An Exact Method for the Stability Analysis of Linear Consensus Protocols With Time Delay

This technical note presents a methodology for the stability analysis of linear consensus protocols with time-delayed communications. Second order agent dynamics with a fixed and undirected communication topology and uniform delays are considered. This class of group dynamics is very complex and is not fully explored to date. The proposed technique takes advantage of the general structure of the control protocols in performing a state transformation that allows a decomposition of the characteristic equation into a set of factors. These factors distribute the imprint of the delay in the characteristic equation in a much simpler form to achieve the stability analysis in parts. The procedure also prepares the characteristic equation for the deployment of the Cluster Treatment of Characteristic Roots paradigm, a recent method which declares the stability features of the system for various compositions of the time delay and other control parameters. In order to show the effectiveness of this approach, it is applied to different consensus protocols under the assumptions of fixed and undirected communication topologies and uniform communication time delays.