On the Exponential Stability of Koopman Model Predictive Control

TL;DR

This paper establishes local exponential stability guarantees for Koopman MPC, demonstrating improved convergence and cost performance over traditional methods through theoretical analysis and experiments on an inverted pendulum.

Contribution

It provides the first explicit local exponential stability guarantees for Koopman MPC under Lipschitz conditions, extending stability analysis beyond asymptotic results.

Findings

Koopman LQR with zero modeling error is globally optimal and stabilizes the nonlinear plant.

Under Lipschitz conditions, Koopman LQR and MPC are locally exponentially stable.

Experiments show Koopman MPC outperforms Taylor-linearized MPC in convergence and cost on an inverted pendulum.

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.