âTime Driven-Event Driven-Parameters Changes

Development of New Systems Paradigm Based on the “Time Driven-Event Driven-Parameters Changes” with Applications by

Eng. Ahmed Rabie Ginidi Ginidi A Thesis Submitted to the Faculty of Engineering, Cairo University, in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY in Electrical Power and Machines Engineering

Faculty of Engineering, Cairo University Giza, Egypt

2015




Under the Supervision of Prof. Dr. Hassen Taher Dorrah

Faculty of Engineering, Cairo University Giza, Egypt 2015

Engineer : Date of Birth : Nationality : E-mail : Phone. : Address : Registration Date : Awarding Date : Degree : Department :

Supervisors :

Ahmed Rabie Ginidi Ginidi 1 / 1 / 1986 Egyptian [email protected]

84448884840 -84441141110 Bahabshin – Naser - Beni-suef 1 / 3 / 2011 / / 2015 Philosophy of Doctor Electrical Power and Machines Engineering

Prof. Dr. Hassen Taher Dorrah

Examiners : Prof. Dr. Hassen Taher Dorrah Prof. Dr. Osama Yousef Abo Elhagag Prof. Dr. Ahmed Mohamed El Garhy (Helwan University)

Title of Thesis: Development of New Systems Paradigm Based on the “Time Driven-Event Driven-Parameters Changes” with Applications Key Words: Consolidity-based theory of change, event-driven, time-driven system configuration, Parameters changes, Simulation analysis.

Summary: New research study indicated modelling and simulation analysis for “Event Driven – Time Driven –Parameters Changes” Paradigm. This paradigm is based on two-level system. The first one is the time driven level or basic level which is governed by the system physical equation, whereas the second upper level is the event driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. This thesis has concentrated on developing a modeling and simulation analysis for the changing of parameters and the consolidity if there is events occurrences it. The simulation analysis can be executed by calculating the affecting system changes of the parameters of the system throughout developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state. Three real life case studies can elaborate this paradigm and demonstrate the concept. The first case study is the investigation of prey-predator model operating under external environmental influences. The second case study is the analysis of HIV/AIDS Epidemic problem model subject to variability of awareness and educational programs. The third case study is the analysis of spread of infectious disease problem subject to variability of level of hygiene. The three cases studies confirm the suitability of the suggested paradigm to handle real life system under external influences “on and above” their normal conditions.

Acknowledgment I would like to forward my greatest thanks to my supervisor Prof. Dr. Hassen Taher Dorrah for his guidance, sincere efforts and encouragement throughout this work. I owe him lots of gratitude for having shown me this way of research. Thank you, to Prof. Dorrah, not only for his invaluable mentorship but also for having made me rediscover the true pleasure of learning, which I reckoned to have totally lost in the last years. Thanks, to Prof. Dorrah, for explaining to me that flowers don’t grow faster if you pull them. He could not even realize how much I have learned from him. I would like to express my deepest appreciation and gratitude to my uncle Saad, who died on Tuesday 26 November 2013, for his encouragement throughout this work. Thank you, to my father, for having taught me with his life example that real success comes only from honest work. My admiration for your person is limitless. Thank you, to my mother, for having shown me the importance of humility and that good behavior finds its reward in itself and not in external compensation. I feel a deep sense of gratitude to my parents who encouraged and taught me the good things that really matter in life. Thank you, to my sister Saher, for her encouragement during the work. It feels reassuring to have a person you know you can always rely on. I am grateful to my brothers Usama and Eslam, my sisters: Asmaa, Elham and my wife. Thank you, to all the people who have dedicated me their precious time and enriched me, even if only with a wise word of experience.

i

Table of Contents Page No. Acknowledgment Table of Contents List of Tables List of Figures List of Symbols and Abbreviations Abstract

i ii vi vii xi xiii

Chapter 1: Introduction

1

1.1 Introduction 1.2 Thesis Objectives and Scope 1.3 Thesis Organization

1 4 4

Chapter 2: Conventional Time Driven-Event Driven Paradigm 2.1 Introduction 2.2 Time Driven 2.2.1 Time-Driven Simulation 2.3 Event Driven System 2.3.1 Event Definitions 2.3.2 Event-Driven Simulation 2.4 System Dynamics and Discrete Event Simulation 2.5 Time-Driven and Event-Driven Systems 2.6 The Major Properties of Discrete Event Systems (DES) 2.7 Event Modeling 2.7.1 Overview and Formal Model 2.8 Differences between Conventional Time-Driven and Event Driven 2.9 Components of Event-Driven System in Computer Science 2.10 Event-Driven Paradigm 2.10.1 Implementation 2.10.2 Event-Driven Simulation and Output Analyses 2.11 Comparison of the Two Models 2.12 Examples for Event Systems 2.12.1 Event-Driven Control for Reducing Resource Usage 2.12.2 Sensor-Based Event-Driven Control 2.12.3 Sensor Networks Integrated 2.13 Disadvantages of the Conventional Event Driven 2.14 Concluding Remarks

ii

6 6 6 6 7 8 8 9 10 10 11 11 15 15 18 19 20 21 22 22 22 23 23 24

Chapter 3: System Change Paradigm “Time Driven-Event Driven-Parameters Changes”

25

3.1 Introduction 3.2 The Relation between Time Driven and Event Driven 3.3 New Definition for Event 3.4 Classifications of Events 3.5 The Difference between Events and Disturbances 3.6 System Change Pathway Conceptual Graph 3.7 Calculating the System Parameter Change 3.8 The Consolidity-Based Theory of Change 3.9 Changes the Parameters of the System 3.10 Effect of Time in the relation between Consolidity and Theory of Change 3.11 The integrated Basic System and Upper System Layers Configuration 3.12 Main Advantages of the Combined Time-Driven and Event-Driven 3.13 Incremental Parameters Changes versus Affected Varying Environment 3.14 Systems Change Procedure 3.15 The Changeability Coefficient 3.16 Time-Event-Change System Paradigm 3.17 Concluding Remarks

25 25 26 26 33 33 35 35 37 39 39 41 42 43 44 45 48

Chapter 4 : Consolidity Theory

49

4.1 Introduction 4.2 Consolidity Definition 4.3 System Consolidity Index 4.4 Basics of the System Consolidity 4.4.1 Basic Definition of System Consolidity 4.4.2 Different Classifications of System Consolidity 4.4.3 The strong and weakness Ranking of Systems Consolidity 4.5 Implementation Strategy of Consolidity for Existing and New Systems 4.5.1 Consolidity Implementation Strategy for Existing Systems 4.5.2 Consolidity Implementation Strategy for New Systems 4.5.3 Essential Rules for Implementation of Consolidity 4.6 The consolidity for 2nd Order of State Space Problems using the stability coefficients 4.7 The consolidity for 3rd Order of State Space Problems using the stability coefficients 4.8 Consolidity Index for 2nd Order of State Space Problems using the Pole Placement 4.9 Consolidity Index for 3rd Order of State Space Problems using the Pole Placement 4.10 Consolidity Applications and Implementations 4.11 Concluding Remarks

iii

49 49 50 52 52 53 54 55 55 56 56 58 59 60 65 69 72

Chapter 5: System Change Pathway of Linear Relationship Paradigm 5.1 Introduction 5.2 A Model for HIV/AIDS Infectious Problem 5.3 Consolidity Analysis of the HIV/AIDS Infectious Problem 5.4 Consolidity Proposed Change Paradigm for the HIV/AIDS Infectious Problem 5.5 The Influence of Media Campaigns on the Number of HIV/AIDS Infected People 5.6 Changes in the Number of HIV/AIDS Infected People with Low Level of Media Campaigns 5.7 Changes in the Number of HIV/AIDS Infected People with Media Campaigns 5.8 Concluding Remarks Chapter 6: System Change Pathway of Exponential Relationship Paradigm 6.1 Introduction 6.2 Model for Event Driven and Parameters Changes for Prey Predator Application 6.3 Problem Description 6.4 Prey Predator Population Problem Formulation 6.5 Mathematical Module for Consolidity of Prey Predator Population Problem 6.6 Consolidity Based Change for Prey Predator Population Problem 6.7 Spread of Infectious Disease Problem Formulation 6.8 Concluding Remarks Chapter 7: Modeling and Simulation Analysis for “Event-Consolidity-Parameters Changes” Paradigm

73 73 73 74 77 79 80 83 85 86 86 86 87 88 89 91 95 99

100

7.1 Introduction 100 7.2 The Events and Parameter Changes 100 7.3 The Methodology for the Simulation Analysis and Modeling of Event Driven – Consolidity – Parameter Changes Paradigms 102 7.4 Simulation Analysis for Prey-predator Problem 103 7.4.1 The Steps of Simulation Analysis for Prey-Predator 103 7.4.2 Simulation Analysis for Prey-Predator with Increase the Rainfall Level 104 7.4.3 Simulation Analysis for Prey-Predator with Decrease the Rainfall Level 107 7.4.4 Simulation Analysis for Prey-Predator with Fluctuating the Rainfall Level 109 7.5 Simulation Analysis for HIV/AIDS Infectious Disease 110 7.5.1 The Steps of Simulation Analysis for HIV/AIDs Infectious Disease 111 7.5.2 Simulation Analysis for HIV/AIDS Infected People with Low Level of Media Campaigns 112 7.5.3 Simulation Analysis for HIV/AIDS Infected People with High Level of Media Campaigns 115 7.5.4 Simulation Analysis for HIV/AIDS Infected People with Fluctuation of the Level of Media Campaigns 117

iv

7.6 Simulation Analysis for Spread of Infectious Diseases 119 7.6.1 The Steps of Simulation Analysis for Spread of Infectious Diseases 120 7.6.2 Simulation Analysis for Spread of Infectious Diseases with Low Level of Hygiene of the Dorm 121 7.6.3 Simulation Analysis for Spread of Infectious Diseases with High Level of Hygiene of the Dorm 123 7.6.4 Simulation Analysis for Spread of Infectious Diseases with Fluctuation in Level of Hygiene of the Dorm 125 7.7 Concluding Remarks 127 Chapter 8: Conclusions and Recommendations for Further Research

128

8.1 Conclusions 8.2 Suggestions for Further Research

128 130

Refernces

131

Appendix A: Consolidity Index for the 2nd Order of State Space Using Stability Coefficients

138

A.1 Derivation of the Output for the 2nd Order of State Space Problems A.2 Derivation of the Input for the 2nd Order of State Space Problems A.3 Consolidity Index for the 2nd Order of State Space Problems A.4 Special Case: Consolidity Index for Equal Roots

138 139 139 139

Appendix B: Consolidity Index for the 3rd Order of State Space Using Stability Coefficients B.1 Derivation of the Output for the 3rd Order of State Space Problems B.2 Derivation of the Input for the 3rd Order of State Space Problems B.3 Consolidity Index for the 3rd Order of State Space Problems B.4 Special Cases: Consolidity Index for Equal Roots for 3rd Order Systems

140 140 143 143 144

Appendix C: Consolidity Index for the 3rd Order of State Space Using Pole Placement 147 C.1 Consolidity Index Using Pole Placement C.1.1 Ackermann’s Formula C.1.2 Derivation of the Input for the 3rd Order of State Space Problems C.1.3 Consolidity Index for the 3rd Order of State Space Problems C.2.1 Direct Comparison Method C.2.2 Derivation of the Input for the 3rd Order of State Space Problems C.2.3 Consolidity Index for the 3rd Order of State Space Problems

147 147 155 155 155 156 156

‫الملخص العربي‬ v

List of Tables Table No

Table Title

Page No.

2.1

Comparison between Time-driven and Event-driven System

15

2.2

Functional Specifications of the Control Machine

21

3.1

Searching the Relationship the Time and Event

25

3.2

Main Features of the Joint Time-Driven and Event-Driven Configuration

42

Average Consolidity, Controllability and Stability for 2nd Order State Space when Changing Parameter (a) where n=2, b=4 and c=2

58

Average Consolidity, Controllability and Stability for 3rd Order State Space when Changing (a where n=3, b=6, c=5 and d=1

60

Average Consolidity, Controllability and Closed Loop Feedback Gains for 2nd Order State Space when Changing Parameter (a) where n=2, b=4, c=2and the Poles are (-3, -3).

62

Average Consolidity, Controllability for 2nd Order State Space for Different Cases and Changing a where n=2, b=4, c=2.

63

Average Consolidity, Controllability and Stability for 3rd order State Space when Changing a where n=3, b=6, c=5, d=1and the Poles are (4+4i, -4-4i, -10)

66

Average Consolidity, Controllability for 3rd Order State Space for Different Cases and Changing a where n=3, b=6, c=5, d=1

67

The Influence of Media Campaigns on the Number of HIV/AIDS Infected People and the Consolidity Index of the System

80

5.2

Least Squares Data for the HIV/AIDS with Low Level of Media Campaigns

81

5.3

Least Squares Data for the HIV/AIDS with High Level of Media Campaigns

83

6.1

The Effect Rainfall on both of Rate of Growth of Prey and Consolidity Index

91

6.2 6.3

Least Squares Data for Prey Predator The Influence of Hygiene on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System

93

6.4

Least Squares Data for the Spread of Infectious Diseases

97

4.1 4.2 4.3

4.4 4.5 4.6

5.1

vi

95

7.1 7.2 7.3 7.4 7.5 7.6 7.7

7.8

7.9

Influence of the Rainfall Level on the Increase in the Number of Prey and Consolidity (Scenario I)

106

Influence of the Rainfall Level on the Decrease in the Number of Prey and Consolidity (Scenario II)

108

Influence of the Rainfall Level on the Fluctuation of the Number of Prey and Consolidity (Scenario III)

110

The Influence of Decrease the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario I)

114

The Influence of Increase the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario II)

117

The Influence of Fluctuation of the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario III)

118

The Influence of the Low Level of Hygiene of the Dorm on the Increase of the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to (Scenario I)

123

The Influence of the High Level of Hygiene of the Dorm on Decrease of the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario II)

125

The Influence of the Fluctuation of Hygiene of the Dorm on the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario III)

126

vii

List of Figures Figure No

Figure Title

Page No.

2.1

Illustration of the Notions Event Domain, Event Type and Event Instance.

12

2.2

General Model of Event-based Visualization – The Major Components of Event-based Visualization – Event Specification, Event Detection and Event Representation – are Attached to the Classic Visualization Pipeline.

13

2.3

Example of Conventional DES in Software Architectures

14

2.4

Schematic Diagram of Event-Driven Paradigm

19

3.1

Event Types and Classification

27

3.2

Event Originating from Solid Earth (Geophysical)

28

3.3

Events Caused by Short-lived/ Small to Meso Scale Atmospheric Processes (in the Spectrum from Minutes to Days) (Meteorological)

29

3.4

Events Caused by Deviations in the Normal Water Cycle and/or Overflow of Bodies of Water Caused by Wind Set-up (Hydrological)

3.5

Events Caused by Long-lived/Meso to Macro Scale Processes (in the Spectrum from intra-seasonal to Multi-decadal Climate Variability) (Climatological)

3.6

Disaster caused by the Exposure of Living Organisms to Germs and Toxic Substances (Biological)

3.7

A New Conceptual Graph Demonstrating the Successive Change Pathway of (State-Space Systems) when Subjected to Events, Varying Environments, or Activities.

3.8

Systems Pillars: Controllability (Determinant of Controllability Matrix), Stability (Eigenvalues), and Consolidity (Consolidity Index)

3.9

System Parameters Changes Scenarios: Parameter 1 (Decrease), Parameter 2 (Fluctuate) and Parameter 3 (Increase)

3.10

Change of the State of Parameters According to Negative Discrete Events Occurrence

3.11 3.12 3.13

30

31 32

34 36 36 37

Change of the State of Parameters According to Positive Discrete Events Occurrence

38


38

Systems Change Pathways with the Two Layers (Basic Layer and the Upper Layers) and the Changes in the Parameters of the System [3]

41

viii

3.14

A Sample of Linear Patterns Relating the Incremental System Parameter Change versus Affected Varying Environment or Event Strength, for Different Values of System Consolidity Index C and Certain Specified Changeability Coefficient 

44

3.15

Event Driven-Time Driven-Parameter Change Paradigm

45

3.16

Time of Discrete Events Occurrence

46

3.17

A Chart of System Parameters Changes under Events Occurrence

47

4.1

Basic Definition of System Consolidity

53

4.2

Sketch Showing Different Classes of System Consolidity

54

4.3

Sketch Showing Possible Ranking within the Superior–Inferior Consolidity Scale

55

Consolidity versus Controllability for 2nd Order State Space for Different Cases when Changing a where n=2, b=4, c=2

64

Consolidity versus Controllability for 3rd Order State Space for Different Cases when Changing a where n=3, b=6, c=5, d=1

68

4.6

Suggested Areas for the Scope of Applications of Consolidity Theory

70

5.1

Changing the Number of HIV/AIDS Infected People due to the Influence of Media Campaigns

78

5.2

The Influence of Media Campaigns on Consolidity Index

79

5.3

Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Low Level of Media Campaigns

82

Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Media Campaigns

84

6.1

Example of a Real Life Predator-Prey Population Dynamic Problem [11]

88

6.2

Changing the Number of Prey and Consolidity Index due to the Influence of Changing the Rainfall Level

6.3

The Rate of Parameter Change (Number of Prey) due to the Influence of Changing the Rainfall Level and Consolidity Index

94

The influence of Room Cleaning on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System

96

Parameter Change (the Infection Rate of the Spread of Infectious Diseases) due to the Influence of Changing the Event Level and Consolidity

98

4.4 4.5

5.4

92

6.4 6.5

ix

7.1 7.2 7.3 7.4 7.5

7.6

7.7 7.8

7.9

7.10

System Parameters Change Forms: (a) Parameter 1 (Decrease) and (b) Parameter 2 (Increase) c) Parameter 3 (Fluctuate).

101

Simulation Analysis for the Increase in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (Scenario I)

105

Simulation Analysis for the Decrease in the Number of Prey and Consolidity Index due to the Influence of the Rainfall Level (Scenario II)

107

Simulation Analysis for the Variation in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (scenario III)

109

Simulation Analysis for the Increase in Number of HIV/AIDS Infected People and Consolidity Index with Decrease the Level of Media Campaigns (Scenario I)

113

Simulation Analysis for the Decrease in Number of HIV/AIDS Infected People and Consolidity Index with Increase the level of Media Campaigns (Scenario II)

116

Simulation Analysis for Changing the Number of HIV/AIDS Infected People with Different Levels of Media Campaigns (Scenario III)

119

Simulation Analysis for the Increase the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the Low Level of Hygiene of the Dorm (Scenario I)

122

Simulation Analysis for the Decrease the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the High Levels of Hygiene of the Dorm (Scenario II)

124

Simulation Analysis for the Fluctuation in Infection Rate and the Consolidity index of Spread of Infectious Diseases due to Different Levels of Hygiene of the Dorm (Scenario III)

126

x

List of Symbols and Abbreviations a

average number of different sexual partners per year

a1

interaction term that describes how the hares are diminished as a function of the lynx population

A

entity

A(t)

number of men with AIDS

AIDS

acquired immunodeficiency syndrome



changeability coefficient of the system

b

fuzzy parameter

b1

growth coefficient of the lynxes



constant parameter

cr

removal rate

c

controls the prey consumption rate for low hare population

d

mortality of the lynxes

D

data sets



incremental

DES

discrete event system

P

parameter change

EDS

event driven system

ET

event type

f

final (or end) event state

CO /( I  S )

consolidity index

CO

output fuzziness factor

C( I S )

input fuzziness factor

CO /( I  S ) 

consolidity index at event state step l (dimensionless)

HIV

human immunodeficiency virus

i

index

k

maximum population of the hares (in the absence of lynxes)

k1

constant and parameter

k2, k3, k4, k5

fuzzy parameters

xi

kr

infection rate

m

intermediate event state



event state or event number (index)

N(t)

population size

P

power set

 Ii

corresponding fuzzy level for I i

 Oi

corresponding fuzzy level for the output

 Si

corresponding fuzzy level for S j

V

set of visual representations

VOi

deterministic value of output component O j

VS i

deterministic value of system component S j

VI i

deterministic value of input component I i

r

growth rate of the hares

t

time state or time factor

TDS

time driven system

x(t)

those uninfected with disease but may become infected

X(t)

number of susceptible males HIV virus in the population

y(t)

those who are presently infected with the disease

Y(t)

number of males infected with HIV virus

Z(t)

number infected with virus but is non-infections

xii

Abstract

New research study indicated the system change pathway depends on the Time driven - Event driven – parameter Change paradigm. This paradigm is based on two-level system. The first is the time driven level or basic level which is governed by the system physical equation, where the second upper level is the event –driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. The parameters of the system can be changed due to the influence of event and consequently the consolidity also will be changed. These events can be positive or negative that can increase, decrease the parameters of the system and consolidity of the system, therefore entity will be changed to be built more or destroyed. This thesis has concentrated on developing modeling and simulation analysis for “Event Driven– Time Driven – Parameters Change” Paradigm if there are events occurrences on the parameters. The modeling concentrates on developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state. The model shows the relationship between the Event multiplied by the Consolidity Index and the parameter change has been developed and the parameter change can be calculated through the least squares approximations. Therefore, the developed equations for any system depend on the application as it can be linear, parabolic, exponential, logarithmic …etc. In the simulation, we will calculate the consolidity of the system, and then if we have an event, we can calculate the parameter change throughout relationship between the incremental parameters changes versus affected varying environment and events. Second, after the parameter changed and we can calculate the consolidity and so on. This simulation analysis is developed by three scenarios. In these scenarios, the events can influence the parameter and therefore can be increased or decreased. Three real life applications will be discussed throughout the work to study the influence of event driven system based consolidity theory on the system parameters. The first application is the studying of prey predator population problem that is a famous problem that describes the nature of life between prey and predator. In this context, a model has been studied throughout increase the rate of growth of prey and this can be achieved by variability of the climate. In addition, simulation analysis has been developed through three scenarios. The first scenario shows that if the rainfall increased, the rate of growth of prey will be increased and the Consolidity would be improved. The second scenario shows that if the rainfall has decreased, the number of prey will be decreased and the consolidity index will be higher which means that the system is unconsolidated. The third scenario shows the variation in the rainfall which will affect the increase and decrease of number of prey.

xiii

The second application is the studying of reducing the number of HIV/AIDS infectious people through radio campaigns on knowledge and use of condoms. In this problem, a modelling has been studied throughout increase and decrease the media campaigns. A simulation analysis also has been developed through three scenarios. The first scenario shows the increase in the number of HIV/AIDS infected people and the increase in the value of consolidity index as the level of media campaigns is high. The second scenario shows the decrease in the number of HIV/AIDS infected people and the decrease in the value of consolidity index as the level of media campaigns is low. The third scenario shows the fluctuation in the level of media campaigns and its influence on the number of HIV/AIDS infectious people and the consolidity index. The third application is the analysis of variability of level of hygiene and its influence on spread of infectious disease. In this problem, a model has been studied throughout changing the infection rate. A simulation analysis also has been developed through three scenarios. The first scenario shows the increase in the infection rate and also the increase in the consolidity index due to the low level of dorm cleaning. The second scenario shows the decrease in the infection rate and the decrease in the consolidity index due to the increase in percentage of hygiene of the dorm. The third scenario illustrates the fluctuation of the consolidity index and the infection rate of the spread of infectious diseases due to the influence of hygiene of the dorm. The three case studies confirm the suitability of the suggested paradigm to handle real life systems under external influences “on and above” their normal conditions.

xiv




Under the Supervision of Prof. Dr. Hassen Taher Dorrah

Faculty of Engineering, Cairo University Giza, Egypt 2015




Faculty of Engineering, Cairo University Giza, Egypt

2015

Engineer : Date of Birth : Nationality : E-mail : Phone. : Address : Registration Date : Awarding Date : Degree : Department :

Supervisors :

Ahmed Rabie Ginidi Ginidi 1 / 1 / 1986 Egyptian [email protected]

84448884840 -84441141110 Bahabshin – Naser - Beni-suef 1 / 3 / 2011 / / 2015 Philosophy of Doctor Electrical Power and Machines Engineering

Prof. Dr. Hassen Taher Dorrah

Examiners : Prof. Dr. Hassen Taher Dorrah Prof. Dr. Osama Yousef Abo Elhagag Prof. Dr. Ahmed Mohamed El Garhy (Helwan University)

Title of Thesis: Development of New Systems Paradigm Based on the “Time Driven-Event Driven-Parameters Changes” with Applications Key Words: Consolidity-based theory of change, event-driven, time-driven system configuration, Parameters changes, Simulation analysis.

Summary: New research study indicated modelling and simulation analysis for “Event Driven – Time Driven –Parameters Changes” Paradigm. This paradigm is based on two-level system. The first one is the time driven level or basic level which is governed by the system physical equation, whereas the second upper level is the event driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. This thesis has concentrated on developing a modeling and simulation analysis for the changing of parameters and the consolidity if there is events occurrences it. The simulation analysis can be executed by calculating the affecting system changes of the parameters of the system throughout developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state. Three real life case studies can elaborate this paradigm and demonstrate the concept. The first case study is the investigation of prey-predator model operating under external environmental influences. The second case study is the analysis of HIV/AIDS Epidemic problem model subject to variability of awareness and educational programs. The third case study is the analysis of spread of infectious disease problem subject to variability of level of hygiene. The three cases studies confirm the suitability of the suggested paradigm to handle real life system under external influences “on and above” their normal conditions.

