ABSTRACT - studentsrepo.um.edu.mystudentsrepo.um.edu.my/8241/3/SECOND_PART_OF_THESIS.pdfSatu contoh yang biasa adalah fungsi gelombang kecil Haar yang digunakan dalam tesis ini untuk

i

ABSTRACT

Several computational methods have been proposed to solve optimal control problems.

These methods are classified either as direct or indirect methods. This thesis is based on

solving optimal control problems by using both direct and indirect methods.

Orthogonal functions have been used to solve various problems of dynamic systems. A

typical example is the Haar wavelet function, which is used in this work to convert the

underlying differential equations in an optimal control problem into a system of linear

algebraic equations.

To establish an indirect method, we propose a novel feedback control method that uses

only linear systems to solve affine nonlinear control system with quadratic cost function

and infinite time horizon. This method is a numerical technique that is based on the

combination of Haar wavelets operational matrices and successive Generalized

Hamilton-Jacobi-Bellman (GHJB) equation. This method improves the closed-loop

performance of stabilizing controls and reduces the problem of solving a nonlinear

Hamilton-Jacobi-Bellman (HJB) equation to solve the corresponding GHJB equation.

An interesting fact is that when the process of improving the controls and solving GHJB

equation is iterated, the solution to the GHJB equation converges uniformly to the

solution of the HJB equation which is in the form of the gradient of the Lyapunov

function )(xV . The Lyapunov function )(xV is the measure of the performance index,

which can be determined by integrating )(xV parallel to the axes. In the process of

establishing this novel feedback control method, we have to define new operational

matrices of integration for a chosen stabilizing domain ),[ and a new operational

matrix for the product of two dimensional Haar wavelet functions.

ii

To establish a direct method, an efficient new algorithm is proposed to solve nonlinear

optimal control problems with a finite time horizon under inequality constraints. In this

technique, we parameterize both the states and the controls by using Haar wavelet

functions and Haar wavelet operational matrix. The nonlinear optimal control problem

is converted into a quadratic programming (QP) problem through the quasilinearization

iterative technique. The inequality constraints for trajectory variables are transformed

into quadratic programming constraints by using the Haar wavelet collocation method.

The quadratic programming problem with linear inequality constraints is then solved by

using standard QP solver.

Both proposed numerical methods have been applied to several examples. The proposed

methods obtain better or comparable results compared with other established methods.

Moreover, the methods are attractive, stable, convergent and easily coded.

The direct method has been applied in this thesis to solve a practical optimal control

problem. This problem is the multi-item production-inventory model with stock-

dependent deterioration rates and deterioration due to self-contact and the presence of

the other stock. The problem is addressed by using four different types of demand rates

namely, constant, linear, logistic and periodic demand rates. The solution to the model

is discussed numerically and displayed graphically. By enhancing the resolution of the

Haar wavelet, we can improve the accuracy of the states, controls and cost. Simulation

results were also compared with those obtained by other researchers.

iii

ABSTRAK

Beberapa kaedah pengiraan telah dicadangkan untuk menyelesaikan masalah kawalan

optimum. Kaedah-kaedah ini dikelaskan sama ada sebagai kaedah langsung atau kaedah

tidak langsung. Tesis ini adalah berdasarkan kepada menyelesaikan masalah kawalan

optimum dengan menggunakan kedua-dua kaedah langsung dan tidak langsung.

Fungsi ortogon telah digunakan untuk menyelesaikan berbagai masalah dalam sistem

dinamik. Satu contoh yang biasa adalah fungsi gelombang kecil Haar yang digunakan

dalam tesis ini untuk menukar asas persamaan pengamiran sandaran dalam masalah

kawalan optimum kepada suatu sistem persamaan aljabar linear.

Bagi mewjudkan kaedah langsung, kami mencadangkan kaedah kawalan suap balik

yang asli yang menggunakan hanya sistem linear untuk menyelesaikan sistem kawalan

afin tak linear dengan fungsi kuadratik dan ufuk masa tak terhingga. Kaedah ini adalah

satu teknik berangka yang berasaskan gabungan matriks operasi gelombang Haar dan

persamaan teritlak Hamilton-Jacobi-Bellman (HJB) berturutan. Kaedah ini

meningkatkan prestasi gelung tertutup kawalan stabil dan menurunkan masalah

menyelesaikan persamaan tak linear Hamilton-Jacobi-Bellman (HJB) kepada

menyelesaikan persamaan GHJB yang sepadan. Satu fakta menarik ialah apabila proses

