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On the accuracy of the model predictive control method

Georgi Angelov, Alberto Domínguez Corella, Vladimir Veliov

TL;DR

The paper analyzes the online accuracy of Model Predictive Control for finite-horizon Bolza problems with uncertain parameters, introducing a two-metric extension of strong metric sub-regularity via an optimality map. By bounding the MPC-generated trajectory against the ideal reference solution, the authors show error bounds that depend on the averaged step errors, with a linear rate when the switching set is empty and a sqrt-rate correction plus discretization term otherwise. The results are specialized to coercive and affine bang-bang problems, with a numerical spacecraft-stabilization example demonstrating sharpness and practical robustness of the MPC scheme under perturbations. This work provides a rigorous, general framework for assessing MPC accuracy beyond coercive cases and offers insights for designing robust online optimal-control algorithms.

Abstract

The paper investigates the accuracy of the Model Predictive Control (MPC) method for finding online approximate optimal feedback control for Bolza type problems on a fixed finite horizon. The predictions for the dynamics, the state measurements, and the solution of the auxiliary open-loop control problems that appear at every step of the MPC method may be inaccurate. The main result provides an error estimate of the MPC-generated solution compared with the optimal open-loop solution of the ``ideal'' problem, where all predictions and measurements are exact. The technique of proving the estimate involves an extension of the notion of strong metric sub-regularity of set-valued maps and utilization of a specific new metric in the control space, which makes the proof non-standard. The result is specialized for two problem classes: coercive problems, and affine problems.

On the accuracy of the model predictive control method

TL;DR

The paper analyzes the online accuracy of Model Predictive Control for finite-horizon Bolza problems with uncertain parameters, introducing a two-metric extension of strong metric sub-regularity via an optimality map. By bounding the MPC-generated trajectory against the ideal reference solution, the authors show error bounds that depend on the averaged step errors, with a linear rate when the switching set is empty and a sqrt-rate correction plus discretization term otherwise. The results are specialized to coercive and affine bang-bang problems, with a numerical spacecraft-stabilization example demonstrating sharpness and practical robustness of the MPC scheme under perturbations. This work provides a rigorous, general framework for assessing MPC accuracy beyond coercive cases and offers insights for designing robust online optimal-control algorithms.

Abstract

The paper investigates the accuracy of the Model Predictive Control (MPC) method for finding online approximate optimal feedback control for Bolza type problems on a fixed finite horizon. The predictions for the dynamics, the state measurements, and the solution of the auxiliary open-loop control problems that appear at every step of the MPC method may be inaccurate. The main result provides an error estimate of the MPC-generated solution compared with the optimal open-loop solution of the ``ideal'' problem, where all predictions and measurements are exact. The technique of proving the estimate involves an extension of the notion of strong metric sub-regularity of set-valued maps and utilization of a specific new metric in the control space, which makes the proof non-standard. The result is specialized for two problem classes: coercive problems, and affine problems.

Paper Structure

This paper contains 12 sections, 6 theorems, 96 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strong sub-regularity
  4. The optimality map
  5. The accuracy of the model predictive control method
  6. The model predictive control method
  7. The main theorem
  8. Sufficient conditions for strong sub-regularity of the optimality map
  9. The case of coercive problems
  10. The case of affine problems with bang-bang optimal controls
  11. A numerical example
  12. Proof of the main theorem

Key Result

Proposition 2.2

Assume that $\mathcal{Z}$ is a linear space and $d_\mathcal{Z}$ is a shift-invariant metric in $\mathcal{Z}$. Assume, in addition, there exists a number $\gamma > 0$ such that $d(y_1,y_2) \leq \gamma d^*(y_1,y_2)$ for every $y_1, y_2 \in \mathcal{Y}$. Let $\Phi : \mathcal{Y} \rightrightarrows \mat Then for every function $\varphi : \mathcal{Y} \to \mathcal{Z}$ that satisfies the conditions the

Figures (1)

  • Figure 1: Red: optimal open-loop control and trajectory; Blue and Black: MPC generated solutions with N=160 and N=960, correspondingly.

Theorems & Definitions (13)

  • Remark 1.1
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • ...and 3 more