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Data-based Low-conservative Nonlinear Safe Control Learning

Amir Modares, Bahare Kiumarsi, Hamidreza Modares

Abstract

This paper develops a data-driven safe control framework for nonlinear discrete-time systems with parametric uncertainty and additive disturbances. The proposed approach constructs a data-consistent closed-loop representation that enables controller synthesis and safety certification directly from data. Unlike existing methods that treat unmodeled nonlinearities as global worst-case uncertainties using Lipschitz bounds, the proposed approach embeds nonlinear terms directly into the invariance conditions via a geometry-aware difference-of-convex formulation. This enables facet- and direction-specific convexification, avoiding both nonlinearity cancellation and the excessive conservatism induced by uniform global bounds. We further propose a vertex-dependent controller construction that enforces convexity and contractivity conditions locally on the active facets associated with each vertex, thereby enlarging the class of certifiable invariant sets. For systems subject to additive disturbances, disturbance effects are embedded directly into the verification conditions through optimized, geometry-dependent bounds, rather than via uniform margin inflation, yielding less conservative robust safety guarantees. As a result, the proposed methods can certify substantially larger safe sets, naturally accommodate joint state and input constraints, and provide data-driven safety guarantees. The simulation results show a significant improvement in both nonlinearity tolerance and the size of the certified safe set.

Data-based Low-conservative Nonlinear Safe Control Learning

Abstract

This paper develops a data-driven safe control framework for nonlinear discrete-time systems with parametric uncertainty and additive disturbances. The proposed approach constructs a data-consistent closed-loop representation that enables controller synthesis and safety certification directly from data. Unlike existing methods that treat unmodeled nonlinearities as global worst-case uncertainties using Lipschitz bounds, the proposed approach embeds nonlinear terms directly into the invariance conditions via a geometry-aware difference-of-convex formulation. This enables facet- and direction-specific convexification, avoiding both nonlinearity cancellation and the excessive conservatism induced by uniform global bounds. We further propose a vertex-dependent controller construction that enforces convexity and contractivity conditions locally on the active facets associated with each vertex, thereby enlarging the class of certifiable invariant sets. For systems subject to additive disturbances, disturbance effects are embedded directly into the verification conditions through optimized, geometry-dependent bounds, rather than via uniform margin inflation, yielding less conservative robust safety guarantees. As a result, the proposed methods can certify substantially larger safe sets, naturally accommodate joint state and input constraints, and provide data-driven safety guarantees. The simulation results show a significant improvement in both nonlinearity tolerance and the size of the certified safe set.

Paper Structure

This paper contains 19 sections, 8 theorems, 123 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Setup
  4. Data-based Representation of Closed-loop Systems
  5. Data-based Nonlinear Safe Control using Nonlinearity Minimization
  6. A Low-Complexity Data-based Nonlinear Safe Control Design
  7. Systems with no Disturbances
  8. Systems with Additive Disturbances
  9. Discussion
  10. Feasibility
  11. Computational complexity
  12. Simulation Results
  13. System Description and Data Collection
  14. System Description
  15. Data Collection
  16. ...and 4 more sections

Key Result

Lemma 1

Consider the system system with collected input--state data given by data-u and data-x--data-xx. Let the controller be given by contNew. Then, under Assumption 5, the data-based closed-loop representation of the system is given by where with $G_{K,1} \in \mathbb{R}^{T \times n}$ and $G_{K,2} \in \mathbb{R}^{T \times N}$. Moreover, under Assumption 5, the solutions $G_{K,1}$ and $G_{K,2}$ exist a

Figures (6)

  • Figure 1: Theorem 1: closed-loop state trajectories in $(x_1,x_2,x_3)$ from box-vertex initial conditions.
  • Figure 2: Theorem 3 (unstructured curvature): closed-loop trajectories in $(x_1,x_2,x_3)$ from box-vertex initial conditions.
  • Figure 3: Theorem 3 (structured curvature): closed-loop trajectories in $(x_1,x_2,x_3)$ from box-vertex initial conditions.
  • Figure 4: Control input trajectories $u(t)$ for the vertex-initialized simulations.
  • Figure 5: Theorem 3 (Face-only with certification): closed-loop trajectories in $(x_1,x_2,x_3)$ from box-vertex initial conditions.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Theorem 1
  • ...and 13 more