linear quadratic regulator (lqr) controller design

LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER DESIGN FOR

INVERTED PENDULUM

NOR AKMAL BINTI ALIAS

A project report submitted in partial fulfillment of the

requirement for the award of the degree

Master of Electrical Engineering

Faculty of Electrical and Electronic Engineering

University Tun Hussein Onn Malaysia

JUNE 2013

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ABSTRACT

The Inverted Pendulum System is an under actuated, unstable and nonlinear

system. The challenge of this project is to keep the inverted pendulum balanced and

track the linear cart to a commanded position. This project presents investigations of

performance comparison between conventional Proportional Integral Derivatives

(PID) and modern control Linear Quadratic Regulator (LQR) schemes for an

inverted pendulum system. The goal is to determine which control strategy delivers

better performance with respect to pendulum’s angle and cart’s position. LQR

algorithm needed to compute what the steady-state value of the states should be by

multiply that by the chosen gain K, and a new value as the reference is used for

computing the input. Gain k is the state feedback gain matrix, so the system becomes

a closed loop control system. The LQR should produce a better response compared

to PID control strategies in both discrete time and continuous time control.

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ABSTRAK

Sistem Bandul Songsang adalah satu sistem di bawah gerakkan, tidak stabil dan tidak

linear. Cabaran projek ini adalah untuk memastikan bandul terbalik seimbang dan

menjejaki kereta itu dalam keadaan linear kepada kedudukan yang ditetapkan. Projek

ini membentangkan siasatan perbandingan prestasi antara Proportional Integral

Derivatives (PID) dan kawalan moden Linear Quadratic Regulator (LQR) skim

untuk sistem bandul terbalik. Matlamatnya adalah untuk menentukan strategi

kawalan menyampaikan prestasi yang lebih baik berkenaan dengan sudut bandul dan

kedudukan kereta ini. Algoritma LQR yang diperlukan untuk mengira nilai keadaan

mantap suatu keadaan hendaklah dengan meggandakan gandaan K yang dipilih, dan

nilai baru sebagai rujukan yang digunakan untuk mengira input. Gandaan K adalah

gandaan matriks maklum balas suatu keadaan, oleh itu sistem akan menjadi satu

sistem kawalan gelung tertutup. LQR akan menghasilkan tindak balas yang lebih

baik berbanding dengan strategi kawalan PID dalam kedua-dua masa yang

berasingan dan kawalan masa yang berterusan.

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TABLE OF CONTENTS

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURE xii

LIST OF SYMBOLS AND ABBREVIATION xv

LIST OF APPENDICES xvi

CHAPTER 1 INTRODUCTION

1.1 Project Background 1

1.2 Problem Statements 2

1.3 Project Objectives 2

1.4 Project Scopes 2

CHAPTER 2 LITERATURE REVIEW

2.1 PID Controller 4

2.2

2.3

LQR Controller

Inverted Pendulum

2.3.1 Rotary Position Sensor

2.3.2 Motor with encoder (SPG30-20K)

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7

9

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2.4

2.5

2.6

2.3.2 Motor Driver L293

MATLAB

2.4.1 MATLAB Simulink

2.4.2 M-File

Arduino MEGA 2560

Inverted Pendulum Example in Real Life and

Previous Case Study

2.6.1 Inverted Pendulum Example in Real Life

2.6.2 Previous Case Study

2.6.2. 1 LQG/LTR Controller Design for Rotary

Inverted Pendulum Quanser Real-Time

Experiment

2.6.2.2 DC Motor Controller Using Linear Quadratic

Regulator (LQR) Algorithm Implementation

on PIC

2.6.2.3Modeling and Controller Design for an

Inverted Pendulum System

2.6.2.4 Real-Time Optimal Control for Rotary

Inverted Pendulum

2.6.2.5 Controller Design of Inverted Pendulum Using

Pole Placement and LQR

2.6.2.6 Stabilization of Real Inverted Pendulum Using

Pole Separation Factor

2.7.7 Modeling and Controller Design of Inverted

Pendulum

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CHAPTER 3 METHODOLOGY 20

3.1

3.2

Familiarization of the System (Phase1)

3.1.1 System Overview

Project Implementation (Phase 2)

3.2.1 Inverted Pendulum

3.2.1.1 Equations of Motion

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21

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3.2.1.2 Transfer Function

3.2.1.3 State space

3.2.2 Control Method

3.2.2.1 PID Controller

3.2.2.1.1 PID Tuning

3.2.2.2 LQR Controller

3.2.2.2.1 LQR Controller Block Diagram

3.2.2.2.2 Mathematical Model

3.2.3 The Hardware Implementation

3.2.3.1 Inverted Pendulum

3.2.3.2 Rotary Position Sensor

3.2.3.3 DC Motor with Encoder

3.2.3.4 Arduino Mega

3.3 System Operation (Phase Three)

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CHAPTER 4 RESULT AND ANALYSIS

