Role of Integral Control for Enlarging Second-Order Delay Consensus Margin Under PID Protocols: None

Proportional, integral, and derivative (PID) feedback control, as a popular control law, plays a central role in industrial processes and traditional control applications. In the context of multiagent systems, one may also wonder what the fundamental capability and limitation of PID control may be. This article attempts to provide an answer from the viewpoint of consensus robustness against uncertain delay. We consider robust consensus of second-order unstable agents under PID feedback protocols, subject to a constant but unknown time delay over an undirected graph. The issue concerns the so-called delay consensus margin (DCM), which is the largest delay range within which robust consensus can be achieved. The specific problem under study investigates the role of integral control on the robust consensus, seeking to understand whether integral control can be employed to enhance consensus robustness. Our result shows that there is none; that is, in a PID protocol, the integral control action has no improving effect on the DCM, and that PID and proportional-derivative (PD) protocols achieve the same DCM. As a byproduct of this finding, the DCM under PID and PD protocols is found to be computable by solving a quasiconcave, albeit nonsmooth, unimodal optimization problem.