meningkatkan kawalan dan menyelesaikan persamaan GHJB dilelarkan, penyelesaian

untuk persamaan GHJB itu menmpu secara seragam ke penyelesaian bagi persamaan

HJB dalam bentuk kecerunan fungsi Lyapunov )(xV . Fungsi Lyapunov )(V x adalah

ukuran bagi indeks prestasi, yang boleh ditentukan dengan mengkanirkan )(xV selari

kepada paksi- paksi. Dalam proses mewujudkan kaedah kawalan suap balik yang asli ini

kami perlu menentukan matriks operasi pengamiran yang untuk domain perstabilan

, menstabilkan yang dipilih dan matriks operasi baru bagi hasil darab fungsi

gelombang kecil Haar dalam dua dimensi.

iv

Untuk mewujudkan kaedah langsung, satu algoritma baru yang cekap telah dicadangkan

untuk menyelesaikan masalah kawalan optimum tak linear dengan tempoh masa yang

terhad di bawah kekangan ketahsamaan. Dalam teknik ini, kami berparameter kedua-

dua keadaan dan kawalan menggunakan fungsi gelombang kecil Haar dan matriks

operasi gelombang kecil Haar. Masalah kawalan optimum tak linear ditukar menjadi

masalah pengaturcaraan kuadratik (QP) melalui teknik lelaran kuasilinear. Selain itu,

Kekangan ketahsamaan bagi pembolehubah trajektori diubah menjadi kekangan

pengaturcaraan kuadratik menggunakan kaedah kolokasi gelombang kecil Haar.

Masalah pengaturcaraan kuadratik dengan kekangan ketaksamaan linear kemudiannya

diselesaikan menggunakan solver QP yang biasa.

Kedua-dua kaedah berangka yang dicadangkan telah digunakan pada beberapa contoh.

Kami mendapati bahawa kaedah yang dicadangkan mendapatkan keputusan yang lebih

baik atau yang setanding berbanding dengan kaedah biasa yang lain. Juga, kaedah ini

adalah menarik, stabil, menumpu dan mudah dikodkan.

Dalam tesis ini, kaedah langsung telah digunakan untuk menyelesaikan suatu masalah

kawalan optimum praktis. Masalahnya ialah model inventori-pengeluaran pelbagai item

dengan kadar kemerosotan bergantung kepada stok dan kemerosotan disebabkan saling

bersantuhan dan kehadiran stok lain. Masalah ini diatasi dengan menggunakan empat

jenis kadar permintaan iaitu kadar permintaan tetap, linear, logistik dan berkala.

Penyelesaian kepada model dibincangkan secara berangka dan dipaparkan secara grafik.

Dengan meningkatkan resolusi gelombang kecil Haar, kami boleh meningkatkan

ketepatan keadaan, kawalan dan kos. Keputusan simulasi juga dibandingkan dengan

hasil yang diperolehi oleh penyelidik lain

v

ACKNOWLEDGEMENTS

Alhamdulillah, Praise to Allah for His Blessings and without His Will, this study will

never be completed.

I would like to express my utmost gratitude to my supervisor, Dr. Amran Hussin,

for guided me in the genesis of this thesis and had shown great dedication in helping me

through the completion of this study. His kindness and patience in supplementing my

knowledge is unparalleled.

With deep sense of gratitude, I would like to thank my parents and family members

who have always supported me throughout the period. Without their tremendous love

and support, I would not have been able to concentrate on my study and endure some

difficult times through all these years.

Special thank also goes to my friends; Huda Abdualrazzaq, Walaa Najm, Manal

Sharbatily, Dr. Rabha Ibrahim, and others whom have always directly or indirectly

motivate me all along this journey.

I also would like to gratefully acknowledge the support given by the staff members

of the Institute of Mathematical Sciences, University of Malaya especially Mrs. Puan

Budiyah, Mr. Malik for their support and assistance.

Finally, I would like to thank everybody who was important to the successful

realization of thesis, as well as expressing my apology that I could not mention

personally one by one.