4.1 Simulation Result

4.1.1 Inverted Pendulum Design

4.1.2 Proportional Integral and Derivatives (PID)

Controller

4.1.2.1 Discrete PID

4.1.1.2 Continuous PID

4.1.3 Linear Quadratic Regulator Controller

4.1.2.1 Discrete LQR

4.1.2.2 Continuous LQR

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4.2

4.2

Comparison between Two Controllers

Hardware Result

4.2.1 Inverted Pendulum System setup

4.3.2 Rotary Position Sensor

4.3.3 DC Motor with encoder (SPG30-20K)

4.3.3.1 Quadrature encoder with Hall

Effect Sensor

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x

4.3.3.2 Motor movement 85

CHAPTER 5 CONCLUSION AND RECOMMENDATIONS

5.1 Conclusion

5.2 Recommendations

89

89

90

REFERENCES 91

APPENDIX

VITA

xi

LIST OF TABLES

2.1 Inverted Pendulum Example in Real Life 15

3.1 Effects of increasing parameter 25

4.1 Simulation result for discrete position 74

4.2 Simulation result for discrete angle 74

4.3 Simulation result for continuous position 75

4.4 Simulation result for continuous angle 77

4.5 Differences between LQR and PID 77

4.6 Description of the analysis of sensor 80

4.7 Motor speed measured across voltage 87

xii

LIST OF FIGURE

3.1 The Block Diagram of the system 21

3.2 Single inverted pendulum model 23

3.3 MATLAB code to generate transfer function 26

3.4 Discrete function code 27

3.5 MATLAB code to generate continuous state space 28

3.6 Block diagram of linear time invariant discrete time 29

control system represented in state space

3.7 MATLAB code to generate discrete state space 30

3.8 PID controller with kp, ki and kd feedback gains 32

3.9 Closed loop control system with u(k) = -Kx(k) 33

3.10 Solution P of steady state Riccati equation when it 37

reached constant

3.11 Inverted pendulum on a rail 38

3.12 Wheel attached to the motor 39

3.13 Pendulum attached to propeller adapter 40

3.14 Resistive sensor being mounted on the cart 41

3.15 The algorithm of Sensor’s Output Result Subroutine 41

3.16 The algorithm of Motor Operation Subroutine 41

3.17 DC Motor with Encoder 42

3.18 Circuit of the project with Arduino 43

3.19 Clearer version of the Circuit 43

3.20 Encoder pin connections 44

3.21 The algorithm of Arduino Usage Subroutine 45

3.22 Flowchart of the Basic LQR and PID Controller Design 46

for Inverted Pendulum Operation

3.23 Flowchart of the PID and LQR controller 47

3.24 Flowchart for the continuous LQR 48

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3.25 Flowchart for the discrete LQR 49

3.26 Flowchart for the continuous PID 50

3.27 Flowchart for the discrete PID 51

4.1 Subsystem of inverted pendulum 53

4.2 Continuous Transfer Function 54

4.3 Discrete Transfer Function 54

4.4 Continuous state space for inverted pendulum 55

4.5 Discrete state space for inverted pendulum 56

4.6 Open Loop impulse response 57

4.7 State space output response 58

4.8 Simulink block for angle of pendulum 59

4.9 Figure 4.9: Result in pendulum position 60

4.10 Pole zero mapping 61

4.11 Simulink block for position of cart 62

4.12 Result in cart’s position 63

4.13 Pole zero mapping 64

4.14 Simulink block for continuous PID using subsystem 65

4.15 Output response for PID 66

4.16 Pole zero mapping 67

4.17 Simulink design for discrete LQR 68

4.18 Results for the response 69

4.19 Pole and Zero mapping 70

4.20 Simulink design for continuous LQR 71

4.21 Output response for LQR 54

4.22 Pole zero mapping 73

4.23 Comparison between continuous LQR and PID position 75

4.24 Comparison between continuous LQR and PID angle 76

4.25 Inverted pendulum on a rail 78

4.26 Rotary position sensor schematic 60

4.27 Sensor being attached to the pendulum 61

4.28 Graph Speed versus voltage 80

4.29 Waveform from channel A and B 82

4.30 Circular state table 82

4.31 Pulse generated along the whole rail 83

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4.32 Bigger division of pulse with CH1 information 83