Acknowledgment I would like to forward my greatest thanks to my supervisor Prof. Dr. Hassen Taher Dorrah for his guidance, sincere efforts and encouragement throughout this work. I owe him lots of gratitude for having shown me this way of research. Thank you, to Prof. Dorrah, not only for his invaluable mentorship but also for having made me rediscover the true pleasure of learning, which I reckoned to have totally lost in the last years. Thanks, to Prof. Dorrah, for explaining to me that flowers don’t grow faster if you pull them. He could not even realize how much I have learned from him. I would like to express my deepest appreciation and gratitude to my uncle Saad, who died on Tuesday 26 November 2013, for his encouragement throughout this work. Thank you, to my father, for having taught me with his life example that real success comes only from honest work. My admiration for your person is limitless. Thank you, to my mother, for having shown me the importance of humility and that good behavior finds its reward in itself and not in external compensation. I feel a deep sense of gratitude to my parents who encouraged and taught me the good things that really matter in life. Thank you, to my sister Saher, for her encouragement during the work. It feels reassuring to have a person you know you can always rely on. I am grateful to my brothers Usama and Eslam, my sisters: Asmaa, Elham and my wife. Thank you, to all the people who have dedicated me their precious time and enriched me, even if only with a wise word of experience.

i

Table of Contents Page No. Acknowledgment Table of Contents List of Tables List of Figures List of Symbols and Abbreviations Abstract

i ii vi vii xi xiii

Chapter 1: Introduction

1

1.1 Introduction 1.2 Thesis Objectives and Scope 1.3 Thesis Organization

1 4 4

Chapter 2: Conventional Time Driven-Event Driven Paradigm 2.1 Introduction 2.2 Time Driven 2.2.1 Time-Driven Simulation 2.3 Event Driven System 2.3.1 Event Definitions 2.3.2 Event-Driven Simulation 2.4 System Dynamics and Discrete Event Simulation 2.5 Time-Driven and Event-Driven Systems 2.6 The Major Properties of Discrete Event Systems (DES) 2.7 Event Modeling 2.7.1 Overview and Formal Model 2.8 Differences between Conventional Time-Driven and Event Driven 2.9 Components of Event-Driven System in Computer Science 2.10 Event-Driven Paradigm 2.10.1 Implementation 2.10.2 Event-Driven Simulation and Output Analyses 2.11 Comparison of the Two Models 2.12 Examples for Event Systems 2.12.1 Event-Driven Control for Reducing Resource Usage 2.12.2 Sensor-Based Event-Driven Control 2.12.3 Sensor Networks Integrated 2.13 Disadvantages of the Conventional Event Driven 2.14 Concluding Remarks

ii

6 6 6 6 7 8 8 9 10 10 11 11 15 15 18 19 20 21 22 22 22 23 23 24

Chapter 3: System Change Paradigm “Time Driven-Event Driven-Parameters Changes”

25

3.1 Introduction 3.2 The Relation between Time Driven and Event Driven 3.3 New Definition for Event 3.4 Classifications of Events 3.5 The Difference between Events and Disturbances 3.6 System Change Pathway Conceptual Graph 3.7 Calculating the System Parameter Change 3.8 The Consolidity-Based Theory of Change 3.9 Changes the Parameters of the System 3.10 Effect of Time in the relation between Consolidity and Theory of Change 3.11 The integrated Basic System and Upper System Layers Configuration 3.12 Main Advantages of the Combined Time-Driven and Event-Driven 3.13 Incremental Parameters Changes versus Affected Varying Environment 3.14 Systems Change Procedure 3.15 The Changeability Coefficient 3.16 Time-Event-Change System Paradigm 3.17 Concluding Remarks

25 25 26 26 33 33 35 35 37 39 39 41 42 43 44 45 48

Chapter 4 : Consolidity Theory

49

4.1 Introduction 4.2 Consolidity Definition 4.3 System Consolidity Index 4.4 Basics of the System Consolidity 4.4.1 Basic Definition of System Consolidity 4.4.2 Different Classifications of System Consolidity 4.4.3 The strong and weakness Ranking of Systems Consolidity 4.5 Implementation Strategy of Consolidity for Existing and New Systems 4.5.1 Consolidity Implementation Strategy for Existing Systems 4.5.2 Consolidity Implementation Strategy for New Systems 4.5.3 Essential Rules for Implementation of Consolidity 4.6 The consolidity for 2nd Order of State Space Problems using the stability coefficients 4.7 The consolidity for 3rd Order of State Space Problems using the stability coefficients 4.8 Consolidity Index for 2nd Order of State Space Problems using the Pole Placement 4.9 Consolidity Index for 3rd Order of State Space Problems using the Pole Placement 4.10 Consolidity Applications and Implementations 4.11 Concluding Remarks

iii

49 49 50 52 52 53 54 55 55 56 56 58 59 60 65 69 72

Chapter 5: System Change Pathway of Linear Relationship Paradigm 5.1 Introduction 5.2 A Model for HIV/AIDS Infectious Problem 5.3 Consolidity Analysis of the HIV/AIDS Infectious Problem 5.4 Consolidity Proposed Change Paradigm for the HIV/AIDS Infectious Problem 5.5 The Influence of Media Campaigns on the Number of HIV/AIDS Infected People 5.6 Changes in the Number of HIV/AIDS Infected People with Low Level of Media Campaigns 5.7 Changes in the Number of HIV/AIDS Infected People with Media Campaigns 5.8 Concluding Remarks Chapter 6: System Change Pathway of Exponential Relationship Paradigm 6.1 Introduction 6.2 Model for Event Driven and Parameters Changes for Prey Predator Application 6.3 Problem Description 6.4 Prey Predator Population Problem Formulation 6.5 Mathematical Module for Consolidity of Prey Predator Population Problem 6.6 Consolidity Based Change for Prey Predator Population Problem 6.7 Spread of Infectious Disease Problem Formulation 6.8 Concluding Remarks Chapter 7: Modeling and Simulation Analysis for “Event-Consolidity-Parameters Changes” Paradigm

73 73 73 74 77 79 80 83 85 86 86 86 87 88 89 91 95 99

100

7.1 Introduction 100 7.2 The Events and Parameter Changes 100 7.3 The Methodology for the Simulation Analysis and Modeling of Event Driven – Consolidity – Parameter Changes Paradigms 102 7.4 Simulation Analysis for Prey-predator Problem 103 7.4.1 The Steps of Simulation Analysis for Prey-Predator 103 7.4.2 Simulation Analysis for Prey-Predator with Increase the Rainfall Level 104 7.4.3 Simulation Analysis for Prey-Predator with Decrease the Rainfall Level 107 7.4.4 Simulation Analysis for Prey-Predator with Fluctuating the Rainfall Level 109 7.5 Simulation Analysis for HIV/AIDS Infectious Disease 110 7.5.1 The Steps of Simulation Analysis for HIV/AIDs Infectious Disease 111 7.5.2 Simulation Analysis for HIV/AIDS Infected People with Low Level of Media Campaigns 112 7.5.3 Simulation Analysis for HIV/AIDS Infected People with High Level of Media Campaigns 115 7.5.4 Simulation Analysis for HIV/AIDS Infected People with Fluctuation of the Level of Media Campaigns 117

iv

7.6 Simulation Analysis for Spread of Infectious Diseases 119 7.6.1 The Steps of Simulation Analysis for Spread of Infectious Diseases 120 7.6.2 Simulation Analysis for Spread of Infectious Diseases with Low Level of Hygiene of the Dorm 121 7.6.3 Simulation Analysis for Spread of Infectious Diseases with High Level of Hygiene of the Dorm 123 7.6.4 Simulation Analysis for Spread of Infectious Diseases with Fluctuation in Level of Hygiene of the Dorm 125 7.7 Concluding Remarks 127 Chapter 8: Conclusions and Recommendations for Further Research

128

8.1 Conclusions 8.2 Suggestions for Further Research

128 130

Refernces

131

Appendix A: Consolidity Index for the 2nd Order of State Space Using Stability Coefficients

138

A.1 Derivation of the Output for the 2nd Order of State Space Problems A.2 Derivation of the Input for the 2nd Order of State Space Problems A.3 Consolidity Index for the 2nd Order of State Space Problems A.4 Special Case: Consolidity Index for Equal Roots

138 139 139 139

Appendix B: Consolidity Index for the 3rd Order of State Space Using Stability Coefficients B.1 Derivation of the Output for the 3rd Order of State Space Problems B.2 Derivation of the Input for the 3rd Order of State Space Problems B.3 Consolidity Index for the 3rd Order of State Space Problems B.4 Special Cases: Consolidity Index for Equal Roots for 3rd Order Systems

140 140 143 143 144

Appendix C: Consolidity Index for the 3rd Order of State Space Using Pole Placement 147 C.1 Consolidity Index Using Pole Placement C.1.1 Ackermann’s Formula C.1.2 Derivation of the Input for the 3rd Order of State Space Problems C.1.3 Consolidity Index for the 3rd Order of State Space Problems C.2.1 Direct Comparison Method C.2.2 Derivation of the Input for the 3rd Order of State Space Problems C.2.3 Consolidity Index for the 3rd Order of State Space Problems

147 147 155 155 155 156 156

‫الملخص العربي‬ v

List of Tables Table No

Table Title

Page No.

2.1

Comparison between Time-driven and Event-driven System

15

2.2

Functional Specifications of the Control Machine

21

3.1

Searching the Relationship the Time and Event

25

3.2

Main Features of the Joint Time-Driven and Event-Driven Configuration

42

Average Consolidity, Controllability and Stability for 2nd Order State Space when Changing Parameter (a) where n=2, b=4 and c=2

58

Average Consolidity, Controllability and Stability for 3rd Order State Space when Changing (a where n=3, b=6, c=5 and d=1

60

Average Consolidity, Controllability and Closed Loop Feedback Gains for 2nd Order State Space when Changing Parameter (a) where n=2, b=4, c=2and the Poles are (-3, -3).

62

Average Consolidity, Controllability for 2nd Order State Space for Different Cases and Changing a where n=2, b=4, c=2.

63

Average Consolidity, Controllability and Stability for 3rd order State Space when Changing a where n=3, b=6, c=5, d=1and the Poles are (4+4i, -4-4i, -10)

66

Average Consolidity, Controllability for 3rd Order State Space for Different Cases and Changing a where n=3, b=6, c=5, d=1

67

The Influence of Media Campaigns on the Number of HIV/AIDS Infected People and the Consolidity Index of the System

80

5.2

Least Squares Data for the HIV/AIDS with Low Level of Media Campaigns

81

5.3

Least Squares Data for the HIV/AIDS with High Level of Media Campaigns

83

6.1

The Effect Rainfall on both of Rate of Growth of Prey and Consolidity Index

91

6.2 6.3

Least Squares Data for Prey Predator The Influence of Hygiene on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System

93

6.4

Least Squares Data for the Spread of Infectious Diseases

97

4.1 4.2 4.3

4.4 4.5 4.6

5.1

vi

95

7.1 7.2 7.3 7.4 7.5 7.6 7.7

7.8

7.9

Influence of the Rainfall Level on the Increase in the Number of Prey and Consolidity (Scenario I)

106

Influence of the Rainfall Level on the Decrease in the Number of Prey and Consolidity (Scenario II)

108

Influence of the Rainfall Level on the Fluctuation of the Number of Prey and Consolidity (Scenario III)

110

The Influence of Decrease the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario I)

114

The Influence of Increase the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario II)

117

The Influence of Fluctuation of the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario III)

118

The Influence of the Low Level of Hygiene of the Dorm on the Increase of the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to (Scenario I)

123

The Influence of the High Level of Hygiene of the Dorm on Decrease of the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario II)

125

The Influence of the Fluctuation of Hygiene of the Dorm on the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario III)

126

vii

List of Figures Figure No

Figure Title

Page No.

2.1

Illustration of the Notions Event Domain, Event Type and Event Instance.

12

2.2

General Model of Event-based Visualization – The Major Components of Event-based Visualization – Event Specification, Event Detection and Event Representation – are Attached to the Classic Visualization Pipeline.

13

2.3

Example of Conventional DES in Software Architectures

14

2.4

Schematic Diagram of Event-Driven Paradigm

19

3.1

Event Types and Classification

27

3.2

Event Originating from Solid Earth (Geophysical)

28

3.3

Events Caused by Short-lived/ Small to Meso Scale Atmospheric Processes (in the Spectrum from Minutes to Days) (Meteorological)

29

3.4

Events Caused by Deviations in the Normal Water Cycle and/or Overflow of Bodies of Water Caused by Wind Set-up (Hydrological)

3.5

Events Caused by Long-lived/Meso to Macro Scale Processes (in the Spectrum from intra-seasonal to Multi-decadal Climate Variability) (Climatological)

3.6

Disaster caused by the Exposure of Living Organisms to Germs and Toxic Substances (Biological)

3.7

A New Conceptual Graph Demonstrating the Successive Change Pathway of (State-Space Systems) when Subjected to Events, Varying Environments, or Activities.

3.8

Systems Pillars: Controllability (Determinant of Controllability Matrix), Stability (Eigenvalues), and Consolidity (Consolidity Index)

3.9

System Parameters Changes Scenarios: Parameter 1 (Decrease), Parameter 2 (Fluctuate) and Parameter 3 (Increase)

3.10

Change of the State of Parameters According to Negative Discrete Events Occurrence

3.11 3.12 3.13

30

31 32

34 36 36 37


38


38

Systems Change Pathways with the Two Layers (Basic Layer and the Upper Layers) and the Changes in the Parameters of the System [3]

41

viii

3.14

A Sample of Linear Patterns Relating the Incremental System Parameter Change versus Affected Varying Environment or Event Strength, for Different Values of System Consolidity Index C and Certain Specified Changeability Coefficient 

44

3.15

Event Driven-Time Driven-Parameter Change Paradigm

45

3.16

Time of Discrete Events Occurrence

46

3.17

A Chart of System Parameters Changes under Events Occurrence

47

4.1

Basic Definition of System Consolidity

53

4.2

Sketch Showing Different Classes of System Consolidity

54

4.3

Sketch Showing Possible Ranking within the Superior–Inferior Consolidity Scale

55

Consolidity versus Controllability for 2nd Order State Space for Different Cases when Changing a where n=2, b=4, c=2

64

Consolidity versus Controllability for 3rd Order State Space for Different Cases when Changing a where n=3, b=6, c=5, d=1

68

4.6

Suggested Areas for the Scope of Applications of Consolidity Theory

70

5.1

Changing the Number of HIV/AIDS Infected People due to the Influence of Media Campaigns

78

5.2

The Influence of Media Campaigns on Consolidity Index

79

5.3

Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Low Level of Media Campaigns

82

Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Media Campaigns

84

6.1

Example of a Real Life Predator-Prey Population Dynamic Problem [11]

88

6.2

Changing the Number of Prey and Consolidity Index due to the Influence of Changing the Rainfall Level

6.3

The Rate of Parameter Change (Number of Prey) due to the Influence of Changing the Rainfall Level and Consolidity Index

94

The influence of Room Cleaning on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System

96

Parameter Change (the Infection Rate of the Spread of Infectious Diseases) due to the Influence of Changing the Event Level and Consolidity

98

4.4 4.5

5.4

92

6.4 6.5

ix

7.1 7.2 7.3 7.4 7.5

7.6

7.7 7.8

7.9

7.10

System Parameters Change Forms: (a) Parameter 1 (Decrease) and (b) Parameter 2 (Increase) c) Parameter 3 (Fluctuate).

101

Simulation Analysis for the Increase in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (Scenario I)

105

Simulation Analysis for the Decrease in the Number of Prey and Consolidity Index due to the Influence of the Rainfall Level (Scenario II)

107

Simulation Analysis for the Variation in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (scenario III)

109

Simulation Analysis for the Increase in Number of HIV/AIDS Infected People and Consolidity Index with Decrease the Level of Media Campaigns (Scenario I)

113

Simulation Analysis for the Decrease in Number of HIV/AIDS Infected People and Consolidity Index with Increase the level of Media Campaigns (Scenario II)

116

Simulation Analysis for Changing the Number of HIV/AIDS Infected People with Different Levels of Media Campaigns (Scenario III)

119

Simulation Analysis for the Increase the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the Low Level of Hygiene of the Dorm (Scenario I)

122

Simulation Analysis for the Decrease the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the High Levels of Hygiene of the Dorm (Scenario II)

124

Simulation Analysis for the Fluctuation in Infection Rate and the Consolidity index of Spread of Infectious Diseases due to Different Levels of Hygiene of the Dorm (Scenario III)

126

x

List of Symbols and Abbreviations a

average number of different sexual partners per year

a1

interaction term that describes how the hares are diminished as a function of the lynx population

A

entity

A(t)

number of men with AIDS

AIDS

acquired immunodeficiency syndrome



changeability coefficient of the system

b

fuzzy parameter

b1

growth coefficient of the lynxes



constant parameter

cr

removal rate

c

controls the prey consumption rate for low hare population

d

mortality of the lynxes

D

data sets



incremental

DES

discrete event system

P

parameter change

EDS

event driven system

ET

event type

f

final (or end) event state

CO /( I  S )

consolidity index

CO

output fuzziness factor

C( I S )

input fuzziness factor

CO /( I  S ) 

consolidity index at event state step l (dimensionless)

HIV

human immunodeficiency virus

i

index

k

maximum population of the hares (in the absence of lynxes)

k1

constant and parameter

k2, k3, k4, k5

fuzzy parameters

xi

kr

infection rate

m

intermediate event state



event state or event number (index)

N(t)

population size

P

power set

 Ii

corresponding fuzzy level for I i

 Oi

corresponding fuzzy level for the output

 Si

corresponding fuzzy level for S j

V

set of visual representations

VOi

deterministic value of output component O j

VS i

deterministic value of system component S j

VI i

deterministic value of input component I i

r

growth rate of the hares

t

time state or time factor

TDS

time driven system

x(t)

those uninfected with disease but may become infected

X(t)

number of susceptible males HIV virus in the population

y(t)

those who are presently infected with the disease

Y(t)

number of males infected with HIV virus

Z(t)

number infected with virus but is non-infections

xii

Abstract

New research study indicated the system change pathway depends on the Time driven - Event driven – parameter Change paradigm. This paradigm is based on two-level system. The first is the time driven level or basic level which is governed by the system physical equation, where the second upper level is the event –driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. The parameters of the system can be changed due to the influence of event and consequently the consolidity also will be changed. These events can be positive or negative that can increase, decrease the parameters of the system and consolidity of the system, therefore entity will be changed to be built more or destroyed. This thesis has concentrated on developing modeling and simulation analysis for “Event Driven– Time Driven – Parameters Change” Paradigm if there are events occurrences on the parameters. The modeling concentrates on developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state. The model shows the relationship between the Event multiplied by the Consolidity Index and the parameter change has been developed and the parameter change can be calculated through the least squares approximations. Therefore, the developed equations for any system depend on the application as it can be linear, parabolic, exponential, logarithmic …etc. In the simulation, we will calculate the consolidity of the system, and then if we have an event, we can calculate the parameter change throughout relationship between the incremental parameters changes versus affected varying environment and events. Second, after the parameter changed and we can calculate the consolidity and so on. This simulation analysis is developed by three scenarios. In these scenarios, the events can influence the parameter and therefore can be increased or decreased. Three real life applications will be discussed throughout the work to study the influence of event driven system based consolidity theory on the system parameters. The first application is the studying of prey predator population problem that is a famous problem that describes the nature of life between prey and predator. In this context, a model has been studied throughout increase the rate of growth of prey and this can be achieved by variability of the climate. In addition, simulation analysis has been developed through three scenarios. The first scenario shows that if the rainfall increased, the rate of growth of prey will be increased and the Consolidity would be improved. The second scenario shows that if the rainfall has decreased, the number of prey will be decreased and the consolidity index will be higher which means that the system is unconsolidated. The third scenario shows the variation in the rainfall which will affect the increase and decrease of number of prey.

xiii

The second application is the studying of reducing the number of HIV/AIDS infectious people through radio campaigns on knowledge and use of condoms. In this problem, a modelling has been studied throughout increase and decrease the media campaigns. A simulation analysis also has been developed through three scenarios. The first scenario shows the increase in the number of HIV/AIDS infected people and the increase in the value of consolidity index as the level of media campaigns is high. The second scenario shows the decrease in the number of HIV/AIDS infected people and the decrease in the value of consolidity index as the level of media campaigns is low. The third scenario shows the fluctuation in the level of media campaigns and its influence on the number of HIV/AIDS infectious people and the consolidity index. The third application is the analysis of variability of level of hygiene and its influence on spread of infectious disease. In this problem, a model has been studied throughout changing the infection rate. A simulation analysis also has been developed through three scenarios. The first scenario shows the increase in the infection rate and also the increase in the consolidity index due to the low level of dorm cleaning. The second scenario shows the decrease in the infection rate and the decrease in the consolidity index due to the increase in percentage of hygiene of the dorm. The third scenario illustrates the fluctuation of the consolidity index and the infection rate of the spread of infectious diseases due to the influence of hygiene of the dorm. The three case studies confirm the suitability of the suggested paradigm to handle real life systems under external influences “on and above” their normal conditions.

xiv

Chapter 1 Introduction

1.1 Introduction Most research in digital feedback control considers periodic or time-driven control systems, where continuous-time signals are represented by their sampled values at a fixed frequency. Time-driven system performs control calculations all the time at a fixed rate, so also when nothing significant has happened in the process. This is clearly an unnecessary waste of resources like processor load and communication busload and thus not optimal if these aspects are considered as well. However, when the system is under operation, it faces many more changes due to event occurrences and these events are discrete. A discrete event system (DES) is a dynamic system with discrete states that are event-driven. The principal features of such systems are that they are discrete, asynchronous, and possibly nondeterministic. Event Driven system enables the designers to make better balanced multidisciplinary trade-offs resulting in a better overall system performance and reduced cost price. Another option is to perform a control update when new measurement data arrives. DESs arise in a variety of contexts ranging from computer operating systems to the control of complex multimodal processes. DES theory, particularly on modeling and control, has been successfully employed in many areas such as concurrent program semantics, monitoring and control of complex systems, and communication protocols. The behavior of a DES is described as the sequences of occurrences of events involved. Event-driven and time-driven models exhibit a complementary nature, in the sense that they reflect the nature of the contrasting requirements that any real-time system must meet - i.e., performing actions "as soon as possible" or "at the right time". Even if each of the models is sufficient to model a system, they should be intermixed to improve the expressiveness. The event-driven paradigm, which relies on an object oriented model, focuses on external events which control the execution of actions performed by agents (or objects, or actors, or the like). It supports a separation between the actions performed by an agent and the control performed by the environment. However, the concept of time is still missing, and the actual execution of actions is often left to the runtime support. In real-time contexts, event-driven languages can be exploited for systems with sporadic actions that must be executed as soon as possible. The time-driven paradigm is algorithm-oriented, but it includes the concept of time to specify when actions should be executed. This paradigm, though widely used in the process-control area, does not capture the concept of an "event". Usually, reactions to unpredictable external stimuli are implemented by means of polling mechanisms.

