Output consensus robustness and performance of first-order agents

In this article, we study consensus robustness and performance problems for continuous-time multi-agent systems. We consider first-order unstable agents coordinated by an output feedback protocol over a network subject to an unknown, uncertain constant delay. Our objectives are twofold. First, we seek to determine the largest range of delay permissible so that the agents may achieve robustly consensus despite variation of the delay length, herein referred to as the delay consensus margin. Second, we attempt to determine consensus error performance quantified under an Script capital H2$$ {\mathscr{H}}_2 $$ norm criterion, which measures the disruptive effect of random nodal noises on consensus. We consider both undirected and directed graphs. For undirected graphs, we obtain analytical results for the delay consensus margin and the consensus error performance, while for directed graphs, we develop computational results and analytical bounds. The results provide conceptual insights and exhibit how the agents' unstable pole, nonminimum phase zero, as well as the network topology and network delay may limit fundamentally the consensus robustness and performance of first-order agents.