4.33 Bigger division of pulse with CH2 information 84

4.34 Pulse generated along the half of the rail 84

4.35 Bigger division of pulse with CH1 and CH2 information 85

4.36 H-Bridge Motor 86

4.37 Motor Driver circuit connection 86

4.38 Graph of speed versus voltage 87

xv

LIST OF SYMBOLS AND ABBREVIATION

PID - Proportional-Integral-Derivative

LQR - Linear Quadratic Regulator

Kp - Proportional gain

Ki - Integral gain

Kd - Derivative gain

V - Volt

PWM - Pulse Width Modulator

PIC - Programmable Integrated Circuit

MATLAB - Matrix Laboratory

GUI - Graphical User Interface

AC - Alternating Current

DC - Direct Current

V - Voltage

m - meter

b - friction

l - length

F - force

x - cart position

θ - angle

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LIST OF APPENDICES

A PSM 2 Gantt Chart 51

PSM 1 Gantt Chart 52

B MATLAB Coding 53

C Arduino Coding

D The Experiments

CHAPTER 1

INTRODUCTION

1.1 Project Background

One-dimensional inverted pendulum is a nonlinear problem, which has been considered

by many researchers, Omatu and Yashioka, 1998 [1] and Magana and Holzapfel, 1998

[2], most of which have used linearization theory in their control schemes. In general,

the control of this system by classical methods is a difficult task. This is mainly because

this is a nonlinear problem with two degrees of freedom (the angle of the inverted

pendulum and the position of the cart), and only one control input. The inverted

pendulum is used for control engineers to verify a modern control theory since its

characteristics as marginally stable as a control. This system is popularly known as a

model for the attitude control especially in aerospace field. However, it also has its own

deficiency due to its principles, highly non linear and open-loop unstable system. Thus,

causing the pendulum falls over quickly whenever the system is simulated due to the

failure of standard linear techniques to model the non-linear dynamics of the system.

Moreover, it makes the identification and control become more challenging. The

common control approaches to overcome the problem by this system namely Linear

Quadratic Regulator (LQR) control and Proportional-Integral-Derivative (PID) control

that require a good knowledge of the system and accurate tuning to obtain good

performance. Nevertheless, it attribute to difficulty in specifying an accurate

mathematical model of the process. This project presents investigations of performance

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comparison between conventional (PID) and modern control (LQR) schemes for an

inverted pendulum system. The dynamic model and design requirement have been taken

from Carnegie Mellon, University of Michigan. Performance of both control strategy

with respect to pendulum’s angle and cart’s position is examined. Comparative

assessment of both control schemes to the system performance is presented and

discussed.

1.2 Problem Statement

The challenge of this project is to keep the inverted pendulum balanced and track the

linear cart to a commanded position. In practice, it is interesting to point out that similar

dynamics and control problem apply to rudder roll stabilization of ships. During this

project, the controllers that will be used are the PID and the LQR. So the proposed

controllers with proper design and tuning will be able to overcome the above mentioned

challenge.

1.3 Objectives

Based on above problem statement, the objectives are:

1.3.1 To design LQR and PID controllers for inverted pendulum.

1.3.2 To compare the performance result for both controllers in order to get the best

controller design.

1.3.3 To balance the inverted pendulum by applying a force to the cart that the pendulum

is attached to.

1.4 Scopes

The scopes for this Linear Quadratic Regulator Controller Design for Inverted Pendulum

are as stated below:

1.4.1 This project uses the long single inverted pendulum as a default configuration.

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1.4.2 Design a controller using modern technique which is LQR and conventional

technique PID.

1.4.3 Simulate the controllers using Matlab and conclude the best controller based on

the simulation results.

1.4.4 Design inverted pendulum system using Matlab.

1.4.5 Control the pendulum's position that should be return to the vertical position after

given the initial disturbance.

1.5 Report Outline

This report contains five chapters which are Introduction, Literature Review,

Methodology, Result and Analysis and lastly Conclusion and Recommendation.

The Introduction chapter is about the project background, problem statement,

objectives and scopes. These topics are the basic information about the whole system.

The main idea of the system is stated in this chapter.

Chapter two contains about the PID and LQR controller, the hardware

components as well as the software, and lastly the inverted pendulum example in real

life and the previous case study. This chapter is aiming the critical points of knowledge

and theoretical to a particular topic. The controllers and the inverted pendulum are the

core in this project. The past researches are as the guidance in implementing this system.

Chapter three which is Methodology is about the system overview, hardware

design, software parts and system operation. This chapter is concern about the process of

designing and implementing this whole system. Every process has the role that have

been specify for the system to be function as needed.

Chapter four is basically regarding the results and analysis for this project. The

results are displayed and then being analyzed in this chapter so that readers have better

understanding about whole process.

The last chapter in this report is Conclusion and Recommendation. This chapter

is summary of the whole project. The idea is conclude in this last chapter and the

recommendation part is to upgrade the system to a higher level.

CHAPTER 2

LITERATURE REVIEW

2.1 PID Controller

PID stands for Proportional-Integral-Derivative. This is a type of feedback controller

whose output, a control variable, is generally based on the error between some user

defined set point and some measured process variable. Each element of the PID

controller refers to a particular action taken on the error. The error is then used to adjust

some input to the process in order to its defined set point. The schematic for this

problem is depicted as below.

Figure 2.1: Schematic diagram for controller

5

The design can be easier to be analyzed by using below figure.

