PID Controller Design Based on the Stabilization and Bifurcation of a Desired Equilibrium for a Delayed Complex System with a Variable Parameter

This paper proposes a design solution of PID (Proportional-Integral-Derivative) bifurcation control for controlling the stabilization and bifurcation of a desired equilibrium on a delayed complex system with a variable parameter. Firstly, we stabilize the controlled system to the desired equilibrium by the function of the integrating element of the controller. Afterwards, we can achieve some conditions on the stability and Hopf bifurcation of the controlled system by means of linearizing the system at an ideal state and discussing the associated characteristic equation. Meanwhile, it is found that the regulation of controller is very effective for optimizing the performance of dynamics on the system. Finally, we set up several sets of experiments for validating the validity of theoretical analysis. In addition, we also provide some simulation diagrams, which can illustrate the relationship curve between various parameters and the bifurcation point on the controlled system.