### NONLINEAR STOCHASTIC OPERATORS TO CONTROL THE CONSENSUS PROBLEM IN MULTI-AGENT

### SYSTEMS

### BY

### RAWAD A. A. ABDULGHAFOR

### A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy in Information Technology

### Kulliyyah of Information & Communication Technology International Islamic University Malaysia

### 2017

ii

**ABSTRACT **

Consensus problem in multi-agent systems (MAS) has attracted growing interest in recent time. Traditional approaches to controlling consensus problem are often based on linear models, which take their origin from the well known DeGroot model. Various researchers in recent time have proposed nonlinear models simply due to the fact that linear models converge slowly, are characterized with higher number of iterations and at the same time incapable of converging to optimal consensus. Nonlinear models on the other hand, are more efficient, converging faster with lesser number of iterations and to approximate optimal consensus.

However, the downside of nonlinear models is that they are often of higher complexity and are setup with restricted conditions. The present concern is to investigate possible nonlinear models with faster convergence to optimal consensus yet with relatively low complexity and more flexible system conditions. The main aim of this research is to control the consensus problem in MAS via doubly stochastic quadratic operator (DSQO). In this research, a nonlinear model of DSQO using majorization theory is investigated to control consensus problem under more flexible conditions as well as lower complexity for specific subclass of the DSQO. The extreme points of doubly stochastic quadratic operator (EDSQO) is then examined for lower complexity. The EDSQO is considered in this case as a special subclass of DSQO sets points in space. Therefore, the vertices of this class are in turn examined to formulate a general solution for the convergence problem. The study focuses on the EDSQO on finite-dimensional simplex (FDS). The general theorems for the DSQO on finite dimensions are derived and presented particularly for finite number of agents. The work carries out a study to define the extreme points of the set of EDSQO on two-dimensional simplex (2DS) and examines the limit behaviour of the trajectories of the EDSQO on FDS. The work then follows up with the dynamic classifications of DSQO on FDS, so as to elaborate a platform class of positive DSQO (PDSQO) which are suitable for reaching a consensus for MAS. The DSQO algorithms are then evaluated based on various kinds of transition matrices and the results are compared to the existing classical consensus algorithms. This presentation also includes the study of higher DSQO degrees (HDSQO) as well as fractional DSQO degrees (FDSQO) for consensus problem in MAS. The research work derives novel low-complexity nonlinear convergence models MPDSQO and MEDSQO which are modified from the DSQO and EDSQO for consensus problem in MAS. In general, the proposed novel protocols of the DSQO have shown and proved to be advantageous over the existing linear and other nonlinear models.

iii

**ثحبلا ةصلاخ **

### تبذتجا ارد

### ( ليكولا ةددعتم مظن في قفاوتلا ةلكشم ةس MAS

### مظعم .ةيملعلا ثابحلاا في ةيرخلأا ةنولآا في ديازتلما مامتهلاا )

### تيلاو ،ةيطلخا ةيضيارلا جذامنلا ساسأ ىلع ةينبم ةقباسلا تاساردلا في ةحترقلما قفاوتلا ةلكشم ىلع مكحتلل ةيديلقتلا بيلاسلأا جذونم لىا اهلصأ دوعي DeGroot

### . قا جذامنلا نلأ ارظن ةيطلخا يرغلا ةيضيارلا جذامنلا ةيرخلأا ةنولآا في نوثحابلا ضعب حتر

### ةهج نم ةيطلخا يرغلا ةيضيارلا جذامنلا .لثمأ زكرتم لىإ براقتلا زجعو ،تاراركتلا نم بركأ ددع عم ،ءطبب براقتت ةيطلخا ةيضيارلا قأ ددع عم عرسأ لكشب قفتتو زكرمتت ،ةءافك رثكأ يه ،ىرخأ كنه نإف ،كلذ عمو .لثمأ زكرتم لىا براقتتو تاراركتلا نم ل

### ليالحا قلقلا .ةديقم طورش عمو اديقعت رثكأ نوكت ام ابلاغ انهأ وه ةيطخ يرغلا ةيضيارلا جذامنلا في ايبلس ابناج تيلا تيادحتلاو

### براقت عم ةيطخ يرغ ةيضيار جذانم قيقتح ةيفيك وه ةلكشلما هذه لح في ينثحابلا هجاوت اديقعت لقا ةيلمع عم لثما قفاوت لىا عرسا

### فعاضلما يئاوشعلا ةيناثلا ةجردلا نم لغشلما ىعدت ةيطخ يرغ ةيضيار جذانم قيقتح تم ،ثحبلا اذه في .ةنورم رثكأ ماظن طورشو ( DSQO ةيرظن مادختسبا )