1

Time-driven paradigms are especially suitable for defining cyclic tasks which must be executed under an environment-independent timing. Some paradigms widely used in the expert-systems area support the dynamic planning of sequences of actions. However, the definition of the agenda usually depends on the internal status of the system, and expresses neither a response to asynchronous events nor the explicit management of time. The researchers have developed the time driven and event driven paradigm in the software architectures, but they didn’t use it in the real life application, consequently they have ignored the system parameter changes due to the variation of time and event that can influence on the parameters of the system. The scheme of the problem depends on the events and time that influence on the system parameters. These events can be positive or negative that can increase or decrease the parameters of the system, therefore the system will be changed to be built more or destroyed. Incorporating the even-driven (or activity-driven) nature of the problem jointly with the conventional time-driven platforms will be applied throughout consolidity -based theory of change. Consolidity is a new term introduced by Dorrah and Gabr [1-5] that means ‘‘the act or quality of consolidation’’. Consolidity theory provides an effective tool towards scrutinizing the inner behavior of systems. Such property is an essential pre-requisite for proper operation of systems working in a varying environment. Moreover, consolidity is an inherent built-in property that can be defined within a fully fuzzy environment. For systems operating in a fully fuzzy environment, the implementation of the theory to some representative case studies was successfully presented. For real life systems, consolidity property is accompanied in the system through the natural physical laws governing their existence. For man-made systems, this property still till now is overlooked by researchers and developers, and is kept just as a by-product of the designed system. Consolidity is measured by the systems output reactions versus combined input/system parameters reaction when subjected to varying environments and events [1]. System consolidity is expressed by the consolidity index denoted in this thesis by CO /( I  S ) . The system is defined to be consolidated if CO /( I  S ) < 1, neutrally consolidated if CO /( I  S ) = 1 and unconsolidated if CO /( I  S ) > 1. Such consolidity index plays an essential role in governing system parameters changes due to the events ‘‘on and above’’ normal operation during the system change pathway and acts mainly at the upper event-driven level. Stack-based systems change pathway theory elaborated, System analysis and design using consolidity theory with applications and a new generalized stacking programming (GDSP) model has been introduced in [6-8]

2

The system joint ‘‘time-driven’’ and ‘‘event-driven’’ configuration has recently been revealed based on a two-level form where the basic level represents the time scale ‘‘t’’ governed physical system layer(s), and the upper level is the influencing event layer(s). While the basic level is usually subject to normal operation disturbances that can be absorbed by the physical system behavior, the upper level is subject to events, activities and varying environments ‘‘on and above’’ the normal system operation. Other examples of the upper level effects are any external or internal excessive influences and happenings affecting the system such as accidents, collisions, impacts, breaks, shocks, collapses, eruptions, and destructions. Such upper level operation leads at each event occurrence to corresponding parameter changes in the physical system layer(s) at the lower level [2–4]. The first target of this thesis is to concentrate on developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state by the system self-consolidity; following

the general relationship at any event step  as:  Parameter change (  ) = Function [consolidity (  ) , varying environment or event (  ) ]. This represents the main core of modeling the system change in the suggested theory. In this respect, there are many foreseen illustrative examples in natural sciences, biology, medicine and engineering that can lead such experimentations. The second target is to develop a simulation analysis and modeling for the consolidity if there are events occurrences on the parameter, after choosing relationship between the incremental parameters changes versus affected varying environment and events of various values. In this simulation, we can calculate the change and then calculate the consolidity. The simulation analysis includes three scenarios for each application. The first scenario is the increase in the parameter due to the influence of events. The second scenario is the decrease in the parameter due to the influence of events. The third scenario is the fluctuation in the parameter due to the influence of increase and decrease the events. The thesis also elaborates the consolidity to the second and third order state space systems in hypothetical forms. In this respect, the three case studies will be modeled to further elaborate this paradigm and demonstrate the concept using three real life case studies. The first case study is the study of investigation of prey-predator model operating under external environmental influences. The prey predator population problem is a famous problem that describes the nature of life between prey which is a weak vegetation animal and predator which is feeding on the vegetation animal. In this context, the consolidity of prey predator population problem has been studied throughout increase the rate of growth of prey and this can be achieved by variability of the climate. In this problem, the climate is rainfall which can influence on the number of prey that can increase or decrease the consolidity of the system. The second is the study of the analysis of HIV/AIDS Epidemic problem model subject to variability of awareness and educational programs. This problem discusses the studying of reducing the number of HIV/AIDS Epidemic problem through radio campaigns on knowledge and use of condoms for prevention of HIV/AIDS. In this problem, the consolidity has been improved rather than without radio campaigns. 3

The third is the study of spread of infectious disease problem subject to variability of level of hygiene. The three case studies confirm the suitability of the suggested paradigm to handle real life system under external influences “on and above” their normal conditions. In this chapter, the thesis objectives and scope are presented. Then, a brief description of the contents of each chapter of this thesis is provided.

1.2 Thesis Objectives and Scope The main objective of this thesis is to develop an efficient paradigm that will be able to measure the external influences “on and above” their normal conditions of the parameters of the system. To achieve this goal, the following sub goals are to be achieved. 1. Developing a research study demonstrating that the system change pathway depends on the Time driven - Event driven – Parameter Change paradigm. 2. Choosing relationship between the incremental parameters changes versus affected varying environment and events of various values. 3. Applying the paradigm to real life application to confirm the suitability of the suggested paradigm to handle real life system under external influences “on and above” their normal conditions. 4. Developing simulation analysis for the model that can calculate the change and then calculate the consolidity if there are events occurrences on the parameter. 5. Applying simulation analysis for the paradigm to real life application.

1.3 Thesis Organization The thesis is organized in eight chapters and three appendices. The first chapter gives a general introduction and the thesis objectives and scope as well as the thesis organization. Chapter two introduces concepts for time-event system paradigm. The first is the time driven which is governed by the system physical equation, where the second is the event –driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. A detailed description for Event and Time has been represented in this chapter. Chapter three introduces concepts for time driven-event driven- parameter changes system paradigm. The first is the time driven level or basic level which is governed by the system physical equation, where the second upper level is the event –driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes in the system parameters. This chapter shows how the event occurrence can influence on the parameter status. In addition, developing a methodology to measure the change in the parameters if there was an event occurrence has been developed. 4

Chapter four presents a detailed description of the definitions of consolidity and consolidity index calculations for state space system. This is achieved by introducing the representation of the uncertainties of the system parameters, presenting the mathematical proofs for the formula used in the proposed technique and illustrating the propagation of the input parameters uncertainty. This chapter consists of two main parts. The first part includes measuring the degree of consolidity index of 2nd and 3rd order state space system that has been checked by using the roots of the system as the output of the system and the results for this analogy. The second part is to measure the consolidity index of 2nd and 3rd order state space system by using state feedback of pole placement as the output of the system to adjust the parameters of the systems has been applied to the state space system. Chapter five elaborates the paradigm of Consolidity based change for linear regression In order to get feeling and being familiar with the suggested paradigm and the event occurrence on the parameters and the calculations of changes of this parameter due to the event occurrence on the parameters. The case study which familiar with this paradigm is study in the analysis of HIV/AIDS Epidemic problem model subject to variability of awareness and educational programs. Chapter six demonstrates two real life applications. This chapter elaborates the paradigm of Consolidity based change for exponential function. This paradigm is applied for two real life problems. The first can study in the investigation of prey-predator model operating under external environmental influences. The second can study in the analysis of spread of infectious disease problem subject to variability of level of hygiene. Chapter seven presents simulation analysis for the model if there are events occurrences on the parameter after choosing relationship between the incremental parameters changes versus affected varying environment and events of various values. In this simulation analysis, we can calculate the change and then calculate the consolidity. In this respect, the three case studies will be modeled and simulated. These case studies are prey predator, HIV/AIDs infectious disease and the spread of infectious diseases. Concluding remarks and suggestions for further research are given in chapter eight. Appendix A provides derivations for the 2nd order state space equations with the stability coefficient as the output of the system. Appendix B presents derivations for the 3rd order state space equations with the stability coefficient as the output of the system with special case studies. Appendix C introduces derivations for the special cases for 3rd order state space equations with the state feedback coefficients as the output of the system using the Ackremann’s formula and direct comparison method.

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Chapter 2 Conventional Time Driven-Event Driven Paradigm

2.1 Introduction Time-driven systems change state in response to a uniformly progressing physical time. However Event-driven systems change state in response to the occurrence of asynchronous discrete events that result in instantaneous state transitions. The state of an event-driven system is unchanged between event occurrences. To eliminate the time-consuming testing procedures and to discover possible design imperfections at early project stage, the systems with discrete states and event driven behavior [commonly known as discrete event systems (DES)] can be introduced. Conventional system theory provides very well developed tools for the design of continuous and discrete time systems. However, the equivalent system theory for systems with event driven dynamics is still far from being developed mature enough for practical use. This chapter presents a detailed description for the conventional time driven and event driven and classifications of the types of events that can affect on the parameters of the system.

2.2 Time Driven The time is governing the system state equation and consequently the system response. It is also crucial in determining the order of the sequence of the occurrence of events. This is beside the intrinsic role of time in the synchronized coordination of various affecting events of the whole living universe [9, 10]. 2.2.1 Time-Driven Simulation

For continuous systems, time-driven simulations advance time with a fixed increment. With this approach the simulation clock is advanced in increments of exactly t time units. Then after each update of the clock, the state variables are updated for the time interval [t , t ] . This is the most widely known approach in simulation of natural systems [8]. Less widely used is the time-driven paradigm applied to discrete systems. In this case we have specifically to consider whether [11]:  The time step t is small enough to capture every event in the discrete system. This might imply that we need to make t arbitrarily small, which is certainly not acceptable with respect to the computational times involved.

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 The precision required can be obtained more efficiently through the event-driven execution mechanism. This primarily means that we have to trade efficiency for precision.

2.3 Event Driven System The term “discrete event system” was introduced to identify an increasingly important class of dynamic systems in terms of their most critical feature: the fact that their behavior is governed by discrete events occurring asynchronously over time and solely responsible for generating state transitions. In between event occurrences, the state of such systems is unaffected. Examples of such behavior abound in technological environments such as computer and communication networks, automated manufacturing systems, air traffic control systems, C3I (Command, Control, Communication, and Information) systems, advanced monitoring and control systems in automobiles or large buildings, intelligent transportation systems, distributed software systems, and so forth [12,13]. The operation of such environments is largely regulated by human-made rules for initiating or terminating activities and scheduling the use of resources through controlled events, such as hitting a keyboard key, turning a piece of equipment “on”, or sending a message packet. In addition, there are numerous uncontrolled randomly occurring events, such as a spontaneous equipment failure or a packet loss, which may or may not be observable through sensors. A discrete event system (DES) is a dynamic system with discrete states that are event-driven. The principal features of such systems are that they are discrete, asynchronous, and possibly nondeterministic. DESs arise in a variety of contexts ranging from computer operating systems to the control of complex multimodal processes. DES theory, particularly on modeling and control, has been successfully employed in many areas such as concurrent program semantics, monitoring and control of complex systems, and communication protocols. The behavior of a DES is described as the sequences of occurrences of events involved. For more details, refer to [14-16]. A discrete event model is different from continuous-time models. In continuous-time models, the state of the system changes continuously over time, and is usually represented by differential equations. In a discrete event model, the changes in the state of the system occur at separate instances of time. Even though discrete event and continuous-time models are defined in an analogous manner, a discrete event simulation model is not always used to model a discrete system nor is a continuous model always used for a continuous system. The characteristic of the system, and/or the objectives of the simulation study play a major role for the modelers to prefer the simulation model as discrete event or continuous-time [17-19]. A discrete-event simulation model is discrete, dynamic and stochastic in nature. Discreteevent simulation is the modeling of a system in which the state of the system changes only at a discrete set of points in time.

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2.3.1 Event Definitions

An event is anything significant that happens or is contemplated as happening. An event happens completely or not at all and is significant because it may affect some action. It is contemplated as happening because it could be a fact becoming true or could be a transition of an entity in the real world. It might be part of a business process; for example, a trade order has been issued, an aircraft on a specific flight has landed, a reading of sensor data has been taken; or it might be monitoring information about IT infrastructure, middleware, applications, and business processes [20]. An event should be thought of as occurring instantaneously and causing transitions from one system state value to another. An event may be identified with a specific action taken (e.g., somebody presses a button). It may be viewed as a spontaneous occurrence dictated by nature (e.g., a computer goes down for whatever reason too complicated to Figure out) or it may be the result of several conditions which are suddenly all met (e.g., the fluid level in a tank exceeds a given value). Event-driven world view concentrates on the events and their effect on the state of the system, the process interaction world view focuses on the processes, entities and their lifecycle in the system, while activity scanning concentrates on the activities of the model and the conditions that allow them to begin. 2.3.2 Event-Driven Simulation

In event-driven simulation the next-event time advance approach is used. For the case of discrete systems this method consists of the following phases [21]: Step 1: The simulation clock is initialized to zero and the times of occurrence of future events are determined. Step 2: The simulation clock is advanced to the time of the occurrence of the most imminent (i.e. first) of the future events. Step 3: The state of the system is updated to account for the fact that an event has occurred. Step 4: Knowledge of the times of occurrence of future events is updated and the first step is repeated. The advantage of this approach is that periods of inactivity can be skipped over by jumping the clock from event time to the next event time. This is perfectly safe since by definition all state changes only occur at event times. Therefore causality is guaranteed. The event-driven approach to discrete systems is usually exploited in queuing and optimization problems. However, as we will see next, it is often also a very interesting paradigm for the simulation of continuous systems [22-24].

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The most significant difference between a static system and a dynamic system is that a dynamic system has an internal state that evolves over time and that determines the output. The structure of the internal state space as well as its evolution function can be difficult to construct. While the input and output variables of a dynamic system are “visible” outside the system, the state is, by its nature, hidden within the system. Sometimes it can be helpful to think of each element q  Q of the state space as being that part of the history of the system that is sufficient for computing the current output of the system.

2.4 System Dynamics and Discrete Event Simulation Traditionally, the most important distinction is the purpose of the modeling. The discrete event approach is to find, e.g., how many resources the decision maker needs such as how many trucks, and how to arrange the resources to avoid bottlenecks, i.e., excessive waiting lines, waiting times, or inventories. While the system dynamics approach is to prescribe for the decision making to, e.g., timely respond to any changes, and how to change the physical structure, e.g., physical shipping delay time [25]. System dynamics is the rigorous study of problems in system behavior using the principles of feedback, dynamics and simulation. In more words, system dynamics is characterized by [26]:  Searching for useful solutions to real problems, especially in social systems (businesses, schools, governments...etc) and the environment.  Using computer simulation models to understand and improve such systems.  Basing the simulation models on mental models, qualitative knowledge and numerical information.  Using methods and insights from feedback control engineering and other scientific disciplines to assess and improve the quality of models.  Seeking improved ways to translate scientific results into achieved implemented improvement.  Systems dynamics approach looks at systems at a very high level so is more suited to strategic analysis. Discrete event approach may look at subsystems for a detailed analysis and is more suited, e.g., to process re-engineering problems.  Systems dynamics is indicative, i.e., helps us understand the direction and magnitude of effects (i.e., where in the system do we need to make the changes), whereas discrete event approach is predictive (i.e., how many resources do we need to achieve a certain goal).  Systems dynamics analysis is continuous in time and it uses mostly deterministic analysis, whereas discrete event process deals with analysis in a specific time horizon and uses stochastic analysis. We have seen that in continuous systems the state variables change continuously with respect to time, whereas in discrete systems the state variables change instantaneously at separate points in time.

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Unfortunately for the computational experimentation there are but a few systems that are either completely discrete or completely continuous state, although often one type dominates the other in such hybrid systems. The challenge here is to find a computational model that mimics closely the behavior of the system, specifically the simulation time-advance approach is critical. More details refer to [27-29]. If we take a closer look into the dynamic nature of simulation models, keeping track of the simulation time as the simulation proceeds, we can distinguish between two time-advance approaches: time-driven and event-driven.

2.5 Time-Driven and Event-Driven Systems The distinction between these two notions is the result of two fundamentally different concepts of clocks and time as follows [30]:  A clock ticks at a uniform rate (which could be continuous as in physical time or discrete as if the clock were a metronome). At every clock tick, an input (also called an event) e is selected from the input set I. If events occur for only some clock ticks and not others, then simply introduce a “null event” to represent the case of a clock tick having no corresponding event. In this case, state transitions are synchronized with or driven by the clock ticks. The clock alone is responsible for (drives) state transitions. For this reason, such a system is said to be time-driven.  A sequence of events is presented to the system. These events are not necessarily known in advance and need not occur at a clock tick. Indeed, there need not be a clock at all in the usual sense. In such a system, events occur asynchronously. The notion of time, if it is explicitly defined at all, is simply an index that determines the position of an event occurrence within the sequence of event occurrences. In other words, the event stream drives the clock rather than the other way around. Such a system is said to be event-driven.

The notions of time- and event-driven system are not complementary. A system can be both time-driven and event-driven. It is helpful to consider some examples to illustrate the distinctions among the three kinds of systems introduced above. In time-driven systems, the time is a global variable that is the same for every component of a system. In event-driven systems, on the other hand, every component has a different notion of time, since the time for a component is determined by the event stream that is presented to it. So not only is the notion of time different on different levels of a dynamic system, but the notion of time is different for every component on the same level as well [31].

2.6 The Major Properties of Discrete Event Systems (DES)  A DES is event driven, i.e. the state transitions are caused by the occurrence of events.  As in time driven systems, a DES model can be defined in continuous or discrete time depending on the time (T).  The state space of a DES can be either discrete or continuous depending on the state ( X). In a run of a DES, only a countable number of different states can be accessed.

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 During the run of a discrete event system, between any two time instances there is only a finite number of other time instances, i.e. T(s)  T where s is the union of all signals, is order isomorphic to a subset of the integers.  Ordering and timing of events can be represented in signals (based on sets) or in streams (based on ordered sequences). Processes can be represented as functions acting on signals or streams [32].

2.7 Event Modeling Now, the general model of event-based visualization will be introduced. First, an overview of the model will be given. The main aspects – event specification, event detection and event representation – will be discussed in detail in separate sections [33]. 2.7.1 Overview and Formal Model The basic idea of event-based visualization comprises three steps as follows: 1. Let users specify their interests as event types, 2. Determine if and where these interests match with the data (that is, detect event instances) and 3. Consider detected event instances when generating the visual representation. Let’s clarify this general procedure by an example. A physician who visualizes data that contain the daily numbers of cases of different diseases is assumed. Increases and decreases in the data can be discerned from a suitable visualization. However, there is a special condition that is of interest to the physician: If the number of new cases of influenza increases for three successive days and the absolute number of cases of influenza exceeds 300, then this might be an indication of a possible wave of influenza. In an event-based visualization session, the physician would do the following: He specifies the condition of interest as an event type [34-35]. The system tries to find actual instances of that event type. If no events are detected, the visualization is presented as usual. But if an event is detected, the visual representation is adjusted automatically to highlight the special situation. Charts are enhanced with markers or color scales are altered to emphasize the data record for which the event has been detected. The physician can quickly and easily recognize the importance of the current health situation, which enables him to take steps to prevent a possible epidemic of influenza. The integration of the user into the process of creating visual representations helps increase the relevance in the resulting images. To make this model truly general and flexibly applicable, it is put on a formal basis. First, it is necessary to define event types and event instances as well as their frame of reference on an abstract level. Then, a visualization transformation can be defined that considers event instances so as to achieve event-based visualization.

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Figure 2.1: Illustration of the Notions Event Domain, Event Type and Event Instance. To describe the frame of reference in which events occur, the notion of event domains is introduced. An event domain ED: contains entities with respect to which event types can be specified. Such entities can be, for instance, attributes of a relational data set, or simply time points. We define that event instances occur with regard to entities of an event domain. A distinction between event types and event instances is generally required in event-based systems. In our case, an event type et  ET : is used to express a concrete interest with regard to entities of an event domain. Here, ET denotes the set of all possible event types. To differentiate between event types that address different event domains, the notion of abstract event types is introduced. The abstract event type for an event domain ED is defined as ED  ET: where ED contains only those event types of ET that can be evaluated with respect to the elements of ED. In other words, an abstract event type is a notation to associate an event domain with a set of compatible event types. Actual event instances can be defined as follows. Assumed et  ED is an event type in some abstract event type ED and an entity ed  ED conforms to the interest expressed as et, then the triple e = (ed, et, EP)  E: denotes an event instance (or short event) of type et. The set of all possible event instances is denoted by E. The set EP denotes event parameters that can be assigned to an event instance. Event instances establish a connection between interests (that is, the event type) and concrete entities of some event domain. The introduced terms and their relation are illustrated in Figure 2.1.

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Event-based visualization can now be defined formally as follows. The detection of event instances of some event types in a data set can be modeled as a mapping from P(ET) and D (the set of data sets) to P(E) (P denotes power sets) detect: P(ET) × D → P(E). Visualization in general can be modeled as a mapping from D (the set of data sets) and P (the set of visualization parameters) to V (the set of visual representations) vis : D × P → V. For event-based visualization, the mapping vis is extended so as to take the event instances detected in the data into account evis : D × P × P (E) → V.

Figure 2.2 General Model of Event-based Visualization – The Major Components of Eventbased Visualization – Event Specification, Event Detection and Event Representation – are Attached to the Classic Visualization Pipeline. This formal description is the basis for our general model of event-based visualization, which consists of the three steps event specification, event detection and event representation. In order to combine these event-based concepts with visualization technology, we attached them to the classic visualization pipeline with the stages data analysis, filtering, mapping and rendering. As most visualization approaches follow this pipeline, virtually any visualization technique can be easily enhanced by the advantages of events. Figure 2.2 illustrates the extended visualization pipeline. Next, we will discuss the steps of our model in detail. There are three main categories of events known in the literature specially in computer science systems: – A Time Event, that occurs when the system reaches a certain instant of time; – A Change Event, that happens when an institutional entity changes in some way. – An Action Event, that happens when an agent perform an action (an interesting type of this kind of events is ExchMsg, which represents the act of sending a message).

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It is illustrated from Figure 2.3 that the parameters of the system can be changed if there are event occurrences on the parameter, otherwise the system will be unchanged.

Start

Read the system parameters

Event Existence

Yes

No

No Change of the system Parameter

Change the state of the system

Build the new parameters of the system

End Figure 2.3 Example of Conventional DES in Software Architectures

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2.8 Differences between Conventional Time-Driven and Event Driven Some differences between the conventional Time-Driven and Event Driven [36-39] have been noticed as shown in Table 2.1. Table 2.1 Comparison between Conventional Time-driven and Event-driven System

Aspects Time-driven System Event-driven System Functionality Time-driven systems change state in Event-driven systems change state in response to a uniformly progressing response to the occurrence of physical time. asynchronous discrete events that result in instantaneous state transitions. Variable The time is a global variable that is the Every component has a different same for every component of a notion of time, since the time for a system. component is determined by the event stream that is presented to it. Advantages The simulation clock is advanced in The advantage of this approach is that increments of exactly t time units. periods of inactivity can be skipped Then after each update of the clock, over by jumping the clock from event the state variables are updated for the time to the next event time. time interval [t , t ] . Controllers Perform control calculations all the Perform a control update when new time at a fixed rate, so also when measurement data arrives. The aim of nothing significant has happened in event-driven controllers is to reduce the process. the resource utilization for the controller tasks which resulting in a better overall system performance and reduced cost price. Use Time-driven model would be preferred Event-driven methods (or more when the characteristics of the generally hybrid networks) showed experiment produces high neural good performance and accuracy when activity. working with low rates of activity [13].

2.9 Components of Event-Driven System in Computer Science Decision making is becoming increasingly difficult at the same pace as the increasing complexity of real world systems. It is important to understand the concept of a system and its components. Banks et al. (2005) defines a system as a group of objects that are joined together in some regular interaction or interdependence towards the accomplishment of some purpose.

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The complexity arises from the interrelations and interactions among the various elements of the system. It can be understood that in such a complex system, changing one aspect can influence or change other parts of the system. To model and analyze the system requires clearly defined terms for the components of the system. The following components make up a discrete event simulation [40-42]: 2.9.1 Entity

Kelton et al. (2004) defines entities as the objects of interest that move around, change status, affect and are affected by other entities and the state of the system, and affect the output performance measures. Further, Kelton et al. elaborates on entities by describing them as the dynamic objects of the simulation that are created, move around the model, and are disposed of as they leave the model. As in real-world systems, entities can have many independent copies or realizations of itself at a time. However, as described by Kelton et al., a simulation model can have fake or logic entities which do not correspond to any tangible or real world object. The purpose of their existence in the simulation model is to take care of certain modeling operations. 2.9.2 Events

An instantaneous occurrence that changes the state of a system is called an event. Discrete event simulation, as per definition, models the evolution of a system through time by changing its state at discrete points of time. Event-driven simulation uses events to build the structure of the model. Entities and events are the basic simulation elements around which the process- and event-driven approaches are created. The proposal research focuses on handling of entities and events while integrating the process- and event-driven approaches. 2.9.3 State Variables

State variables are the set of variables that describe the characteristics of a system at any given time, irrespective of the types of entities existing in the system. Similar to the attributes of an entity, state variables are the characteristics of the system but they are not attached or tagged to any entity. Most of them can be accessed or changed by any entity in the system. State variables represent the state of the system that changes during the simulation run.

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2.9.4 Attributes

Attributes are the properties or characteristics of an entity that distinguishes it from other entities. The attribute values of a particular entity are attached or tagged to that entity itself. The value of an attribute will usually vary across the entities, even if all the entities have the same set of attributes. Either the simulation model, by default, or a modeler defines the attributes of an entity. The values or states of an attribute are changed along with the state variables at discrete points of time. 2.9.5 Event List/Event Calendar/Future Event List (FEL)

Event list or Event Calendar or Future Event List (FEL), is the list of events to occur in the simulated future, ordered by time. The general functionality of the event list is to sort the events according to their order of time of occurrence. When the simulation logic calls for the current event to occur during the simulation run, the event list provides the event with smallest or earliest time possible. When an appropriate event is executed, it schedules new events and places them in the event list. After an event occurs, it no longer is found in the event list. 2.9.6 Entity List

Similar to the event list, an entity list is the list of entities that are required to be processed by blocks in a model, indexed by time. During the simulation run, the entity with the earliest time is removed from the entity list and sent to the corresponding block in the model it is mapped to. The detailed functionality of the entity list is discussed in the succeeding sections. 2.9.7 Activity

Activity time is the duration of time of during which an activity takes place (Banks et al. 2005). For example, activity time could represent a service time, an inter arrival time or any other processing time. The duration of an activity can be either deterministic or random variable or a function depending upon certain variables. 2.9.8 Delay

Delay (Banks et al., 2005) is defined as duration of time of unspecified indefinite length.

