Estimation of Dynamic Laplacian Eigenvalues in Dynamic Consensus Networks

In this paper we generalize recent results on networks with static weights by introducing the idea of dynamic networks with real rational weights. Specifically, we consider networks whose nodes are transfer functions (typically integrators) and whose edges are strictly positive real transfer functions representing dynamical systems that couple the nodes. We show that strictly positive realness of the edges is a sufficient condition for dynamic networks to be stable (i.e., to reach consensus). To study the spectral properties of dynamic networks, we introduce the Dynamic Grounded Laplacian matrix, which is used to estimate lower and upper bounds for the real parts of the smallest and largest non-zero eigenvalues of the dynamic Laplacian matrix. These bounds can be used to obtain stability conditions using the Nyquist graphical stability test for undirected dynamic networks controlled using distributed controllers. Numerical simulations are provided to verify the effectiveness of the results.