Koopman-based lifting techniques for nonlinear systems identification

TL;DR

This paper introduces a Koopman-based lifting technique for nonlinear systems identification that linearizes the problem in an infinite-dimensional space, enabling accurate modeling from low-sampling data without derivatives.

Contribution

It presents two novel numerical schemes for Koopman operator approximation, including a parametric method for broad systems and a dual method for high-dimensional data with limited samples.

Findings

Main method accurately reconstructs vector fields including chaotic systems.

Dual method effectively estimates vector fields with small datasets in high dimensions.

Theoretical convergence results support the proposed techniques.

Abstract

We develop a novel lifting technique for nonlinear system identification based on the framework of the Koopman operator. The key idea is to identify the linear (infinitedimensional) Koopman operator in the lifted space of observables, instead of identifying the nonlinear system in the state space, a process which results in a linear method for nonlinear systems identification. The proposed lifting technique is an indirect method that does not require to compute time derivatives and is therefore well-suited to low-sampling rate datasets. Considering different finite-dimensional subspaces to approximate and identify the Koopman operator, we propose two numerical schemes: the main method and the dual method. The main method is a parametric identification technique that can accurately reconstruct the vector field of a broad class of systems (including unstable, chaotic, and system with inputs). The dual method provides estimates of the vector field at the data points and is well-suited to identify high-dimensional systems with small datasets. The present paper describes the two methods, provide theoretical convergence results, and illustrate the lifting techniques with several examples.