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2.9.9 Resources

A resource is a source of supply or support. In manufacturing systems, a resource represents personnel, equipment, machines or a storage space. In a simulation model, an entity is allocated or can seize a unit of resource when it is available and can release the resource when finished. Kelton et al. discusses how resources are given to an entity rather than the entity being assigned to the resource, since an entity can be simultaneously using multiple resources. 2.9.10 Queues

Queues are used in a simulation model to store entities that are waiting for a resource to be seized. Queues usually have an interface that allows the addition and removal of entities. 2.9.11 Simulation Clock

The simulation clock is a variable that keeps track of the simulated time during a simulation run. In a discrete event simulation system, the simulation clock records the time of the events rather than the conventional real time. It keeps track of the time from event to event and ignores the time in between, where nothing interesting happens from the simulation perspective. Usually, the simulation clock works closely with the main simulation program and event list. When the main simulation program removes the next event or event with smallest time from the event list, the simulation clock updates the clock time and moves the time of the system to that time.

2.10 Event-Driven Paradigm Feedback schedulers are usually time triggered, that is, they execute as periodic tasks. The advantage with this mode is that it makes convenient to design and analyze the performance of the feedback schedulers using well-established feedback control theory and technology. This is because sampled data control theory in existence basically originates from periodic sampling [43]. The efficiency of the time-triggered mechanism can be examined in terms of response speed and overhead. To achieve quick response, feedback schedulers prefer small activation intervals so that almost all changes in workload can be treated in a timely fashion. Since the execution of feedback schedulers consumes resources, which are originally limited, the decrease of activation interval yields the increase of feedback scheduling overheads, which could adversely influence the system performance [44].

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When a relatively large activation interval is chosen for the feedback scheduler, on the other hand, it is possible that the system stays in a steady state for quite a long time, when there is actually no need for sampling period adjustment. In this situation, time triggered feedback schedulers could potentially waste resources in periodic executions of the feedback scheduling algorithm and unnecessary updates of system parameters. If the changes in workload are regular, it will be easy to determine the period for the feedback scheduling task. However, this will become very difficult when the changes in workload are irregular and unpredictable. From the above observations, we suggest an event driven mechanism to improve the efficiency of feedback schedulers. Discussed below is how to implement this mechanism [45]. 2.10.1 Implementation

The schematic diagram of the event-driven activation mechanism is depicted in Figure 2.4. Similar to the structure of event-based controllers, there are basically two parts in this paradigm, the event detector and the feedback scheduling algorithm. The event detector is time triggered, while the feedback scheduling algorithm is triggered by the execution-request event issued by the event detector [46-48]. This event-driven activation mechanism is generally applicable to almost all feedback scheduling methods.

Execution Request Event Detector

Feedback Scheduling Algorithm

Time-Triggered

Time-Triggered

Figure 2.4 Schematic Diagram of Event-Driven Paradigm

From Figure 2.4, it can be seen that this paradigm is actually a combination of time-triggered and event triggered approaches. That is why we use the term event-driven rather than eventtriggered. An intuitive advantage with this event-driven paradigm is that both the predictability and flexibility inherent in time-triggered and event-triggered systems respectively can be achieved through the combination.

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The key to implementing the event-driven paradigm is the design of the event detector. The major role of the event detector is deciding under what conditions the system needs to execute the feedback scheduling algorithm. As mentioned previously, the goal of the feedback scheduling loop is to maintain the CPU utilization at a desired level. Intuitively, when the utilization is in or close to steady states, there is no need for executing the feedback scheduling algorithm. On the contrary, if the utilization has significantly deviated from the desired level, then it becomes mandatory to run the feedback scheduler to adjust scheduling parameters [49-50]. 2.10.2 Event-Driven Simulation and Output Analyses

In event-driven simulation models, a modeler has the ability to know the state of everything at any time and flexibility with regard to attributes, variables, and logic flow. Alternatively, the ease of modeling a real system with a commonly understood flow chart approach using process blocks, has allowed process-driven models to be more accepted among the nontechnically oriented simulation user community. The process driven modeling framework’s main strength is drawn from the process analysis approach, which when applied in an orderly manner helps in analyzing the multiplicity of factors affecting a process. Simulation environments based on this approach are very effective in translating the analysts understanding of system structure into a model [51].

If we are simulating what happens within a manufacturing plant during a 10-hour day, the simulation study might take one second of real time to execute, but the simulation clock will show an elapsed time of 10 hours. The movement of the simulation clock is extremely critical, and there are two common methods for updating time: (1) the simulation clock could be updated a fixed amount at each increment or (2) the clock could be updated whenever something happened to change the status of the system. The advantage of the first method is that it is conceptually simple; the disadvantage is that if the increment is too small then there are inefficiencies in “wasted” calculations, and if the increment is too large then there are mathematical difficulties in representing everything that happens during the increment. Therefore, most simulation procedures are based on the second method of time management, called event-driven simulations. In other words, the simulation clock is updated whenever an event occurs that might change the system.

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2.11 Comparison of the Two Models The discussion of the time-driven and event-driven control models can be illustrated by a simple example that allows the models to be understood and compared. The example describes the behavior of a crane which moves items from a conveyor belt to a storage area. The crane moves to location (xo, yo), to grab an item, and then moves to location (x1, y1), to drop the item in the storage area. A further requirement imposes that, at location (x1, y1), the item cannot be dropped unless it is in a steady state. In an object-oriented style, the crane can be modeled as an agent that reacts to commands issued by a control machine. The commands trigger the grab, move, and drop actions. The behavior of the agent can be described by a finite-state automaton, in which each status defines the set of commands that the agent can execute (Rumbaugh, 1991). For instance, only if the crane is in an idle status, can the grab command be executed. Non-valid commands are assumed to be discarded, raising an exception [52-53]. The functional specifications for the control machine of the system can be defined by associating state conditions with commands that must be issued to the agent whenever a state condition becomes true. Table 2.2 shows a set of specifications for the crane system. It is assumed that the conveyor belt starts at 8:00am, while, for simplicity, the termination is not specified. Further non-functional requirements could be added to complete the system's specification. They could deal mainly with performance and safety issues. For instance, a requirement could state that no more than one item out of 100 should be missed, and another that a recovery action must be performed within one second from the occurrence of an alarm. Table 2.2 Functional Specifications of the Control Machine

State

Command

Time= 7:59 am Item available

Move to ( x0 , y0 ) Grap item

Item grapped

Move to ( x1, y1)

Crane at ( x1, y1) Item steady

Drop item

Item stored

Move to ( x0 , y0 )

In general, the choice between time-driven and event driven control clauses depends on the requirements of the application, on the features of the hardware and software environment, and on the taste of the designer. A problem is that the definition of a programming paradigm based on the two models is not straightforward. In practice, every programming paradigm assumes one of the two models as the primitive, hence imposing a restriction that systems will be designed by using only that control model, at the price of a loss of expressiveness. 21

In many control applications the time constraints are specified in terms of timed sequences of actions. In this case, the time-driven approach is possibly the most suitable, since the requirement of acting "at the right time" is met by definition. For instance, the motion of a robot can be specified in a natural way as a timed plan defining a sequence of actions, i.e., elementary steps [54-56]. The use of a time-driven control machine provides several advantages. First, if there are no hidden policies (as in the concurrent and event-driven paradigms), the temporal behavior can be explicitly controlled in a deterministic way. Second, a time driven programming paradigm is highly expressive in many embedded, hard-real-time application domains. Third, the management of the time at the basic level makes efficient implementations easier, and supports a fine-tuned and accurate timing. From this point of view, the proposed approach is expected to be better, or at least not worse, than the best event-driven approaches [57-58].

2.12 Examples for Event Systems If we look to all systems as economics, historical, engineering…etc, we will find that all systems are considered to be events not periods. In the following section, examples in control engineering are developed. Although there are various benefits of using event-driven control like reducing resource usage (e.g. processor and communication load), their application in practice is hampered by the lack of a system theory for event-driven control systems [59]. 2.12.1 Event-Driven Control for Reducing Resource Usage The control update times τk are in conventional time-driven control related through τk+1 = τk + Ts, where Ts is a fixed sample time, meaning that the control value is updated every Ts time units. To reduce the number of control calculations, we propose not to update the control value if the state x(τk) is contained in a set B ,as an example, around the origin.

2.12.2 Sensor-Based Event-Driven Control We can however apply variations on classical design methods if we define our models of the plant and the controller in the (angular) position domain instead of the time domain. This idea is based on the observation that the encoder pulses arrive equally spaced in the position domain. It is shown that, by applying this event-driven controller, we not only decrease the encoder resolution – and therefore the system cost price - but also the average processor load, compared to the originally applied controller in industry. Start-up behavior of event-driven sensor networks for impact load monitoring [60].

22

2.12.3 Sensor Networks Integrated Sensor networks integrated in components made of textile-reinforced composites can be used for monitoring impact loads on these components. A very low power consumption of such systems is achieved by switching on the power supply only when a relevant impact is detected. The sensor network consists of modules with start-up sensor, activation circuit and, application specific integrated circuits (ASICs) that can be integrated into a composite made of glass fibers and polypropylene [61]. A fast start of the data recording after the beginning of the impact event is very important. The effect of several network components on the achievable start-up time is shown and results of simulation and experimental investigations are given. In addition, the power consumption during one measuring cycle is considered. The system can be used for example in a component of a vehicle to monitor the impact load in crash situations. Another application is impact load monitoring in a transport container for sensitive goods to detect improper handling. A sensor network for impact load monitoring with very low standby current can be realized by switching on the power supply only when a relevant impact is detected by a start-up sensor. It is important that the data acquisition starts quickly after the beginning of the impact event. The startup process of a sensor network can be integrated into a textile-reinforced composite component made of glass fibers and polypropylene. In addition, the power consumption in operation mode was investigated with a test system. More applications in the field of event driven and time driven are smart home controller [62], wildfire spread and suppression models [63], creating log files [64], supervisory control [65], automata [66], event processing [67], constraint programming [68], geographically distributed [69], Petri nets [70], spacecraft control [71], ...etc.

2.13 Disadvantages of the Conventional Event Driven The researchers have developed the time driven and event driven paradigm in the software architectures or the module of Event Driven is based on the feedback control system, but they didn’t use it in the real life application, consequently they have ignored the system parameter changes due to the variation of time and event that can influence the parameters of the system. Therefore in the next chapter, system change pathway that includes “Event Driven- Time Driven- parameters Changes” will be developed.

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2.14 Concluding Remarks The impact of the distinction between time-driven and event-driven systems is especially important when one is composing systems from smaller systems. In time-driven systems, the time is a global variable that is the same for every component of a system. In event-driven systems, on the other hand, every component has a different notion of time, since the time for a component is determined by the event stream that is presented to it. So not only is the notion of time different on different levels of a dynamic system, but the notion of time is different for every component on the same level as well. Time driven System Performs the control calculations in all the time with a fixed rate and also if there is nothing significant happened in the process. Event driven System Perform a control calculations when new thing or measurement data arrives. The main task of eventdriven controllers is to reduce the calculations for the controller tasks which will reduced cost price and improve overall system performance. Unfortunately, the conventional time driven and event driven are not sufficient for real life problem as they ignore the internal parameter changes. Therefore in the next chapter, a development of new system change pathway paradigm based on the “Time Driven - Event Driven - Parameter Change” is explained to study the influence of the external events or the environment on the internal parameter changes.

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Chapter 3 System Change Paradigm “Time Driven-Event Driven-Parameters Change”

3.1 Introduction The new system change pathway depends on the “Time driven - Event driven – Parameter Change” paradigm. This paradigm is based on two-level system. The first is the time driven level or basic level which is governed by the system physical equation, whereas the second upper level is the event –driven level causing influence on the system on and above the normal situation. Such influences affecting events cause, at each event step, changes to the system parameters. In this chapter, a complete vision to Time driven - Event driven – parameter Change paradigm has been introduced in a simple way.

3.2 The relation between Time Driven and Event Driven When searching on Google site on 24 May 2014 there are Millions of papers as depicted in Table 3.1 related to Time, Event, Time Driven, Event Driven and both of “Time Driven Event Driven”. However, none of these papers has demonstrated the relationship between the time and event in the real life systems. Table 3.1 Searching the Relationship the Time and Event

Word Searches numbers (Millions)

Time

Event

2770M

677M

Time

Event

Time Driven and

Driven

Driven

Event Driven

363M

104 M

86M

In the next chapter we will develop a new description for time driven and event driven

25

3.3 New Definition for Event Events are defined in this work as any incidental or intentional, external or internal, momentarily or time determinable happenings, occurrences, activities or impacts of significant sizes ‘‘on and above’’ the normal system stand or original set-point. These events should have impetus upon the system and lead to shifting the system or some of its components beyond their normal specified operations [72]. In general, events may be related to each other through time, causality or aggregation. Such relations could effectively influence the overall intensity of such events. Events can also be classified into the following categories: descriptive, prescriptive, factual or assumptive. Additional classifications in other applications are: common, composite, group or cascaded events. In general, each event can equivalently be represented by an overall or effective value that combines its type and strength. Examples of events are the solar radiation, rain storms, winds, gravitational effects, fierce birds attacks, wild life animals feedings, human settlement activities, earthquakes and tectonic movements, etc. On the other hand, nature stores these events into corresponding consolidity-based scaled parameters changes of the ecology of forests, aquaculture, living beings, remnants of plants and animals, weathering, erosions, sedimentation of different types, etc. Real-life systems operations follow the same principle of the two-level multilayer configurations which typically replicate that of the Universe global configuration (of the heavens and earth layers one above another) but with much reduced scale and complexity.

3.4 Classifications of Events There are more types and classifications of events in our life. In this section, we study types of events “on and above” the normal system situation. These events can be geophysical, meteorological, hydrological, climatological, and biological.

26

These types of events are depicted in Figure 3.1. In this figure, the first type is geophysical as earthquake, volcano, and mass movement (dry). The second type is meteorological as storm. The third type is hydrological as flood and mass movement (wet). The fourth type is climatological as extreme temperature, drought and wildfire. The fifth type is biological as epidemic, insect infestation and animal stampede.

Geophysical

Hydrological

Meteorological

Earthquake

Flood Storm Mass Movement (wet)

Volcano

Mass Movement (dry)

Climatological

Biological

Extreme Temperature

Epidemic

Drought

Insect Infestation

Wildfire

Animal Stampede

Figure 3.1 Event Types and Classification

27

We will explain these types of events in details. The first type is geophysical. These types are earthquake, volcano, and mass movement. Earthquake may as ground shaking and Tsunami. Volcano may as volcanic and eruption. Mass movement may as landslide, avalanche, subsidence and rock fall. The avalanche can be snow debris, subsidence can be sudden longlasting and rock fall can be mudslide debris flow as depicted in Figure 3.2.

Earthquake

Ground shaking Tsunami

Volcanic

Volcano

Eruption Geophysical

Rock fall

Mass Movement

Avalanche

Snow Debris

Subsidence

Sudden Long- lasting

Landslide

Mudslide Debris flow

Figure 3.2 Event Originating from Solid Earth (Geophysical)

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The second type is meteorological which is as storm. The storm is consists of local/ convective storm, extra-tropical cyclone storm (winter Storm) and tropical storm. The local/ convective storm is as thunderstorm/ lightening, snowstorm/ blizzard, sandstorm/ duststorm, generic (severe) storm, tornado and organic storm (winds strong) as depicted in Figure 3.3. , Tropical Storm

Extra-Tropical cyclone Storm (winter Storm) Meteorological

Storm

Thunderstorm/ Lightening

Snowstorm/ Blizzard

Sandstorm/ Dust storm Local/ Convective storm Generic (severe) Storm

Tornado

Organic Storm (strong Winds)

Figure 3.3 Events Caused by Short-lived/ Small to Meso Scale Atmospheric Processes (in the Spectrum from Minutes to Days) (Meteorological)

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The third type is Hydro which will be as flood, mass movement (wet), avalanche, landslide, and subsidence. The flood can be storm surge/ costal flood, flash flood, general river flood. mass movement (wet) is as rock fall. Landslide is as debris Avalanche and debris flow. Avalanche is as debris avalanche and snow avalanche. Subsidence is as longlasting subsidence and sudden subsidence as depicted in Figure 3.4.

General river flood Flood

Flash flood

Storm surge/ Costal flood Mass Movement (wet)

Rock fall

Debris flow Landslide Debris Avalanche Hydro Snow Avalanche Avalanche Debris Avalanche Sudden Subsidence Subsidence Long-lasting Subsidence

Figure 3.4 Events Caused by Deviations in the Normal Water Cycle and/or Overflow of Bodies of Water Caused by Wind Set-up (Hydrological).

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The fourth type is Climatological which will be as extreme temperature, drought and wild fire. The extreme temperature can be as heat wave, cold wave as frost and extreme winter Conditions as snow pressure, icing and freezing rain debris avalanche. The wild fire is as forest fire and land fires (grass, scrub, bush, etc.) as depicted in Figure 3.5.

Heat Wave

Cold Wave

Frost

Extreme Temperature Snow Pressure

Extreme winter Conditions

Climatological

Icing

Freezing Rain Debris Avalanche Drought

Drought

Forest Fire Wild Fire Land Fires (grass, scrub, bush, etc.)

Figure 3.5 Events Caused by Long-lived/Meso to Macro Scale Processes (in the Spectrum from intra-seasonal to Multi-decadal Climate Variability) (Climatological).

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The fifth type is biological which is as epidemic. Epidemic includes viral infectious diseases, bacterial infectious diseases, parasitic infectious diseases, fungal infectious diseases, prion infectious diseases, insect infestation as grasshopper/ locust/ worms, and animal stampede as depicted in Figure 3.6.

Viral Infectious Diseases

Bacterial Infectious Diseases

Parasitic Infectious Diseases

Biological

Epidemic

Fungal Infectious Diseases

prion Infectious Diseases

Insect infestation

Grasshopper/ Locust/ Worms

Animal Stampede

Figure 3.6 Disaster caused by the Exposure of Living Organisms to Germs and Toxic Substances (Biological).

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3.5 The Difference between Events and Disturbances A disturbance is a signal that tends to adversely affect the value of the output of a system. If a disturbance is generated within the system, it is called internal, while an external disturbance is generated outside the system and is an input. In the presence of disturbances, feedback control refers to an operation that tends to reduce the difference between the output of a system and some reference input and does so on the basis of this difference. The disturbance can affect the whole system, however it can be overcome by the closed loop controller with the set points that are defined to the system. On the other hand, the events are an external objects or varying environment that can influence on the system and change its parameters itself.

3.6 System Change Pathway Conceptual Graph The current practices for developing built-as-usual physical system are based on giving the sole emphasis on designing each system with strong stability and high controllability. Consolidity in this practice was but a direct by-product of the finished designed system. Thus, the current built-as-usual methods have a high possibility toward mostly moving such systems in the undesirable direction of the inferior consolidity end (high consolidity index). Such current practices pathways have been depicted for typical natural or man-made produced physical systems as illustrated in the new updated conceptual graph of Figure 3.7, based on conventional physical systems of state-space like form of the linear or linearized type. In the majority of real-life situations, however, each system’s change pathway follows a zigzagging pattern with many downs and ups, but the prevailing tendency will always be toward a definite final (or end) state(s). It can be seen that Figure 3.7 illustrates in a simple way the progress of successive system changes under varying environments or events through different event states l = 0, 1, 2, . . ., m, . . ., f, where 0 indicates the original (or initial state), m denotes an intermediate state, and f designates the final (or end) state. In the analysis, the event state l is assumed for simplicity to be of the whole (integer) numbers type.

33

Original System State Superior Stability and Controllability/ Inferior Consolidity) .

X  A0 X  B0u

Controllability and Stability

Events and Environmental Influences

.

X  A1 X  B1u


Intermediate System (Moderate Stability and Controllability/ Moderate Consolidity)

.

X  A2 X  B2u

.


X  Am X  Bmu

.

Consolidity

X  Af X  B f u

Final System State (Inferior Stability and Controllability/ Superior Consolidity

Figure 3.7 A New Conceptual Graph Demonstrating the Successive Change Pathway of (State-Space Systems) when Subjected to Events, Varying Environments, or Activities..

34

The varying environments, events, or activities are considered to be all additional influences ‘‘on and above’’ the system normal stands or original set points. Moreover, such graph is developed based on the opposite mathematical relation between consolidity versus both controllability and stability [3]. The Real-life natural and man-made systems developed using built-as-usual practices usually undergo during their life span into similar cycles based on their status of consolidity. Original normal systems usually exist with superior stability and controllability set points. Such stand, however, will be directly accompanied by inferior system consolidity (high consolidity index) that makes these systems very susceptible to parameters changes under operation in real-life fully fuzzy environment. If the affecting varying environments or events by one way or another are carefully observed and controlled, the system will remain always within its normal stand or original set point. Such condition, however, is not viable as these systems are operating as part of larger systems of the universe and must interact and be interacted by the other systems.

3.7 Calculating the System Parameter Change The continual parameters changes, however, can only be handled through event-driven (or activity-driven) frameworks that are logically influenced by the consolidity behavior of the systems. Such consolidity-based change behavior constitutes an essential part of the systems inner property during their operation pathways. The academic literature together with the current built-as-usual systems practices of most disciplines is giving their strongest emphasis on the immediate behavior (including the dynamics) of natural and man-made systems, thus overlooking the systems unavoidable changes during their course of life. In fact, at present, there is no existing unified theory for mathematically formulating the mechanism of systems parameters changes when subjected to varying environments or events. Only there are some descriptive or evidential isolated models for understanding systems changes in some specific areas, such as organizations, political behavior, education, community-based socio-cultures, evolution of living systems, and very few other areas [3-4].

3.8 The Consolidity-Based Theory of Change Due to the inferior consolidity (high consolidity index) of the natural and man-made systems, the parameters of the systems tend to change very slightly far from their original set points toward more improved consolidity (lower consolidity index), leading to degrading their corresponding levels of controllability and stability as illustrated in the sample pattern graphs of Figure 3.8 and Figure 3.9. The mechanism of the conceptual pathway development is based on that the system behavior change rate is logically conceived to be not accidentally happened but is relatively influenced at the point of progress after being subjected to the varying environments or events with the associated direct (on-the-spot) value of the system consolidity index. The analysis of the suggested conceptual model is based on event-driven system framework, where the periods of inactivity can be passed over by jumping from one changed environment or event state to the next changed environment or event state and so on. 35

Figure 3.8 Systems Pillars: Controllability (Determinant of Controllability Matrix), Stability (Eigenvalues), and Consolidity (Consolidity Index). Figure 3.9 shows the System parameters changes forms. Such changes can influence on the parameter status, where the parameter may be increased, decreased or fluctuating.

Scenario I

Scenario III

Scenario II

Figure 3.9 System Parameters Changes Scenarios: Parameter 1 (Decrease), Parameter 2 (Fluctuate) and Parameter 3 (Increase). 36

3.9 Changes the parameters of the system To illustrate the changes of the parameter due to event influences, let’s take the following examples. If we have an entity A and there is an event and this event is negative change, then the entity A will be changed to entity A1. In addition, if there is another negative event affect on entity A1, then the entity will be changed to entity A2 As depicted in Figure 3.10. The event may be rainfall that affect on the grass and then when the rainfall decreases, the grass decreases and the number of prey can decrease

Figure 3.10 Change of the State of Parameters According to Negative Discrete Events Occurrence. If we have an entity A and there is an event and this event is positive change, then the entity A will be changed to entity A1. In addition, if there is another positive event affect on entity A1, then the entity will be changed to entity A2 As depicted in Figure 3.11. The event may be rainfall that affect on the grass and then when the rainfall increases, the grass increases and the number of prey can increase. If we have an entity A and there is an event and this event is negative change, then the entity A will be changed to entity A1. In addition, if there is another negative event affect on entity A1, then the entity will be changed to entity A2. If there is another positive event affect on entity A2, then the entity will be changed to entity A3 as depicted in Figure 3.12. The event may be rainfall that affect on the grass and then when the rainfall increases, the grass increases and the number of prey can increase. If the rainfall decreases, the grass decreases and the number of prey will decrease.

37

Figure 3.11 Change of the State of Parameters According to Positive Discrete Events Occurrence.

Figure 3.12 Change of the State of Parameters According to Fluctuating Discrete Events Occurrence.

38

3.10 Effect of Time in the relation between Consolidity and Theory of Change In many life sciences and specifically in the field of molecular dynamics, real-life physical systems behave basically on event-driven frameworks rather than the time-driven formulations. Nevertheless, in situations where events have certain time durations, it is apparent that their overall effects can be transformed to the equivalents of events strength. A clear example is the traditional light bulbs; their life expectancy depends mainly on the total number of their affecting events N on the on/off operation cycles, and their life is lower if the light is cycled often. Another manifest example is that the expected life span of civil buildings and structures in identified tectonically (subject to earth shaking) active zones is mostly determined by the type, strength, and sequence form of affected earthquakes events rather than their normal standard designed life durations. Furthermore, most of the technologies and computer-based schemes we have already developed and rely on in our daily life are also event-driven beside their normal time-driven framework. Typical examples are the communication networks, manufacturing facilities, the execution of computer software systems, etc. The time, on the other hand, is governing the system state equation and consequently the system response. It is also crucial in determining the order of the sequence of the occurrence of events. This is beside the intrinsic role of time in the synchronized coordination of various affecting events of the whole living universe (usually referred to as the timeline). At each step of affected event, once the system parameters are changed, the system will not behave similarly once more from the consolidity point of view. This gives rise to the importance of addressing jointly both time and event states in representing any worked out system configuration. If we have an entity A and there is an event, then the entity A will be changed to entity A1. In addition, if there is another event affect on entity A1, then the entity will be changed to entity A2 and so on. This event can be positive (increase) or negative (decrease) depending on the problem therefore, the system parameters can be changed.

