Delay Consensus Margin of First-Order Multiagent Systems With Undirected Graphs and PD Protocols

In this article, we study robust consensus problems for continuous-time first-order multiagent systems (MASs) with time delays. We assume that the agents input is subject to an uncertain constant delay, which may arise due to interagent communication or additionally, also by self-delay in the agent dynamics. We consider dynamic feedback control protocol in the form of proportional and derivative (PD) control, and seek to determine the delay consensus margin (DCM) achievable by PD feedback protocols, whereas the DCM is a robustness measure that defines the maximal range of delay within which robust consensus can be achieved despite the uncertainty in the delay. With an undirected graph, we show that the DCM can be determined exactly by solving a unimodal concave optimization problem, which is one of univariate convex optimization and can be solved using convex optimization or gradient-based numerical methods. The results show how unstable agent dynamics and graph connectivity may limit the range of delay tolerable, so that consensus can or cannot be maintained in the presence of delay variations.