vii
TABLE OF CONTENTS
CHAPTER
TITLE
DECLARATION
DEDICATION
ACKNOWLEDGEMENT
ABSTRACT
ABSTRAK
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
LIST OF ABBREVIATIONS
LIST OF SYMBOLS
LIST OF APPENDICES
LIST OF ALGORITHMS
1
INTRODUCTION
1.1
Introduction
1.2
Motivations
1.3
Background of the Problem
1.4
Optimal Control Problem
1.5
Statement of the Problem
1.6
Objectives of the Research
1.7
Scope of the Research
1.8
Significance of the Study
1.9
Main Contributions
1.10
Thesis Overview
2
LITERATURE REVIEW
2.1
Introduction
2.2
Dynamic Optimization
2.3
Discretization Methods
PAGE
ii
iii
iv
v
vi
vii
x
xi
xiv
xvi
xviii
xix
1
1
2
3
4
6
6
7
7
7
9
10
10
11
15
viii
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
Constraint Handling Methods
System Integration Methods
Global Optimization Methods
Random Numbers
Probability
Direct Stochastic Methodology
Steps for Development of PGSJ
Probabilistic Tools
Steps for Analyzing PGSJ
Improvement and Comparisons
Concluding Remarks
19
22
26
32
33
36
37
38
38
40
41
3
PROBABILISTIC GLOBAL SEARCH METHOD
3.1
Introduction
3.2
The PGSJ Algorithm
3.3
Convergence Analysis
3.4
Numerical Simulation
3.5
Complexity Analysis
3.6
Numerical Direct Strategies
3.7
Control Parameterization Framework
3.8
Implications of Constraints
3.9
Case Studies
3.10
Parameter Selection in PGSJ Algorithm
3.11
Concluding Remarks
42
42
43
55
60
63
64
65
66
66
74
78
4
CONTINUOUS ANT COLONY OPTIMIZATION
4.1
Introduction
4.2
Ant Colony Optimization
4.3
Continuous Ant Colony Optimization
4.4
Diversity in Ant Colony Optimization
4.5
Improvement on Ant Colony Optimization
4.6
Numerical Simulations
4.7
Concluding Remarks
79
79
80
82
85
88
91
102
5
ANALYSIS OF RESULTS AND DISCUSSION
5.1
Introduction
5.2
Comparisons
5.3
Discussions
103
103
103
114
ix
5.4
5.5
5.6
6
Improvements
A Practical Problem
Concluding Remarks
CONCLUSION
6.1
Introduction
6.2
Summary of the Thesis
6.3
Directional Future Research
116
121
125
130
130
130
133
REFERENCES
134
Appendix A
151
x
LIST OF TABLES
TABLE NO.
2.1
3.1
3.2
3.3
4.1
5.1
5.2
TITLE
Some trial parameters and their corresponding value of
performance index regarding Problem (1.3)
The algorithm inputs functions and parameters
Results of testing PGSJ with several benchmark problems
Solution of Example 3.2 using BCP/PGSJ method
Comparing the performance of ACO family
The performance of PGSJ, PGSJ-LS1, PGSJ-LS2, and
PGSJ-LS3 based on the iterative solutions
The performance of PGSJ, PGSJ-LS1, PGSJ-LS2, and
PGSJ-LS3 based on the number of function evaluations
PAGE
17
46
62
70
96
118
119
xi
LIST OF FIGURES
FIGURE NO.
1.1
2.1
2.2
2.3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
TITLE
An illustration of a tunnel-diode oscillator
Illustration of the control graphs corresponding to some
parameters regarding Problem (1.3)
The chart of available approaches for the solution of
constrained OCPs
An illustration of the direct methodology used in this study
A uniform pdf when N = 5, and the domain of the pdf is
[2.5, 5]
An illustration of a general pdf when N = 5, and the
domain of the pdf is [2.5, 5]
An illustration of a pdf after bisecting when N = 5, and
the domain of the pdf is [2.5, 5]
The PGSJ flowchart
The search space
Initialization
Uniform pdf
pdf-updating
Continuing pdf-updating
Bisecting
Continuing the bisecting
The second iteration
pdf-updating
Continuing pdf-updating
Bisecting
The third iteration
The number of the function evaluations while the
dimension is increased
The graph of optimal control for Example 3.1.
