Numerical Analysis of the Dynamic Behavior of Systems with PID: An Approach Using the Python Control Systems Library

Abstract

This work presents a detailed computational analysis of the behavior of PID (Proportional-Integral-Derivative) controllers through numerical simulation, investigating the individual and combined effects of the parameters Kp, Ki, and Kd on the performance of closed-loop control systems. The methodology employs the Python Control Systems Library to simulate a second-order system representative of common industrial processes, analyzing performance metrics such as overshoot, settling time, and steady-state error. The results show that increasing the proportional gain Kp reduces response time but increases overshoot, while the integral term Ki eliminates steady-state error at the cost of increased oscillation. The derivative term Kd proved effective in reducing overshoot and improving the system's relative stability. The parametric analysis revealed complex interactions among the three terms, highlighting the need for simultaneous tuning to optimize performance. The obtained results corroborate classical control theory and provide practical insights for the design of PID controllers in industrial applications, demonstrating the importance of computational simulation as a tool for analysis and design in control engineering.