vi

TABLE OF CONTENTS

Page

ABSTRACT i

ABSTRAK iii

ACKNOWLEDGEMENT v

TABLE OF CONTENTS vi

LIST OF FIGURES xi

LIST OF TABLES xvi

LIST OF SYMBOLS AND ABBREVIATIONS xviii

1 INTRODUCTION

1.1 Overview of Thesis 1

1.2 Motivation 6

1.3 Scope of the Study 8

1.4 Research Objectives 9

1.5 Organization of the Thesis 10

2 LITERATURE REVIEW 13

3 THE HAAR WAVELET METHOD

3.1 Introduction 27

3.2 Haar Wavelet Function 29

3.3 Haar Series Expansion 33

3.4 Matrix of One-dimensional Haar Wavelets 36

vii

3.5 Operational Matrix for Integrating One-dimensional Haar

Wavelets

39

3.6 Operational Matrix of the Product of One-dimensional Haar

Wavelet Vectors

42

3.7 Operational Matrix of the Integral Products of One-dimensional

Haar Wavelet Vectors

44

3.8 Matrix of Two-dimensional Haar Wavelets 46

3.9 Approximation of Function for Two-dimensional Functions 49

3.10 Operational Matrices for Two-dimensional Haar Wavelets 50

3.10.1 Partial Integration of Function Vectors of Two-

dimensional Haar Wavelets with Respect to 1x

51

3.10.2 Partial Integration of Function Vectors of Two-

dimensional Haar Wavelets with Respect to 2x

57

3.11 Operational Matrix of the Product of Two-dimensional Haar

Wavelet Vectors

63

3.12 Algorithm to Compute the Operational Matrix of the Product of

Two-dimensional Haar Wavelet Vectors

71

3.13 Conclusion 74

4 INDIRECT METHOD NONLINEAR OPTIMAL CONTROL

PROBLEMS

4.1 Introduction 75

4.2 Problem Statement 76

4.3 Generalized Hamilton-Jacobi-Bellman Equation 77

4.3.1 Principle of Optimality 77

viii

4.4 Successive Generalized Hamilton-Jacobi-Bellman Equation 82

4.4.1 Algorithm of the Successive GHJB Equation 84

4.5 Approximation Functions via Haar Wavelets Approximation 86

4.6 Successive Haar Wavelet Collocation Method 92

4.7 Numerical Results 102

4.7.1 Example 1 102

4.7.2 Example 2 105

4.7.3 Example 3 107

4.7.4 Example 4 112

4.7.5 Example 5 115

4.8 Numerical Discussions 121

4.9 Conclusion 123

5 DIRECT METHOD CONSTRAINED OPTIMAL CONTROL

PROBLEMS

5.1 Introduction 124

5.2 Problem Statement 125

5.3 Proposed Method 126

5.4 Numerical Solution to the Nonlinear Optimal Control Problem 127

5.4.1 Quasilinearization Technique 127

5.4.2 Optimal Control Problem using Haar Wavelet Method 128

5.4.2.1 Parameterization using Haar Wavelet Functions 128

5.4.2.2 Approximation of the Performance Index 131

ix

5.4.2.3 Approximations of System Dynamics 136

5.4.2.4 Approximations of Equality Constraints 139

5.4.2.5 Approximations of Inequality Constraints 140

5.5 Numerical Results and Discussions 142

5.5.1 Linear Optimal Control Problems 143

5.5.1.1 Example 1 143

5.5.1.2 Example 2 147

5.5.2 Nonlinear Optimal Control Problems 151

5.5.2.1 Example 3 151

5.5.2.2 Example 4 159

5.6 Conclusion 164

6 APPLICATION OF HAAR WAVELET METHOD TO

PRODUCTION-INVENTORY MODEL

6.1 Introduction 166

6.2 Optimal Control of Two-Item Production- Inventory Model 167

6.3 Reformulate of the Optimal Control of Two-Item Production-

Inventory Model

169

6.4 Numerical Solution 170

6.4.1 Constant Demand Rates 170

6.4.2 Linear Demand Rates 175

6.4.3 Logistic Demand Rates 180

6.4.4 Periodic Demand Rates 189

x

6.5 Numerical Discussions and Conclusion 193

7 CONCLUSION AND FUTURE WORK

7.1 Conclusion 196

7.2 Future Work 199

REFERENCES 201

LIST OF PUBLICATIONS AND SEMINARS 212

APPENDIX

A KRONECKER PRODUCT 213

B MATLAB CODE FOR INDIRECT AND DIRECT METHODS 218

xi

LIST OF FIGURES

Figure Title Page

3.1 First four Haar functions in the interval of ( 10 x ) 30

3.2 First four Haar functions in the interval of ( 11 x ) 31

3.3 Collocation point 36

4.1 Phase flow plotted against lines of constant cost (Beard, 1995) 82

4.2

Algorithm of the successive generalized Hamilton-Jacobi Bellman

equation

85

4.3 Optimal feedback control for Example 1 via the SHWCM with

16,8 m and 16

1

8

11 ,x , respectively

104

4.4 Value cost function for Example 1 via the SHWCM with

32,16,8 m and , respectively

105

4.5 Value cost function for Example 2 via the SHWCM with 16,8 m 106


