Consensus and Eigenvalue Estimation in Dynamic Consensus Networks with Strictly Positive Real Systems

Abstract

The use of consensus networks in networking has received great attention due to its wide array of applications in fields such as robotics, transportation, sensor networking, communication networking, biology, and physics. The focus of this paper is to study a generalization of consensus problems whereby the weights of network edges are no longer static gains, but instead are dynamic systems, leading to the notion of dynamic consensus networks. 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 transform each concept of static graph theory into dynamic terms, out of which a generalized dynamic graph theory naturally emerges. We present a framework for dynamic graphs and dynamic consensus networks. This framework introduces the idea of dynamic degree, adjacency, incident, and Laplacian matrices in a way that naturally extends these concepts from the static case. 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.