Delay Effect on First-Order Consensus over Directed Graphs: Optimizing PID Protocols for Maximal Robustness
Dan Ma, Jianqi Chen, Renquan Lu, Jie Chen, Tianyou Chai
Abstract
In this paper we study delay robustness of multi-agent systems (MASs) connected via directed graphs. We consider first-order agents, and we assume that the agents' input is subject to an uncertain constant delay, which may arise due to interagent communication or, additionally, by self-delay in the agent dynamics. We consider dynamic output feedback control protocol in the form of PID control and seek to determine the delay consensus margin (DCM) achievable by PID feedback protocols. The DCM, which generally poses a nonsmooth max-min problem, is a robustness measure that defines the maximal range of delay within which robust consensus can be achieved despite the variation and uncertainty in the delay. We show that the DCM achievable by PID protocols coincides with that by PD protocols and can be computed by solving convex optimization problems. Specifically, unlike with an undirected graph, we show that the DCM with a directed graph can be computed approximately via an iterative algorithm in which each step amounts to solving a quasi-concave optimization problem. The results show how unstable agent dynamics and graph connectivity may fundamentally limit the tolerable range of delay, so that consensus can or cannot be maintained in the presence of delay variations.