Control of a ball and beam apparatus

Abstract

The ball and beam apparatus is a nonlinear and open-loop unstable system which, due to its simplicity, is found in a lot of university control labs, such that it is used to implement and test several methods to control the position of the ball on the beam. The system consists of a ball on a beam rotated by a DC motor together with several sensors for taking measurements through the interface of a dSPACE DS1104 controller board and processed through software algorithms coded through the Simulink environment provided by MATLAB software. This is a continuation of a project where the ball and beam set-up was already designed and assembled. Some modifications are added to it to improve performance and reliability. The goal of this dissertation is to model the ball and beam system using Lagrange equations to obtain a set of nonlinear state-space equations which can then be linearized to implement several computer control methods to control the system. These include discrete-time regulation and tracking, pole-placement and linear quadratic optimal controllers. Each of these are tested using both full state feedback and with state observation, which for the case of the optimal controller, consists of a Kalman filter to obtain a linear quadratic Gaussian (LQG) controller. A theoretical analysis for each mentioned controller is conducted followed by their respective simulation results using a variety of testing strategies, taking into consideration real plant scenarios such as sensor noise and actuator saturation, from which the performance of each can be discussed and evaluated. Ultimately, the LQG regulator and tracking controllers are implemented on the actual physical plant, taking into consideration the unmodeled nonlinearities such as motor and sensor responses. These are tested using similar strategies as that of the corresponding simulations to obtain results that can be used for comparison purposes, highlighting the differences between simulations and real world experiments.