Modeling, Simulation, Design and Comparative Analysis of the PID and LQR Controllers for an Inverted Pendulum

The inverted pendulum is a classic nonlinear unstable system used in control theory for testing control methods. In this article a nonlinear and a linearized model is obtained to design a PID and LQR controller. The MATLAB®/Simulink® software is used for linear and nonlinear model simulation and controller design. WEBOTS software is used for implementation and for comparative analysis between the controllers performance using the standard deviation of the tilt angle and cart position, and the RMS value of the force applied to the cart as performance indices. To validate the robustness of each controller the system is tested against disturbances and reference changes in the cart position. The results of this study show that the LQR controller has minimum steady-state error and a smooth control signal. Nonetheless, according to the performance indices, the PID controller performs better than the LQR, with a position standard deviation of 0.0903m, angle standard deviation of 0.0120 rad and RMS value of the control signal of 1.1007N.