3.11 The integrated Basic System and Upper System Layers Configuration The events and activities operations can be recognized based on the above investigation in the form of high level influencing layer(s) (to be denoted as the ‘‘event-driven or upper system layer(s)’’) acting over the ordinary time-driven state systems state equations to be designated as the ‘‘time-driven or basic system layer(s)’’. Such basic system layer(s) provide essentially the form and nature of the system together with its governing state equations.

39

In general, the two essential pillars of controllability and stability operate at the basic system level. On the other hand, the third pillar of consolidity is prevailing at the event-driven or upper system level and influencing system parameters pathway changes. Such notion is schematized by the joint two-level multilayer system configuration delineated in Figure 3.7. The event clocklike register progress is represented in the Figure as counter-clockwise to be conceptually congruent with the normal rotational direction of the majority of all parts (smallest to biggest) of our master universe. In the same Figure, it is conceptually elucidated the dual links and interactions between both the basic and the upper systems levels responsible for guiding the change pathway of systems when subjected to events or varying environments. The internal stacking of the system parameters changes with event state l is represented in Figure 3.13 by the shown clocklike rotating procedure. It may be observed also that the presented principle of the two-level multilayer system configuration as illustrated in the figure typically replicates that of the universe global configuration (of the heavens and earth layers one above another) but with much reduced scale and complexity [3]. The events happenings in the upper system level are considered of the general form that could be synchronous, semi-synchronous, or asynchronous with time in the basic system layer. Figure 3.13 can be interpreted by the existence of an ‘‘Event clocklike register’’ with state l that runs alongside with the ordinary ‘‘time clock’’ with state t. Different than the ordinary continued operation of the time clock, the event clocklike register begins its operation with the original (or initial) system state, steps forward only with the occurrence of any coming event affecting the system and stops at the final (or end) system state. It can be further stated from Figure 3.13 depicting the joint two level system configuration that while controllability and stability at the time-driven or basic system level govern the original real-life system set point status, the consolidity* at the event-driven or upper system level relatively influences sequentially system parameters changes during its pathway progress when subjected to varying environments or events.

* Consolidity is a new inner system property and will be discussed in chapter four

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Effects on the normal operations

Consolidity

Activities and Events

Upper layer(s)

Eventdriven system (EDS)

changes in the internal parameters of the system

Controllability and Stability

Basic layer

Timedriven System (TDS)

Figure 3.13 Systems Change Pathways with the Two Layers (Basic Layer and the Upper Layers) and the Changes in the Parameters of the System [3].

3.12 Main Advantages of the Combined Time-Driven and Event-Driven The main configuration of real life natural or man-made systems is based on joint two-level operation as shown in the conceptual graph of Figure 3.13. The basic level is governed by the time-driven system physical equations, while the upper level is affected by activities, events and varying environments ‘‘on and above’’ the normal system operation or set points. These events include all happenings and occurrences that may cause significant incremental changes on system parameters. A brief comparison between the time-driven and event-driven operations is summarized in Table 3.2. This is equivalent to following the important overall systems paradigm of ‘‘Time driven-event driven- parameters change’’.

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Table 3.2 Main Features of the Joint Time-Driven and Event-Driven Configuration [3] Ser. 1

Aspect Level

Time-Driven level Basic or lower level Basic physical layer system

Event-Driven level Higher or upper level of

2

Layer(s)

3

Variables

Time scale ‘‘t’’

4

Governing pillar(s)

5

Governing equations

6

Affected influences

Controllability and Stability State (or dynamic) operating equations Conventional system disturbances absorbed by system dynamics

7

System changes

System changes are transferred into changes in physical system layers at time scale t with different stack classes and categories

Upper influencing layer of events Event clocklike register of state ‘‘  ’’ Consolidity System transition equations per influencing event Events ‘‘on and above’’ normal operations leading to consequent changes in system parameters System changes are transferred at each event step l to corresponding parameters changes at basic system level and to changes in consolidity index at the higher level

3.13 Incremental Parameters Changes versus Affected Varying Environment A model is developed to illustrate the explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state by the system self-consolidity; following the general relationship at any event step 

as:

 Parameter change (  ) = Function [consolidity (  ) ,

varying environment or event (  ) ] . The parameter change can be calculated through the least squares approximations and the developed equation for any system depends on the application as it can be linear, parabolic, exponential, logarithmic …etc. By analyzing the model, we can determine the factor and therefore we can calculate the parameter change if there is an event influence on the parameter of the system as shown below in (3.1). This represents the main core of modeling the system change in the suggested theory.

P   .Event .C   Or

P   e

( .Event .C )

(3.1) ,…etc

‘‘  ,  ’’ are the changeability coefficient and will be discussed below.

42

3.14 Systems Change Procedure The operation of the above joint system configuration will lead to parameter changes in the physical system layer(s). For simple analysis, the formed transition (or transformations) procedure can be expressed for linear (or linearized) memory less procedure ‘‘F’’, for  =0, 1, 2. . . m, . . ., f (m is the intermediate state, and f is the final state) as follows: Step 1: Initialization Initialize the procedure for the initial state  =0 for both system parameters and system consolidity index: Initialize:

CO /( I  S ) (0) = System consolidity Index [System parameters (0) ] such that CO /( I  S ) (0) designates the initial value of the consolidity index CO /( I  S ) by applying the system physical equation(s).

Step 2: System parameter(s) change Determine parameter(s) change due to the affected environments or event at any state  assuming the following linear (or linearized) equation:  System Parameter change

( )

=  .CO /( I  S )  . Affected environment or event

( )

(3.2)

such that the term  System Parameter change (  ) describes the incremental or step parameter change at event clocklike register of state  . The term ‘‘Affected environment or event’’ indicates the overall or effective value, which combines varying environment(s) or event(s) type and strength. Moreover, the parameter CO /( I  S )  , designates the associated direct consolidity index based on the complete knowledge of the system physical equations and corresponding to varying environment or event at the state  . The parameter ‘‘  ’’ describes the changeability coefficient of the system. The effect of system consolidity on system parameter change is demonstrated by the example shown in Figure 3.14.

43

As consolidity index is updated at each event state  due to the change in system parameters, the system change pathway operation will move hence and forth among the family of curves of Figure 3.14 based on the present value of consolidity at state  until reaching in the end at the final event state f. Equation 3.2 clearly indicates that any affected activity, event or varying environment ‘‘on and above’’ the system normal situation will never pass without being intelligently recorded through such incremental scaled-based change in system parameters. At the end, all successive changes in the parameter changes under affected events beyond the ordinary normal system behavior (of event state  ) will constitute the overall system change pathway.

Change the Parameter

Linear Exponential

Affected Varying Environment and Consolidity Figure 3.14 A Sample of Linear Patterns Relating the Incremental System Parameter Change versus Affected Varying Environment or Event Strength, for Different Values of System Consolidity Index C and Certain Specified Changeability Coefficient  .

3.15 The Changeability Coefficient The changeability coefficient ‘‘  ,  ’’ describes the durability of the system for confronting changes. In general, the value of this coefficient is essentially related to the form of system nature for natural systems and the type of compositions of the manmade systems. The value of ‘‘  ,  ’’ can be determined in real life situations through extended experimentations and measurements.

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 System Parameter Change (  ) = Function [consolidity index (  ) , varying environment or ( ) event (  ) ].  System Parameter change =  .CO /( I  S )  . Affected environment or event ( )

.

The changeability coefficient will be the best representable slope of the line (or curve) plotted between the parameters change ‘‘  System Parameter Change (  ) ’’ as Y-axis versus the product of the corresponding affected events and the on-the-spot consolidity index “ CO /( I  S )  . Affected environment or event (  ) ” as X-axis. This is carried out using suitable number of events observations. Consolidity index can systematically be calculated from the knowledge of the system physical equations at the required spot as explained in [2–4]. The fitting approach could also be adjusted for nonlinear relations (such as the exponential relation) using relevant mathematical transformations.

3.16 Time-Event-Change System Paradigm The varying environments, events, or activities are considered to be all additional influences ‘‘on and above’’ the system normal stands or original set points. The mechanism of the conceptual pathway development is based on that the system behavior change rate is logically conceived to be not accidentally happened but is relatively influenced at the point of progress after being subjected to the varying environments or events with the associated direct (on-thespot) value of the system consolidity index. The Change pathway theory is essentially based on the joint system time-driven and eventdriven configuration and applying the important overall systems paradigm of ‘‘Time drivenevent driven-parameters change’’, that can be represented by the symbolic inter-related triangle depicted in Figure 3.15.

Event



Time

Parameter Change

t

P

Figure 3.15 Event Driven-Time Driven-Parameter Change Paradigm

45

The time is no longer a suitable independent variable in system modeling and this can be shown in Figure 3.16. This Figure shows that if there is no event, the system would be unchanged. But if there is an event, the state of the system parameters would be changed. The analysis of the suggested conceptual model is based on event-driven system framework, where the periods of inactivity can be passed over by jumping from one changed environment or event state to the next changed environment or event state and so on.

Event

 1

 n

 2 ......

Time

t1

t2

tn

Figure 3.16 Time of Discrete Events Occurrence

We can summarize the algorithm for analyzing the influence of events on the system parameters as depicted in the chart (Figure 3.17). In this chart, we can notice that if there is an event, one or more of the system parameters could be changed; and consequently the consolidity would also be changed. In addition, the parameters can be changed in positive or negative way depending on the nature of events. The factors  ,  can be determined through the least squares approximations by interaction between two parameters. The first parameter is the consolidity of the system multiplied by the event. The second parameter is the change in the parameter status. Therefore the model can be created, and then the factor  ,  will be developed. If there is an event occur we can calculate the change in the parameter.

46

Start

Read the system Parameters

 0

Event Existence

No

Yes

   1

No Change of the system Parameter

Event Model

Calculate the Change of the ( ) system Parameter ( P ) Calculate the New Consolidity ( ) Index of the System C

No

f Yes Calculate  as in eq (3.2)

End

Figure 3.17 A Chart of System Parameters Changes under Events Occurrence. 47

3.17 Concluding Remarks A new concept for time driven and event driven have been introduced in this chapter. The Change pathway theory is essentially based on the joint system time-driven and event-driven configuration and applying the important overall systems paradigm of ‘‘Time driven-event driven-parameters change’’. This paradigm is based on two-level system. The first is the time driven level or basic level which is governed by the system physical equation, where the second upper level is the event – driven level causing influence on the system on and above the normal situation. Such influences affecting events cause at each event step changes the system parameters. In addition, a model is developed to illustrate the explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state by the system self-consolidity.

48

Chapter 4 Consolidity Theory

4.1 Introduction Consolidity is one of the inherent properties of the universe typically operating in fully fuzzy environment. Its creation was based on magnificent physical laws that enable its consolidity regardless of all the ongoing changes and fuzzy occurrences that continuously take place. Such consolidity properties of the universe were lucidly revealed in most sacred books as a manifestation of God’s power. In this chapter, a detailed definition to the Consolidity and System Consolidity index developed by Dorrah and Gabr [2,5] will be introduced. Then, the results for the derivations of the state space consolidity have been developed. First, the roots of state space problems and its corresponding fuzzy levels have been derived and are considered as the output for consolidity index. The consolidity index is calculated as the output (the roots of state space) to the input (parameters of the system). In addition, the results for the derivations of pole placement are developed to choose the parameters of new systems or adjusting the parameters of built as usual systems. The pole placement model for the equations has been derived by taking the coefficients of the pole placement as the output of the system. The results show that the system will be consolidated if we use pole placement as the output of the system instead of the roots of state space problems. Then we can make tuning to the parameters of the system to face any change of the state because of events.

4.2 Consolidity Definition Consolidity represents a new different look to systems; as we all know every system in the universe can be classified as stable, unstable or marginally stable. But is this classification sufficient enough? If it is so, why many stable systems collapse in our daily life? So, it was important to take a deeper look into systems and study the internal relations and interactions between the systems’ parameters, or simply we have to study system consolidity [1]. An ample number of examples of systems can be given which looked very stable and then suddenly collapsed due to their inferior consolidity, such as the deficiency of the human immune system, the transformation of dangerous diseases and viruses, transfer of normal living cells into cancerous tissues, collapse of moving aerospace vehicles, blackout of electricity grids, dissolution of political or financial organizations, etc.

49

The vital point of all these systems is that though they faced the same changes but some vital systems seem to be more bonded and well connected than others. They are therefore consolidated as they adjust themselves for the new changes reacting against huge changes of ‘‘fuzziness’’ of the surrounding environment. Their inner changes ‘‘fuzzy level changes’’ were limited so they restored steadily to their normal operation and absorbed advantageously these varying situations. Consolidity is open in its application to wide classes of systems. Even for the system that thought not to be fuzzy, we can still pretend that these systems are operating in a virtual fuzzy environment and perform typically similar consolidity testing. In fact all the present physical systems in our daily life are subject to continuous alterations (such as aging and deterioration) that make them gradually changing, and thus will behave later equivalently as if they are operating in a fuzzy environment. In the following section, the basics of system consolidity and its different classifications are firstly presented highlighting the new notion of superior (or inferior) consolidated versus natural and built-as-usual systems. Consolidity theory explains many phenomena recurring in our daily lives such as:      

Drifting of countries economic behavior towards crisis. Dissolution of formed political or business organizations. Sudden revolutionary actions in a stable political system. Breakdown of vehicles’ brake system. Blackout of running electricity grids. Abrupt changes in temperature and pressures in nuclear reactors.

4.3 System Consolidity Index The System Consolidity Index is now presented in this section as given by [1]. This index measures the system overall output fuzziness behavior versus the combined input and system parameters variations. It describes the degree of how the systems react against input and system fuzzy variation actions. Let us assume a general system operating in fully fuzzy environment, having the following elements: Input parameters:

I  (VI i ,  I i )

(4.1)

such that VI i , i  1,2,....m describe the value of input component I i and  I i indicates its corresponding fuzzy level.

50

System parameters:

S  (VS i ,  S i )

(4.2)

such that VS i , i  1,2,....n describe the value of system component S j and  S i indicates its corresponding fuzzy level. Output parameters:

O  (VOi ,  Oi )

(4.3)

such that VOi , i  1,2,....k describe the value of output component O j and  Oi indicates its corresponding fuzzy level. We will apply in this investigation, the weighted (or overall) fuzzy levels, first for the combined input and system parameters, and second for output parameters. For the combined input and system parameters, we have for the weighted fuzzy level to be denoted as the combined Input and System Fuzziness Factor C I  S , given as:

CI  S 

 im1VI i . I i  im1VI i



 nj 1VS i . S i

(4.4)

 nj 1VS i

Similarly, for the Output Fuzziness Factor CO , we have

CO 

 ik1VOi . Oi

(4.5)

 ik1VOi

Let the positive ratio CO / C I  S defines the System Consolidity Index, to be denoted as

CO / C I  S . Based on CO / C I  S , the system consolidity state can then be classified as discussed in this chapter.

51

The selection of the fuzzy levels testing scenarios for both the system and input should follow similar consideration. First of all the input and system fuzzy values for system consolidity testing are selected as integer values to be preferably in the range ±8 for open fuzzy environment and in the range ±4 for bounded fuzzy environments. Nevertheless, the output fuzzy level could assume open values beyond these ranges based on the overall consolidity of the system. However, all over implementation procedure in the paper, the exact fraction values of fuzzy levels are preserved all over the calculations and are rounded as integer values only at the final results. Another important aspect in selecting the fuzzy level is to avoid falling near singularities upon calculating the combined input and system consolidity factor C( I  S ) (denominator of the Consolidity Index CO / C I  S ). Finally, it must be pointed out that using the suggested fuzzy approach, it is now amenable to derive the Consolidity Indices in compact mathematical forms for many applications, such as the trigonometric, hyperbolic, and exponential functions, analytic geometry, vector analysis, ordinary differentiation, partial fractions, etc. Similar implementations to various fuzzy matrices operations and to standard fuzzy probabilistic/statistics functions and expressions are also rendered straightforward endeavors.

4.4 Basics of the System Consolidity 4.4.1 Basic Definition of System Consolidity

Systems can be classified according to consolidity into three categories as follows, see Figure 4.1 [1]: 1. Consolidated systems or well connected, under hold, under grasp, well linked, robust or well joined systems. 2. Neutrally consolidated systems, and 3. Unconsolidated systems or weakly connected, separated, non-robust or isolated systems. A system operating at a certain stable original state in fully fuzzy environment is said to be consolidated if it’s overall output is suppressed corresponding to their combined input and parameters effect, and vice versa for unconsolidated systems. Neutrally consolidated systems correspond to marginal or balanced reaction of output, versus combined input and system.

52

Linear, Nonlinear, Multivariable, Dynamic,…etc Systems

Consolidated

Neutral

FO /( I S )  1

FO /( I S )  1

Well Connected

Neutrally Consolidated

Well Linked

Unconsolidated FO /( I S )  1

Weakly Connected Weakly Linked

Robust

Non-robust

Under Hold

Separated

Well Joined

Isolated

Figure. 4.1 Basic Definition of System Consolidity

In order to measure the degree of consolidity of systems, the Consolidity Index CO / C I  S is defined as the positive ratio CO /( I  S ) where C( I  S ) denotes combined Input and System Factor and CO indicates Output Factor. Based on CO /( I  S ) the consolidity state can then be classified [3] as: 1. Consolidated if CO /( I  S )  1, to be referred to as ‘‘Class C’’. 2. Neutrally Consolidated if CO /( I  S )  1 , to be denoted by ‘‘Class N’’. 3. Unconsolidated if CO /( I  S )  1 , to be designated as ‘‘Class U’’. 4.4.2 Classes of Consolidity The various classifications of consolidity classes are defined based on corresponding consolidity zones as shown in Figure 4.2. In the classifications, there are three zones: Consolidated, unconsolidated and mixed class.

53

CO

Unconsolidated (U)

Mixed Class (M) Consolidated (C)

C I S

Figure 4.2 Sketch Showing Different Classes of System Consolidity

4.4.3 The strong and weakness Ranking of Systems Consolidity Systems in real life vary according to their consolidity based on their score of Consolidity Index CO /( I  S ) . For most applications, several systems can be built with a wide variety of this index. These systems could relatively be ranked based on their overall consolidity indices in a stair wise form as shown in Figure 4.3. In this ranking, we have the following types: 1. Superior consolidated system offering the lowest index score CO /( I  S )  1 . 2. Neutrally consolidated system with index score CO /( I  S )  1 3. Inferior consolidated system having the highest score of index CO /( I  S )  1 . 4. Natural or Built-as-usual systems that could assume consolidity values between the superior and inferior consolidated extremes.

54

Superior

Strong

Neutral or Built as Usual I

FO /( I S )  1 Neutral

Neutral or Built as Usual II Weak

Inferior

Figure. 4.3 Sketch Showing Possible Ranking within the Superior–Inferior Consolidity Scale.

In real life, it is the main intention to exert our efforts to move the natural or built-as-usual systems based on their desired consolidity to one of the two extremes of the superior or inferior consolidated points.

4.5 Implementation Strategy of Consolidity for Existing and New Systems 4.5.1 Consolidity Implementation Strategy for Existing Systems

For existing natural or man-made systems, the consolidity situation could be complicated. The testing of these existing systems could reveal poor consolidity or even the unconsolidity of such systems. This is quite expected as we previously built or dealt with all these existing systems not putting into our minds that inner property of system consolidity.

55

Still the developers should not feel discouraged if they discovered that their existing celebrated built-as-usual systems could have been much enhanced if the consolidity concept was taken into their attention during development. In this case, interference in the existing natural or man-made systems is inevitable for performing necessary consolidity adjustment. For existing man-made system, the situation could be specifically much easier by altering parameters of the system within the utmost extend permitted for changes. As for natural systems, the consolidity improvement matter could also be possible by interfering within the system parameters together with controlling the environment and trying to direct the physical process towards better targeted consolidity. 4.5.2 Consolidity Implementation Strategy for New Systems Consolidity of systems as measured by the Consolidity Index can assume a wide range of values. For the same system, it is possible that various prototypes can be developed fulfilling almost the same degree of functionality. These systems can be ranked at the upper end starting from the best consolidated one with the lowest consolidity index score (the superior consolidated prototype). At the bottom end, there is the worst consolidated (inferior) prototype with the biggest consolidity index score. In general, as the generation of these prototypes during the system development process is not completely exhaustive, the terms of superior or inferior of consolidation remain only as relative comparison. Such comparison is sufficient for all real life applications as the system developers could follow later other cycles of improvement to locate a much better superior that surmounts the old superior one. 4.5.3 Essential Rules for Implementation of Consolidity For the proper implementation of consolidity, the following rules are proposed to be used as strict regulations during the systems modeling, analysis, design and building cycles [4]: 1. Rule I: Refrain at all circumstances from any arbitrary assignment of system parameters as this could lead the system to possibly fall into the trap of undesired unconsolidity zone. 2. Rule II: Select always the arbitrary parameters need to be assigned in an exhaustive way that allows the best selection providing the most appropriate consolidity state without sacrificing the proper system functionality. 3. Rule III: Interfere when possible into existing systems by changing parameters and controlling environment to shift consolidity of the system to the most desired state without sacrificing the proper system functionality. 4. Rule IV: Avoid entirely using any empirical, regression, artificial or imaginary models for Consolidity decisions if these models’ coefficients do not correspond as one to one to the parameters of original physical system.

56

The above four Golden Rules for Consolidity implementation are closely related and completely rational. The arbitrary assignment of parameters during system modeling, analysis, design or building could open the door to unavoidable improper choices. This reflects our incomplete understanding of such choices, and overlooks other mathematical justification factors in such choices. If we are forced to do such arbitrary assignment of parameter values, we have to make exhaustive selections and test the impact of each selection on Consolidity. A wiser decision can then be made towards the proper selection achieving the most desired Consolidity (superior or inferior). Of course the number of exhaustive selections could be reduced by excluding infeasible choices and use our best practical experience judgment without sacrificing the proper system functionality. On the other side, for systems having their consolidity far from the desired one, they should be interfered to move their consolidity state into the most desired one. This can be achieved if possible through carrying out some experimental trials for changing both system parameters and environmental conditions. The fourth rule is self-explanatory due to the fact that consolidity is an inherent property of the physical system and is directly affected by the values of the physical system parameters. The empirical, regression, artificial or imaginary models are not founded on physical grounds if their coefficients do not correspond to the physical system parameters. Therefore, they do not carry the same matching consolidity property corresponding to their original physical systems. In general, it is dangerous to use such empirical, regression or imaginary models for key real life applications, as they could lead to possible implementation of consolidity to illustrative problems It is highly advisable that all over the system consolidity analysis and design, the system parameters should be kept as symbols and not substituted with their corresponding values except at the final step of analysis. This is indispensable for easily tracking of these parameters and their corresponding fuzziness at all steps of analysis. It is remarked that the cost of building of superior or inferior consolidated systems is mostly the same, and its selection only depends on the cleverness of the system developer. In fact the only difference between superior or inferior consolidated system is some slight adjustment of the system physical parameters. Of course some exhaustive search must be made to attain the most superior or least inferior consolidated system. In the next section, the system consolidity index is developed and it is applied to the 2 nd and 3rd order state space system.

57

4.6 The consolidity for 2nd Order of State Space Problems using the stability coefficients The derivations for consolidity index for the 2nd order of state space equation are developed in Appendix A using the stability coefficients [73-74] and its fuzzy level as the output of the system. In addition, special cases are developed in this Appendix. Table 4.1 Average Consolidity, Controllability and Stability for 2nd Order State Space when Changing Parameter (a) where b=4 and c=2 a=0.5

a=1

a=5

a=10

1

-0.5

-0.6

-0.4000 + 0.4899i

-0.2 + 0.4i

2

-7.5

-3.4

-0.4000 - 0.4899i

-0.2 - 0.4i

la

lb

lc

1

7

8

1.076

1.233

1.847

2.678

3

8

7

0.884

1.014

1.402

1.953

2

7

6

0.993

1.133

1.600

2.233

1

3

5

0.978

1.109

1.315

1.900

-1

-4

-7

0.939

1.177

1.443

2.100

-2

-7

-4

1.078

1.221

1.696

2.329

-2

-6

-8

1.034

1.167

1.380

1.967

-1

-5

-6

1.000

1.149

1.689

2.424

-1

1

8

1.067

1.137

1.967

3.200

1

-1

-7

0.943

1.124

2.192

3.400

0.999

1.146

1.653

2.418

4.000

1.000

0.040

0.010

Average Consolidity Controllability

Table 4.1 shows the roots of the 2nd order state space system which refer to the stability of the system. In addition, the Table shows different values of average consolidity when changing the corresponding fuzzy level of the parameters of the 2nd order state space system. The Table also shows changing of consolidity, controllability and stability when changing parameter (a).

