On Koopman Resolvents and Frequency Response of Nonlinear Systems

TL;DR

This paper introduces a new way to analyze the frequency response of nonlinear systems using Koopman operator theory, generalizing classical linear methods.

Contribution

It formulates a frequency response framework for nonlinear systems within the Koopman operator approach, extending classical Laplace transform methods.

Findings

Derivation of frequency response via Laplace transform of nonlinear plant output

Introduction of Bode plots for nonlinear systems based on Koopman resolvent

Conditions for the existence of frequency response in three classes of nonlinear dynamics

Abstract

This paper proposes a novel formulation of frequency response for nonlinear systems in the Koopman operator framework. This framework is a promising direction for the analysis and synthesis of systems with nonlinear dynamics based on (linear) Koopman operators. We show that the frequency response of a nonlinear plant is derived through the Laplace transform of the output of the plant, which is a generalization of the classical approach to LTI plants and is guided by the resolvent theory of Koopman operators. The response is a complex-valued function of the driving angular frequency, allowing one to draw the so-called Bode plots, which display the gain and phase characteristics. Sufficient conditions for the existence of the frequency response are presented for three classes of dynamics.