Generalized Optimal and Explicit PI/PID Tuning Formulas for Underdamped Second-order Systems

This study aims to propose generalized formulas in an explicit form to optimally tune the gains of the proportional-integral (PI) and proportional-integral-derivative (PID) controllers of an underdamped second-order system. Unlike other methods used in literature, the controller gains are tuned explicitly based on the known system parameters of the second-order system, namely: the damping ratio and the natural frequency. The proposed accurate and one-step formulas do not need any further effort to effectively control any given second-order system while keeping the desired performance indices within near-optimal specifications. The proposed work relies on a restricted deadbeat response criterion and cascaded surface curve fitting procedure to end by near-optimal controller gains tuning formulas. A particle swarm optimization (PSO) method is employed to optimize the primary tuned gains for wide-range of system frequency. The proposed formulas are tested and simulated to demonstrate their effectiveness with comparisons with the closest common related gains tuning methods. Based on the comprehensive analysis, the proposed work ensures 2% maximum percentage overshoot when the proposed formulas are adopted to tune the gains of the PID controllers. Different comparisons are carried out to show the effectiveness of the proposed work.