Analytic Time Domain Specifications PID Controller Design for a Class of 2^ndOrder Linear Systems: A Genetic Algorithm Method

In this paper, an analytical approach of a Proportional-Integral-Derivative (PID) controller design for a servo motor position control with precise desired performance demands is presented. When designing linear controllers, the resulting closed-loop system is often considered as an approximation of a standard $2^{\mathrm {nd}}$ order system to simplify the gain design process. However, in reality, model discrepancies could exist in such approximation, which inevitably lead to performance mismatches between the actual system's behavior and the desired one. In order to solve this issue, this paper presents an analytical PID controller solution, which is able to achieve desired time domain specifications precisely. In addition, a disturbance is added to the system and the associated PID controller will be designed to reject disturbance. Given a $2^{\mathrm {nd}}$ order linear system along with a PID controller, the related analytical solutions are derived first. Next, the associated fitness functions and constraints are constructed under given prescribed time domain specifications. Lastly, the PID control gains are determined using Genetic Algorithm. To demonstrate the effectiveness of the proposed solution, many numerical simulations are performed. A set of control gains is also found using the standard $2^{\mathrm {nd}}$ order system formulas in order to prove the accuracy of the presented method. Moreover, a comparison study with respect to the MATLAB PID Tuner is considered to illustrate the advantage of the proposed PID design scheme. The main contributions of this study include precisely satisfying given control performance demands with exact analytical solutions, avoiding the existing model discrepancies between the standard $2^{\mathrm {nd}}$ order system and the exact closed-loop model. Simulations firmly prove that the proposed method can fulfill users' different control demands as well as remove the frequently used trial-and-error strategy.