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

TL;DR

This paper introduces a new concept of participation factors for nonlinear dynamical systems within the Koopman operator framework, enabling analysis beyond linear systems and providing a numerical estimation method from time series data.

Contribution

It extends the concept of participation factors to nonlinear systems using Koopman operators and develops a numerical method for their estimation from data.

Findings

Participation factors are defined for the entire domain of attraction in nonlinear systems.

The method leverages dynamic mode decomposition for numerical estimation.

Participation factors generalize the linear case to nonlinear dynamics.

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.