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Exploiting Over-Approximation Errors as Preview Information for Nonlinear Control

Antoine Aspeel, Antoine Girard, Thiago Alves Lima

TL;DR

The paper tackles constrained nonlinear control by reframing over-approximation error as input-dependent preview information. It defines informed policies that depend on both the state and the over-approximation error, and formulates concretization as a fixed-point problem; Brouwer’s theorem guarantees existence in general, while efficient solution methods are provided for input-affine and nonlinear cases via convex programs and contraction-based iterations, respectively. For linear over-approximations with shared input structure, explicit or LP-based solutions are available; for nonlinear dynamics, contraction conditions enable a Banach-fixed-point–based approach to converge to a valid input. Numerical experiments on both an input-affine and a nonlinear system show that informed policies substantially reduce conservatism compared to uninformed policies, highlighting the practical impact for safe, scalable nonlinear control.

Abstract

We study the control of nonlinear constrained systems via over-approximations. Our key observation is that the over-approximation error, rather than being an unknown disturbance, can be exploited as input-dependent preview information. This leads to the notion of informed policies, which depend on both the state and the error. We formulate the concretization problem -- recovering a valid input for the true system from a preview-based policy -- as a fixed-point equation. Existence of solutions follows from the Brouwer fixed-point theorem, while efficient computation is enabled through closed-form, linear, or convex programs for input-affine systems, and through an iterative method based on the Banach fixed-point theorem for nonlinear systems.

Exploiting Over-Approximation Errors as Preview Information for Nonlinear Control

TL;DR

The paper tackles constrained nonlinear control by reframing over-approximation error as input-dependent preview information. It defines informed policies that depend on both the state and the over-approximation error, and formulates concretization as a fixed-point problem; Brouwer’s theorem guarantees existence in general, while efficient solution methods are provided for input-affine and nonlinear cases via convex programs and contraction-based iterations, respectively. For linear over-approximations with shared input structure, explicit or LP-based solutions are available; for nonlinear dynamics, contraction conditions enable a Banach-fixed-point–based approach to converge to a valid input. Numerical experiments on both an input-affine and a nonlinear system show that informed policies substantially reduce conservatism compared to uninformed policies, highlighting the practical impact for safe, scalable nonlinear control.

Abstract

We study the control of nonlinear constrained systems via over-approximations. Our key observation is that the over-approximation error, rather than being an unknown disturbance, can be exploited as input-dependent preview information. This leads to the notion of informed policies, which depend on both the state and the error. We formulate the concretization problem -- recovering a valid input for the true system from a preview-based policy -- as a fixed-point equation. Existence of solutions follows from the Brouwer fixed-point theorem, while efficient computation is enabled through closed-form, linear, or convex programs for input-affine systems, and through an iterative method based on the Banach fixed-point theorem for nonlinear systems.

Paper Structure

This paper contains 18 sections, 9 theorems, 39 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Related work
  3. Contributions
  4. Notation
  5. OVER-APPROXIMATIONS
  6. INFORMED POLICIES
  7. Concretization as a fixed point
  8. Systems with shared input dependency
  9. Input affine systems
  10. Nonlinear systems
  11. COMPUTING LINEAR OVER-APPROXIMATIONS
  12. SYNTHESIZING INFORMED POLICIES
  13. System level synthesis
  14. Contractivity condition
  15. EXPERIMENTS
  16. ...and 3 more sections

Key Result

Theorem 1

Let $f,\hat{f}:\mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R}^{n_x}$ and $\pi:\mathcal{X}\times\mathcal{E}\rightarrow\mathcal{U}$ be three functions, and assume that $\mathcal{E}$ satisfies eq:errorInclusion. For any $x\in\mathcal{X}$, if Problem prob:concretization has a solution $u$, then

Figures (2)

  • Figure 1: Trajectories obtained with informed and uninformed policies for the dynamics \ref{['eq:inputAffineExperiment']}. The informed policy guarantees to be further on the right at the end of the horizon.
  • Figure 2: Trajectories obtained with informed and uninformed policies for the dynamics \ref{['eq:nonlinearExperiment']}. The informed policy guarantees to be further on the right at the end of the horizon.

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Lemma 1: Brouwer fixed-point theorem
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 2: Banach fixed-point theorem
  • ...and 5 more