Finite Dimensional Koopman Form of Polynomial Nonlinear Systems

TL;DR

This paper introduces a systematic method to compute finite-dimensional Koopman embeddings for certain polynomial nonlinear systems, enabling exact linear representations without approximation.

Contribution

It provides the first constructive approach for finite Koopman representations of polynomial systems, addressing a key gap in the theory.

Findings

Exact finite-dimensional Koopman embeddings are achievable for specific polynomial systems.

The method eliminates the need for ad-hoc observable selection and approximation errors.

The approach enhances the practical applicability of Koopman-based analysis.

Abstract

The Koopman framework is a popular approach to transform a finite dimensional nonlinear system into an infinite dimensional, but linear model through a lifting process, using so-called observable functions. While there is an extensive theory on infinite dimensional representations in the operator sense, there are few constructive results on how to select the observables to realize them. When it comes to the possibility of finite Koopman representations, which are highly important form a practical point of view, there is no constructive theory. Hence, in practice, often a data-based method and ad-hoc choice of the observable functions is used. When truncating to a finite number of basis, there is also no clear indication of the introduced approximation error. In this paper, we propose a systematic method to compute the finite dimensional Koopman embedding of a specific class of polynomial nonlinear systems in continuous-time such that, the embedding, without approximation, can fully represent the dynamics of the nonlinear system.