Dynamic consensus networks: spectral properties, consensus, and control

Abstract

The idea of consensus 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 dissertation 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. We transform each concept of static graph theory into dynamic terms, out of which a generalized dynamic graph theory naturally emerges. Three different types of dynamic consensus networks are addressed based on node dynamics and network topology. We investigate stability and consensus for each type of dynamic consensus network and develop conditions under which dynamic networks achieve consensus. For the first type, we use conditions of connectedness, positivity, and diagonal dominance to show consensus. For the second type, we use the condition of strictly-positive realness of edges. For the third type, which is composed of more general dynamics, we use graphical verification based on the relationship of the eigenvalues of the dynamic Laplacian matrix with the Nyquist plots of individual node dynamics. In order to investigate consensus using Nyquist's graphical-stability test, we develop a method to estimate the bounds of the eigenvalues of the dynamic Laplacian matrix by introducing the idea of a Dynamic Grounded Laplacian sub-matrix as a means of reducing the order of complexity of computation that would be necessary in the case of the full dynamic Laplacian matrix. The dissertation finally considers controllability and controller design for dynamic consensus networks. The ideas developed for dynamic graph theory, in conjunction with the behavioral approach, lead to the development of a controllability analysis methodology for dynamic consensus networks. We finally establish a generalized methodology for designing a controller for a dynamic consensus network in the presence of external disturbances, focusing especially on using decentralized controllers that achieve consensus in the absence of disturbances and attenuation of disturbances to a prescribed H[infinity] performance level.