### majorization ضافنخا نع لاضف ةنورم رثكأ فورظ لظ في قفاوتلا ةلكشم ىلع ةرطيسلل

### لا فنصت ،ماع لكشب .تاديقعت DSQO

### و .ةساردلل عساو ادج لاامج EDSQO

### نم ةصاخ ةيعرف ةئف وه DSQO

### ددتح

### ممقلا ةطقن فعاضم يئاوشعلا ةيناثلا ةجردلا نم لغشم نم )هفرطتلما( ممقلا طاقن صحف تم ةيادب ،كلذلو .ءاضفلا في

### ( EDSCO ا هذه نم ممقلا صحف تم دقف .هرابتخلا ديقعتلا ةلق ببسل )

### براقتلا ةلكشلم ماع لح ةغايصل ةئفل DSQO

### .

### ىلع ةساردلا زكرتو EDSQO

### ل ةماعلا تيارظنلا تمدق ثم نم .داعبلأا يئانث لكش ىلع DSQO

### داعبا ةدع لاكشلأ

### .ةددمح ءلامع ةدعل قيبطت جذومنك ةصاخ )ةدودعم(

### ام وا كولس ةسارد لىإ ةجالحا يه ةيطلخا يرغ جذونم في ةيزكرلما ةلكشلما

### ىمسي ةئف في يهتنت لم ةمئاق تلاز ام ةلكشلما هذه .ةمظنلاا هذه كولس ةيانه DSQO

### وه ثحبلا اذه نم يسيئرلا فدلها .

### يضيارلا جذومنلا برع ءلاكولا ددعتم ماظن في قفاوتلا ةلكشم ىلع مكحتلل لثما لح دايجا DSQO

### هذه في ةددلمحا فادهلأا .

### وممج نم ممقلا طاقن فيرعتو ديدتح يه ةساردلا نم ةع

### EDSQO يئانث داعبا لكش ىلع

### 2 DS كولس ةيانه ةساردل ،

### لا تاراسم EDSQO

### نم ةيكيمانيدلا تافينصتلا نع ثحبلبا موقنس ،فادهلاا قيقحتل لاصاوتو . DSQO

### ددعتم لكشل

### نم تافينصتلا نم ةئف عضوو ،داعبلاا DSQO

### ( بجولما PDSQO ءارلآا في قفاوت لىإ لصوتلل ةبسانم نوكت فوس تيلاو )

### ل ثحبلا اذه في ةحترقلما ةيمزراولخا مييقتل كلذكو ،ءلاكولا ددعتم ماظن نأشب DSQO

### قيقحتل ةقباس تايمزراوخ عم ةنراقبم

### تايئانه ةساردل ىرخلأا فادهلأا لمشتو .ةيلاقتنلاا تافوفصلما نم ةفلتمخ عاونأ ىلع ةدمتعم قفاوتلا DSQO

### سأ تاجردب

### ( ىلعأ HDSQO ( ةيرسك سأ ةجردب اضياو ،)

### FDSQO اضيا ثحبلا اذه جتنأ .ءلاكولا ددعت ماظن في قفاوتلا ةلكشلم )

### ديقعتلا ةضفخنم ةديدج ةيطخلا ةيضيار زكرتم جذانم MPDSQO

### و MEDSQO نم ليدعت

### DSQO و

### EDSQO

### نم ةحترقلما ةديدلجا تلاوكوتوبرلا ،ماع لكشب .ءلاكولا ددعت ماظن في قفاوتلا ةلكشلم DSQO

### ياازم ترهظأ ثحبلا اذه في

### .ىرخلأا ةيطلخا يرغ ةيضيارلا جذامنلاو يطلخا يضيارلا جذومنلا نم رثكأ لضفأ

iv

**APPROVAL PAGE **

This thesis of Rawad A. A. Abdulghafor has been approved by the following:

____________________________

Sherzod Turaev Supervisor

____________________________

Akram Zeki Co-Supervisor

____________________________

Azzeddine Messikh Internal Examiner

____________________________

Mohamed Othman Internal Examiner

____________________________

Adel Al-Jumail External Examiner

____________________________

Ismaiel Hassanein Chairman

v

**DECLARATION **

I hereby declare that this thesis is the result of my own investigation, except where otherwise stated. I also declare that it has not been previously or concurrently submitted as a whole for any other degrees at IIUM or other institutions.

Rawad A. A. Abdulghafor

Signature………. Date …...

vi

**COPYRIGHT PAGE **

**INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA **

**DECLARATION OF COPYRIGHT AND AFFIRMATION OF ** **FAIR USE OF UNPUBLISHED RESEARCH **

**NONLINEAR STOCHASTIC OPERATORS TO CONTROL THE ** **CONSENSUS PROBLEM IN MULTI-AGENT SYSTEMS **

I declare that the copyright holders of this dissertation are jointly owned by the student and IIUM.

Copyright © 2017 (Rawad A. A. Abdulghafor) and International Islamic University Malaysia. All rights reserved.

No part of this unpublished research may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission of the

copyright holder except as provided below

1. Any material contained in or derived from this unpublished research may be used by others in their writing with due acknowledgement.

2. IIUM or its library will have the right to make and transmit copies (print or electronic) for institutional and academic purposes.

3. The IIUM library will have the right to make, store in a retrieved system and supply copies of this unpublished research if requested by

other universities and research libraries.

By signing this form, I acknowledged that I have read and understand the IIUM Intellectual Property Right and Commercialization policy.

Affirmed by Rawad A. A. Abdulghafor

……..……….. ………..

Signature Date

vii

*This thesis is dedicated to my beloved parents *

viii

**ACKNOWLEDGEMENT **

Words are indeed inadequate to express my profound gratitude to my supervisor and mentor, Asst. Prof Dr. Sherzod Turaev, for his relentless efforts in motivating, guiding and supporting me in every form morally possible during the course of this study. This work would not have been accomplished without his conscientious guidance and support. May ALLAH reward him abundantly and grant him long life in good health.