Figure 2.2: Easier schematic

Three parameters must be designed in the PID controller and each parameter has

an effect on the error. The transfer function of the PID controller is written as [3]:= + + .Where, Kp is the controller gain, Ki is the integral time and Kd is the derivative

time. It is important to determine appropriate parameters to guarantee stability and

system performance. There are several methods for tuning PID parameters. However, in

this study the three parameters of PID controller values are computed by trial and error.

Proportional: Error multiplied by a gain, Kp. This is an adjustable amplifier. In

many systems Kp is responsible for process stability. Too low and the PV can

drift away. Too high and the PV can oscillate.

Integral: The integral of error multiplied by a gain, Ki. In many systems Ki is

responsible for driving error to zero, but to set Ki too high is to invite oscillation

or instability.

Derivative: The rate of change of error multiplied by a gain, Kd. In many

systems Kd is responsible for system response. Too high and the PV will

oscillate. Too low and the PV will respond sluggishly.

6

2.2 LQR Controller

In order to overcome some problems that faced by PID controller, the other type of

control methods can be developed such as Linear-Quadratic Regulator (LQR) optimal

control. LQR is a control scheme that gives the best possible performance with respect

to some given measure of performance. The performance measure is a quadratic

function composed of state vector and control input.

Linear Quadratic Regulator (LQR) is the optimal theory of pole placement

method. LQR algorithm defines the optimal pole location based on two cost function. To

find the optimal gains, one should define the optimal performance index firstly and then

solve algebraic Riccati equation. LQR does not have any specific solution to define the

cost function to obtain the optimal gains and the cost function should be defined in

iterative manner.

LQR is a control scheme that provides the best possible performance with

respect to some given measure of performance. The LQR design problem is to design a

state feedback controller K such that the objective function J is minimized. In this

method a feedback gain matrix is designed which minimizes the objective function in

order to achieve some compromise between the use of control effort, the magnitude, and

the speed of response that will guarantee a stable system. For a continuous-time linear

system described by [4]:

ẋ = +With a cost functional defined as= ʃ ( ᵀ + ᵀ )Where Q and R are the weight matrices, Q is required to be positive definite or

positive semi-definite symmetry matrix. R is required to be positive definite symmetry

matrix. One practical method is to Q and R to be diagonal matrix. The value of the

elements in Q and R is related to its contribution to the cost function J. The feedback

control law that minimizes the value of the cost is:= −K is given by = ᵀ

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And P can be found by solving the continuous time algebraic Riccati equation:

ᵀ + − ᵀ + = 0The LQR algorithm is, at its core, just an automated way of finding an

appropriate state. As such it is not uncommon to find that control engineers prefer

alternative methods like full state feedback (also known as pole placement) to find a

controller over the use of the LQR algorithm. With these the engineer has a much clearer

linkage between adjusted parameters and the resulting changes in controller behavior.

Difficulty in finding the right weighting factors limits the application of the LQR based

controller synthesis.

2.3 Inverted Pendulum

It is virtually impossible to balance a pendulum in the inverted position without applying

some external force to the system. The problem involves a cart, able to move backwards

and forwards, and a pendulum, hinged to the cart at the bottom of its length such that the

pendulum can move in the same plane as the cart. That is, the pendulum mounted on the

cart is free to fall along the cart's axis of motion. The system is to be controlled so that

the pendulum remains balanced and upright, and is resistant to a step disturbance.

If the pendulum starts off-centre, it will begin to fall. The pendulum is coupled to

the cart, and the cart will start to move in the opposite direction, just as moving the cart

would cause the pendulum to become off centre. As change to one of parts of the system

results in change to the other part, this is a more complicated control system than it

appears at first glance. The inverted pendulum cart runs along a track. A rotary position

sensor measures the cart position from its rotation and another potentiometer measures

the angle of the pendulum.

An inverted pendulum is a pendulum which has its center of mass above its pivot

point. It is often implemented with the pivot point mounted on a cart that can move

horizontally and may be called a cart and pole. Inverted Pendulum is an inherently

unstable system. Force must be properly applied to keep the system intact. To achieve

this, proper control theory is required. The Inverted Pendulum is essential in the

evaluating and comparing of various control theories.

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Figure 2.3: Single inverted pendulum

The inverted pendulum is mounted on a moving cart. A servomotor is controlling

the translation motion of the cart. That is, the cart is coupled with a DC motor through

pulley and belt mechanism. The motor is derived by DC electronics, which also contains

controller circuits. A rotary position sensor is used to feedback the angular motion of the

pendulum to servo electronics to generate actuating-signal. Controller circuits process

the error signal, which then drives the cart through the servomotor. The motion of the

cart applies moments on the inverted pendulum and thus it keeps the pendulum upright.

So briefly, the Inverted Pendulum system is made up of a cart and a pendulum.

