Pêndulo invertido: projeto e implementação de controle PI-DF com identificação ARMAX

Abstract

This undergraduate thesis begins by introducing the essential concepts for the solution of the inverted pendulum control which was designed and built to be tested in real time. Two main areas are approached to solve this problem: Control Systems and Systems Identification. These concepts will be used to analyze and solve the presented problem, whose relevance persists after decades of research in the area of Control Systems. Next, a brief approach to mathematical modeling is presented using Lagrangian mechanics to derive equations that describe the inverted pendulum’s dynamical behavior. In order to overcome the limitations faced in the mathematical modeling, descriptive methods used in Systems Identification are proposed, particularly the AutoRegressive-Moving-Average with eXogenous Input (ARMAX) models are compared in performance with the White Box model obtained by Lagrangean mechanics. To control the inverted pendulum in the upright position, it is proposed to use a digital control system with two Proportional-Integral-Derivative (PID) controllers with two degrees of freedom integrated with a first-order low-pass filter in the derivative action in the feedback branch (PI-DF), whose model is derived from the classic PID. The controllers are arranged in the cascade topology, where the controller of the inner loop keeps the pendulum rod balanced in the upright position, and the other controller of the outer loop is responsible for controlling the position of the cart along the linear guide. The tuning of the digital controllers was implemented by the Root Locus (RL) and by the Ziegler Nichols sustained oscillatory response method. The techniques implemented were compared in different scenarios by the Normalized Root Mean Squared Error (NRMSE) and transient response metrics. By comparing the results obtained on real time, it was found that the tuning of the controllers by the RL method for the system identified with the ARMAX model had the best performance compared to the RL tuning performed in the White Box model.