I owe a deep debt of gratitude to my former supervisor Dr. Farruh Shahidi, for his invaluable guidance, tireless efforts, and patience in supervising this thesis. None of the work presented here would have been possible without his in sightlines and his constant support. What I have learned from him in research, as well as in real life, is absolutely indescribable.

I am also indeed very grateful to my Co-supervisor, Assoc. Prof Dr. Akram Zeki for his suggestion from inception, invaluable advice and constructive contributions in making this research into fruition.

I would also like to express me since gratitude to my colleagues Dr. Adamu Abubakar, Dr.

Akeem Olowolayemo, Mr. Yazan Aljarody, Mr. Hashum Rafiq and Ms. Aiysha Jasem for providing suggestions and advising that helped me significantly in improving my thesis.

I am also grateful to all academic advisors, mentors and well-wishers who were concerned and offered advice in different forms that were useful to me in the process of conducting and completing this work. I am indeed very grateful to Prof. Imad Fakhri, Dr. Mohd Izzuddin, Prof.

Abdul Rahaman Abdul Wahab, Assoc. Prof. Messikh Az Eddine, Prof. Abdul Rahaman Ahalan, Prof. Asadullah Shah, all KICT family and Prof. Khadeejah Al-Ssyaghi and Prof.

Mohammed Al-Shuaibi directors of the University of Taiz, all of whom were very supportive in different forms and offered advice and words of encouragement to me when the going was very tough.

I would also like to show appreciation to all administrative staffs of KICT specifically Sisters Narita and Khairya, and who have provided me with some form of assistance during the course of this work.

I am also very grateful to my parents, Abdulkhaleq and Hamam, including my brothers Mustafa, Ryadh, Safwan, Radhwan and brother-in-law Ayoub and my sister Bushra for giving me the power during my life and encouragement during my study. Many thanks also to my uncles Abdulbaqi, Ali, Mohammed, Abdulssalam, Abdulqader and Mujib Alrahaman for teaching me in secondary school and encourage me to continue for postgraduate study.

I am especially thankful to the Taiz University and the IIUM department of scholarship and financial assistance for giving me the president award scholarship to complete my PhD.

Finally, my thanks and appreciation to all members of my family “Abdulghafour Family”.

ix

**TABLE OF CONTENTS **

Abstract ... ii

**ثحبلا ةصلاخ** ... iii

Approval Page ... iv

Declaration ... v

Copyright Page ... vi

Acknowledgement ... viii

Table of Contents ... ix

List of Tables ... xiii

List of Figures ... xiv

List of Equations ... xxi

List of Symbols ... xxv

List of Abbreviations ... xxvii

**CHAPTER ONE: INTRODUCTION ... 1**

1.1 Overview ... 1

1.2 Problem Statement ... 7

1.3 Research Objectives ... 9

1.4 Scope of The Research ... 11

1.5 Thesis Contribution ... 11

1.6 Research Methodology ... 13

1.7 Thesis Organization ... 16

**CHAPTER TWO: LITERATURE REVIEW ... 17**

2.1 Introduction ... 17

2.2 Agent ... 17

2.3 Muti-Agent Systems (MAS) ... 18

2.4 Consensus Problem ... 20

2.5 Research Background ... 23

2.6 Related Work ... 34

2.6.1 Quadratic Stochastic Operator (QSO) ... 34

2.6.2 Doubly Stochastic Quadratic Operator (DSQO) ... 35

2.6.3 Extreme Doubly Stochastic Quadratic Operator (EDSQO) ... 36

2.6.4 Simplex ... 38

2.6.5 Majorization ... 38

2.6.6 Stochastic Matrix and Doubly Stochastic Matrix ... 40

**CHAPTER THREE: MATHEMATICAL PRELIMINARIES AND FORMULATION **
**OF THE LIMIT BEHAVIOUR OF NONLINEAR STOCHASTIC **
**OPERATORS ... 42**