The graph of error between exact and computational
control for Example 3.1.
PAGE
5
18
35
39
45
45
48
51
52
52
52
52
53
53
54
54
54
54
54
54
64
68
68
xii
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
The graph of state variables for Example 3.1.
The graph of optimal control for Example 3.2.
The graph of state variables for Example 3.2.
The optimal control for Example 3.3
The optimal value for state variable x1 for Example 3.3
The graph of the constraint for Example 3.3
The optimal control for Example 3.4
The optimal value for state variable x1 for Example 3.4
The optimal value for state variable x2 for Example 3.4
The optimal value for state variable x3 for Example 3.4
The graph of the constraint for Example 3.4
Performance of the ACOR algorithm on example 3.1
Solution of example 3.1 using ACOR algorithm
Performance of the DACOR algorithm on example 3.1
Solution of example 3.1 using DACOR algorithm
Performance of the IACOR -LS algorithm on example 3.1
Solution of example 3.1 using IACOR -LS algorithm
The performance of the pdf–based algorithms against the
exact solution on the solution of Example 3.1
The state variable x1 (t) obtained by the pdf-based methods
on the solution of Example 3.1
Solution of Example 3.1 using the pdf–based algorithms
The performance of the pdf–based algorithms on the
solution of Example 3.2
The state variable x2 (t) obtained by the pdf-based methods
on the solution of Example 3.2
The state variable x1 (t) obtained by the pdf-based methods
on the solution of Example 3.2
Solution of Example 3.2 using the pdf-based algorithms
The performance of the pdf–based algorithms on the
solution of Example 3.3
Solution of Example 3.3 using the pdf-based algorithms
The constraint of Example 3.3 obtained by the pdf–based
algorithms
The state variable x1 (t) obtained by the pdf-based methods
on the solution of Example 3.3
The state variable x2 (t) obtained by the pdf-based methods
on the solution of Example 3.3
69
71
71
73
74
75
75
76
76
77
77
98
99
99
100
101
101
104
104
105
106
106
107
107
108
108
109
109
110
xiii
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
5.29
5.30
5.31
5.32
The state variable x4 (t) obtained by the pdf-based methods
on the solution of Example 3.3
The state variable x5 (t) obtained by the pdf-based methods
on the solution of Example 3.3
The performance of the pdf–based algorithms on the
solution of Example 3.4
Solution of Example 3.4 using the pdf-based algorithms
The constraint of Example 3.4 obtained by the pdf–based
algorithms
The state variable x1 (t) obtained by the pdf-based methods
on the solution of Example 3.4
The state variable x2 (t) obtained by the pdf-based methods
on the solution of Example 3.4
The state variable x4 (t) obtained by the pdf-based methods
on the solution of Example 3.4
The iterative solutions used to evaluate the performance of
PGSJ, PGSJ-LS1, PGSJ-LS2, and PGSJ-LS3
The iterative solutions used to evaluate the performance of
PGSJ, PGSJ-LS1, PGSJ-LS2, and PGSJ-LS3
The performance of PGSJ, PGSJ-LS1, PGSJ-LS2, and
PGSJ-LS3 based on the number of function evaluations
The performance of PGSJ, PGSJ-LS1, PGSJ-LS2, and
PGSJ-LS3 based on the number of function evaluations
The control variable u1 (t) obtained by the PGSJ and PGSJLS methods on the solution of Problem (5.1)
The control variable u2 (t) obtained by the PGSJ and PGSJLS methods on the solution of Problem (5.1)
The state variable x5 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
The state variable x1 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
The state variable x2 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
