Data-driven optimal control of unknown nonlinear dynamical systems using the Koopman operator

TL;DR

This paper introduces a theoretically certifiable, data-driven framework combining a modified Koopman operator with reinforcement learning to achieve stabilizing control of unknown high-dimensional nonlinear systems, with proven convergence and low error.

Contribution

It develops a novel approach that relaxes observable function requirements and uses neural networks to solve PDEs, enabling control of systems up to 9 dimensions with convergence guarantees.

Findings

Achieves stabilizing control for systems up to 9 dimensions.

Learned value functions and control laws converge to true system solutions.

Control cost errors are within $10^{-5}$ to $10^{-3}$.

Abstract

Nonlinear optimal control is vital for numerous applications but remains challenging for unknown systems due to the difficulties in accurately modelling dynamics and handling computational demands, particularly in high-dimensional settings. This work develops a theoretically certifiable framework that integrates a modified Koopman operator approach with model-based reinforcement learning to address these challenges. By relaxing the requirements on observable functions, our method incorporates nonlinear terms involving both states and control inputs, significantly enhancing system identification accuracy. Moreover, by leveraging the power of neural networks to solve partial differential equations (PDEs), our approach is able to achieving stabilizing control for high-dimensional dynamical systems, up to 9-dimensional. The learned value function and control laws are proven to converge to those of the true system at each iteration. Additionally, the accumulated cost of the learned control closely approximates that of the true system, with errors ranging from $10^{-5}$ to $10^{-3}$.