Graph Topology and Time Delay Relations in Linear Second-Order Consensus Laws

In this paper, we present an analytical study on the relationship between graph topology and time delay in a family of linear second-order consensus laws. Specifically, after determining the maximum delay the system can tolerate as a function of its parameters for both directed and undirected graphs, we provide insights into the interplay between these parameters and practical guidelines for selecting appropriate gains and communication topology. Additionally, we propose several techniques to enhance system performance by adjusting weights and edges, depending on the permissible modifications within the communication graph. These adjustments aim to improve system convergence in terms of settling time, control input energy, and maximum tolerable delay. Finally, simulation results are presented to illustrate the findings of our study.