Koopman Performance Analysis of Nonlinear Consensus Networks

TL;DR

This paper uses Koopman operator theory to analyze the spectral properties of nonlinear consensus networks, providing a new framework to evaluate their performance based on graph topology.

Contribution

It introduces a spectral decomposition approach for nonlinear networks using Koopman theory, linking nonlinear dynamics to linear spectral analysis for performance assessment.

Findings

Spectral representation enables performance evaluation based on Koopman eigenvalues.

Develops a scalable computational method for Koopman mode decomposition.

Connects network topology with systemic performance measures.

Abstract

Spectral decomposition of dynamical systems is a popular methodology to investigate the fundamental qualitative and quantitative properties of these systems and their solutions. In this chapter, we consider a class of nonlinear cooperative protocols, which consist of multiple agents that are coupled together via an undirected state-dependent graph. We develop a representation of the system solution by decomposing the nonlinear system utilizing ideas from the Koopman operator theory and its spectral analysis. We use recent results on the extensions of the well-known Hartman theorem for hyperbolic systems to establish a connection between the original nonlinear dynamics and the linearized dynamics in terms of Koopman spectral properties. The expected value of the output energy of the nonlinear protocol, which is related to the notions of coherence and robustness in dynamical networks, is evaluated and characterized in terms of Koopman eigenvalues, eigenfunctions, and modes. Spectral representation of the performance measure enables us to develop algorithmic methods to assess the performance of this class of nonlinear dynamical networks as a function of their graph topology. Finally, we propose a scalable computational method for approximation of the components of the Koopman mode decomposition, which is necessary to evaluate the systemic performance measure of the nonlinear dynamic network.