Analysis of the Effects of PID Parameters on the Deflection Angle of a Frusta Pendulum

Abstract

The Frusta pendulum is a representative nonlinear system widely used as a benchmark for 
evaluating control algorithms in automation and robotics. Among various control strategies, 
the Proportional–Integral–Derivative (PID) controller remains one of the most widely adopted 
owing to its simple structure and satisfactory industrial performance. However, the individual 
influence of the proportional, integral, and derivative gains (Kip, Kid, and Kid) on the dynamic 
response of the Frusta pendulum has not been systematically quantified. This paper presents a 
quantitative investigation of these effects on the pendulum angular deviation α through 
mathematical modeling and MATLAB/Semolina simulations. The results show that Kip 
primarily determines response speed but increases overshoot when excessively large; Kid 
eliminates steady-state error but may prolong settling time and degrade stability if improperly 
tuned; and Kid suppresses oscillations and improves transient performance, though excessive 
values reduce response speed and increase noise sensitivity. The optimized PID parameters 
enable a short settling time, low steady-state error, and reduced oscillations. Compared with 
the Linear Quadratic Regulator (LQR), the PID controller offers greater implementation 
simplicity and ease of tuning, whereas LQR provides smoother responses and superior 
stability at the cost of requiring an accurate model and more sophisticated design. This work 
establishes a systematic quantitative framework for evaluating individual PID parameters, 
identifies an effective parameter set, and provides practical guidelines for controller tuning in 
balancing systems such as self-balancing robots.