Koopman Operator Dynamical Models: Learning, Analysis and Control

TL;DR

This paper reviews and unifies data-driven methods for approximating Koopman operators, emphasizing their theoretical foundations, relation to control systems, and future challenges in nonlinear dynamical modeling.

Contribution

It provides a comprehensive review and categorization of Koopman operator approximation methods, connecting theory with practical modeling and control applications.

Findings

Categorizes existing Koopman approximation methods.

Highlights the connection between Koopman theory and system control.

Discusses challenges and future directions in the field.

Abstract

The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigm's relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.