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Participation Factors for Nonlinear Autonomous Dynamical Systems in the Koopman Operator Framework

Kenji Takamichi, Yoshihiko Susuki, Marcos Netto

TL;DR

This work develops globally defined modal participation factors (PFs) and generalized PFs (GPs) for nonlinear dynamical systems using the Koopman operator formalism. By reinterpret­ing classical LTI PFs through variational dynamics and leveraging Koopman eigenfunctions and Koopman Mode Decomposition (KMD), it defines PFs/GPs that are valid over the entire basin of attraction for systems with a stable equilibrium or limit cycle. A data-driven numerical method based on Dynamic Mode Decomposition is proposed to estimate PFs from time-series without requiring explicit eigenfunction computation, with extensions to higher-order modes. Two 2D nonlinear examples (an EP and a LC) illustrate analytic PF/GP structures and demonstrate the method's accuracy and practical feasibility, highlighting potential applications in stability analysis and control of nonlinear power systems.

Abstract

We devise a novel formulation and propose the concept of modal participation factors to nonlinear dynamical systems. The original definition of modal participation factors (or simply participation factors) provides a simple yet effective metric. It finds use in theory and practice, quantifying the interplay between states and modes of oscillation in a linear time-invariant (LTI) system. In this paper, with the Koopman operator framework, we present the results of participation factors for nonlinear dynamical systems with an asymptotically stable equilibrium point or limit cycle. We show that participation factors are defined for the entire domain of attraction, beyond the vicinity of an attractor, where the original definition of participation factors for LTI systems is a special case. Finally, we develop a numerical method to estimate participation factors using time series data from the underlying nonlinear dynamical system. The numerical method can be implemented by leveraging a well-established numerical scheme in the Koopman operator framework called dynamic mode decomposition.

Participation Factors for Nonlinear Autonomous Dynamical Systems in the Koopman Operator Framework

TL;DR

This work develops globally defined modal participation factors (PFs) and generalized PFs (GPs) for nonlinear dynamical systems using the Koopman operator formalism. By reinterpret­ing classical LTI PFs through variational dynamics and leveraging Koopman eigenfunctions and Koopman Mode Decomposition (KMD), it defines PFs/GPs that are valid over the entire basin of attraction for systems with a stable equilibrium or limit cycle. A data-driven numerical method based on Dynamic Mode Decomposition is proposed to estimate PFs from time-series without requiring explicit eigenfunction computation, with extensions to higher-order modes. Two 2D nonlinear examples (an EP and a LC) illustrate analytic PF/GP structures and demonstrate the method's accuracy and practical feasibility, highlighting potential applications in stability analysis and control of nonlinear power systems.

Abstract

We devise a novel formulation and propose the concept of modal participation factors to nonlinear dynamical systems. The original definition of modal participation factors (or simply participation factors) provides a simple yet effective metric. It finds use in theory and practice, quantifying the interplay between states and modes of oscillation in a linear time-invariant (LTI) system. In this paper, with the Koopman operator framework, we present the results of participation factors for nonlinear dynamical systems with an asymptotically stable equilibrium point or limit cycle. We show that participation factors are defined for the entire domain of attraction, beyond the vicinity of an attractor, where the original definition of participation factors for LTI systems is a special case. Finally, we develop a numerical method to estimate participation factors using time series data from the underlying nonlinear dynamical system. The numerical method can be implemented by leveraging a well-established numerical scheme in the Koopman operator framework called dynamic mode decomposition.
Paper Structure (19 sections, 8 theorems, 80 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 19 sections, 8 theorems, 80 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Linear Systems
  4. Nonlinear Systems
  5. Summarized theory of Koopman operators
  6. Main results
  7. Illustrative Examples
  8. Nonlinear system with stable equilibrium point
  9. Nonlinear system with stable limit cycle
  10. Numerical Method
  11. Proposed algorithm
  12. Demonstration
  13. Nonlinear system \ref{['EX1_System']}
  14. Nonlinear system \ref{['EX2_System']}
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 1

Consider the LTI system LTI with the PFs ${P_{j}^{k}}$ and mode-in-state GP $P_{j}^{k(\ell)}$ from Definitions def_MinSPF_Linear to def_SinMPF_Linear. For a given initial change $\delta{\bf x}\in\mathbb{R}^n$, the variational dynamics in xk_vari and zk_vari of the state ($x_k$) and the modal ($z_j$) where $\delta z_j^0$ is the initial change to the $j$-th mode given in eqn:delta-z_j_LTI.

Figures (8)

  • Figure 1: $P^{1}_1(r,\theta)$
  • Figure 2: $P^1_{{\langle 11\rangle}}(r,\theta)$
  • Figure 3: $P^{1(2)}_1(r,\theta)$
  • Figure 5: Computational result of the mode-in-state GP $\hat{P}_{1}^{1(2)}{({\bf x}^0_{n})}$ for the nonlinear system \ref{['EX1_System']} with the stable equilibrium point. The initial change $\Delta$ is set to $10^{-6}$. The result is consistent with the analytical form $\frac{2+4\sqrt{2}}{7}x_2$.
  • Figure 6: Result of $\hat{P}_{1}^{1}(r^{0}_{n},\theta^{0}_{n})$
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Remark 4
  • Definition 4
  • Remark 5
  • Remark 6
  • ...and 28 more