Interplay between performance and communication delay in noisy linear consensus networks

This work investigates performance of noisy time-delayed linear consensus networks from a graph topological point of view. Performance of the network is measured by the square of the H2-norm of the system. The focus of this paper is on noisy consensus networks with homogeneous time delays affecting both the agent and all its neighbors. We derive an exact expression for the performance measure of the network in terms of time delay parameter and Laplacian eigenvalues of the underlying graph of the network. It is shown that the performance measure is a convex and Schur-convex function of Laplacian eigenvalues. We characterize the network topology with optimal performance. Furthermore, we quantify a fundamental limit on the best achievable performance based on performance of the optimal topology.