Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Exponential Stability of Koopman Model Predictive Control

Xu Shang, Jorge Cortés, Yang Zheng

TL;DR

The paper addresses the challenge of guaranteeing exponential stability for Koopman MPC when controlling constrained nonlinear systems. It derives local exponential stability for the Koopman-LQR controller under Lipschitz lifting and one-step prediction error, then designs terminal ingredients in the lifted space to enable a stabilizing Koopman MPC (S-KMPC). The authors prove local exponential stability for S-KMPC under small modeling error, and demonstrate through an inverted pendulum example that S-KMPC achieves faster convergence and lower accumulated cost than Taylor-linearized MPC. This work provides explicit stability guarantees for data-driven Koopman-based MPC and informs terminal-setup design in lifted coordinates, with practical implications for scalable nonlinear control.

Abstract

Koopman Model Predictive Control (MPC) uses a lifted linear predictor to efficiently handle constrained nonlinear systems. While constraint satisfaction and (practical) asymptotic stability have been studied, explicit guarantees of local exponential stability seem to be missing. This paper revisits the exponential stability for Koopman MPC. We first analyze a Koopman LQR problem and show that 1) with zero modeling error, the lifted LQR policy is globally optimal and globally asymptotically stabilizes the nonlinear plant, and 2) with the lifting function and one-step prediction error both Lipschitz at the origin, the closed-loop system is locally exponentially stable. These results facilitate terminal cost/set design in the lifted Koopman space. Leveraging linear-MPC properties (boundedness, value decrease, recursive feasibility), we then prove local exponential stability for a stabilizing Koopman MPC under the same conditions as Koopman LQR. Experiments on an inverted pendulum show better convergence performance and lower accumulated cost than the traditional Taylor-linearized MPC approaches.

On the Exponential Stability of Koopman Model Predictive Control

TL;DR

The paper addresses the challenge of guaranteeing exponential stability for Koopman MPC when controlling constrained nonlinear systems. It derives local exponential stability for the Koopman-LQR controller under Lipschitz lifting and one-step prediction error, then designs terminal ingredients in the lifted space to enable a stabilizing Koopman MPC (S-KMPC). The authors prove local exponential stability for S-KMPC under small modeling error, and demonstrate through an inverted pendulum example that S-KMPC achieves faster convergence and lower accumulated cost than Taylor-linearized MPC. This work provides explicit stability guarantees for data-driven Koopman-based MPC and informs terminal-setup design in lifted coordinates, with practical implications for scalable nonlinear control.

Abstract

Koopman Model Predictive Control (MPC) uses a lifted linear predictor to efficiently handle constrained nonlinear systems. While constraint satisfaction and (practical) asymptotic stability have been studied, explicit guarantees of local exponential stability seem to be missing. This paper revisits the exponential stability for Koopman MPC. We first analyze a Koopman LQR problem and show that 1) with zero modeling error, the lifted LQR policy is globally optimal and globally asymptotically stabilizes the nonlinear plant, and 2) with the lifting function and one-step prediction error both Lipschitz at the origin, the closed-loop system is locally exponentially stable. These results facilitate terminal cost/set design in the lifted Koopman space. Leveraging linear-MPC properties (boundedness, value decrease, recursive feasibility), we then prove local exponential stability for a stabilizing Koopman MPC under the same conditions as Koopman LQR. Experiments on an inverted pendulum show better convergence performance and lower accumulated cost than the traditional Taylor-linearized MPC approaches.

Paper Structure

This paper contains 25 sections, 11 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Exponential stability
  4. Nonlinear MPC basics and Koopman linearization
  5. Problem statement
  6. Infinite-horizon Koopman LQR
  7. Koopman optimal control
  8. Local exponential stability of Koopman LQR
  9. Stabilizing Koopman MPC
  10. Design of the terminal ingredients
  11. Asymptotic stability for an exact Koopman model
  12. Local exponential stability of stabilizing Koopman MPC
  13. Numerical experiments
  14. Experiment setup
  15. Closed-loop performance
  16. ...and 10 more sections

Key Result

Lemma 1

rawlings2017model Consider the autonomous system $x_{t+1}=g(x_t)$ with $g(\mathbb{0})=\mathbb{0}$. Suppose there exist constants $\alpha_1,\alpha_2,\alpha_3>0$, an invariant set $\mathcal{D}\subseteq\mathbb R^n$ with $\mathbb{0} \in \mathrm{int}(D)$, and a function $V: \mathcal{D} \to \mathbb{R}_{+} Then, the system is locally exponentially stable, and $\mathcal{D}$ is a region of attraction (ROA)

Figures (2)

  • Figure 1: LQR performance: exact Koopman model vs. first-order (Taylor) linearization. Left: closed-loop trajectory; right: accumulated cost.
  • Figure 2: MPC performance for inverted pendulum: approximated Koopman model vs. first-order (Taylor) linearization. Left: closed-loop trajectory; right: accumulated cost.

Theorems & Definitions (18)

  • Definition 1: Local exponential stability
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Example 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 3
  • ...and 8 more