3.1 Introduction ... 42

3.2 Formulation of the Limit Behaviour of DSQO. ... 42

3.2.1 Defining EDSQO on 2DS ... 43

3.2.1.1 The Theory of QSO ... 43

3.2.1.2 The Theory of DSQO ... 45

3.2.1.3 The Theory of EDSQO ... 47

3.2.2 Limiting Behavior of the Trajectories of EDSQO and DSQO ... 49

x

3.2.2.1 Limiting Behavior of Trajectories of EDSQO ... 50

3.2.2.2 Dynamic Classifications and Limit Behavior of the Trajectories of EDSQO on 2DS, 3DS and 4DS ... 50

3.2.2.3 Dynamics of DSQO on a FDS ... 51

3.3 Reaching a Consensus via DSQO for MAS ... 53

3.3.1 EDSQO for Consensus Problem in MAS ... 54

3.3.2 Reaching a Nonlinear Consensus for MAS via Positive DSQO ... 54

3.4 The Evaluation of Linear and Nonlinear Stochastic Distribution for Consensus Problem in MAS ... 55

3.4.1 Transition Matrices ... 55

3.4.1.1 Stochastic Matrix (SM) ... 56

3.4.1.2 Doubly Stochastic Matrix (DSM) ... 56

3.4.1.3 Сubic Stochastic Matrix for QSO, DSQO and EDSQO (CSM) ... 56

3.4.1.4 Сubic Doubly Stochastic Matrix for QSO, DSQO and EDSQO (CDSM) ... 56

3.4.2 Linear Stochastic Distribution of DeGroot Model ... 57

3.4.3 Nonlinear Stochastic Distributing of QSO Saburov Model ... 58

3.5 The Consensus of Higher Degrees of Nonlinear Stochastic Operators for MAS ... 59

3.5.1 The Distribution of HDG Consensus ... 60

3.5.2 Nonlinear Distribution with Higher Degrees QSO, DSQO and EDSQO Protocols ... 60

3.6 The Consensus of Fractional Degrees of Nonlinear Stochastic Operators for MAS ... 61

3.6.1 The Distribution of FDG Protocol ... 61

3.6.2 Nonlinear Distribution of FDSQO, EDSQO and FQSO Protocols ... 61

3.7 Low-Complexity Nonlinear Protocols for Distributed Consensus Modified from DSQO for MAS ... 62

3.7.1 New Notions for MPDSQO Protocol ... 62

3.8 A Novel Low-Complexity Nonlinear Convergence Protocol Modified from EDSQO for MAS ... 63

3.8.1 New Notions for MEDSQO ... 63

3.9 Summary ... 64

**CHAPTER FOUR: RESULT AND DISCUSSION ... 65**

4.1 Introduction ... 65

4.2 The EDSQO on 2DS ... 65

4.2.1 The Extreme Points of the DSQO ... 66

4.2.2 The Transition Matrices of EDSQO on 2DS ... 67

4.2.3 The Operators of EDSQO on 2DS ... 71

4.2.4 Summary ... 75

4.3 Limit Behaviour and Dynamic Classifications of Trajectories of DSQO on FDS ... 75

4.3.1 Limit Behavior of the Trajectories of EDSQO on 2DS ... 75

4.3.1.1 The Existence of the Limit of EDSQO ... 76

4.3.1.2 Simulation Experiments of Existence of Limit Behaviour of EDSQO on 2DS ... 76

4.3.1.3 Summary ... 91

4.3.2 The Limit Behaviour of Trajectories of EDSQO on FDS ... 91

4.3.2.1 The Limit Behaviour of EDSQO on FDS... 91

xi

4.3.2.2 Simulation Experiments of Limit Behaviour of EDSQO on

2DS, 3DS and 4DS ... 95

4.3.2.3 Summary ... 100

4.3.3 Dynamic Classification of DSQO on a Finite-Dimensional Simplex ... 101

4.3.3.1 The Limit Behavior of DSQO ... 102

4.3.3.2 The Classification of Extreme Points of DSQO on 𝑆2 ... 107

4.3.3.3 Summary ... 111

4.4 Nonlinear Consensus of DSQO based on Majorization Technique for Consensus Problem in MAS ... 111

4.4.1 The Consensus Problem in MAS via DSQO based on Majorization Technique ... 112

4.4.2 Simulation of the EDSQO for Consensus Problem in MAS ... 118

4.4.3 Reaching a Consensus for MAS via EDSQO and DSQO ... 129

4.4.4 Reaching a Consensus for MAS via PDSQO ... 130

4.4.5 Simulation of DSQO for Consensus Problem in MAS ... 132

4.4.6 Summary ... 137

4.5 The Comparison of Linear Consensus and Nonlinear Consensus of Stochastic Distribution for MAS ... 138

4.5.1 DeGroot Linear Protocol ... 139

4.5.1.1 Using Stochastic Matrix ... 139

4.5.1.2 Using Doubly Stochastic Matrix ... 140

4.5.2 QSO Nonlinear Protocol of Saburov ... 140

4.5.2.1 Using Stochastic Matrix ... 141

4.5.2.2 Using Doubly Stochastic Matrix ... 141

4.5.3 DSQO Nonlinear Protocol ... 142

4.5.3.1 Using CSM ... 143

4.5.3.2 Using Doubly Cubic Stochastic Matrix ... 143

4.5.4 EDSQO Nonlinear Protocol ... 144

4.5.4.1 Using Cubic Stochastic Matrix ... 144

4.5.4.2 Using Doubly Cubic Stochastic Matrix ... 145

4.5.5 The Comparison of the Consensus Protocols of DeGroot, QSO, DSQO and EDSQO ... 146

4.5.6 Summary ... 153

**CHAPTER FIVE: THE CONSENSUS OF HIGHER AND FRACTIONAL DEGREE**
** ... 154**

5.1 Introduction ... 154

5.2 The Consensus of Higher Degrees of DSQO, EDSQO, QSO and DG for MAS ... 154

5.2.1 The Consensus of Higher Degree ... 155

5.2.2 The Initial Statuses and Transition Matrix to Simulate the Consensus of HDG, HQSO, HDSQO and HEDSQO Protocols ... 157

