On Koopman Resolvents and Frequency Response of Nonlinear Systems
This work addresses a foundational challenge in control theory for analyzing nonlinear systems, though it appears incremental as it builds on existing Koopman operator methods.
The paper tackles the problem of defining frequency response for nonlinear systems by proposing a novel formulation within the Koopman operator framework, generalizing classical linear time-invariant approaches and deriving it through Laplace transforms and resolvent theory.
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.