On Koopman Resolvents and Frequency Response of Nonlinear Systems

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.

💡 Research Summary

This paper introduces a rigorous frequency‑response concept for continuous‑time nonlinear systems by exploiting the Koopman operator framework and its resolvent theory. Classical linear time‑invariant (LTI) systems admit a frequency response defined as the transfer function G(iω), which can be visualized with Bode plots. For nonlinear systems, existing approaches such as harmonic balance, describing functions, Volterra series, and other time‑domain techniques lack a systematic frequency‑domain formulation. The authors address this gap by representing a nonlinear plant

  ẋ = F(x, u), y = g(x, u)

as a linear (but infinite‑dimensional) evolution on a space of observables via the Koopman semigroup Kₜ f = f ∘ Sₜ, where Sₜ is the flow of the autonomous dynamics ẋ = F₀(x) (with u = 0). The infinitesimal generator L of this semigroup satisfies L f = F₀·∇ₓ f, and its resolvent R(s; L) = (sI − L)⁻¹ exists for complex s in the resolvent set ρ(L).

To capture the effect of a sinusoidal forcing, the authors embed the input u(t) = u₀e^{iωt} into the state vector, forming a skew‑product system

  ẋ = F(x, u),  ẋᵤ = iω u,  y = g(x, u).

The Koopman generator for this augmented system is

  L_forced = F(x, u)·∇ₓ + (iω u) ∂/∂u.

If the observable space contains the monomials uⁿ (for integer n) or u^{1/n} (for subharmonic responses), then i nω and i ω/n belong to the point spectrum σₚ(L_forced) with eigenfunctions ϕ_{i nω}=uⁿ and ϕ_{i ω/n}=u^{1/n}, respectively.

The paper defines two types of frequency response: (i) fundamental and harmonic responses, where a steady‑state periodic output yₙ(t) = Hₙ(ω) (u₀e^{iωt})ⁿ exists, and (ii) subharmonic responses, where y_{1/n}(t) = H_{1/n}(ω) (u₀e^{iωt})^{1/n}. The Laplace transform of the output for the forced system can be expressed as

  ĥy(s; x₀, u₀) =