The goal of the controller is to move the cart to its commanded position without causing

the pendulum to tip over. In open loop this system is unstable.

The various stages of the work for accomplishing the task of controlling the

Inverted Pendulum are as follows:

Modeling the IP and linearizing the model for the operating range.

Analyzing the uncompensated closed loop response with the help of a root locus

plot.

Designing the PID controller and simulating it in MATLAB for proper tuning

and verification.

Analyzing the compensated closed loop response of the system.

Implementing the controller on the physical IP model.

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The problem involves a cart, able to move backwards and forwards, and a

pendulum, hinged to the cart at the bottom of its length such that the pendulum can

move in the same plane as the cart, shown below. That is, the pendulum mounted on the

cart is free to fall along the cart's axis of motion. The system is to be controlled so that

the pendulum remains balanced and upright, and is resistant to a step disturbance [5].

2.3.1 Rotary Position Sensor

The model rotary position sensor is a conductive polymer based electronic component

which translates angular position into an electrical signal. This sensor can be used in

various applications where a user needs to control a variable output such as frequency or

volume. It is a 2.1 mm low profile rotary position sensor with a linearity specification of

±2% before and after 1 million rotational cycles. The sensor can function from a

temperature range of -40 °C to 120 °C which exceeds the current temperature range

available for similar products.

Figure 2.4: Resistive sensor

Rotary Position Sensor offers application for both feedback and control where

frequent adjustment or feedback is required. The shaftless design allows for employment

of different engagement mechanisms, such as a customized shaft, a motor control, a

human interface (such as knob) adjustment, or a mechanical arm position locator. The

sensor can also control variable outputs including frequency, change in motor speed or

volume. Typical applications include test and measurement equipment, consumer

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electronics, appliances, small engines, robotics, motion controllers, automation

equipment, electronic instrumentation and medical equipment control panels.

2.3.2 Motor with encoder (SPG30-20K)

This DC geared motor with encoder is formed by a quadrature hall effect encoder board

which is designed to fit on the rear shaft of Cytron's SPG-30 geared motor series. Two

hall effect sensors are placed 90 degree apart to sense and produce two output A and B

which is 90 degree out of phase and allowing the direction of rotation to be determined.

Please note that the encoder is mounted at the rear shaft, the minimum

resolution depends on the motor's gear ratio.

Figure 2.5: DC Geared Motor with Encoder

The disc is painted half white and half black to vary the reflectivity of the surface

as it rotates. A reflective pair of light sensors detects this change in light as the motor

rotates the disc. An attached microcontroller reads the sensors and can determine the

direction and speed of the motor.

Since the encoder disc is attached to the motor shaft, the resolution of the

encoder will be significantly better than it would be if the encoder disc were attached to

the wheel. Not only is the rear of the motor exposed to less ambient light than the wheel,

but the wheel rotates more slowly than the actual motor speed due to gear reduction. For

example, with a 60:1 ratio gear head, an encoder disc mounted on the motor shaft gets

60 counts for every 1 rotation of the wheel.

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2.3.2 Motor Driver L293

The L293 comes in a standard 16-pin, dual-in line integrated circuit package. There is an

L293 and an L293D part number. The D version should be picked because it has built in

flyback diodes to minimize inductive voltage spikes. The L293 is an integrated circuit

motor driver that can be used for simultaneous, bi-directional control of

two small motors.

Figure 2.6 shows Motor driver L293D this circuit can be control and drive servo

motor by apply the input voltage of 4.8V or 6.0 V. It can be used to control two motor in

one time by using principle of H-bridge circuit. The circuit consist two PWM which is

right PWM and left PWM to control the speed of servo motor. Each PWM have two

motor to produce direction of wheel of robot either forward or backward. When motor 1

is on (bit 1) and motor 2 off (bit 0) the direction of wheel is forward while when the

motor 1 is off (bit 0) and motor 2 on (bit 1) the direction of wheel is backward.

Figure 2.6: Motor Driver L293D

The L293D IC has four input signals, labeled a, b, c and d. The driver boosts the

current from these signals. The voltage signal at each output (a, b, c and d) is high or low

when the corresponding input signal is high or low. But the current available from the

output pin is much higher. So the L293D is able to drive output devices requiring quite

high currents of up to 600mA from each output pin. The main advantage of the L293D is

that, unlike a transistor, Darlington driver or MOSFET, it can drive a DC motor in

forward or reverse. It is particularly useful for work with PICs. Current can only flow in

one direction (the direction of the arrow on the circuit symbol) through a transistor,

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Darlington driver or n-channel MOSFET, which is why they cannot be used to reverse a

motor.

2.4 MATLAB

MATLAB (matrix laboratory) is a numerical computing environment and fourth-

generation programming language. MATLAB is a weakly typed programming language.

