Nonparametric Control Koopman Operators

TL;DR

This paper introduces a nonparametric Koopman operator framework in RKHSs for control systems, enabling accurate, scalable, and finite-dimensional models without explicit dictionaries, and demonstrates improved prediction and control capabilities.

Contribution

It develops a novel, dictionary-free Koopman operator representation in RKHSs, unifying control operator learning with infinite-dimensional regression, and enhances scalability with sketching techniques.

Findings

Superior prediction accuracy over bilinear EDMD in high dimensions

Enables finite-dimensional predictors without predefining function spans

Facilitates integration with linear-parameter-varying control methods

Abstract

This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to the well-understood learning theory in RKHSs under one unified framework. Consequently, our proposed framework allows for arbitrarily accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without apriori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.