vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xii
LIST OF FIGURES
xiii
LIST OF SYMBOLS
xvi
INTRODUCTION
1
1.1
General Introduction
1
1.2
Problem Statement
2
1.3
Objective of Project
3
1.4
Scope of Project
4
1.5
Thesis Outline
5
LITERATURE REVIEW
7
2.1
Introduction
7
2.2
Literature Research
7
viii
2.3
3
4
2.2.1 Fuzzy Logic Controller
8
2.2.2 Hybrid Controller
10
2.2.3 Adaptive Neural Network Fuzzy Controller
11
2.2.4 Fusion Function Based Controller
12
Summary of Chapter 2
14
RESEARCH METHODOLOGY
15
3.1
Introduction
15
3.2
Understanding the Inverted Pendulum System
16
3.3
Mathematical Modeling of Nonlinear Plant
16
3.3.1 Newton’s Law of Motion
16
3.3.2 The Energy Method
17
3.4
Deriving the Dynamics of the Linearized System
17
3.5
Controller Design
18
3.6
Simulation and Evaluation of Performance
19
3.7
Comparative Assessment of Results
19
3.8
Methodology
20
3.9
Summary of Chapter 3
21
MODELING OF AN INVERTED PENDULUM
22
4.1
Introduction
22
4.2
Inverted Pendulum Parameters
23
4.3
Mathematical Modeling
24
4.3.1 Generalized Coordinate System
25
4.3.2 Kinetic and Potential Energy Functions
26
4.3.3 Lagrangian Function
28
4.3.4 Lagrange’s Equation
28
4.4
Linear Model of the Inverted Pendulum System
31
4.5
Transfer Function
32
4.6
State-Space
34
4.7 Summary of Chapter 4
35
ix
5
CONTROLLER DESIGN FOR INVERTED PENDULUM
SYSTEM
36
5.1
Introduction
36
5.2
State-space Controller Theory and Design Implementation 37
5.2.1 Introduction
37
5.2.2 State Variable
37
5.2.3 State Space Representation for Linear System
38
5.2.4 State Space Controllability and Observability
39
5.2.5 Controller Design Using Full State Feedback
40
5.2.6 Full State Feedback with Reference Input
41
5.2.7 Full State Feedback with Reference Input for LQR 42
5.3
LQR Controller Design for Inverted Pendulum System
5.4
Fuzzy Logic Controller Theory and Design Implementation 45
5.5
5.4.1 Introduction
45
5.4.2 Components of Fuzzy Logic Controller
46
5.4.3 Methodology of Designing Fuzzy Controller
47
5.4.3.1 Fuzzy Logic Control Variables
48
5.4.3.2 Fuzzification
49
5.4.3.3 Knowledge Base Design
50
5.4.3.4 Inference Encoding Procedure
52
5.4.3.5 Deffuzification
53
5.4.3.6 Tuning Parameters
54
Fuzzy Logic Controller Design for Inverted Pendulum
System
5.6
43
55
Adaptive Neural Network Fuzzy Controller Theory and
Design Implementation
61
5.6.1 Introduction
61
5.6.2 Fuzzy Inference System
62
5.6.2.1 Sugeno Type Fuzzy Inference System
5.6.3 Adaptive Networks
62
64
x
5.6.3.1 Adaptive Network Architecture and Basic
Learning Rule
65
5.6.4 Adaptive Neuro Fuzzy Inference System (ANFIS) 66
5.6.4.1 ANFIS Architecture
67
5.6.5 ANFIS Learning Algorithm
70
5.6.5.1 Learning of Premise Parameters
70
5.6.5.2 Learning of Consequent Parameters
70
5.6.5.3 Hybrid Learning Algorithm
71
5.6.6 ANFIS Editor GUI
5.7
74
5.7.1 State Variables Fusion
75
5.7.2 Adaptive Neural Network Fuzzy Control
76
Summary of Chapter 5
81
RESULTS AND ANALYSIS
82
6.1
Introduction
82
6.2
Results of Different Designed Controllers for Inverted Pendulum
6.3
6.4
7
Adaptive Neural Network Fuzzy Logic Controller Design
for Inverted Pendulum System
5.8
6
72
System
83
6.2.1 Results for State Feedback Controller: LQR
83
6.2.2 Results for Fuzzy Logic Controller
85
6.2.3 Results for Adaptive Neural Fuzzy Controller
87
Overall Comparison of the Controllers’ Performance
89
6.3.1 Comparison of Output Response for Position
89
6.3.2 Comparison of Output Response for Angle
90
Summary of Chapter 6
92
CONCLUSION AND FUTURE WORK
93
7.1
Introduction
