Exact Delay Consensus Margin of First-Order Agents under PID Protocol

Abstract

In this paper we study robust consensus problems for continuous-time first-order multi-agent systems (MAS) connected by an undirected network. We derive analytical expressions of the delay consensus margin achievable by the PID feedback protocols; the delay consensus margin is a robustness measure that defines the maximal range of delay within which robust consensus can be achieved despite the uncertainty in the delay. The results show how the agent dynamics and graph connectivity may fundamentally limit the range of delay tolerable, so that consensus can and cannot be maintained in the presence of uncertainty in the delay. Somewhat surprisingly but yet in an intuitively consistent manner, we find that the delay consensus margin achieved by PID feedback coincides with that by PD protocols, which can be determined by solving efficiently a convex optimization problem in one variable, or alternatively, a unimodal optimization problem.