Cart-Inverted Pendulum

CHAPTER ONE
1.0 INTRODUCTION
1.1 Project Overview
Cart-Inverted pendulum has been classified as a tool for learning the control of nonlinear systems [1]. Its behavior resembles that of a real world systems such as the control of the Segway vehicle, unicycles, rockets, skyscrapers and many more [2]. This control problem has been of interest to many researchers [3], [4], [5], because it is highly nonlinear, unstable and has two degrees of freedom with one control input. Further the control problem is made more complex by including physical constraints such as limited track length of the cart [6].
The non-linearity aspect of this system has been extensively used by researchers in verifying some of the control theories and in demonstrating ideas

This is to be achieved by designing two separate controllers, one for swinging up the pendulum from a down position (pi) by the application of a control force to the cart and the other for stabilizing the controller when the pendulum is close to the upright position (0 or 2pi). Further another controller will be required for keeping the cart within the track length as the swing up controller swings the pendulum close to the upright position (0 or 2pi).
1.2 Problem Statement
The control problem is to use the force generated by the cart to swing the pendulum close to upright position (0 or 2pi) while keeping the cart within the limited track length of 0.4m then stabilize the pendulum in the upright position (0 or 2pi).
The other issue is applying the various tuning methods such as Zeigler Nichols for PID controllers, calculation of the gains for LQR controllers and the fuzzy rules for the fuzzy controllers in tuning this controllers for the best performance results. Besides the trial and error method will also be used in determining the gains to be used in the swing up controllers for it to apply an appropriate force that will always swing the pendulum close upright position (0 or 2pi) while keeping the cart within the limited track.
1.3 Project Objectives
Based on the problem statement discussed above,

 To compare the simulation performance of the PID controllers, LQR, Fuzzy controllers on the cart-inverted pendulum.
 To apply the developed controllers on a real cart-inverted pendulum
 To use the developed controllers in a real time application
1.4 Methodology
This is the step by step approach taken in order to realize the objective of this project. The approach is as follows.
• Background reading of existing work related to the control of nonlinear systems- It entails looking at what other people have done in regards to the control of nonlinear systems and analyzing their findings. This will help me to narrow to some few controllers with better results that be used in a real time application.
• Simulation – Simulate the nonlinear system and the controller. This is useful in checking if my designed controller can work in a real time application.
• Gathering of resources - This section entails identification of components needed and determination of their equivalents or alternatives.
• Pre-implementation – This entails testing of the various functional parts before assembling them