Koopman performance analysis of a class of nonlinear dynamical networks

This paper builds upon the Koopman spectral analysis tools to develop a method for assessment of the performance of a class of first-order nonlinear consensus networks. This class of networks is defined over an interconnected graph with state-dependent weights that are nonlinear functions of the state of the network. The mean energy of the output of the system with respect to random initial conditions is utilized as the performance measure. We quantify this performance measure in terms of the Koopman eigenfunctions of the nonlinear dynamics, and the eigenvalues of the corresponding linearized system at the equilibrium of the network, where the eigenvalues of the linearized system are indeed Laplacian eigenvalues of the underlying graph with opposite sign. Our results reveal that the performance measure of the nonlinear network depends on the interconnection topology of the underlying graph. We illustrate effectiveness of our results using several examples, including a Cucker-Smale type consensus network and first-order network of identical Kuramoto oscillators.