16,8 m

107


32,16,8 m and 32

1

16

1

8

11 , ,x , respectively

108



109

4.9 Approximate solution for cost function with 32m and 001.0

for Example 3

110

4.10 Exact solution for cost function with 32m for Example 3 110

4.11 Approximate solution for optimal feedback control via the SHWCM

with 32m , 001.0 and 14 iterations for Example 3

111

4.12 Exact solution for optimal feedback control with 32m for

Example 3

113

32

1

16

1

8

11 , ,x

32

1

16

1

8

11 , ,x

xii


32,16,8 m and 32

1

16

1

8

11 , ,x , respectively

113



114

4.15 Some state trajectories for Example 4 115

4.16 First optimal feedback control for Example 5 via the SHWCM with

64,32,16,4 m and 63

1

31

1

15

1

3

11 , , ,x , respectively

117

4.17 Second optimal feedback control for Example 5 via the SHWCM wit

64,32,16,4 m and 63

1

31

1

15

1

3


118


64,32,16,4 m and 63

1

31

1

15

1

3


119

4.19 Approximate solution for cost function with 64m , 01.0 and

31 iterations for Example 5

120

4.20 Approximate solution for first optimal feedback control with

64m , 01.0 and 31 iterations for Example 5

120

4.21 Approximate solution for second optimal feedback control with

64m , 01.0 and 31 iterations for Example 5

121

5.1 State variable )(1 tx for Haar wavelet resolutions 8643 ,,, 2 2 2 2m

and 10ft obtained from Example 1

145

5.2 State variable )(2 tx with for Haar wavelet resolutions

8643 ,,, 2 2 2 2m and 10ft obtained from Example 1

146

5.3 Control variable )(tu with Haar wavelet resolutions 643 ,, 2 2 2m

and 10ft obtained from Example 1

147

5.4 State variables )(1 tx , )(2 tx and inequality constraint )(tr obtained

in Example 2 for 128m and 1ft

149

32

1

16

1

8

11 , ,x

xiii

5.5 Control variable )(tu obtained in Example 2 for 128m and 1ft 150

5.6 State variable )(1 tx of Example 3 for 5 quasilinearization iterations

with 64m and 5ft

153

5.7 State variable )(2 tx of Example 3 for 5 quasilinearization iterations

with 64m and 5ft

154

5.8 Control variable )(tu of Example 3 for 5 quasilinearization iterations

with 64m and 5ft

155

5.9 State variable )(1 tx obtained in Example 3 for 543 ,, 2 2 2m and 6

2

with 5ft

156

5.10 State variable )(2 tx obtained in Example 3 for 543 ,, 2 2 2m and

62 with 5ft

157

5.11 Control variable )(tu obtained in Example 3 for 543 ,, 2 2 2m and

62 with 5ft

158

5.12 States variables )(1 tx and )(2 tx obtained in Example 4 for 128m

and 5ft

162

5.13 Control variable )(tu obtained in Example 4 for 128m and 5ft 163

6.1 First inventory level against time 5ft m = 256 using constant

demand rates and for k=1, 2, 3 quasilinearization iterations

172

6.2 Second inventory level against time 5ft and m = 256 using

constant demand rates for k=1, 2, 3 quasilinearization iterations

173

6.3 First production rate against time 5ft and m = 256 using constant

demand rates for k=1, 2, 3 quasilinearization iterations

174

6.4 Second production rate against time 5ft and m = 256 using

constant demand rates for k=1, 2, 3 quasilinearization iterations

175

6.5 First inventory level against time 5ft and m = 256 of the

application of the direct method using linear demand rates for k=1, 2,

3, 4, 5, 6 quasilinearization iterations

177

xiv

6.6 Second production rate against time 5ft and m = 256 of the



178

6.7 First production rate against time 5ft and m = 256 of the



179




180

6.9 First inventory level against time 5ft and m = 256 of the

application of the direct method using logistic demand rates for k=1,

2, 3, 4, 5, 6, 7 quasilinearization iterations

182

6.10 Second inventory level against time 5ft and m = 256 of the



183

6.11 First production rate against time 5ft and m = 256 of the


2, 3, 4, 5, 6,7 quasilinearization iterations

184




185

6.13 First inventory level against time with 5ft and 6543 ,,, 2 2 2 2m

and 72 using logistic demand rates

186

6.14 Second inventory level against time with 5ft and m = 8, 16, 32,

64, 128 using logistic demand rates.

