Consensus Performance of First-Order Agents

This article concerns consensus problems of linear time-invariant systems with stochastic noises and deterministic disturbances. We consider first-order dynamic agents interconnected by a directed graph network subject to interagent time delay, and we seek to determine the error performance achievable by the consensus feedback protocol, whereas the performance quantifies the disruption of consensus by noises and disturbances. The
$\mathcal {H}_{2}$
and
$\mathcal {H}_\infty$
norms of the multiagent system transfer function matrices are employed as measures of the consensus error. For the
$\mathcal {H}_{2}$
consensus performance, we obtain an analytical expression of the consensus error, while for the
$\mathcal {H}_\infty$
consensus performance, we show that it can be determined by solving a sequence of independent unimodal quasi-convex problems. The results help demonstrate how, in the presence of stochastic noises and deterministic disturbances, the agent's unstable dynamics may interact over a delayed network to limit the consensus performance attainable.