It is a weakly typed language because types are implicitly converted. It is a dynamically

typed language because variables can be assigned without declaring their type, except if

they are to be treated as symbolic objects, and that their type can change. Values can

come from constants, from computation involving values of other variables, or from the

output of a function.

2.4.1 MATLAB Simulink

Simulink is a block diagram environment for multi domain simulation and Model-Based

Design. It supports system-level design, simulation, automatic code generation, and

continuous test and verification of embedded systems. Simulink is an input/output

device GUI block diagram simulator which contains a Library Editor of tools from

which we can build input/output devices and continuous and discrete time model

simulations. It contains continuous and discontinuous system model elements.

Simulink, developed by MathWorks, is a commercial tool for modeling,

simulating and analyzing multi domain dynamic systems. Its primary interface is a

graphical block diagramming tool and a customizable set of block libraries. It offers

tight integration with the rest of the MATLAB environment and can either drive matlab

or be scripted from it. Simulink is widely used in control theory and digital signal

processing for multi domain simulation and Model-Based Design.

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2.4.2 M-File

M-files are macros of MATLAB commands that are stored as ordinary text files with the

extension ‘m’, that is filename.m. An M-file can be either a function with input and

output variables or a list of commands.

There are four ways of doing code in MATLAB. One can directly enter code in a

terminal window. This amounts to using MATLAB as a kind of calculator, and it is good

for simple, low-level work. The second method is to create a script M-file. Here, one

makes a file with the same code one would enter in a terminal window. When the file is

run, the script is carried out. The third method is the function M-file. This method

actually creates a function, with inputs and outputs. The fourth method will not be

discussed here. It is a way to incorporate C code or FORTRAN code into MATLAB and

this method uses .mex files.

2.5 Arduino MEGA 2560

The Arduino Mega 2560 is a microcontroller board based on the ATmega2560. It has 54

digital input/output pins (of which 14 can be used as PWM outputs), 16 analog inputs, 4

UARTs (hardware serial ports), a 16 MHz crystal oscillator, a USB connection, a power

jack, an ICSP header, and a reset button. It contains everything needed to support the

microcontroller which simply connect it to a computer with a USB cable or power it

with an AC-to-DC adapter or battery to get started. The Mega is compatible with most

shields designed for the Arduino Duemilanove or Diecimila.

Figure 2.7: Arduino Mega 2560

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The Arduino Mega 2560 has a number of facilities for communicating with a

computer, another Arduino, or other microcontrollers. The Arduino Mega 2560 can be

programmed with the Arduino software. The Atmega 2560 on the Arduino Mega comes

preburned with a bootloader that allows you to upload new code to it without the use of

an external hardware programmer.

2.6 Inverted Pendulum Example in Real Life and Previous Case Study

There are many projects that have the same idea of the inverted pendulum nowadays.

There may be some robotic project that also uses the concept of the inverted pendulum

as well as the controller that is being fed to the system. The idea of it is applicable on the

system that is being design. These references help so much in implementing this project.

2.6.1 Inverted Pendulum Example in Real Life

There are so many things in the surrounding that has the same idea as the

inverted pendulum. All of these things has also contribute in the making this project.

The application of it can give clear information about the process of handling the

inverted pendulum.

Table 2.1: Inverted Pendulum Example in Real Life

Types Description

Mechanical metronome

A metronome is any device that produces regular,

metrical ticks settable in beats per minute.

These ticks represent a fixed, regular aural pulse.

Some metronomes also include synchronized

visual motion for example the pendulum-swing.

15

Segway PT The Segway PT is a two-wheeled, self-

balancing, battery-powered electric

vehicle invented by Dean Kamen. It is produced

by Segway Inc. of New Hampshire, USA.

The Segway detects, as it balances, the change in

its center of mass, and first establishes and then

maintains a corresponding speed, forward or

backward. Gyroscopic sensors and fluid-based

leveling sensors detect the weight shift.

Loading Crane Crane is a type of machine, generally equipped

with a hoist, wire ropes or chains, and sheaves,

that can be used both to lift and lower materials

and to move them horizontally.

It is mainly used for lifting heavy things and

transporting them to other places. It uses one or

more simple machines to create mechanical

advantage and thus move loads beyond the

normal capability of a man.

2.6.2 Previous Case Study

There are several sources that take part in the project implementation. Many sources

have contributed to be the references for this project for example, conference paper and

thesis. By taking the other project as a reference, it may be helpful in order to

implementing this whole project.

16

2.6.2.1 LQG/LTR Controller Design for Rotary Inverted Pendulum Quanser Real-

Time Experiment

This experiment consists of a rigid link (pendulum) rotating in a vertical plane. The rigid

link is attached to a pivot arm, which is mounted on the load shaft of a DC-motor [6].

