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On Koopman Resolvents and Frequency Response of Nonlinear Systems

Yoshihiko Susuki, Natsuki Katayama, Alexandre Mauroy, Igor Mezić

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.

On Koopman Resolvents and Frequency Response of Nonlinear Systems

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.
Paper Structure (17 sections, 6 theorems, 36 equations, 1 figure)

This paper contains 17 sections, 6 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System model
  4. Brief introduction to Koopman resolvents
  5. Formulation of frequency response
  6. The skew-product form of the forced nonlinear plant
  7. Main proposal
  8. Solvable examples
  9. One-dimensional linear case
  10. Two-dimensional nonlinear case
  11. Spectral characterizations
  12. LTI dynamics
  13. Globally stable dynamics
  14. Ergodic dynamics
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Lemma 1

If the mapping $(x,u)\mapsto u^n$$(n\in\mathbb{N})$ belongs to $\mathscr{F}$, then ${\rm i} n\omega$ belongs to the point spectrum $\sigma_{\rm p}(\mathcal{L}_{\rm forced})$ of the Koopman generator $\mathcal{L}_{\rm forced}$. The associated Koopman eigenfunction $\phi_{{\rm i} n\omega}(x,u)$ is giv

Figures (1)

  • Figure 1: Bode plots for frequency responses of the 2d nonlinear example: blue for $H_1(\omega; x_2)$ and orange for $H_2(\omega; x_1)$.

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • ...and 14 more