93
7.2
Conclusion
94
7.3
Suggestion for the Future Works
95
xi
REFERENCES
96-99
xii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
4.1
Properties of the inverted pendulum system
24
4.2
Characteristics of the inverted pendulum system
24
5.1
State space model representation
39
5.2
Inputs and outputs of FLC
56
5.3
Standard labels of quantization
56
5.4
Fuzzy rule matrix for position/angle control
60
5.5
Two passes in hybrid learning algorithm
71
6.1
Summarization of the performance characteristics for
cart’s position
6.2
90
Summarization of the performance characteristics for
pendulum’s angle
91
xiii
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
Flow chart of research methodology
20
4.1
Free body diagram of the inverted pendulum system
23
4.2
Velocity analysis of the pendulum
27
5.1
State feedback control configuration
40
5.2
State feedback control configuration with the input gain,
Nbar
42
5.3
Components in the fuzzy logic controller
47
5.4
Fuzzy logic control in the closed-loop system
48
5.5
Input and output for the fuzzy controller
49
5.6
Fuzzification procedure
50
5.7
Example of fuzzy inference using Mamdani method
52
5.8
Defuzzification by center of gravity
54
5.9
Fuzzy logic controllers in the feedback loop of inverted
pendulum system
55
xiv
5.10
Fuzzy set for the input ‘x’
57
5.11
Fuzzy set for the input ‘delx’
57
5.12
Fuzzy set for the output ‘Force-x’
58
5.13
Fuzzy set for the input ‘theta’
58
5.14
Fuzzy set for the input ‘deltheta’
59
5.15
Fuzzy set for the output ‘Force-theta’
59
5.16
Simulink implementation of FLC for the inverted pendulum
system
61
5.17
First order Sugeno fuzzy model
64
5.18
Adaptive Network
65
5.19
ANFIS architecture
67
5.20
ANFIS editor GUI
73
5.21
Simulation model of inverted pendulum based on ANFIS
77
5.22
Initialization of membership functions
78
5.23
The error curve of ANFIS training
79
5.24
The membership functions of variable E
79
5.25
The membership functions of variable EC
80
5.25
The ANFIS network structure
80
6.1
Step response of the pendulum’s angle with LQR controller 84
6.2
Step response of the cart’s position with LQR controller
84
xv
6.3
Step response of the pendulum’s angle with Fuzzy Logic
Controller
6.4
Step response of the cart’s position with Fuzzy Logic
Controller
6.5
86
86
Step response of the pendulum’s angle with ANFIS
Controller
88
6.6
Step response of the cart’s position with ANFIS controller 88
6.7
Comparison of output response of cart’s position
90
6.8
Comparison of output response of pendulum’s angle
91
xvi
LIST OF SYMBOLS
b
-
Friction of cart
l
-
Length to pendulum centre of mass
x
-
Cart position coordinate
x
-
cart acceleration
θ
-
Pendulum angle from the vertical

-
Pendulum angular acceleration
r(s)
-
Reference signal
e(s)
-
Error signal
u(s)
-
Plant input
x
-
State vector
u
-
Input vector
y
-
Output vector
Ts
-
Settling time
Tr
-
Rising time
ess
-
Steady state error
NS
-
Negative small
NM
-
Negative medium
NL
-
Negative large
ZE
-
Zero
PS
-
Positive small
PM
-
Positive medium
PL
-
Positive Large
xvii
LQR
-
Linear Quadratic Regulator
FLC
-
Fuzzy Logic Controller
ANFIS
Adaptive Neural Fuzzy Inference System
GUI
Graphic User Interface
AI
-
Artificial Intelligence
M
-
Mass of cart
m
-
Mass of pendulum
CV
-
Control variable
E
-
Error
SP
-
Set point
PV
-
Process variable
R
-
Step input to the cart
A
-
State matrix
B
-
Input matrix
C
-
Output matrix
D
-
Direct transmission matrix
%OS
-
Percent overshoot
I
-
Inertia of the pendulum
F
-
Force applied to cart