The state variable x3 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
The state variable x4 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
The state variable x6 (t) obtained by the PGSJ and PGSJ-LS
methods on the solution of Problem (5.1)
110
111
111
112
112
113
113
114
120
120
120
124
125
126
126
127
127
128
128
129
xiv
LIST OF ABBREVIATIONS
ACO
-
Ant Colony Optimization
ACK
-
Ackleys Problem
ARS
-
Adaptive Random Search
BB
-
Branch and Bound
BC
-
Bee Colony
BCP
-
Bernstein based control parameterization
CRS
-
Controlled Random Search
CM
-
Cosine Mixture Problem
EA
-
Evolutionary Algorithms
GA
-
Genetic Algorithm
GARS
-
Generalized Adaptive Random Search
GW
-
Griewank Problem
HAS
-
Hesitant Adaptive Searches
IDP
-
Iterative Dynamic Programming
IVP
-
Initial Value Problem
LHS
-
Latin Hypercube Sampling
LM1
-
Levy and Montalvo 1 Problem
LM2
-
Levy and Montalvo 2 Problem
LMM
-
Linear Multistep Methods
MS
-
Monkey Search
NLP
-
Nonlinear Programming Problem
NP
-
Nested Partitions
NLP
-
Nonlinear Programming Problem
OCP
-
Optimal Control Problem
pdf
-
probability density function
pdfs
-
probability density functions
PGSJ
-
Probabilistic Global Search Johor
PGSL
-
Probabilistic Global Search Lausanne
xv
PRS
-
Pure Random Searches
PSO
-
Particle Swarm Optimization
SA
-
Simulated Annealing
SIN
-
Sinusoidal Problem
SIVP
-
Stiff Initial Value Problem
TS
-
Taylor Series
xvi
LIST OF SYMBOLS
A
-
The acceptable probability density
b
-
The number of bisecting procedure
d
-
Dimension of the problem
D
-
The box of feasible controls
f
-
The objective function,
H
-
Hamiltonian function
Ini
-
The ith interval in the nth iteration
Inij
-
The jth subinterval of the ith interval in the nth iteration
M
-
Maximum number of iterations
N
-
The number of partitions on each interval
P
-
Probability of sampling from complementary search space
S
-
The number of samples in each iteration
t0
-
Initial time
tf
-
Final time
u
-
The control curve
u∗
-
The optimal control curve
x
-
The state variable
-
The optimal state variable
ẋ
-
The first derivative of the state variable
x0
-
Initial value of the state variable
Ω
-
The box of feasible region
σ
-
The scale factor
ϵ
-
The accuracy required
ξ
-
Increment in probability updating procedure
κ
-
The maximum number of updating iterations
λ
-
Adjoint variable
µ
-
Multiplier variable
λ∗
-
The optimal adjoint variable
x
∗
xvii
µ∗
-
The optimal multiplier variable
χ
-
The characteristic function
Ωc
-
The Complementary search space
π
-
Projection function
xviii
LIST OF APPENDICES
APPENDIX
A
TITLE
List of Publications
PAGE
151
xix
LIST OF ALGORITHMS
ALGO. NO.
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
5.1
5.2
5.3
5.4
5.5
TITLE
The sampling operator of PGSJ algorithm
The pdf-updating operator of PGSJ algorithm
The bisecting operator of PGSJ algorithm
The Scaling operator of PGSJ algorithm
The PGSJ algorithm
The GARS algorithm
The ACO framework
The ACOR algorithm
The construct solutions operator of the DACOR algorithm
The DACOR algorithm
The line search operator of the MTS-LS1 algorithm
The MTS-LS1 algorithm
The Line search operator of the IACOR -LS algorithm
The construct solutions operator of the IACOR -LS
algorithm
The archive growth operator of the IACOR -LS algorithm
The IACOR -LS algorithm
The line search operator of the MTS-LS2 algorithm
The MTS-LS2 algorithm
The line search operator of the gaussian line search
algorithm
The gaussian line search algorithm
The PGSJ–LS algorithm
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49
49
50
56
81
86
88
89
91
92
92
93
93
94
116
117
121
122
123