\Delta$, in the $y-(y + l_1\theta)$ plane.
  • Figure 4: The CIS synthesis process is conducted with $d_{\min} = (w - l_2)/32$ m and $u_{\max} = 5^\circ$. On the left, the proposed method computes $(i+1)$-Step Admissible sets $\tilde{\mathcal{A}}_{i+1}(\mathcal{X}_s)$ (represented by unions of red cubes) using $i$-Step Admissible sets $\tilde{\mathcal{A}}_{i}(\mathcal{X}_s)$ (represented by unions of blue cubes), ultimately producing a nested sequence of $i$-Step Admissible sets. The iterative process from iteration 1 to 6 is visualized, and the process terminates at iteration 27, generating a CIS represented by unions of purple cubes. On the right, as a byproduct of this process, we obtain a control law $u_k = \pi(x_k ; \mathcal{C})$ according to Remark \ref{['rmk:u_cis']}. This control law is visualized in a heatmap, empirically demonstrating how it renders the trajectories (red dashed lines) of the closed-loop NNDS $f$ forward invariant with respect to $\mathcal{C}$.
  • Figure 5: Synthesizing CISs with different configurations of the SSQ resolution $d_{\min}$ in the SSQ process and steering limits $u_{\max}$. Similarly, control laws $u_k = \pi(x_k ; \mathcal{C})$ are visualized in heatmaps, with trajectories of the closed-loop NNDS $f$ represented by red dashed lines.
  • ...and 1 more figures

Theorems & Definitions (61)

  • Definition 1: Forward Invariant Set blanchini1999set
  • Definition 2: Control Invariant Set blanchini1999set
  • Remark II.1
  • Definition 3: One-Step Returnable Set blanchini1994ultimate
  • Definition 4: One-Step Returnable Subset
  • Definition 5: $i$-Step Admissible Set dorea1999b
  • Definition 6: $i$-Step Admissible Subset
  • Remark II.2
  • Definition 7: SSQ Process
  • Remark III.1
  • ...and 51 more