Multi-objective parametric design of PI/PID controllers via multi-level game-theoretic optimization for systems with time delay

In this research study, a new methodology of multi-objective optimal design is presented for control systems with time delay. Stackelberg method as an approach of game theory is implemented to develop the multi-objective optimization of proportional–integral (PI) and proportional–integral–derivative (PID) controllers. The surface area of control effort and tracking error are put in order of precedence together as the more important conflicting objective functions of the players in Stackelberg game design. Besides, the desirability of other effective criteria including frequency and time domains are measured as constraints in the procedure of the game-theoretic design. The main purpose of the proposed method is to optimally set and formulate controller parameters in the Stackelberg game-control system (SG-CS) design. Mapping the leader’s variables into the optimized values of the follower’s variables obtained via differential evolution (DE), the effectiveness of the proposed method is demonstrated in constructing a simple and explicit approximate model for rational reaction sets (RRSs). These obtained optimal models for the RRS allow the user to access the set of optimal controller parameters in the entire accessible strategic space of design variables in a formulated form. Utilizing the optimal synthesis for two case studies is validated the superiority of SG-CS design in establishing the explicit relationship between the controller’s parameters.