5.2.2.1 The Consensus of HDG Protocol ... 157

5.2.2.2 The Consensus of HQSO Protocol ... 158

5.2.2.3 The Consensus of HDSQO Protocol ... 159

5.2.2.4 The Consensus of HEDSQO Protocol ... 159

5.2.3 Simulation Results of the Consensus of HDG, HQSO, HDSQO and HEDSQO Protocols when the Degree n is Greater or Equal to Two (𝑛 ≥ 2) ... 160

xii

5.2.3.1 The Consensus of HDG, HQSO, HDQSO and HEDSQO

Protocols when 𝑛 = 2 ... 161

5.2.3.2 The Consensus of HDG, HQSO, HDQSO and HEDSQO Protocols when 𝑛 = 10 ... 162

5.2.3.3 The Consensus of HDG, HQSO, HDQSO and HEDSQO Protocols when 𝑛 = 100 ... 164

5.2.4 Summary ... 166

5.3 The consensus of Fractional Degrees of Nonlinear Dynamics Stochastic Operators for MAS ... 167

5.3.1 The Fractional Consensus of FDSQO, FEDSQO, FQSO and FDG Protocols ... 167

5.3.2 Simulation Results of the Fractional Consensus of FDG, FQSO and FDSQO Protocols ... 171

5.3.2.1 The Consensus of FDG Protocol ... 171

5.3.2.2 The Consensus of FQSO and FDSQO Protocols. ... 174

5.3.3 Summary ... 185

**CHAPTER SIX: TWO NOVEL NONLINEAR CONSENSUS PROTOCOLS OF **
**MPDSQO AND MEDSQO ... 186**

6.1 Introduction ... 186

6.2 A Novel Nonlinear Protocol for Consensus Modified from the DSQO in Networks of Dynamic Agents ... 186

6.2.1 The Consensus of MPDSQO ... 186

6.2.2 Simulation Result of MPDSQO Protocol ... 188

6.2.3 Summary ... 195

6.3 A Novel Low-Complexity Nonlinear Consensus Class Modified from the EDSQO (MEDSQO) for MAS ... 196

6.3.1 The Consensus of MEDSQO ... 196

6.3.2 Simulation Result of MEDSQO ... 200

6.3.3 Summary ... 207

**CHAPTER SEVEN: CONCLUSION AND FUTURE WORK ... 208**

7.1 General Remarks ... 208

7.2 Contribution ... 208

7.3 Future Work ... 212

**REFERENCES ... 214**

**PUBLICATIONS ... 224**

xiii

**LIST OF TABLES **

Table No. Page No.

Table 3.1 The General, Higher, Fractional Models of QSO, DSQO, EDSQO. 64 Table 3.2 The General, Higher, Fractional Models of DeGroot. 64 Table 3.3 The Nonlinear Protocols of QSO, DSQO, EDSQO, PDSQO, MPDSQO,

MEDSQO.

64

Table 4.1 The Comparison of the Number Iterations between DeGroot, QSO and DSQO under SM and DSM.

152

Table 4.2 The Comparison of the Time Sec of the Convergence Computation between DeGroot, QSO-Saburov and DSQO under SM and DSM.

152

xiv

**LIST OF FIGURES **

Figure No. Page No.

Figure 1.1 The Structure of Research Objectives 10

Figure 1.2 Paradigm of the Research Methodology 15

Figure 2.1 The Interaction of Agent with Environment 18

Figure 2.2 A Single Agent and MAS 20

Figure 2.3 The Architecture Structure of Consensus Problem in MAS 22 Figure 2.4 The Convergence of Consensus Problem in MAS 22

Figure 2.5 The Dimensional Simplex 38

Figure 3.1 The Structure of DeGroot Linear Distribution for MAS 58 Figure 3.2 The Structure of QSO, DSQO and EDSQO Nonlinear Distribution for

MAS

59

Figure 4.1 Limit Behavior of V1, V2, and V3 of the EDSQO on 2DS 77 Figure 4.2 Limit Behavior of V4, V5, and V6 of the EDSQO on 2DS 78 Figure 4.3 Limit Behavior of V7, V8, and V9 of the EDSQO on 2DS 79 Figure 4.4 Limit Behavior of V10, V11, and V12 of the EDSQO on 2DS 80 Figure 4.5 Limit Behavior of V13, V14, and V15 of the EDSQO on 2DS 81 Figure 4.6 Limit Behavior of V16, V17, and V18 of the EDSQO on 2DS 82 Figure 4.7 Limit Behavior of V19, V20, and V21 of the EDSQO on 2DS 83 Figure 4.8 Limit Behavior of V22, V23, and V24 of the EDSQO on 2DS 84 Figure 4.9 Limit Behavior of V25, V26, and V27 of the EDSQO on 2DS 85 Figure 4.10 Limit Behavior of V28, V29, and V30 of the EDSQO on 2DS 86 Figure 4.11 Limit Behavior of V31, V32, and V33 of the EDSQO on 2DS 87 Figure 4.12 Limit Behavior of V34, V35, and V36 of the EDSQO on 2DS 88

Figure 4.13 Limit Behavior of V37 of the EDSQO on 2DS 89

xv

Figure 4.14 Limit Behavior of the Trajectories for Each EDSQO of V3.1.F, V3.1.P,V3.1.C,V4.1.P,V4.1.F,V4.1.C,V5.1.F,V5.1.P,V5.1.C of the Initial Extreme Exterior Values

97

Figure 4.15 Limit Behavior of Fixed Points for EDSQO of V3.1.F,V4.1.F, V5.1.F On 2DS, 3DS and 4DS, Respectively

98

Figure 4.16 Limit Behavior of Periodic Points for EDSQO V3.2.F,V4.2.F, V5.2.F On 2DS, 3DS and 4DS, Respectively

99

Figure 4.17 Limit Behavior of Convergence Points for EDSQO V3.3.F,V4.3.