58

The results show that the controllability and the consolidity for the 2nd order state space system are changed when the parameter (a) changed and (b=4, c=2 are constant) as depicted in Table 4.1. It is noticed from the Table 4.1 that when a= 0.5, the system is highly controllable and neutral consolidated. When increase parameter (a) to be 10, the system will be less controllable and more unconsolidated. Therefore, it is noticed from Table 4.1 that if the stability and controllability are increased, the consolidity will be decreased (C > 1) then the system will be unconsolidated. Hence, we will think of the pole placement to adjust the parameters of the systems.

4.7 The consolidity for 3rd Order of State Space Problems using the stability coefficients The derivations for consolidity index for the 3rd order of state space equation are developed in Appendix B using the stability coefficients and its fuzzy level as the output of the system. In addition, special cases are developed in this Appendix. Table 4.2 shows the roots of the 3rd order state space system which refer to the stability of the system. In addition, the Table shows different values of average consolidity when changing the corresponding fuzzy level of the parameters of the 3rd order state space system. The Table also shows changing of consolidity, controllability and stability when changing parameter (a). The results show that the controllability and the consolidity for the 3rd order state space system are changed when the parameter (a) changed and (b=6, c=5, d=1 are constant) as depicted in Table 4.2. It is noticed from the Table 4.2 that when a= 0.5, the system is highly controllable and neutral consolidated. When increase parameter (a) to be 10, the system will be less controllable and unconsolidated. Therefore, it is noticed from Table 4.2 that if the stability and controllability are increased, the consolidity will be decreased (Consolidityindex > 1) then the system will be unconsolidated. Hence, we will think of the pole placement to adjust the parameters of the systems.

59

Table 4.2 Average Consolidity, Controllability and Stability for 3rd Order State Space when Changing a where b=6, c=5 and d=1 a=0.5

a=1

a=5

a=10

1

-11.1

-5.0

-0.4669 + 0.7304i

-0.1790 + 0.6175i

2

-0.6

-0.6

-0.4669 - 0.7304i

-0.1790 - 0.6175i

3

-0.3

-0.3

-0.3

-0.2

la

lb

lc

ld

2

6

5

6

1.235

1.754

2.483

3.012

-2

8

8

2

1.405

1.777

2.125

3.143

-2

-7

-3

2

1.116

1.940

2.508

3.467

1

7

4

5

1.111

1.747

2.417

3.714

1

6

5

4

1.154

1.985

2.614

3.467

-2

7

6

1

1.563

1.648

2.429

3.736

3

-7

-6

2

1.825

2.140

3.091

3.500

1

8

-2

-1

1.833

2.095

2.833

3.277

3

-6

-5

2

1.757

2.089

2.877

3.828

1

5

6

4

1.375

1.800

2.986

3.989

Average Consolidity

1.437

1.898

2.636

3.513

Controllability

8.000

1.000

0.008

0.001

4.8 Consolidity Index for 2nd Order of State Space Problems using the Pole Placement Table 4.3 shows the state feedback gain of the 2nd order state space system where (b=4, c=2) when the desired locations of the closed-loop poles are (-3, -3). In addition, the Table shows different values of average consolidity when changing the corresponding fuzzy level of the parameters of the 2nd order state space system. The Table also shows changing of consolidity and controllability when changing parameter (a).

60

The table shows that when the closed-loop poles for the 2nd order state space system are (-3, -3), and the consolidity index will be 0.744, the system will be consolidated at a=0.1 and will be unconsolidated when a>0.1 and will be unstable and unconsolidated when a>30. However, Note that increase the parameter a means that decrease the all other parameters (b, c). If we choose more closed loop poles, we will find that the values of controllability and consolidity of the 2nd order state space system are changed when the parameter (a) changed and (b=4, c=2) as shown in Table 4.4. It is noticed from this Figure that when a= 0.1, the system is highly controllable and highly consolidated. The results can be summarized in Table 4.4 and Figure 4.5 in 4 cases as follows: Case 1: When the poles are (-3, -3), the consolidity index at (a= 0.1) will be 0.744 Case 2: When the poles are (-5, -5), the consolidity index at (a= 0.1) will be 0.703 Case 3: When the poles are (-8, -8), the consolidity index at (a= 0.1) will be 0.642 Case 4: When the poles are (-10-10i, -10+10i), the consolidity index at (a= 0.1) will be 0.543 It is noticed from Figure 4.4 and Table 4.4 that:     

The best case is case 4, where C= 0.543. When a>0.1, the system will be unconsolidated When the poles are less than -3& -3, and -10-10i & -10+10i above the system will be unconsolidated When a>30, the system will be unstable When the poles are less than (-3, -3), and more than (-10-10i, -10+10i), the system will be unconsolidated

61

Table 4.3 Average Consolidity, Controllability and Closed Loop Feedback Gains for 2nd Order State Space when Changing a where n=2, b=4, c=2 and the Poles are (-3, -3) a 0.1 0.5 1 3 5 7 10 12 15 17 20 25 30 K1

-31.00

1.00

5.00

7.67

8.20

8.43

8.60

8.67

8.73

8.76

8.80

8.84

8.87

K2

-14.00

2.00

4.00

5.33

5.60

5.71

5.80

5.83

5.87

5.88

5.90

5.92

5.93

la

lb

lc

1

4

7

0.847

1.466

1.478

1.854

2.225

2.680

3.183

3.503

3.863

4.158

4.486

4.869

6.022

3

5

8

0.543

1.093

1.098

1.346

1.594

1.933

2.278

2.502

2.731

2.947

3.168

3.395

4.415

-2

-7

-4

0.735

0.927

1.175

1.508

1.805

2.185

2.583

2.838

3.107

3.347

3.599

3.869

4.924

1

7

8

0.923

1.374

1.502

1.927

2.338

2.832

3.397

3.758

4.179

4.514

4.901

5.376

6.614

2

6

8

0.751

1.246

1.315

1.643

1.959

2.358

2.781

3.051

3.341

3.593

3.861

4.154

5.229

3

8

7

0.668

0.993

1.151

1.451

1.728

2.092

2.468

2.709

2.961

3.190

3.428

3.678

4.717

3

7

4

0.565

0.747

0.979

1.254

1.500

1.836

2.177

2.398

2.625

2.839

3.057

3.281

4.299

2

7

6

0.761

1.087

1.259

1.598

1.911

2.306

2.726

2.993

3.279

3.529

3.795

4.084

5.156

1

3

5

0.774

1.363

1.379

1.715

2.045

2.459

2.903

3.185

3.491

3.753

4.035

4.348

5.440

1

5

6

0.869

1.333

1.443

1.834

2.210

2.668

3.177

3.501

3.867

4.166

4.501

4.894

6.055

0.744

1.163

1.278

1.613

1.932

2.335

2.767

3.044

3.344

3.604

3.883

4.195

5.287

100.000

4.000

1.000

0.111

0.040

0.020

0.010

0.007

0.004

0.003

0.003

0.002

0.001

Average Consolidity Controllability

62

Table 4.4 Average Consolidity, Controllability for 2nd Order State Space for Different Cases and Changing a where n=2, b=4, c=2.

Parameter (a)

0.1

0.5

1

3

5

7

10

12

15

17

20

25

30

Controllability

100.000

4.000

1.000

0.111

0.040

0.020

0.010

0.007

0.004

0.003

0.003

0.002

0.001

0.744

1.163

1.278

1.613

1.932

2.335

2.767

3.044

3.344

3.604

3.883

4.195

5.287

0.703

1.120

1.259

1.596

1.912

2.313

2.743

3.018

3.316

3.574

3.851

4.160

5.250

0.642

1.111

1.249

1.584

1.898

2.298

2.725

2.998

3.295

3.551

3.827

4.134

5.160

0.543

1.024

1.169

1.475

1.750

2.115

2.503

2.753

3.028

3.276

3.527

3.786

4.855

Case 1: Consolidity Index Case 2: Consolidity Index Case 3: Consolidity Index Case 4: Consolidity Index

63

Figure 4.4 Consolidity versus Controllability for 2nd Order State Space for Different Cases when Changing a where n=2, b=4, c=2.

64

4.9 Consolidity Index for 3rd Order of State Space Problems using the Pole Placement The derivations for consolidity index for the 2nd order of state space equation are developed in Appendix C using the Pole Placement method with Ackermann’s Formula and Direct Comparison Method. In these methods the state feedback gains and its fuzzy level are considered as the output of the system. Table 4.5 shows the state feedback gain of the 3rd order state space system where (b=6, c=5, d=1) when the desired locations of the closed-loop poles are (-4+ 4i, -4-4i, -10). In addition, the Table shows different values of average consolidity when changing the corresponding fuzzy level of the parameters of the 3rd order state space system. The Table also shows changing of consolidity and controllability when changing parameter (a). The table shows that when the closed-loop poles for the 3rd order state space system are (-4+ 4i, 4-4i, -10), and the consolidity index will be 0.668, the system will be consolidated at a=0.3 and will be unconsolidated when a27. However, Note that increase the parameter (a) means that decrease the all other parameters. If we choose more closed loop poles, we will find that the values of controllability and consolidity of the 3rd order state space system are changed when the parameter (a) changed and (b=6, c=5, d=1) as shown in Table 4.6. It is noticed from this Figure that when a= 0.3, the system is highly controllable and highly consolidated. The results can be summarized in Table 4.6 and Figure 4.6 in 4 cases as follows: Case 1: When the poles are (-4+ 4i, -4-4i, -10), the consolidity index at (a= 0.3) will be 0.668 Case 2: When the poles are (-6+6i, -6-6i, -20), the consolidity index at (a= 0.3) will be 0.639 Case 3: When the poles are (-8-8i, -8+8i, -30), the consolidity index at (a= 0.3) will be 0.535 Case 3: When the poles are (-10-10i, -10+10i, -50), the consolidity index at (a= 0.3) will be 0.499. It is noticed from Figure 4.5 and Table 4.6 that:     

The best case is case 4, where C= 0.499. When a>0.3, the system will be unconsolidated When the poles are less than (-4+ 4i, -4-4*i, -10), and (-10-10i ,-10+10i , -50) above the system will be unconsolidated When a>27, the system will be unstable When the poles are less than (-4+ 4i, -4-4i, -10), and more than (-10-10i, -10+10i), the system will be unconsolidated.

65

Table 4.5 Average Consolidity, Controllability and Stability for 3rd order State Space when Changing a where n=3, b=6, c=5, d=1and the Poles are (4+4i, -4-4i, -10) 0.3

0.8

1

3

5

7

10

12

15

17

20

23

27

K1

316.7

318.8

319.0

319.7

319.8

319.9

319.9

319.9

319.9

319.9

320.0

320.0

320.0

K2

95.3

105.8

107.0

110.3

111.0

111.3

111.5

111.6

111.7

111.7

111.8

111.8

111.8

K3

-2.0

10.5

12.0

16.0

16.8

17.1

17.4

17.5

17.6

17.6

17.7

17.7

17.8

la

lb

lc

ld

2

6

5

6

0.7122

1.0235

1.1285

1.3769

1.6198

1.9577

2.3071

2.5359

2.7740

2.9965

3.2268

3.4534

4.4844

-2

8

8

2

0.8383

1.2114

1.3317

1.7178

2.1140

2.6297

3.2987

3.7818

4.4770

5.0660

5.5661

6.0571

7.2071

-2

-7

-3

2

0.7374

0.9806

1.0775

1.3001

1.5335

1.8649

2.2066

2.4311

2.6638

2.8831

3.1092

3.3321

4.3589

1

7

4

5

0.7868

1.1184

1.2285

1.5195

1.8036

2.1829

2.5940

2.8634

3.1610

3.4222

3.7088

3.9895

5.0892

1

6

5

4

0.7070

1.0403

1.1498

1.4322

1.7070

2.0774

2.4756

2.7368

3.0228

3.2767

3.5527

3.8235

4.9107

-2

7

6

1

0.8022

1.1817

1.3030

1.6968

2.1049

2.6397

3.3575

3.8925

4.3118

4.9304

5.3439

5.8078

7.2976

3

-7

-6

2

0.6013

0.9987

1.1211

1.5120

1.9267

2.4913

3.3350

4.0569

4.5087

4.9017

5.4001

5.9661

7.3620

1

8

-2

-1

0.8507

1.0860

1.1834

1.4217

1.6732

2.0222

2.3891

2.6295

2.8844

3.1178

3.3634

3.6045

4.6534

3

-6

-5

2

0.6480

1.0547

1.1800

1.6056

2.0731

2.7218

3.8132

4.3841

4.6890

5.0729

5.3837

5.8981

7.2200

1

5

6

4

0.7619

1.0934

1.2032

1.4904

1.7706

2.1459

2.5511

2.8166

3.1086

3.3662

3.6473

3.9229

5.0159

Average Consolidity

0.745

1.079

1.191

1.507

1.833

2.273

2.833

3.213

3.560

3.903

4.230

4.586

5.760

Controllability

37.0370

1.9531

1.0000

0.0370

0.0080

0.0029

0.0010

0.0006

0.0003

0.0002

0.0001

0.0001

0.0001

66

Table 4.6 Average Consolidity, Controllability for 3rd Order State Space for Different Cases and Changing a where n=3, b=6, c=5, d=1

Parameter (a)

0.3

0.8

1

3

Controllability

37.037

1.953

1.000

0.037

0.0080

0.0029

0.0010

0.0006

0.0003

0.0002

0.745

1.079

1.191

1.507

1.833

2.273

2.833

3.213

3.560

0.628

1.055

1.166

1.471

1.783

2.207

2.730

3.083

0.545

0.967

1.076

1.366

1.663

2.069

2.562

0.509

0.930

1.038

1.321

1.611

2.009

2.466

Case 1: Consolidity Index Case 2: Consolidity Index Case 3: Consolidity Index Case 4: Consolidity Index

5

7

67

10

12

15

17

20

23

27

0.0001

0.0001

0.0001

3.903

4.230

4.586

5.760

3.530

3.843

4.188

4.432

5.565

2.900

3.247

3.574

3.914

4.279

5.405

2.809

3.179

3.500

3.825

4.138

5.261

Figure 4.5 Consolidity versus Controllability for 3rd Order State Space for Different Cases when Changing a where n=3, b=6, c=5, d=1

68

4.10 Consolidity Applications and Implementations The applications of consolidity cover almost all facets of existing sciences and disciplines as shown in Figure 4.6. In general, consolidity is an internal property of physical systems that enables providing an in-depth look inside such systems, regardless of their field of applications. Such property will lead to giving a new forum for better understanding of various sciences. With the developed consolidity and its methods of calculations, new classes of superior systems developed with strong consolidity will arise and will be taken for granted as the future standard of systems in various disciplines. It is interesting to note that the needed method of calculations for performing consolidity analysis lie within basic college mathematics and statistics and can be applied in a straightforward manner, enabling wide classes of scientists, researches and developers get full benefit of consolidity each in his own field. In fact, using the suggested fuzzy approach the derivations of compact form expressions for consolidity indices of basic fuzzy mathematical/probabilistic functions and expressions are now rendered systematic operations. We present now some specific applications of consolidity that can play real important roles. The first specific application is the consolidity of the human immune systems [15]. The human body is often described as being ‘at war’. By this, it is meant that the body is constantly under attack from things that are trying to do it harm. These include toxins, bacteria, fungi, parasites and viruses. All of these can, under the right conditions, cause damage and destruction to parts of the body and if these were left unchecked, the human body would not be able to function. It is the purpose of the immune system to act as the body’s own army, in defense against this continued stream of possible infections and toxins. This is done by the recognition of self and response to non-self. Although we all have the same immune system, in times of invasion of diseases and epidemics some people die very fast while others survive, why? What is the key point and where is the secret? Why do people react differently when exposed to infection? If we can simulate the immune system and define its physical parameters and use the consolidity index to examine its consolidation we can then adjust the systems parameters on basis of consolidity to get more superior consolidated ones. We can also do the opposite to the attackers, by studying the way every attacker work and then decrease its system’s unconsolidity to lower inferior consolidated values.

69

Engineering

Biology and Medicine

Control and Robotics

Genetics

Industrial Engineering

Bio-statistics

Evolutionary Systems Evolutionary Theory Evolutionary Models

Aeronautics and Space

Bioinformatics Global Modeling

Nuclear Engineering

Medicine Global Optimization

Aerospace Engineering

Biomedical Engineering

Material Engineering

Ecology

Chemical Engineering

Pharmacology Political and Management Science

Basic Science Mathematics

Develoment Studies Physics Organizations Chemistry Operational Research Biology Management Models Astronomy Behavior Science Geology Political Theory Economics and Finance

Social Sciences and Humanities

Financial Systems

Education

Econometrics

Physiology

Business

Psychology Commerce Mass Communication

Accounting

Literature

Marketing

Social Science Operation Management

Figure. 4.6 Suggested Areas for the Scope of Applications of Consolidity Theory. 70

Pharmacology is another important field of the application of consolidity. One of the problems that can be handled is the drug dosing. It can be made more precise by using pharmacokinetic and pharmacodynamic modeling. Pharmacokinetics is the study of the concentration of drugs in tissue as a function of time and dose schedule, while Pharmacodynamics is the study of the relationship between drug concentration and drug effect. By relating dose to resultant drug concentration (pharmacokinetics) and concentration to effect (pharmacodynamics), models for drug dosing can be generated. These models follow the standard state space approach where advanced control theory can easily be applied for their analysis and design. Closed-loop, adaptive, and nonlinear control for clinical pharmacology are still in their infancy, with numerous challenges and opportunities ahead. The integration of advanced control theory and consolidity theory will give a real impetus in fostering the progress of the development of clinical pharmacology systems. Such implementation should only be carried out for physicalbased models avoiding making any consolidity decisions for any types of the empirical-based pharmacodynamic models if their parameters are not related as one to one to the original physical parameters. The discipline of engineering is another excellent field for the wide spread of the consolidity theory. Many engineering design applications such as electric circuits, mechanical machines, civil structural systems, chemical processes, and other engineering systems depend at one step or another on some assumptions or empirical formulas made to the best of engineers knowledge. For this, the designing engineers feel self-importance for the freedom they got in making such arbitrary selection of physical parameters during the design cycle. At the end of the cycle, the engineers will provide good designs with high external functionalities. Nonetheless, the engineers could get much improved superior (or inferior) consolidated systems if they exerted additional efforts to ingeniously adjust the physical parameters (rather than selecting them in arbitrary manner) to satisfy system consolidity target requirements. In this case, engineers will gain their peace of mind that their systems can withstand fully future failures or malfunctions when subjected to any wearing or deteriorations or upon operating in varying fuzzy environment. The discipline of economics and econometrics is also among the important areas for system consolidity implementations. In dealing with this field, two types of models are commonly developed. The first is theoretical models based on the physical laws and rules governing the original system, and the second is artificial or imaginary models developed based on empirical or regression analysis. According to the Golden Rule IV of the system consolidity, it is dangerous to use these artificial or imaginary models if their coefficients are not related as one to one of the original economic or econometric system. For the theoretical models based on physical grounds, there are many kinds of models in finance, labor, economics, macroeconomics, microeconomics and political economics where system consolidity can be successfully implemented to ensure the consolidity of their developed physical models. 71

Last but not least, social sciences and humanities are also other important fields for the implementation of the consolidity theory. These include very potential applications areas in sociology, psychology, philosophy, literature, mass communication, education, humanities, etc. Most of these branches are dealing with formulating models and hypotheses in linguistic or verbal and only small sum are modeled in a numeric manner. These models and hypothesis are based on processing ample number of real life data through fuzzy probability and statistics. They usually seek assuming certain probability densities functions such as fuzzy normal, lognormal, uniform, and exponential, distributions, or calculating fuzzy means, standard deviations and correlations. The present availability of compact forms of consolidity indices for most standard fuzzy probability functions and expressions will naturally foster the rapid progress of consolidity in these indispensable fields. For further simplifying the processed data presentations, the Visual Fuzzy Logic-based Representation will be very appropriate, where data can be attractively represented in graphs by positive or negative color coding the level of their fuzziness.

4.11 Concluding Remarks This chapter gives a description for the consolidity theory. The consolidity is considered as an inner hidden property of natural and man-made systems. Under this notion, the systems are classified into consolidated, quasi-consolidated, neutrally consolidated, unconsolidated, quasiunconsolidated and mixed types based on their output reaction to combined input and parameters action when operating in fully fuzzy environment. Consolidity could provide necessary profound foundations that could give guidance towards solving the enigma of many real life problems. In this context, the consolidity theory has applied for 2nd and 3rd order state space system. The degree of consolidity for 2nd and 3rd order state space system has been checked by using the roots of the system as the output of the system. We have changed the most effective parameter of state space that will affect the other parameters to check consolidity, controllability and stability. In addition, the 2nd and 3rd order state space system has been checked by the pole placement and the state feedback gain has been taken as the output of the system to adjust the parameters of the systems. In these applications, we have choose the most effective parameter of the state space system.

72

Chapter 5 System Change Pathway of Linear Relationship Paradigm 5.1 Introduction System change pathway will be discussed depending on the time driven- event driven systemparameter changes. The parameter of the system can be changed due to the influence of event and consequently the consolidity also will be changed. These events can be positive or negative that can increase or decrease the parameter of the system and consolidity of the system, therefore entity will be changed to be built more or destroyed. In this chapter, a model that shows the relationship between the Event multiplied by the Consolidity Index and the parameter change has been developed and the parameter change can be calculated through the least squares approximations. Therefore, the developed equations for any system depend on the application as it can be linear, parabolic, exponential, logarithmic …etc. In this chapter, two cases in this paradigm are implemented. One of these cases is the change in the number of the people infected with HIV/AIDs with increase the level of media campaigns. The other case is the change in the number of the people infected with HIV/AIDs with decrease the level of media campaigns. The number of HIV/AIDS Epidemic is being reduced through radio campaigns on knowledge and use of condoms for prevention of HIV/AIDS [75-77].

5.2 A Model for HIV/AIDS Infectious Problem The problem of the HIV/AIDS Epidemic is considered for a specific geographical region with high homosexual males’ population. The model describing the physical system can be expressed as [78]: .

X  k1  k 2 X  ak3 X ,

(5.1)

.

Y  ak 3 X  (k 2  k 4 ).Y ,

(5.2)

.

Z  (1  b).k 4Y  k 2 Z ,

(5.3)

73

.

A  bk 4Y  (k 2  k 5 ).A,

(5.4)

N (t )  X (t )  Y (t )  Z (t )  A(t ),

(5.5)

such that N(t) is the population size, X(t) denotes the number of susceptible males in the population, Y(t) designates the number of males infected with HIV virus, Z(t) represents the number infected with virus but is non-infections, and A(t) is the number of men with HIV/AIDS . The parameters k2, k3, k4, k5 and b are modeled as fuzzy parameters, k1 is a constant and parameter a representing Average number of different sexual partners per year will be assumed several scenarios.