The pivot arm can be rotated in the horizontal plane by the DC-motor. The DC-motor is

instrumented with a potentiometer. In addition, a potentiometer is mounted on the pivot

arm to measure the pendulum angle. The principal objective of this experiment is to

balance the pendulum in the vertical upright position and to position the pivot arm. Since

the plant has two degrees of freedom but only one actuator, the system is under-actuated

and exhibits significant nonlinear behavior for large pendulum excursion. The purpose is

to design a robust controller in order to realize a real-time control of the pendulum

position using a Quanser PC board and power module and the appropriate WinCon real-

time software. For the controller design is used a well-known robust method, called

LQG/LTR (Linear Quadratic Gauss Ian/Loop Transfer Recovery) which implements an

optimal statefeedback. The real-time experiment is realized in the Automatic Control

laboratory.

2.6.2.2 DC Motor Controller Using Linear Quadratic Regulator (LQR) Algorithm

Implementation on PIC

Linear Quadratic Regulator (LQR) algorithm is one of the controller methods to control

a system [7]. In this project, the LQR was implemented on the PIC microcontroller to

control the dc motor. The main objective of this controller is to minimize the deviation

of the speed of DC motor. DC motor speed is controlled by its driving voltage. The

higher the voltage, the higher the motor speed will be. The speed of the motor is

specifying that will be the input voltage of the motor and the output will be compare

with the input. As the result, the output must be the same as or approximately the same

as the input voltage. In this project, the LQR algorithm was implemented on the PIC

microcontroller so the result can be shown. Before the implementation on the PIC, the

dc motor state space has to be derived. Then, from the state-space, it can be design the

17

LQR controller by using the MATLAB software. The stable system is got by tuning the

Q and R value that can be seen by the simulation.

2.6.2.3 Modeling and Controller Design for an Inverted Pendulum System

The Inverted Pendulum System is an under actuated, unstable and nonlinear system [8].

Therefore, control system design of such a system is a challenging task. To design a

control system, this thesis first obtains the nonlinear modeling of this system. Then, a

linearized model is obtained from the nonlinear model about vertical (unstable)

equilibrium point. Next, for this linearized system, an LQR controller is designed.

Finally, a PID controller is designed via pole placement method where the closed loop

poles to be placed at desired locations are obtained through the above LQR technique.

The PID controller has been implemented on the experimental set up.

2.6.2.4 Real-Time Optimal Control for Rotary Inverted Pendulum

The rotary inverted pendulum system was a highly nonlinear model, multivariable and

absolutely unstable dynamic system. It was used for testing various design control

techniques and in teaching modern control [9]. The objectives of this study were to

develop a real rotary inverted pendulum which derived the mechanical model by using

Euler-Lagrange and design controller algorithm for self-erecting and balancing of a

rotary inverted pendulum. Research showed a convenient way to implement a real-time

control in self-erecting a pendulum from downward position and balancing the

pendulum in vertical-upright position. An Energy based on PD controller was applied in

self-erecting of the pendulum while LQR controller was applied to balance the

pendulum. Results of both control techniques from computer simulation and experiment

were given to show the effectiveness of these controllers. Both simulations and

experiments were confirmed the control efficiency of the method.

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2.6.2.5 Controller Design of Inverted Pendulum Using Pole Placement and LQR

In this paper modeling of an inverted pendulum is done using Euler – Lagrange energy

equation for stabilization of the pendulum [10]. The controller gain is evaluated through

state feedback and Linear Quadratic optimal regulator controller techniques and also the

results for both the controller are compared. The SFB controller is designed by Pole-

Placement technique. An advantage of Quadratic Control method over the pole-

placement techniques is that the former provides a systematic way of computing the

state feedback control gain matrix. LQR controller is designed by the selection on

choosing. The proposed system extends classical inverted pendulum by incorporating

two moving masses. The motion of two masses that slide along the horizontal plane is

controllable.

2.6.2.6 Stabilization of Real Inverted Pendulum Using Pole Separation Factor

Based on the pole placement design technique, a full state feedback controller using

separation factor is proposed to stabilize a real single inverted pendulum [11]. The

strategy is to start with selection two dominant poles that achieve a certain desired

performance, using a separation factor between the selected dominant poles and the

other poles to eliminate their effect on the system performance, and finally Ackermann’s

formula can be used to calculate the feedback gain matrix to place the system poles at

the desired locations. Simulation and experimental results demonstrate the effectiveness

of the proposed controller, which offers an excellent stabilizing and also ability to

overcome the external resistance acting on the pendulum system.

2.7.7 Modeling and Controller Design of Inverted Pendulum

In this paper modeling of an inverted pendulum has been done and then three different

controllers (PID & SFB, LQR) have been used for stabilization of the pendulum [12].

The proposed system extends classical inverted pendulum by incorporating two moving

masses. The motion of two masses that slide along the horizontal plane is

controllable .The results of computer simulation for the system with Proportional,

19

Integral and Derivative (PID) & State Feedback Controllers are shown. Main focus is to

derive the mathematical model and the analysis of its system performance, then design a

LQR controller in order to get better control.