F,V5.3.F On 2DS, 3DS and 4DS, Respectively

99

Figure 4.18 The Selfish Interaction of x1,x2 and x3 112

Figure 4.19 The Selfish Interaction of X2 for X1, of X3 for X2 and of X1 for X3 113 Figure 4.20 The Convergence Operators of V1,V2,V3 of EDSQO for 3,4,5 Agents,

Respectively

120

Figure 4.21 The Fixed Operators of V1,V2,V3 of EDSQO for 3,4,5 Agents, Respectively

122

Figure 4.22 The Selfish-Periodic Operators of V1,V2,V3 of EDSQO for 3,4,5 Agents, Respectively

124

Figure 4.23 The Convergence and Selfish Operators Of V1,V2,V3 of EDSQO for 3,4,5 Agents, Respectively

126

Figure 4.24 The Convergence of Some Operators of EDSQO for 3 Agents 127 Figure 4.25 The Convergence of Some Operators of EDSQO of 3 Agents 128

Figure 4.26 The Consensus of PDSQO of Case 1 135

Figure 4.27 The Consensus of PDSQO of Case 2 135

Figure 4.28 The Consensus of PDSQO of Case 3 135

Figure 4.29 The Consensus of PDSQO of Case 4 135

Figure 4.30 The Consensus of PDSQO of Case 5 135

Figure 4.31 The Consensus of PDSQO of Case 6 135

Figure 4.32 The Consensus of PDSQO of Case 7 135

Figure 4.33 The Consensus of PDSQO of Case 8 135

Figure 4.34 The Consensus of PDSQO of Case 9 135

Figure 4.35 The Consensus of PDSQO of Case 10 135

xvi

Figure 4.36 The Consensus of 100 Agents via PDSQO 136

Figure 4.37 The Consensus of 500 Agents via PDSQO 137

Figure 4.38 The Convergence of DeGroot Linear Distribution with SM 139 Figure 4.39 The Convergence of DeGroot Linear Distribution with DSM 140 Figure 4.40 The Convergence of QSO Nonlinear Distribution with SM 141 Figure 4.41 The Convergence of QSO Nonlinear Distribution with DSM 142 Figure 4.42 The Convergence of DSQO Nonlinear Distribution with CSM 143 Figure 4.43 The Convergence of DSQO Nonlinear Distribution with CDSM 144 Figure 4.44 The Convergence of EDSQO Nonlinear Distribution with SM 145 Figure 4.45 The Convergence of EDSQO Nonlinear Distribution with CSM 145 Figure 4.46 The Convergence of the DeGroot Linear Distribution and Nonlinear

Distribution of QSO, DSQO and EDSQO with SM and DSM

147

Figure 4.47 The Convergence of the DeGroot Linear Distribution and Nonlinear Distribution of QSO, DSQO and EDSQO with SM and DSM

148

Figure 4.48 The Convergence of the DeGroot Linear Distribution and Nonlinear Distribution of QSO, DSQO and EDSQO with SM and DSM

149

Figure 4.49 The Convergence of the DeGroot Linear Distribution and Nonlinear Distribution of QSO, DSQO and EDSQO with SM and DSM

150

Figure 4.50 The Convergence of the DeGroot Linear Distribution and Nonlinear Distribution of QSO, DSQO and EDSQO with SM and DSM

151

Figure 5.1 The Convergence of HDG Distribution when n=2 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

160

Figure 5.2 The Convergence of HQSO Distribution when n=2 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

161

Figure 5.3 The Convergence of HDSQO Distribution when n=2 with CSM- nonsym, SM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

161

Figure 5.4 The Convergence of HEDSQO Distribution when n=2 with CSM-sym with One Linear Function, CSM-sym and DSM-sym

162

Figure 5.5 The Convergence of HDG Distribution when n=10 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

162

Figure 5.6 The Convergence of HQSO Distribution when n=10 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

163

xvii

Figure 5.7 The Convergence of HDSQO Distribution when n=10 with CSM- nonsym, CSM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM- sym

163

Figure 5.8 The Convergence of HEDSQO Distribution when n=10 with CSM-sym with One Linear Function, CSM-sym and CSM-sym

164

Figure 5.9 The Convergence of HDG Distribution when n=100 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

164

Figure 5.10 The Convergence of QSO Distribution when n=100 with NM-nonsym, NM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM-sym

165

Figure 5.11 The Convergence of HDSQO Distribution when n=100 with CSM- nonsym, CSM-sym, SM-nonsym, SM-sym, DSM-nonsym and DSM- sym

165

Figure 5.12 The Convergence of HEDSQO Distribution when n=100 with CSM- sym with One Linear Function, CSM-sym and DSM-sym