5.3 Consolidity Analysis of the HIV/AIDS Infectious Problem First, the equations are transferred to matrices as in (5.6) as follows: . X   .   (k 2  ak3 ) Y   ak3   0 .   Z   0 .   A   

0  (k 2  k 4 ) (1  b)k 4 bk 4

0 0  k2 0

0   0   0   (k 2  k 5 )

 X  k1  Y  0      U  Z  0       A  0 

(5.6)

Then the controllability matrix is calculated by [ B AB A2 B A3 B] as follows:

 (k 2  ak3 )  ak3 AB    0  0 

0  (k 2  k 4 ) (1  b)k 4 bk4

0 0  k2 0

0   0   0   (k 2  k 5 )

74

k1   k1 (k 2  ak3 ) 0    ak1k 3    0    0     0  0  

(5.7)

  k1 (k 2  ak 3 ) 2    ak1k 3(k 2  ak 3 )  ak1k 3(k 2  k 4 ) 2  A B   ak1k 3k 4 (1  b)   abk1k 3k 4  

(5.8)

  k1 (k 2  ak 3 ) 2    ak1k 3(ak 3  2k 2  k 4 ) 2  A B   ak1k 3k 4 (1  b)   abk1k 3k 4  

0 0  (k 2  ak3 )  ak3  (k 2  k 4 ) 0 A3 B    0 (1  b)k 4  k2  0 bk 4 0 

(5.9)

0   0   0   (k 2  k 5 )

  k1 (k 2  ak3 ) 2    ak1k 3(ak3  2k 2  k 4 )   ak1k 3k 4 (1  b)   abk1k 3k 4  

   k1 (k 2  ak 3 ) 3   ak1k 3((k 2  ak 3 ) 2  (k 2  k 4 )(ak 3  2k 2  k 4 ))  3  A B  ak k k (1  b)(ak  3k  k )  ak k k k (1  b) 1 3 4 3 2 4 1 2 3 4    abk k k ( ak  2 k  k )  abk k  1 3 4 3 2 4 1 3k 4 ( k 2  k 5 )  

   k1 (k 2  ak 3 ) 3   ak1k 3((k 2  ak 3 ) 2  (k 2  k 4 )(ak 3  2k 2  k 4 )) 3  A B    ak1k 3k 4 (1  b)(ak 3  4k 2  k 4 )    abk1k 3k 4 (ak 3  3k 2  k 4  k 5 )  

75

(5.10)

(5.11)

(5.12)

The determinant of controllability matrix [M] a11  0 M   0   0

a12 a 22 0 0

a13 a 23 a33 a 43

a14  a 24  a34   a 44 

(5.13)

where

a11  k1, a12  k1(k2  ak3 ) , a13  k1(k2  ak3 )2 , a14  k1(k2  ak3 )3

a14  k1(k2  ak3 )3 , a23  ak1k 3(ak3  2k2  k4 ) , a24  ak1k 3((k2  ak3 )2  (k2  k4 )(ak3  2k2  k4 )) a33  ak1k 3k4 (1  b) , a34  ak1k 3k4 (1  b)(ak3  3k2  k4 )

a43  abk1k 3k4 , a44  abk1k 3k4 (ak3  3k2  k4  k5 ) thus M  a 3b(1  b)k14 k33k 42 k5

(5.14)

M  a 3bk14 k33 k 42 k5  a 3b 2 k14 k33 k 42 k5

(5.15)

Then the consolidity is calculated as follows:

a 3bk14 k 33 k 42 k 5 (3 a   b  4 k1  3 k 3  2 k 4   k 5 ) M  M a 3b 2 k14 k 33 k 42 k 5 (3 a  2 b  4 k1  3 k 3  2 k 4   k 5 )  M

76

(5.16)

 M  (3 a   b  4 k1  3 k3  2 k4   k5 )  b(3 a  2 b  4 k1  3 k3  2 k4   k5 ) (5.17)  M  (1  b)(3 a   b  4 k1  3 k3  2 k4   k5 )  b b

I/P 

(5.18)

b b  k 2  k2  k 3  k3  k 4  k4  k 5  k5 a  b  k1  k 2  k 3  k 4  k 5

(5.19)

 I / P is the fuzzy level for the input of the system

C =M /  I /P

(5.20)

5.4 Consolidity Proposed Change Paradigm for the HIV/AIDS Infectious Problem The campaign uses condom through radio campaigns. This study examined the influences of Behavior Change Communication (BCC) campaigns on knowledge and use of condoms for prevention of HIV/AIDS and other sexually transmitted infections in target areas of Uganda. Data were drawn from Improved Services for Health (DISH) evaluation surveys, which collected information from representative samples of women and men of reproductive age in the districts served by the DISH project. The effectiveness of condoms in reducing the risk of sexual transmission of the human immunodeficiency virus (HIV) and certain other sexually transmitted infections (STIs) has been well established. This study was carried out on Number of HIV/AIDS infectious people is approximately 3463 in 1992. The event in this problem is the media campaigns (how to help decrease the number of HIV/AIDS infectious people). If the media campaigns have increased across years, the number of HIV/AIDS infectious people will decrease and the consolidity will be better. However if there is no media campaigns, the number of HIV/AIDS infectious people will increase with time as depicted in Figure 5.1.

77

Figure 5.1 Changing the Number of HIV/AIDS Infected People due to the Influence of Media Campaigns. In Figure 5.1, the upper line represents the increase in the number of HIV/AIDS infected people without media campaigns at each year. However the lower line shows the decrease in HIV/AIDS infected people when using media campaigns. In the analysis of HIV/AIDS, it is observed that the media campaigns can help in reducing the number of HIV/AIDS infected people and consequently the consolidity of the system will be influenced by the changing the number of HIV/AIDS infected people. That is shown in Figure 5.2; the upper line illustrate how the increase in the number of HIV/AIDS infected people can influence on the consolidity index for model if there is no media campaigns over years. In this case the consolidity index will be higher which means that the system is inferior consolidated. In addition, the bottom line illustrate how the decrease in the number of HIV/AIDS infected people can influence on the consolidity index for model if there is media campaigns over years. In this context, the consolidity index will be low which means that the system is highly consolidated. 78

Figure 5.2 The Influence of Media Campaigns on Consolidity Index

5.5 The Influence of Media Campaigns on the Number of HIV/AIDS Infected People Table 5.1 shows a comparison between number of HIV/AIDS infected people with and without media campaigns over years and the influence of variation of this number on the Consolidity of the model.

79

Table 5.1 The Influence of Media Campaigns on the Number of HIV/AIDS Infected People and the Consolidity Index of the System

without Media Campaigns

with Media Campaigns

No of HIV/AIDS infected People

CO /( I  S )

No of HIV/AIDS infected People

CO /( I  S )

1

10000

0.92

10000

0.92

2

10107

0.97

9271

0.86

3

10546

1.82

8542

0.81

4

11089

2.73

7813

0.79

5

11695

4.22

7083

0.75

6

12792

6.65

6354

0.71

7

13959

8..87

5625

0.67

8

15160

11.32

4896

0.62

9

16633

13.78

4167

0.59

10

17976

16.90

3438

0.56

11

20069

18.33

2708

0.51

year

5.6 Changes in the Number of HIV/AIDS Infected People with Low Level of Media Campaigns Now, we will develop a model that illustrates the relationship between the Event multiplied be the Consolidity Index and the Change in the number of HIV/AIDS infected people without media campaigns. The value of the event represents the number of infected people/ 10000. In this example, the factor  is developed by least squares approximations throughout the Table 5.2.

80

Table 5.2 Least Squares Data for the HIV/AIDS with Low Level of Media Campaigns No of Consolidity HIV/AIDS Index* infected Event(x) (y)

Event (No of HIV/AIDS infected/1000)

Consolidity Index

1.000

0.920

0.920

1.011

0.970

1.055



x.y

x2

10000

9200.000

0.846

10431.163

0.980

10107

9908.691

0.961

10447.576

1.820

1.919

10546

20241.697

3.684

10702.825

1.109

2.730

3.027

11089

33569.696

9.165

11003.994

1.170

4.220

4.935

11695

57718.217

24.357

11522.648

1.279

6.650

8.507

12792

108817.451

72.364

12493.466

1.396

8.870

12.382

13959

172835.215

153.305

13546.802

1.516

11.320

17.161

15160

260162.579

294.504

14846.020

1.663

13.780

22.920

16633

381232.917

525.339

16411.542

1.798

16.900

30.379

17976

546100.813

922.910

18439.182

2.007

18.330

36.786

20069

738267.807

1353.245 20180.820

x=

y=

 xy =

139.918

150026

 x2 =

2338055.083 3360.680

y



y= 150026

In this method, we will calculate the sum of consolidity multiplied by the event (x) and Changing Number of HIV/AIDS infected people (y) and the sum of the square of the consolidity multiplied by the event (x2) and also the sum of Number of HIV/AIDS infected people(y). Table 5.2 shows how this analytical method can be modeled. First we will calculate the summation of each parameter and put the data in (5.21) and (5.22) to get the value of  and  . By the analytical method,  can be predicted as follows:

n xY  x Y    2  2

(5.21)

n x  ( x)

81

In addition,  can be predicted as follows:



Y   x n

(5.22)

By analyzing the model using linear regression method [79], the factor  will be 271.832 and therefore we can calculate the parameter change (the number of HIV/AIDS infected people) if there is an event influence on the parameter of the system as in (5.23).

P  271.832.Event .C  10181.078

(5.23)

By substituting in (5.23), we will obtain the curve that shows the relation between the consolidity multiplied by the event and Changing Number of HIV/AIDS infected people as depicted in Figure 5.3. In this relation we can predict the Number of HIV/AIDS infected people by substituting in the event and consolidity.

Figure 5.3 Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Low Level of Media Campaigns. 82

5.7 Changes in the Number of HIV/AIDS Infected People with Media Campaigns we will develop another model that illustrate the relationship between the Event (media Campaigns) multiplied be the Consolidity Index and the Change in the number of HIV/AIDS infected people with high level media campaigns as depicted in Figure 5.5. By analyzing the model, the factor  is 9368.224 and therefore we can calculate the parameter change (the number of HIV/AIDS infected people) if there is an event influence on the parameter of the system as in (5.24). In this example, the factor  is developed by least squares approximations throughout the Table 5.3. Table 5.3 Least Squares Data for the HIV/AIDS with High Level of Media Campaigns

Event (No of HIV/AIDS infected/1000)

No of Consolidity HIV/AIDS Consolidity Index* infected Index Event(x) (Y)

x.y

x2



y

1.000

0.920

0.920

10000

9200

0.8464

10488.461

0.927

0.860

0.797

9271

7391.8239

0.6356969

9339.036

0.854

0.810

0.692

8542

5910.2269

0.4787284

8351.588

0.781

0.790

0.617

7813

4822.3946

0.3809692

7652.016

0.708

0.750

0.531

7083

3762.6667

0.2822

6846.330

0.635

0.710

0.451

6354

2866.5054

0.2035219

6096.019

0.563

0.670

0.377

5625

2119.9219

0.1420348

5400.344

0.490

0.620

0.304

4896

1486.1906

0.0921438

4713.438

0.417

0.590

0.246

4167

1024.4695

0.0604437

4172.901

0.344

0.560

0.193

3438

661.91126

0.037067

3673.340

0.271

0.510

0.138

2708

373.99646

0.0190738

3163.522

x=

y=

 xy =

 x2 =

y=

5.266

69897

39620.107

3.1782794

69896.996

83



The equation is as follows: P  9368.224 xEvent .C  1869.695

(5.24)

By substituting in (5.24), we will obtain the curve that shows the relation between the consolidity multiplied by the event and changing number of HIV/AIDS infected people as depicted in Figure 5.4. In this relation we can forecast the Number of HIV/AIDS infected people by substituting in the event and consolidity.

Figure 5.4 Parameter Change (the Number of HIV/AIDS Infected People) due to the Influence of Changing the Event Level and Consolidity with Media Campaigns.

84

5.8 Concluding Remarks In this chapter, A background about the HIV/AIDS Epidemic problem and a model subject to variability of (awareness and educational programs) and without (awareness and educational programs) have been discussed. In addition, a model that shows the relationship between the parameter changes versus (the Event multiplied be the Consolidity Index) have been developed and the parameter change can be calculated through the linear regression method. The results of these problems confirm the suitability of the suggested paradigm to handle real life systems under external influences “on and above” their normal conditions.

85

Chapter 6 System Change Pathway of Exponential Relationship Paradigm

6.1 Introduction System change pathway will be discussed depending on the time driven- event driven systemparameter changes. The parameter of the system can be changed due to the influence of event and consequently the consolidity also will be changed. These events can be positive or negative that can increase or decrease the parameter of the system and consolidity of the system, therefore entity will be changed to be built more or destroyed. In this chapter, a model that shows the relationship between the Event multiplied by the Consolidity Index and the parameter change has been developed and the parameter change can be calculated through the least squares approximations. Therefore, the developed equations for any system depend on the application as it can be linear, parabolic, exponential, logarithmic …etc. To approve this theory, two real life problems will be discussed. The first problem is the Prey Predator Population Problem that is famous ecological systems. The second problem is the influence of hygiene on the infection rate of the problem of spread of infectious diseases in a dorm.

6.2 Model for Event Driven and Parameters Changes for Prey Predator Application Ecological systems are characterized by the interactions of different species within a fluctuating natural environment. Studies of mathematical models are informative in understanding predator– prey interactions. Ecological systems are characterized by the interactions of different species within a fluctuating natural environment. Studies of mathematical models are informative in understanding predator– prey interactions in these systems; as a result, predator–prey models have been important in ecological science since the early days of this discipline [80]. The increase influence of mathematical modeling in theoretical ecology, under both stochastic and deterministic environments, can be attributed not only to the growing need for a quantitative understanding of ecological systems but also to improvements in mathematical capability. The last few decades have been active in the development of different kinds of predator– prey model within the traditional territory of population biology.

86

6.3 Problem Description Predation is a straight-forward interspecies population interaction. One species uses another as a food resource as shown in Figure 6.1. Predators play an important role in controlling prey population numbers in some systems. In simple systems, the predator-prey relationship results in coupled population oscillations as follows: 

Prey numbers increase, predator numbers increase…to a point where the predation causes population decline in the prey item.



Very often, but not always, an increase in prey density results in a straight-forward increase in predator population size.



The action of predators in the face of increasing prey availability can take different forms.



In the top panel, as the number of prey items increases, the number killed by the predator increases in a linear fashion.



Predator-prey relationships often have ramifications for other parts of the ecosystem.



The hare-lynx relationship is an example. Hares eat twigs, more hares = more damage to trees. More lynx = fewer hares and less damage to trees.



Predator-prey relationships are dynamic.



They are influenced by climate dynamics, changes in food availability for the prey species, and dynamics in other areas of the food web.



Predator-prey relationships also are dynamic through evolutionary time.



Often involve an evolutionary “arms race.” Natural selection simultaneously driving the predators toward greater hunting efficiency and the prey toward traits that help them avoid being eaten.

87

Figure 6.1 Example of a Real Life Predator-Prey Population Dynamic Problem [11].

6.4 Prey Predator Population Problem Formulation The Predator–Prey Population dynamics problem includes the effects of competing population, where one species may feed on another. This problem is represented by two ordinary differential equations. The problem of Prey Predator has been modeled in [81].

88

Let H(t) represents the number of hares (prey) and L(t) denotes the number of lynxes (predator). There are three possible equilibrium points for this problem for xe . = (Le, He), namely [13]:

0  xe    , 0 

H *  k  xe   , and xe   e ,  L*e  0 

(61)

where H e* and L*e are given as

L*e 

rH e (c  H e )  H  b cr .(a1b1k  cd  dk ) .1  e   1 a1H e k   (a1b1  d ) 2 k

(6.2)

and

H e* 

cd a1b1  d

(6.3)

In the above, we have r represents the growth rate of the hares, k designates the maximum population of the hares (in the absence of lynxes), a1 is the interaction term that describes how the hares are diminished as a function of the lynx population and c controls the prey consumption rate for low hare population. In addition, b1 denotes the growth coefficient of the lynxes and d indicates the mortality of the lynxes. All these coefficients are modeled as fully fuzzy parameters.

6.5 Mathematical Module for Consolidity of Prey Predator Population Problem

The mathematical equations are developed below. In theses equations, the consolidity index has been developed as follows: assume

X  a1b12crk  b1c 2 rd  b1cdrk

(6.4)

The fuzzy level for X is

a1b12 crk ( a1  2 b1   c   r   k )  b1c 2 rd ( b1  2 c   r   d ) X  X b1cdrk( b1   c   d   r   k )  X 89

(6.5)

and

Y a1b1  d ,  Y 

a1b1 ( a1   b1 )  d d Y

(6.6)

For the Parameter Z

( Z ,  Z )  (Y 2 k , (2 Y  k ))

(6.7)

The crisp value for L and the fuzzy level for L are indicated in (6.8)

( L,  L )  ( X / Z , ( X  Z ))

(6.8)

The crisp value for H and the fuzzy level for H are indicated in (6.9)

(H ,  H )  (

cd , ( c  d Y )) Y

(6.9)

The crisp value for the output and the fuzzy level for the output are indicated in (6.10)

(O / P,  O / P )  ( L  H ,

L L  H H ) LH

(6.10)

The crisp value for the input and the fuzzy level for the input are indicated in (6.11) . ( I / P,  I / P )  (a1  b1  c  d  k  r ,

a1 a1  b b1 c c  d d  k k  r r a1  b1  c  d  k  r

)

(6.11)

Therefore the consolidity index will be

FO /( I  S ) 

O / P I /P

(6.12)

In these mathematical equations, the consolidity indices will be calculated with different values of growth rate of the hares. Table 6.1 shows the different values for the consolidity indices with variation of the growth rate of the hares (r) with different values of fuzzy levels for the parameters of the problem.

90

6.6 Consolidity Based Change for Prey Predator Population Problem Now, we will discuss the events that can influence on the number of prey in this problem and know how the change in the rainfall can increase or decrease the consolidity of the system. In this problem, the event is rainfall that influence on the vegetation density and consequently would affect on the prey number [82]. Table 6.1 The Effect of Rainfall on both of Rate of Growth of Prey and Consolidity Index

Rainfall (mm)

Vegetation density(woody plants/km2)

Number of Prey

Rate of growth of prey (r)

FOl ( I  S )

250

50000

12

1.39

4.62

300

62000

14

1.45

4.36

440

90000

18

1.52

3.81

500

100000

21

1.6

3.58

600

115000

25

1.69

3.37

730

130000

28

1.77

2.91

820

145000

32

1.82

2.72

930

155000

35

1.91

2.33

1060

166667

37

2

2.02

In this problem, x= Vegetation density (woody plants/km2), y= Prey biomass (kg/km2), and the number of prey = Prey biomass/kg, km2=1. It is noticed from Table 6.1 that as the rainfall (mm) increases, the Vegetation density increases and consequently Rate of growth of prey increases and the consolidity will be improved. This confirms the fact that if the environment improved then the consolidity will be improved.

91

This table describes that if the rainfall in (mm) has increased the Vegetation density (woody plants/km2) will be increased, therefore Number of Prey will be increased and consequently Rate of growth of prey (r) will be increased. In this case the consolidity will be improved and vice versa. Figure 6.2 shows the number of prey versus the rainfall. It is shown from the figure that the number of prey increases when the amount of rainfall in mm increases. It is noticed that if the rainfall increased, the grass will grow more and the rate of growth of prey will be increased. When the amounts of rainfall (event) increases, the rate of growth of prey will increases and consequently the consolidity index will be low which means the system Consolidity would be improved. The red line in this figure represents the number of prey which increases through the increase of rainfall and the blue line represents the consolidity index which increases in the value when the number of prey increases which means that the consolidity can be improved by increase the rate of rainfall and the number of prey.

Figure 6.2 Changing the Number of Prey and Consolidity Index due to the Influence of Changing the Rainfall Level.

92

Now, we will develop a model that illustrates the relationship between the Event multiplied by the Consolidity Index and the parameter Change. In this example, the factor  is developed by least squares approximations. In this method, we will calculate the sum of consolidity multiplied by the event (x) and Changing Number of Prey (y) and the sum of the squares of the consolidity multiplied by the event (x2) and also the sum of the logarithm of Number of Prey (Y=ln(y)). Table 6.2 shows how this analytical method can be modeled. First we will calculate the summation of each parameter and put the data in (6.13) and (6.14) to get the value of  and  .

Table 6.2 Least Squares Data for Prey Predator

Event Consolidity Changing Consolidity Rainfall Index * Number of Index (mm) Event(x) Prey (y)

Y=ln( y)

x.Y



x2

y

250

4.62

1.155

12

2.48490665 2.87006718

1.334025

11.753369

300

4.36

1.308

14

2.63905733 3.45188699

1.710864

13.542298

440

3.81

1.6764

18

2.89037176 4.84541921

2.810317

19.04789

500

3.58

1.79

21

3.04452244 5.44969516

3.2041

21.16079

600

3.37

2.022

25

3.21887582 6.50856692

4.088484

26.23205

730

2.91

2.1243

28

3.33220451 7.07860204 4.5126505 28.838513

820

2.72

2.2304

32

3.4657359

930

2.45

2.2785

35

3.55534806 8.10086056 5.1915623 33.264839

1060

2.23

2.3638

37

3.61091791 8.53548776 5.5875504 35.998913

x=

y=

16.9484

222

Y =

7.72997736 4.9746842 31.815715

 xY =

x

2



=

y=

28.2419404 54.5705632 33.414237 221.65438

93

By the analytical method,

 can be predicted as follows:

n xY  x Y    2  2

(6.13)

n x  ( x)

In addition,  can be predicted as follows:



Y   x n

(6.14)

Figure 6.3 The Rate of Parameter Change (Number of Prey) due to the Influence of Changing the Rainfall Level and Consolidity Index.

94

By analyzing the model, we found that the best equation for this model is the simple exponential curve as depicted in Figure 6.3, and the factor  is 0.933 and therefore we can calculate the parameter change if there is an event influence on the parameter of the system as shown below in (6.15). (6.15)

P  4.033401e (0.933.Event .C )

6.7 Spread of Infectious Disease Problem Formulation The system describing the Spread of Infectious Disease is of the nonlinear type expressed as [10]:

.

x  k r x. y

(6.16)

and .

y  k r .x. y  cr . y

(6.17)

such as x(t) represents those uninfected with disease but may become infected, y(t) indicates those who are presently infected with the disease, kr represents the infection rate, and cr is the removal rate. The system is firstly linearized around its operating points and its consolidity testing was carried out for different scenarios of the infection rate k. In this problem, we will study the influence of hygiene on the infection rate k of the problem of spread of infectious diseases in a dorm as illustrated in [83]. The results show that if the dorm is not cleaned, the infectious rate will be higher and the consolidity will be inferior. However, if the dorm is cleaned, the infectious rate will be lower and the system will be consolidated as depicted in Table 6.3 and Figure 6.4.

Table 6.3 The Influence of Hygiene on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System % Clean of the Dorm

Infection Rate

Consolidity Index

25%

0.250

30.254

30%

0.200

24.976

35%

0.150

20.456

40%

0.100

16.520

95

% Clean of the Dorm

Infection Rate

Consolidity Index

45%

0.050

9.238

50%

0.010

5.000

55%

0.005

1.805

70%

0.003

1.321

80%

0.001

0.933

Figure 6.4 The influence of Room Cleaning on the Infection Rate of the Spread of Infectious Diseases and the Consolidity Index of the System

96

A model is developed that illustrate the relationship between the Event (% of dorm room cleaning) multiplied be the Consolidity Index and the Change in the infection rate of the spread of infectious diseases. Now, we will develop a model that illustrate the relationship between the Event multiplied be the Consolidity Index and the parameter Change. In this example, the factor  is developed by least squares approximations. In this method, we will calculate the sum of consolidity multiplied by the event (x) and Changing Number of Prey (y) and the sum of the square of the consolidity multiplied by the event (x2) and also the sum of the logarithm of Number of Prey (Y=ln(y)). Table 6.4 shows how this analytical method can be modeled. First we will calculate the summation of each parameter and put the data in (6.13) and (6.14) to get the value of  and  . Table 6.4 Least Squares Data for the Spread of Infectious Diseases

Consolidity Index* Change(y) Event(x)

Event

Consolidity Index

0.250

30.254

7.564

0.250

0.300

24.976

7.493

0.350

20.456

0.400



xY

x2

-1.386

-10.485

57.207

0.230504

0.200

-1.609

-12.059

56.142

0.2194472

7.160

0.150

-1.897

-13.583

51.260

0.1740675

16.520

6.608

0.100

-2.303

-15.215

43.664

0.11861

0.450

9.238

4.157

0.030

-3.507

-14.577

17.281

0.0215819

0.500

5.000

2.500

0.010

-4.605

-11.513

6.250

0.0068189

0.550

1.805

0.993

0.005

-5.298

-5.260

0.986

0.002391

0.700

1.321

0.925

0.003

-5.809

-5.372

0.855

0.0022806

0.800

0.933

0.746

0.001

-7.601

-5.673

0.557

0.0020147

x=

y=

Y =

 xY 

 x2 

y=

38.145

0.749

-34.016

-93.737

234.202

0.749

97

Y=ln (y)

y



Figure 6.5 Parameter Change (the Infection Rate of the Spread of Infectious Diseases) due to the Influence of Changing the Event Level and Consolidity. By analyzing the model using the least squares approximation, we found that the best equation for this model is the simple exponential curve as depicted in Figure 6.5, and the factor  is 0.695284 and therefore we can calculate the parameter change (infection rate) if there is an event influence on the parameter of the system as in (6.20). (6.20)

P  0.001199 e(0.695284.Event .C )

98

6.8 Concluding Remarks In this chapter, two case studies have been discussed to show the influence of events on the parameters of the system. The first case study is the investigation of prey-predator model operating under external environmental influences. In this case study, a model that show the relationship between the parameter changes versus (the Event multiplied be the Consolidity Index) have been developed and the parameter change can be calculated through the least squares. In this context, the best equation was performed through exponential curve. The second case study is the analysis of spread of infectious disease problem subject to variability of level of hygiene. In this case study, a model that show the relationship between the parameter changes versus (the Event multiplied be the Consolidity Index) have been developed and the parameter change can be calculated through the least squares. In this context, the best equation was performed through exponential curve.

99

Chapter 7 Simulation Analysis for “Time Driven-Event DrivenParameters Change” Paradigm

7.1 Introduction The previous chapter has concentrated on developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state by the system self-consolidity; following the general relationship at any event step. In this chapter, we will develop a simulation analysis for the model if there are events occurrences on the parameter. In this simulation, we can calculate the change and then calculate the consolidity, after choosing relationship between the incremental parameters changes versus affected varying environment and events of various values. In this respect, the three case studies will be modeled. These case studies are prey predator, HIV/AIDs infectious disease with and without media campaigns and the spread of infectious diseases.

7.2 The Events and Parameter Changes

To obvious the event that can influence on an entity or system parameters, let us consider this example: if we have an entity A and there is an event, then the entity A will be changed to entity A1. In addition, if there is another event effect on entity A1, then the entity will be changed to entity A2 and so on. This event can be positive (increase) or (decrease) negative depending on the problem. Therefore, the system parameters can be changed as depicted in Figure 7.1.

