Koopman Operator Dynamical Models: Learning, Analysis and Control
arXiv:2102.02522 · doi:10.1016/j.arcontrol.2021.09.002
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.
This is an authors' version of the work that is published in Annual Reviews in Control journal. Changes were made to this version by the publisher prior to publication
References in corpus (7)
- Applied Koopmanism
- Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
- Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator
- Renormalization Group Functional Equations
- Data-driven feedback stabilization of nonlinear systems: Koopman-based model predictive control
- Koopman principle eigenfunctions and linearization of diffeomorphisms
- A Solution to Schroeder's Equation in Several Variables