On the Existence of Koopman Linear Embeddings for Controlled Nonlinear Systems

TL;DR

This paper characterizes when controlled nonlinear systems can be exactly represented by finite-dimensional Koopman linear models, revealing the necessary system structures and providing a symbolic method to verify such conditions.

Contribution

It establishes necessary and sufficient conditions for exact Koopman embeddings in controlled nonlinear systems and introduces a symbolic procedure to identify these conditions.

Findings

Controlled nonlinear systems must be transformable into a control-affine preserved structure.

The autonomous subsystem must admit a finite-dimensional Koopman model.

A symbolic method can determine the existence of Koopman linear embeddings.

Abstract

Koopman linear representations have become a popular tool for control design of nonlinear systems, yet it remains unclear when such representations are exact. In this paper, we establish sufficient and necessary conditions under which a controlled nonlinear system admits an exact finite-dimensional Koopman linear representation, which we term Koopman linear embedding. We show that such a system must be transformable into a special control-affine preserved (CAP) structure, which enforces affine dependence of the state on the control input and isolates all nonlinearities into an autonomous subsystem. We further prove that this autonomous subsystem must itself admit a finite-dimensional Koopman linear model with a sufficiently-rich Koopman invariant subspace. Finally, we introduce a symbolic procedure to determine whether a given controlled nonlinear system admits the CAP structure, thereby elucidating whether Koopman approximation errors arise from intrinsic system dynamics or from the choice of lifting functions.