CHAPTER 3

METHODOLOGY

This research is adopting methods approach involving development and improvement to

enhance the overall process of project implementation. The project’s Gantt chart is given

in Appendix A. The research is conducting in three phase’s basis. This phase will make

sure that the work will be organized based on the scheduled task.

This project is divided into two parts which is hardware and software. Therefore,

this chapter will discuss about the process for each parts includes designing, testing,

troubleshooting and integrating in order to create one whole system of LQR and PID

Controller Design for Inverted Pendulum.

3.1 Familiarization of the System (Phase 1)

In order to implement this project, the first phase that is needed to be concerned is the

familiarization of the system. The whole system should be identified and clarified in

order to execute the correct process flow of this system.

3.1.1 System Overview

The inverted pendulum is a highly nonlinear and open-loop unstable system. The control

of inverted pendulum is to make the cart and swinging rod achieve desired equilibrium

position as soon as possible. LQR and PID are the controllers that will be used in order

21

to achieve its target. In the meanwhile it will not have too strong oscillation range, too

fast speed and angular velocity. When the inverted pendulum system achieves the

desired position, it can overcome a range of disturbance and keep balance.

Arduino Inverted Pendulum

Figure 3.1: The Block Diagram of the system

Based on the above block diagram, it is shown that there are two elements that

need to be controlled in using the inverted pendulum. The first one is the position

control by the cart or the system meanwhile for the other one is by controlling the angle

of the pendulum. The Arduino Mega in order to interface the software to the hardware

so that the inverted pendulum can work as desired.

Angle control

(Pendulum’s angle)

Position control (cart’s position)

22

3.2 Project Implementation (Phase 2)

The project implementation will covers all of the controllers that will be done in this

project. The mathematical model for each of the controller is very important in order to

design the controller. The controller itself plays a big role in order to stabilize the

inverted pendulum in upright position.

3.2.1 Inverted Pendulum

In this section, two control methods are proposed and explained in details which are PID

and Linear Quadratic Regulator (LQR) controllers. Furthermore, the following design

specifications have been made to evaluate the performance of both control schemes. The

parameters that are used in designing the inverted pendulum are as below:

M mass of the cart 0.208 kg

m mass of the pendulum 0.08 kg

b friction of the cart 0.16 N/m/sec

l length to pendulum center of mass 0.382 m

I inertia of the pendulum 12.5e-6 kg*m^2

L

F

Length of the rail

force applied to the cart0.894m

x cart position coordinate

theta pendulum angle from vertical

23

Figure 3.2: Single inverted pendulum model

The cart with an inverted pendulum, shown in Figure 3.1, is bumped with an

impulse force, F. The dynamic equations of motion for the system, and the pendulum's

angle, theta = 0 is linearized. In other words, the pendulum are assume does not move

more than a few degrees away from the vertical.

3.2.1.1 Equations of Motion

To derive the suitable mathematical model for an Inverted Pendulum system, Figure 3.2

is considered. Below is the mathematical equation for the system.

Adding all the forces on the cart in the horizontal direction,ẍ + ẋ + = (1)

Adding all the forces on the pendulum in the horizontal direction,ẍ + Ӫ − à ² sin = N (2)

Substituting equation (2) in equation (1),( + )ẍ + ẋ + Ӫ − Ӫ = (3)

Adding all the forces along the vertical direction of the pendulum,+ cos − = Ӫ + ẍ cos (4)

Considering sum of the moments about the center of gravity (COG) of the pendulum,− − cos = Ӫ (5)

Now, from equation (4) & (5)( + )Ӫ + sin = − ẍ cos Ӫ (6)

24

The system under consideration is a non-linear system. For ease of modeling and

simulation, it has to take a small case approximation such that the system will be a linear

one. The linearization point will be θ = Π= Π +ϕ is the angle between the pendulum and vertical upward direction. If it is chosen that,

ϕ≈0, then cos = -1, sin = -ϕ

So, after linearization equation (6) becomes,( + ) − = ẍ (7)

And equation (3) becomes, ( + )ẍ + ẋ − =Here, F is the mechanical force to be applied on the moving cart system. But in real time

model it needs to input voltage proportional to the force F. If the input voltage is u, then

equation (8) becomes, ( + )ẍ + ẋ − = (8)

3.2.1.2 Transfer Function

The transfer function of a linear, time-invariant, differential equation system is defined

as the ratio of the Laplace transform of the output (response function) to the Laplace

transform of the input (driving function) under the assumption that all initial conditions

are zero.

Transfer function = G(s) =[ ][ ]

By using the concept of transfer function, it is possible to represent system dynamics by

algebraic equation in s. If the highest power of s in the denominator of the transfer

function is equal to n, the system is called an nth-order system [11].

Laplace transform of equation (7)( + ) ( ) ² − ( ) = − ( ) ² (9)

91

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