Bifurcation control for the collective behavior of delayed-coupled agents via PD controller

How to design control algorithms to effectively regulate the collective behavior with sensing delay remains a formidable challenge. In this paper, we propose to use bifurcation control with PD controller to regulate the collective behavior under sensing delay. The controlled system exhibits the emergence of three distinct states. Bifurcation analysis reveals that the Hopf bifurcation and pitchfork bifurcation curves in the parameter space can be effectively manipulated by selecting the appropriate parameters, thereby enabling state transitions within the system. Furthermore, our findings suggest that the PD controller can influence the convergence rate towards the desired system state. Theoretical properties of these three states, in relation to physical parameters, are theoretically derived, and particle simulations are conducted to validate these theoretical findings.