Simulation and Design of PID Position Control for a Three-Gear Torque Transmission System Using Root Locus

Abstract

This paper presents the design and simulation of a PID position control system for a three-gear torque transmission mechanism using the root locus method. Unlike many previous studies that focus on simple electromechanical plants, this work applies classical PID control to a multi-gear mechanical transmission system with load disturbance, where gear interactions influence the transient response. The system model was derived using first-principles modeling and represented by a second-order transfer function with an integrator. The uncompensated system exhibits poor transient performance, with 45.52% overshoot and a steady-state error of 0.5, failing to meet the design specifications of maximum 8% overshoot, zero steady-state error, and a settling time three times faster than the original system. A PID controller was designed using the root locus technique by placing the dominant poles at a location corresponding to a damping ratio of 0.626, resulting in controller parameters Kp = 1.989, Ki = 0.913, and Kd = 0.325. Simulation results show that the controller eliminates overshoot and achieves zero steady-state error under a 0.5 Nm load disturbance, although the settling time of 4.286 s remains longer than the desired 1.366 s. A comparison of PI, PD, and PID controllers reveals trade-offs between response speed, damping, and steady-state accuracy in multi-gear transmission systems.