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Stability and Performance Analysis of Model Predictive Control of Uncertain Linear Systems

Changrui Liu, Shengling Shi, Bart De Schutter

TL;DR

This work addresses stability and performance guarantees for certainty-equivalent MPC applied to uncertain discrete-time LTI systems with input constraints, where model mismatch is quantified by the Frobenius norm. It leverages relaxed dynamic programming to bound the suboptimality gap between the nominal MPC using an estimated model and the true infinite-horizon optimal controller, and derives a horizon-dependent condition under which the closed-loop system is stable. A key result is an upper bound of the nominal MPC value $\hat{V}_N(x)$ in terms of the true optimal value $V_\infty(x)$ plus error terms that depend on $(\delta_A,\delta_B)$ and the horizon $N$, along with an RDP-based energy-decrease inequality that yields stability and a global performance bound. Numerical simulations on an unstable system illustrate how modeling error and horizon length interact to affect the bounds and actual performance, providing practical guidelines for identification accuracy and horizon design, and the work suggests extensions to terminal costs and online learning-based MPC.

Abstract

Model mismatch often poses challenges in model-based controller design. This paper investigates model predictive control (MPC) of uncertain linear systems with input constraints, focusing on stability and closed-loop infinite-horizon performance. The uncertainty arises from a parametric mismatch between the true and the estimated system under the matrix Frobenius norm. We examine a simple MPC controller that exclusively uses the estimated system model and establishes sufficient conditions under which the MPC controller can stabilize the true system. Moreover, we derive a theoretical performance bound based on relaxed dynamic programming, elucidating the impact of prediction horizon and modeling errors on the suboptimality gap between the MPC controller and the Oracle infinite-horizon optimal controller with knowledge of the true system. Simulations of a numerical example validate the theoretical results. Our theoretical analysis offers guidelines for obtaining the desired modeling accuracy and choosing a proper prediction horizon to develop certainty-equivalent MPC controllers for uncertain linear systems.

Stability and Performance Analysis of Model Predictive Control of Uncertain Linear Systems

TL;DR

This work addresses stability and performance guarantees for certainty-equivalent MPC applied to uncertain discrete-time LTI systems with input constraints, where model mismatch is quantified by the Frobenius norm. It leverages relaxed dynamic programming to bound the suboptimality gap between the nominal MPC using an estimated model and the true infinite-horizon optimal controller, and derives a horizon-dependent condition under which the closed-loop system is stable. A key result is an upper bound of the nominal MPC value in terms of the true optimal value plus error terms that depend on and the horizon , along with an RDP-based energy-decrease inequality that yields stability and a global performance bound. Numerical simulations on an unstable system illustrate how modeling error and horizon length interact to affect the bounds and actual performance, providing practical guidelines for identification accuracy and horizon design, and the work suggests extensions to terminal costs and online learning-based MPC.

Abstract

Model mismatch often poses challenges in model-based controller design. This paper investigates model predictive control (MPC) of uncertain linear systems with input constraints, focusing on stability and closed-loop infinite-horizon performance. The uncertainty arises from a parametric mismatch between the true and the estimated system under the matrix Frobenius norm. We examine a simple MPC controller that exclusively uses the estimated system model and establishes sufficient conditions under which the MPC controller can stabilize the true system. Moreover, we derive a theoretical performance bound based on relaxed dynamic programming, elucidating the impact of prediction horizon and modeling errors on the suboptimality gap between the MPC controller and the Oracle infinite-horizon optimal controller with knowledge of the true system. Simulations of a numerical example validate the theoretical results. Our theoretical analysis offers guidelines for obtaining the desired modeling accuracy and choosing a proper prediction horizon to develop certainty-equivalent MPC controllers for uncertain linear systems.
Paper Structure (21 sections, 23 theorems, 108 equations, 3 figures)

This paper contains 21 sections, 23 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. System Description & Definitions
  5. Problem Formulation
  6. Theoretical Analysis
  7. Evaluation of the MPC value function
  8. Closed-loop Stability and Performance Analysis
  9. Numerical Example
  10. Conclusions
  11. Matrix Inequalities
  12. Technical Lemmas
  13. Proof of Lemma \ref{['lm:local_stabilization_property']}
  14. Linear bound on square root function
  15. Difference between the optimal inputs
  16. ...and 6 more sections

Key Result

Corollary 1

Under Assumption ass:initial_controllability, $\mathcal{X}_{\mathrm{ROA}}$ is a control invariant set, i.e., for all $x \in \mathcal{X}_{\mathrm{ROA}}$, there exists $u \in \mathcal{U}$ such that $Ax + Bu \in \mathcal{X}_{\mathrm{ROA}}$.

Figures (3)

  • Figure 1: The behavior of $\alpha_N$ (upper left), $\beta_N$ (upper right), $\xi_N$ (lower left), and $J_{\text{bound}}$ (lower right) for a varying modeling error $\delta \in [10^{-3}, 10^{-2}]$. For a given error level $\delta$, $100$ estimated systems are simulated, and the mean, variance, and max (min) of each of the four quantities are shown, respectively, using a solid line, dashed lines, and dotted lines.
  • Figure 2: The behavior of $\alpha_N$ (upper left), $\beta_N$ (upper right), $\xi_N$ (lower left), and $J_{\text{bound}}$ (lower right) for a varying prediction horizon $N \in \{6,7,8,9,10\}$. For the specified $\delta = 5\cdot 10^{-3}$, $100$ estimated systems are simulated, and the mean, variance, and max (min) of each of the four quantities are shown, respectively, using a solid line, dashed lines, and dotted lines.
  • Figure 3: The behavior of the true performance as a function of the modeling error (left) and prediction horizon (right). For a given error level $\delta$, $100$ estimated systems are simulated, and the mean, variance, and max (min) of each of the four quantities are shown, respectively, using a solid line, dashed lines, and dotted lines.

Theorems & Definitions (44)

  • Definition 1: Error-consistent function
  • Corollary 1
  • Remark 1: Nullified input at stage $N$
  • Remark 2: State constraints & region of attraction
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • ...and 34 more