187

6.15 First production rate against time with 5ft and 6543 ,,, 2 2 2 2m

and 72 using logistic demand rates.

188

6.16 Second production rate against time with 5ft and

6543 ,,, 2 2 2 2m and 72 using logistic demand rates.

189

6.17 First inventory level against time with 5ft and 6543 ,,, 2 2 2 2m

and 72 using periodic demand rates.

190

xv

6.18 Second inventory level against time with 5ft and

6543 ,,, 2 2 2 2m and 72 using periodic demand rates.

191

6.19 First production rate against time with 5ft and 6543 ,,, 2 2 2 2m

and 72 using periodic demand rates

192

6.20 Second production rate against time with 5ft and

6543 ,,, 2 2 2 2m and 72 using periodic demand rates

193

xvi

LIST OF TABLES

Table Title Page

4.1 Iteration results )(iu for Example 1 when 8m and

8

11 x

103

4.2 Iteration results )(V i for Example 1 when 8m and

8

11 x

103

4.3 Numerical comparison of feedback control methodologies in

Example 5 at initial condition )1,1(0 x , (Beeler at el., 2000)

116

5.1 Results of the performance index for Example 1 with resolution of

Haar wavelet 8m ,16 ,32 , 64 ,128 and 256

144

5.2 Results of the performance index for Example 2 for Haar wavelet

resolutions 8m ,16 ,32 , 64 ,128 and 256

149

5.3 Simulation results of the performance index for Example 2

(Vlassenbroeck, 1988)

151

5.4 Optimal value of the performance index and convergence error for

Example 3 with Haar wavelet resolution 64m

153

5.5 Results of the performance index for Example 3 with Haar wavelet

resolutions 128,64,32,16,8 m and 256

156

5.6 Simulation results of the performance index for Example 3

(Vlassenbroeck, 1988)

159

5.7 Results of the performance index for Example 4 with Haar wavelet

resolution for 8m ,16 ,32 , 64 ,128 and 256

161


Example 4 with resolution of Haar wavelet 128m

161

5.9 Simulation results of the performance index for Example 4 163

6.1 Values of system parameters and the initial states (El-Gohary &

Elsayed, 2008)

170

6.2 Simulation results of the application of the direct method using

constant demand rates for m = 8, 16, 32, 64, 128, and 256

172

xvii

6.3 Simulation results of the application of the direct method using linear

demand rates for m = 8, 16, 32, 64, 128, and 256

176


the application of the direct method using linear demand rates for

resolution of Haar wavelet m =256

176


logistic demand rates for m = 8, 16, 32, 64, 128, and 256

181


application of the direct method using logistic demand rates for

resolution of Haar wavelet 256m

181


periodic demand rates for m = 8, 16, 32, 64, 128, and 256

190

6.8 Summarize result obtained from El-Gohary and Elsayed, (2008)

method for application with four types of demand rates

195

xviii

LIST OF SYMBOLS AND ABBREVIATIONS

u Optimal control

jx Collocation points

x State variable

x Optimal infinite horizon state trajectory corresponding to V*

J Performance index

0x Initial condition vector

fx Final condition vector

u An arbitrary admissible control

m Level of Haar wavelet

t Time

ft Finite time

0t Initial time

1n

Number of state variables

2n Number of control variables

T Transpose operation

mH Haar wavelet matrix

essH Hessian matrix

)(0 th Haar scaling function

)(1 th

Haar mother wavelet function

mP Haar wavelet operational matrix

)(tmh Vector of Haar wavelet functions

xix

ic Haar series coefficient

id Haar series coefficient

Q Positive semi-definite matrix

nI Identity matrix

R

Symmetric positive definite matrix

M(c) Operational matrix of product one dimension

D)N( Operational matrix of product two dimensions

LQP Linear quadratic programming problem

Eqn. Equation

HJB Hamilton-Jacobi-Bellman Equation

GHJB Generalize Hamilton-Jacobi-Bellman Equation

V(i) thi performance index in successive GHJB equation

)(iu thi control in successive GHJB equation

V*

Solution of the HJB equation

Compact subset of n containing a ball around the origin

x

V Row vector of partial derivatives of V ,

nxxx

V , ,

V ,

V

21

SHWCM Successive Haar wavelet collocation method

Documents

ABSTRACT - studentsrepo.um.edu.mystudentsrepo.um.edu.my/8241/3/SECOND_PART_OF_THESIS.pdfSatu contoh yang biasa adalah fungsi gelombang kecil Haar yang digunakan dalam tesis ini untuk