166

Figure 5.13 The Consensus of FDG with SM, where n=2 171

Figure 5.14 The Consensus FDG with DSM, where n=2 171

Figure 5.15 The Consensus of FDG with SM, where n=3 172

Figure 5.16 The Consensus of FDG with DSM, where n=3 172

Figure 5.17 The Consensus of FDG with SM, where n=4 172

Figure 5.18 The Consensus of FDG with DSM, where n=4 172

Figure 5.19 The Consensus of FDG with SM, where n=5 172

Figure 5.20 The Consensus of FDG with DSM, where n=5 172

Figure 5.21 The Consensus of FDG with SM, where n=10 173

Figure 5.22 The Consensus of FDG with DSM, where n=10 173

Figure 5.23 The Consensus of FDG with SM, where n=100 173

Figure 5.24 The Consensus of FDG with DSM, where n=100 173

Figure 5.25 The Consensus of FDG with SM, where n=1000 173

Figure 5.26 The Consensus of FDG with DSM, where n=1000 173

Figure 5.27 The Consensus of FDG with SM, where n=∞ 174

Figure 5.28 The Consensus of FDG with DSM, where n=∞ 174

Figure 5.29 The Consensus of FQSO and FDSQO with NM, where n=2 175

xviii

Figure 5.30 The Consensus of FQSO and FDSQO with NM, where n=2 175 Figure 5.31 The Consensus of FQSO and FDSQO with SM, where n=2 175 Figure 5.32 The Consensus of FQSO and FDSQO with SM, where n=2 175 Figure 5.33 The Consensus of FQSO and FDSQO with DSM, where n=2 175 Figure 5.34 The Consensus of FQSO and FDSQO with DSM, where n=2 175 Figure 5.35 The Consensus of FQSO and FDSQO with NM, where n=3 176 Figure 5.36 The Consensus of FQSO and FDSQO with SM, where n=3 176 Figure 5.37 The Consensus of FQSO and FDSQO with DSM, where n=3 176 Figure 5.38 The Consensus of FQSO and FDSQO with NM, where n=4 176 Figure 5.39 The Consensus of FQSO and FDSQO with SM, where n=4 176 Figure 5.40 The Consensus of FQSO and FDSQO with DSM, where n=4 176 Figure 5.41 The Consensus of FQSO and FDSQO with NM, where n=5 177 Figure 5.42 The Consensus of FQSO and FDSQO with SM, where n=5 177 Figure 5.43 The Consensus of FQSO and FDSQO with SDM, where n=5 177 Figure 5.44 The Consensus of FQSO and FDSQO with NM, where n=10 177 Figure 5.45 The Consensus of FQSO and FDSQO with SM, where n=10 177 Figure 5.46 The Consensus of FQSO and FDSQO with DSM, where n=10 177 Figure 5.47 The Consensus of FQSO and FDSQO with NM, where n=100 178 Figure 5.48 The Consensus of FQSO and FDSQO with SM, where n=100 178 Figure 5.49 The Consensus of FQSO and FDSQO with DSM, where n=100 178 Figure 5.50 The Consensus of FQSO and FDSQO with NM, where n=1000 178 Figure 5.51 The Consensus of FQSO and FDSQO with DSM, where n=1000 178 Figure 5.52 The Consensus of FQSO and FDSQO with DSM, where n=1000 178 Figure 5.53 The Consensus of FQSO and FDSQO with NM, where n=∞ 179 Figure 5.54 The Consensus of FQSO and FDSQO with SM, where n=∞ 179 Figure 5.55 The Consensus of FQSO and FDSQO with DSM, where n=∞ 179 Figure 5.56 The Consensus of Negative FQSO and FDSQO with NM, where n=2 181

xix

Figure 5.57 The Consensus of Negative FQSO and FDSQO with SM, where n=2 181 Figure 5.58 The Consensus of Negative FQSO and FDSQO with DSM, where n=2 181 Figure 5.59 The Consensus of Negative FQSO and FDSQO with NM, where n=3 181 Figure 5.60 The Consensus of Negative FQSO and FDSQO with SM, where n=3 181 Figure 5.61 The Consensus of Negative FQSO and FDSQO with DSM, where n=3 181 Figure 5.62 The Consensus of Negative FQSO and FDSQO with NM, where n=4 182 Figure 5.63 The Consensus of Negative FQSO and FDSQO with SM, where n=4 182 Figure 5.64 The Consensus of Negative FQSO and FDSQO with DSM, where n=4 182 Figure 5.65 The Consensus of Negative FQSO and FDSQO with NM, where n=5 182 Figure 5.66 The Consensus of Negative FQSO and FDSQO with SM, where n=5 182 Figure 5.67 The Consensus of Negative FQSO and FDSQO with DSM, where n=5 182 Figure 5.68 The Consensus of Negative FQSO and FDSQO with NM, where n=2 183 Figure 5.69 The Consensus of Negative FQSO and FDSQO with SM, where n=10 183 Figure 5.70 The Consensus of Negative FQSO and FDSQO with DSM, where n=10 183 Figure 5.71 The Consensus of Negative FQSO and FDSQO with NM, where n=100 183 Figure 5.72 The Consensus of Negative FQSO and FDSQO with SM, where n=100 183 Figure 5.73 The Consensus of Negative FQSO and FDSQO with DSM, where

n=100

183

Figure 5.74 The Consensus of Negative FQSO and FDSQO with NM, where n=1000

184

Figure 5.75 The Consensus of Negative FQSO and FDSQO with SM, where n=1000 184 Figure 5.76 The Consensus of Negative FQSO and FDSQO with DSM, where

n=1000 184

Figure 5.77 The Consensus of Negative FQSO and FDSQO with NM, where n=∞ 184 Figure 5.78 The Consensus of Negative FQSO and FDSQO with SM, where n=∞ 184 Figure 5.79 The Consensus of Negative FQSO and FDSQO with DSM, where n=∞ 184

Figure 6.1 The Consensus of MDSQO of Case 1 191

Figure 6.2 The Consensus of MDSQO of Case 2 191

Figure 6.3 The Consensus of MDSQO of Case 3 191

xx

Figure 6.4 The Consensus of MDSQO of Case 4 191

Figure 6.5 The Consensus of MDSQO of Case 5 191

Figure 6.6 The Consensus of MDSQO of Case 6 191

Figure 6.7 The Consensus of MDSQO of Case 7 191

Figure 6.8 The Consensus of MDSQO of Case 8 191

Figure 6.9 The Consensus of MDSQO of Case 9 191

Figure 6.10 The Consensus of MDSQO of Case 1 191

Figure 6.11 The Consensus of MPDSQO for 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50 ,60, 70, 80, 90, 100, 200, 300,400 and 500 Agents, Respectively

195

Figure 6.12 The Consensus of 3 Agents via MEDSQO using SM of Initial Status of (0,0,1)

201

Figure 6.13 The Consensus of 3 Agents via MEDSQO using SM of Initial Status of (0.28,0.324,0.396) (Random)

201

Figure 6.14 The Consensus of 3 Agents via MEDSQO using DSM of Initial Status of (0,0,1)

202

Figure 6.15 The Consensus of 3 Agents via MEDSQO using DSM of Initial Status of (0.28,0.324,0.396) (Random)

202

Figure 6.16 The Consensus of 4 Agents via MEDSQO using SM of Initial Status of (1,0,0,0)

203

Figure 6.17 The Consensus of 4 Agents via MEDSQO using SM of Initial Status of (0.28,0.16,0.54,0.2) (Random)

203

Figure 6.18 The Consensus of 4 Agents via MEDSQO using DSM of Initial Status of (1,0,0,0)

204

Figure 6.19 The Consensus of 4 Agents via MEDSQO using DSM of Initial Status of (0.28,0.16,0.54,0.2) (Random)

204

Figure 6.20 The Consensus of 5 Agents via MEDSQO using SM of Initial Status of (0,0,0,1,0)

205

Figure 6.21 The Consensus of 5 Agents via MEDSQO using SM of Initial Status of (0.18,0.6,0.9,0.12,0.55) (Random)

205

Figure 6.22 The Consensus of 5 Agents via MEDSQO using DSM of Initial Status of (0,0,0,1,0)

206

Figure 6.23 The Consensus of 5 Agents via MEDSQO using DSM of Initial Status of (0.18,0.6,0.9,0.12,0.55) (Random)

206

xxi

**LIST OF EQUATIONS **

Equation No. Page No.

Equation 3.1 43

Equation 3.2 44

Equation 3.3 44

Equation 3.4 44

Equation 3.5 45

Equation 3.6 45

Equation 3.7 46

Equation 3.8 46

Equation 3.9 47

Equation 3.10 47

Equation 3.11 47

Equation 3.12 48

Equation 3.13 48

Equation 3.14 54

Equation 3.15 57

Equation 3.16 57

Equation 3.17 57

Equation 3.18 58

Equation 3.19 60

Equation 3.20 60

Equation 3.21 61

Equation 3.22 61

Equation 3.23 62

Equation 3.24 63

xxii

Equation 4.1 66

Equation 4.2 67

Equation 4.3 71

Equation 4.4 71

Equation 4.5 93

Equation 4.6 93

Equation 4.7 93

Equation 4.8 93

Equation 4.9 93

Equation 4.10 94

Equation 4.11 94

Equation 4.12 94

Equation 4.13 94

Equation 4.14 95

Equation 4.15 95

Equation 4.16 95

Equation 4.17 105

Equation 4.18 105

Equation 4.19 105

Equation 4.20 105

Equation 4.21 105

Equation 4.22 106

Equation 4.23 106

Equation 4.24 106

Equation 4.25 106

Equation 4.26 108

Equation 4.27 108

xxiii

Equation 4.28 108

Equation 4.29 110

Equation 4.30 110

Equation 4.31 115

Equation 4.32 115

Equation 4.33 115

Equation 4.34 116

Equation 4.35 116

Equation 4.36 116

Equation 4.37 116

Equation 4.38 117

Equation 4.39 117

Equation 4.40 117

Equation 4.41 117

Equation 4.42 118

Equation 4.43 118

Equation 4.44 129

Equation 4.45 129

Equation 4.46 131

Equation 4.47 131

Equation 4.48 131

Equation 4.49 131

Equation 4.50 131

Equation 4.51 131

Equation 4.52 131

Equation 4.53 131

Equation 4.54 132

xxiv

Equation 5.1 155

Equation 5.2 155

Equation 5.3 168

Equation 5.4 168

Equation 5.5 168

Equation 5.6 168

Equation 5.7 169

Equation 5.8 169

Equation 5.9 170

Equation 5.10 171

Equation 6.1 187

Equation 6.2 187

Equation 6.3 187

Equation 6.4 188

Equation 6.5 197

Equation 6.6 197

Equation 6.7 197

Equation 6.8 197

Equation 6.9 197

Equation 6.10 197

Equation 6.11 198

Equation 6.12 198

Equation 6.13 198