100

Figure 7.1 System Parameters Change Forms: (a) Parameter 1 (Decrease) and (b) Parameter 2 (Increase) c) Parameter 3 (Fluctuate).

101

7.3 The Methodology for the Simulation Analysis and Modeling of Time Driven – Event Driven –Parameter Changes Paradigms In this section, we will develop a simulation analysis and modeling for Event and consolidity for the problems after choosing relationship between the incremental parameters changes versus affected varying environment and events of various values. In this simulation, we can calculate the change and then calculate the consolidity. In the following steps, we can apply the simulation analysis and modeling to the real life problems. First we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is an event, we will calculate the change in the parameter P through the relation between the parameter changes and the consolidity index multiplied by the event, and consequently we will have the new parameter after changing its state. Second we will calculate the consolidity index CO /( I  S ) for the new system parameters. If there is another event, we will calculate the new change in the parameter P through the relation between the parameter changes and the consolidity index multiplied by the event, and consequently we will have the new parameter after changing its state and so on. We can summarize the steps as follows:

Step 1: Calculate CO /( I  S )

( 0)

with the parameter (A0).

At Event (1) we will calculate P (1) , and then we will calculate the parameter (A1) after the changes that change its state. Step 2: Calculate CO /( I  S )

(1)

with the new parameter (A1).

At Event (2), we will calculate P (2) , and then we will calculate the parameter (A2) after the changes that change its state. Step 3: Calculate CO /( I  S )

( 2)


At Event (3), we will calculate P (3) , and then we will calculate the parameter (A3) after the changes that change its state. Step 4: Calculate CO /( I  S )

(3)


At Event (4), we will calculate P (4) , and then we will calculate the parameter (A4) after the changes that change its state.

102

Step (  -1): Calculate CO /( I  S ) (   2) with the new parameter ( A 1 ). At Event (  -1), we will calculate P  1 , and then we will calculate the parameter ( A ). Step (  ): Calculate CO /( I  S ) (  1) with the new parameter ( A ). At Event (  ), we will calculate P  , and then we will calculate the parameter ( A 1 ). Repeat steps until   f , f is the final state event. 7.4 Simulation Analysis for Prey-predator Problem In this section, we will develop a simulation analysis and modeling for Event and consolidity for Prey-predator Problem dynamic problem after choosing relationship between the incremental parameters changes (prey number due to the increase in the vegetation when the rainfall increases) versus affected varying environment or events of various values (the increase in the rainfall) and the consolidity of the system. In this simulation, we can calculate the change and then calculate the consolidity in each step. In the following steps, we can apply the simulation analysis and modeling to this problems. First we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is an event (the rainfall), we will calculate the change in the number of prey P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new number of prey after changing its state. Second we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is another event, we will calculate the new change in the number of prey P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new parameter after changing its state and so on. We can summarize the steps as follows: 7.4.1 The Steps of Simulation Analysis for Prey-Predator


( 0)

with the parameter (N0).

(1) ( 0) At Event (1), P (1)   e( .Event .C ) , then we will calculate the parameter (N1) after the change that change its state

103


(1)

with the new parameter (N1).

At Event (2), we will calculate P (2)   e( .Event .C parameter (N2) after the changes that change its state. ( 2)


( 2)

(1)

),

and then we will calculate the


( 3) ( 2) At Event (3), we will calculate P (3)   e( .Event .C ) , and then we will calculate the parameter (N3) after the changes that change its state.


(3)


At Event (4), we will calculate P (4)   e( .Event parameter (N4) after the changes that change its state.

( 4)

.C ( 3) )

, and then we will calculate the

Step 5: as the previous steps and so on. In this section, we will develop a simulation analysis for number of prey and consolidity for Prey-Predator dynamic problem with effect of variation in level of rainfall level. 7.4.2 Simulation Analysis for Prey-Predator with Increase the Rainfall Level

These steps will be simulated at each event. If there is an event (rainfall) will affect on the number of prey and that is shown as follows:

Step 1: at event =0.25, CO / I  S ) Step 2: at event =0.3, CO /( I  S )

( 0)

(1)

Step 3: at event =0.35, CO /( I  S ) Step 4: at event =0.4, CO / I  S )

Step 5: at event =0.45, CO /( I  S ) Step 6: at event =0.5, CO /( I  S )

=4.62, P(2) =14.552

( 2)

(3)

=4.62, P(1) =11.753

=4.36, P(3) =16.565

=4.14, P(4) =18.684

( 4)

(5)

=3.96, P(5) = 20.996

=3.8, P(6) = 23.419

Step 7: so on

104

We will simulate the paradigm to show how the parameter can changes if there are events occurrences on it. In this paradigm we can simulate each step. At each step of the simulation, there will amounts of rainfall. As these amounts of rainfall increase, the vegetation will grow up and the rate of growth of prey will increase therefore we can calculate the consolidity for the system which mean that the number prey will grow up. This fact is noticed while modeling the problem. The study has covered variation of increase in the rainfall. In this context, we will develop a model that illustrates the relationship between the parameters Change versus the Event multiplied by the Consolidity Index. By analyzing the model, we found that the best equation for this model is the simple exponential curve, and the factor  is 0.926 and consequently we can calculate the parameter change if there is an event influence on the parameter of the system as shown below in (7.1). P  4.033401 e(0.926 xEvent xC )

(7.1)

Figure.7.2 Simulation Analysis for the Increase in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (Scenario I).

105

After developing relationship between the incremental parameters changes (prey number due to increase in the vegetation) versus affected varying environment or events of various values (the increase in the rainfall) and the consolidity of the system, a simulation analysis is developed for the number of prey and consolidity of the Problem in different scenarios. The first scenario shows that if the rainfall increased, the rate of growth of prey will be increased and the consolidity would be improved as depicted in Figure 7.2. Table 7.1 shows the influence of the rainfall level on the number of the prey and consolidity index. The table shows that if the rainfall level is increased, the number of the prey will increase and the consolidity will be improved as the consolidity index will be decreased. In this table, the number of the prey is rounded to the higher integer number. Table 7.1 Influence of the Rainfall Level on the Increase in the Number of Prey and Consolidity (Scenario I). Event state ()

Value of Event (Rainfall Level)

Consolidity Index

Number of Prey

1

0.25

4.62

12

2

0.3

4.36

15

3

0.35

4.14

17

4

0.4

3.96

19

5

0.45

3.8

21

6

0.5

3.58

24

7

0.55

3.47

25

8

0.6

3.37

28

9

0.65

3.13

31

10

0.7

2.98

31

11

0.75

2.89

32

12

0.8

2.78

35

13

0.85

2.7

36

14

0.9

2.58

39

15

0.95

2.47

39

16

1

2.33

40

106

7.4.3 Simulation Analysis for Prey-Predator with Decrease the Rainfall Level

The second scenario of the simulation analysis shows that if the rainfall has decreased, the number of prey will be decreased and the consolidity index will be higher which means that the system is unconsolidated as depicted in Figure 7.3.

Figure 7.3 Simulation Analysis for the Decrease in the Number of Prey and Consolidity Index due to the Influence of the Rainfall Level (Scenario II).

Table 7.2 shows the influence of the rainfall level on the number of the prey and consolidity index. The table shows that if the rainfall level is decreased, the number of the prey will decrease and the consolidity will not be improved as the consolidity index will be increased. In this table, the number of the prey is rounded to the higher integer number.

107

Table 7.2 Influence of the Rainfall Level on the Decrease in the Number of Prey and Consolidity (Scenario II). Event State (  )


Consolidity Index

Number of Prey

1

1.1

2.22

32

2

1

2.33

32

3

0.95

2.47

31

4

0.9

2.58

31

5

0.85

2.7

31

6

0.8

2.78

30

7

0.75

2.89

28

8

0.7

2.98

27

9

0.65

3.13

25

10

0.6

3.37

23

11

0.55

3.47

23

12

0.5

3.58

21

13

0.45

3.8

18

14

0.4

3.96

17

15

0.35

4.14

15

16

0.3

4.36

13

17

0.25

4.62

11

108

7.4.4 Simulation Analysis for Prey-Predator with Fluctuating the Rainfall Level

The third scenario of the simulation analysis shows the variation in the rainfall which will affect on the increase and decrease of number of prey as depicted in Figure 7.4. The figure shows that if the rainfall has decreased, the number of prey will be decreased and the consolidity index will be high and if the rainfall has increased, the number of prey will be increased and the consolidity index will be low.

Figure 7.4 Simulation Analysis for the Variation in the Number of Prey and Consolidity due to the Influence of the Rainfall Level (scenario III).

Table 7.3 shows the influence of the rainfall level on the number of the prey and consolidity index. The table shows the variation of the consolidity of the system when changing the rainfall. In this table, if the rainfall has decreased, the number of prey will be decreased and the consolidity index will be high and if the rainfall has increased, the number of prey will be increased and the consolidity index will be low which means that the system Consolidity would be improved and therefore if the environment has improved, consolidity would be improved. As depicted in the table, at each event we can calculate the consolidity of the system and if there are events, we can calculate the change in the parameter P . In this table, the number of the prey is rounded to the higher integer number.

109

Table 7.3 Influence of the Rainfall Level on the Fluctuation of the Number of Prey and Consolidity (Scenario III). Event State ()


Consolidity Index

Change in the Prey Number

1

1.1

2.22

32

2

0.9

2.58

26

3

0.8

2.78

28

4

0.85

2.7

36

5

0.65

3.13

22

6

0.7

2.98

31

7

0.6

3.37

22

8

0.35

4.14

12

9

0.25

4.62

11

10

0.3

4.36

15

11

0.55

3.32

37

12

0.75

2.89

41

13

0.45

3.8

14

14

0.4

3.96

17

15

0.5

3.58

26

16

0.55

3.47

25

17

0.7

2.98

38

7.5 Simulation Analysis for HIV/AIDS Infectious Disease

In this section, we will develop a simulation analysis for Event and consolidity for HIV/AIDS infectious disease effect high and low level of media campaigns after choosing relationship between the incremental parameters changes (number of HIV/AIDS infected people with effect of media campaigns) versus affected varying environment or events of various values (low or high level of media campaigns) and the consolidity of the system. In this simulation analysis, we can calculate the change and then calculate the consolidity in each step.

110

In theses following steps, we can apply the simulation analysis to this problem. First we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is an event, we will calculate the change in the number of HIV/AIDS infected people P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new number of HIV/AIDS Infected people after changing its state. Second we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is another event, we will calculate the new change in the number of HIV/AIDS infected people P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new parameter after changing its state and so on. We can summarize the steps as follows: 7.5.1 The Steps of Simulation Analysis for HIV/AIDs Infectious Disease


( 0)

with the number of HIV/AIDs infected people (N0).

At Event (1) : P (1)   .Event (1) .C (0)   , then we will calculate the parameter (N1) after the change that change its state Step 2: Calculate CO /( I  S )

(1)


At Event (2), we will calculate P (2)   .Event (2) .C (1)   , and then we will calculate the parameter (N2) after the changes that change its state. Step 3: Calculate CO /( I  S )

( 2)


At Event (3), we will calculate P (3)   .Event (3) .C (2)   , and then we will calculate the parameter (N3) after the changes that change its state. Step 4: Calculate CO /( I  S )

(3)


At Event (4), we will calculate P (4)   .Event (4) .C (3)   , and then we will calculate the parameter (N4) after the changes that change its state. Step 5: as the previous steps and so on. In this section, we will develop a simulation analysis for number of HIV/AIDS Infected people and consolidity for HIV/AIDS Infectious Disease with effect of variation in level of Media Campaigns.

111

7.5.2 Simulation Analysis for HIV/AIDS Infected People with Low Level of Media Campaigns These steps will be simulated at each year. In each year, there is an event will affect on the number of HIV/AIDS infected people and that is shown as follows:

Step 1: at year=1, Event = 1, CO /( I  S )

( 0)

=0.920, P(1) = 10431.16

Step 2: at year=2, Event = 1.01, CO /( I  S ) Step 3: at year=3, Event = 1.05, CO /( I  S ) Step 4: at year=4, Event = 1.1, CO /( I  S )

(1)

( 2)

(3)


=0.920, P(2) = 10446.02 =0.965, P (3) = 10703.97

=1.832, P (4) = 11000.08

( 4)

(5)

( 6)

=2.739, P (5) = 11538.42 =4.342, P (6) = 11954.62 =5.437, P (7) = 12503.54

Step 8: so on

In this paradigm we can simulate each step. At the first year of the simulation, we can calculate the consolidity for the system. As in this study, there are low levels of media campaigns, therefore the event will increases which mean that the number of HIV/AIDs infected people will increases and the changes in the number of HIV/AIDs infected people will be grow up. This fact is noticed while modeling the problem. The study has covered 23 year without media campaigns. This simulation analysis includes different scenarios of changing the effect of media campaigns. The first scenario shows the incremental changes in the number of HIV/AIDS Infected people and the consolidity of the system versus events of various values (media campaigns) as depicted in Figure 7.5. The figure shows the lack of the level of media campaigns which result in increase the number of HIV/AIDS infectious people and increase in the consolidity index of the system.

112

Figure 7.5 Simulation Analysis for the Increase in Number of HIV/AIDS Infected People and Consolidity Index with Decrease the Level of Media Campaigns (Scenario I). As depicted in Figure 7.5, at each year we can calculate the consolidity and if there are events, we can calculate the change in the parameter P . If there are low levels of media campaigns over years, the number of HIV/AIDS infected people will increase and the consolidity index will be higher which means the system is unconsolidated. Table 7.4 shows the variation of the consolidity of the system and the number of HIV/AIDs infected people if there is lack of media campaigns over years. The table also shows the increase in the number of HIV/AIDS infected people and the increase in the consolidity index which means the system is unconsolidated at each event over years. In this table, the number of HIV/AIDs infected people is rounded to the higher integer number.

113

Table 7. 4 The Influence of Decrease the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario I) Event State (  )

Value of Event (number of infected/ 10000)

Consolidity Index

Change in Number of HIV/AIDS Infected People

1

1

0.920

10431

2

1.01

0.965

10446

3

1.05

1.832

10704

4

1.1

2.739

11000

5

1.15

4.342

11539

6

1.2

5.437

11955

7

1.25

6.835

12504

8

1.3

7.564

12854

9

1.35

8.132

13165

10

1.4

8.927

13578

11

1.45

10.162

14186

12

1.5

11.412

14834

13

1.55

12.589

15485

14

1.6

13.928

16239

15

1.65

14.523

16695

16

1.7

15.378

17288

17

1.75

16.159

17868

18

1.8

16.841

18421

19

1.85

17.291

18877

20

1.9

17.714

19330

21

1.95

18.001

19723

22

2

18.330

20147

23

2.05

18.512

20497

114

7.5.3 Simulation Analysis for HIV/AIDS Infected People with High Level of Media Campaigns These steps will be simulated at each year. In each year, there is an event will affect on the number of HIV/AIDS infected people and that is shown as follows:

Step 1: at year=1, Event = 1, CO /( I  S )

(1)

=0.920, P = 10431.16


( 2)

(3)

( 4)

=0.920, P = 9799.429 =0.891, P = 9053.249 =0.852, P = 8367.496

(5) Step 4: at year=5, Event = 0.80, CO /( I  S ) =0.816, P = 7812.897 ( 6) Step 5: at year=6, Event = 0.75, CO /( I  S ) =0.793, P = 7286.871 (7) Step 6: at year=7, Event = 0.70, CO /( I  S ) =0.771, P = 6840.475

Step 7: so on

We will simulate the paradigm to show how the parameter can changes if there are events occurrences on it. In this paradigm we can simulate each step. At the first year of the simulation, we can calculate the consolidity for the system. As in this study, there are media campaigns, therefore the event will decrease which mean that the number of HIV/AIDs infected people will decrease and the changes in the number of HIV/AIDs infected people will be grow down. This fact is noticed while modeling the problem. The study has covered 17 year with media campaigns.

115

Figure7.6 Simulation Analysis for the Decrease in Number of HIV/AIDS Infected People and Consolidity Index with Increase the level of Media Campaigns (Scenario II). The second scenario shows the decrease in the number of HIV/AIDS infected people and the decrease in the value of consolidity index as the level of media campaigns is high as depicted in Figure 7.6. As depicted in Figure 7.6, at each year we can calculate the consolidity and if there are events, we can calculate the change in the parameter P . If there are media campaigns over years, the number of HIV/AIDS infected people will decrease and the consolidity index will be lower which means the system is consolidated. Table 7.5 shows the variation of the consolidity of the system and the number of HIV/AIDs infected people if there is increase in the media campaigns over years. The table also shows decrease in the number of HIV/AIDS infected people and the decrease in the consolidity index which means the system is consolidated at each event over years. In this table, the number of HIV/AIDs infected people is rounded to the higher integer number.

116

Table 7.5 The Influence of Increase the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario II) Event State ()

Event (number of infected/ 10000)

Consolidity Index


1

1

0.920

10431

2

0.95

0.891

9799

3

0.9

0.852

9053

4

0.85

0.816

8368

5

0.8

0.793

7813

6

0.75

0.771

7287

7

0.7

0.748

6841

8

0.65

0.721

6321

9

0.6

0.682

5529

10

0.55

0.641

5173

11

0.5

0.626

4802

12

0.45

0.601

4403

13

0.4

0.592

4088

14

0.35

0.561

3709

15

0.3

0.536

3376

16

0.25

0.492

3022

17

0.2

0.451

2715

7.5.4 Simulation Analysis for HIV/AIDS Infected People with Fluctuation of the Level of Media Campaigns If there are variations in the levels of media campaigns such that high or low level, the number of HIV/AIDs infected people will be increased or decreased. The study has covered 17 year with media campaigns. 117

As depicted in Table 7.6, at each year we can calculate the consolidity and if there are events, we can calculate the change in the parameter P . If there are media campaigns over years, the number of HIV/AIDS infected people will decrease and the consolidity index will be lower which means the system is consolidated. In this table, the number of HIV/AIDs infected people is rounded to the higher integer number. Table 7.6 The Influence of Fluctuation of the Level of Media Campaign on the Number of HIV/AIDS Infected People (Scenario III)

Event State ()

Event (number of infected/ 10000)

Consolidity Index


1

0

1.800

10431

2

0.95

1.300

9799

3

1.2

4.437

11955

4

0.85

0.716

8367

5

1.15

4.242

11538

6

0.5

1.600

10335

7

1.3

8.027

13018

8

1.35

7.032

12762

9

1.01

0.965

10446

10

1.25

5.835

12164

11

1.35

7.132

12798

12

0.8

0.871

8397

13

1.4

7.134

12896

14

1.01

0.965

10446

118

Figure 7.7 shows the variation of the consolidity of the system and the number of HIV/AIDs infected people. If there is increase in the media campaigns over years, the number of HIV/AIDS infected people will be decreased and also the consolidity index will be high at each event over years. However if there is decrease in the level of media campaigns over years, the number of HIV/AIDS infected people will be increased and also the consolidity index will be increased at each event over years. This is noted in the third scenario that shows the fluctuation in the level of media campaigns and its influence on the number of HIV/AIDS infectious people and the consolidity index as depicted in Figure 7.7.

Figure 7.7 Simulation Analysis for Changing the Number of HIV/AIDS Infected People with Different Levels of Media Campaigns (Scenario III). 7.6 Simulation Analysis for Spread of Infectious Diseases

In this section, we will develop a simulation analysis for Event (hygiene of the dorm) and consolidity for spread of infectious diseases after choosing relationship between the incremental parameters changes (infection rate of spread of infectious diseases) versus affected varying environment or events of various values (infection rate) and the consolidity of the system. 119

In this simulation analysis, we can calculate the change and then calculate the consolidity in each step. In theses following steps, we can apply the simulation analysis to this problem. First we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is an event, we will calculate the change in the infection rate of Spread of Infectious Diseases P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new infection rate of Spread of Infectious Diseases after changing its state. Second we will calculate the consolidity index CO /( I  S ) for the system parameters. If there is another event, we will calculate the new change in the infection rate of spread of infectious diseases P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new parameter after changing its state and so on. We can summarize the steps as follows: 7.6.1 The Steps of Simulation Analysis for Spread of Infectious Diseases


( 0)

with the parameter (K0).

(1) ( 0) At Event (1), P (1)   e( .Event .C ) , then we will calculate the parameter (K1) after the change that change its state


(1)

with the new parameter (K1).

At Event (2), we will calculate P (2)   e ( .Event parameter (K2) after the changes that change its state.

( 2)


( 2)

.C (1) )



( 3) ( 2) At Event (3), we will calculate P (3)   e( .Event .C ) , and then we will calculate the parameter (K3) after the changes that change its state.


(3)


At Event (4), we will calculate P (4)   e( .Event parameter (K4) after the changes that change its state.

( 4)

.C ( 3) )


Step 5: as the previous steps and so on. In this section, we will develop a simulation analysis for infection rate and consolidity index in the problem of Spread of Infectious Diseases with effect of variation in level of Hygiene of the Dorm.

120

7.6.2 Simulation Analysis for Spread of Infectious Diseases with Low Level of Hygiene of the Dorm These steps will be simulated at each year. In each year, there is an event (hygiene of the dorm) will affect on the infection rate of the spread of infectious diseases and that is shown as follows:

Step 1: at Event=0.25, CO /( I  S ) Step 2: at Event =0.3, CO /( I  S )

(1)

( 2)

Step 3: at Event =0.35, CO /( I  S ) Step 4: at Event =0.4, CO /( I  S )

=30.254, P =0.659

(3)

( 4)

Step 5: at Event =0.45, CO /( I  S )

=30.254, P =0.731

=24.976, P =0.522

=20.456, P =0.354

(5)

=16.520, P =0.210

Step 6: so on.

We will simulate the paradigm to show how the parameter can changes if there are events occurrences on it. In this paradigm we can simulate each step. At the first step of the simulation, we can calculate the consolidity for the system. As in this study, if the event (hygiene of the dorm) increased, the infection rate of the spread of infection diseases will decreased therefore the rate of transferring the diseases will be decreased. This fact is noticed while modeling the problem. The study has covered variable changes in the infection rate in the dorm. In this section, we will develop a simulation analysis for infection rate of Spread of Infectious Diseases and the consolidity of the system due to the influence of (hygiene of the dorm). This simulation analysis includes different scenarios of changing the influence of cleaning of the dorm. The first scenario shows the increase in the infection rate and also the increase in the consolidity index due to the low level of dorm cleaning as depicted in Figure 7.8.

121

Figure 7.8 Simulation Analysis for the Increase the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the Low Level of Hygiene of the Dorm (Scenario I) Figure 7.8 shows the scenario that illustrates the fluctuation of the consolidity index and the infection rate of the spread of infectious diseases due to the influence of hygiene of the dorm. The results show that if the percentage of hygiene of the dorm is low, the infectious rate will be low and the consolidity will be inferior. However, if the percentage of hygiene of the dorm increased, the infectious rate will be high and the system will be superior consolidated. Table 7.7 shows the variation of the consolidity of the system and the infection rate if there is increase the percentage of hygiene of the dorm. The table also shows the increase in the consolidity index (which means the system is consolidated at each event step) as the percentage of hygiene of the dorm decreases and therefore the infection rate of the spread of infectious diseases will increase.

122

Table 7.7 The Influence of the Low Level of Hygiene of the Dorm on Increase the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to (Scenario I) Event State ()

Event (level of Hygiene)

Consolidity Index

Change in the Inflation Rate

1

0.85

0.88

0.002

2

0.8

0.933

0.0020056

3

0.78

1.005

0.0019886

4

0.75

1.142

0.0020249

5

0.735

1.215

0.0021491

6

0.7

1.321

0.0021658

7

0.67

1.381

0.0022184

8

0.65

1.491

0.0022379

9

0.612

1.572

0.0022611

10

0.6

1.665

0.0023099

11

0.58

1.784

0.0023464

12

0.55

2.050

0.0023718

13

0.534

2.200

0.0025665

7.6.3 Simulation Analysis for Spread of Infectious Diseases with High Level of Hygiene of the Dorm Figure 7.9 shows the scenario that illustrates the decrease of the consolidity index and the infection rate of the spread of infectious diseases due to the influence of hygiene of the dorm. The results show that if the percentage of hygiene of the dorm is high, the infectious rate will be low and the consolidity will be improved in the second scenario. 123

Figure 7.9 Simulation Analysis for the Decrease the Consolidity Index and Infection Rate of Spread of Infectious Diseases due to the High Levels of Hygiene of the Dorm (Scenario II) Table 7.8 shows the variation of the consolidity of the system and the infection rate if there is increase the percentage of hygiene of the dorm. The table also shows the decrease in the consolidity index (which means the system is consolidated at each event step) as the percentage of hygiene of the dorm increases and therefore the infection rate of the spread of infectious diseases will decreases.

124

Table 7.8 The Influence of the High Level of Hygiene of the Dorm on Decrease the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario II) Event State ()


Consolidity Index

Change in the Infection Rate

1

0.58

1.734

0.0024826

2

0.6

1.665

0.0024714

3

0.612

1.572

0.0024349

4

0.65

1.491

0.0024397

5

0.67

1.381

0.0024013

6

0.7

1.321

0.002348

7

0.735

1.215

0.0023549

8

0.75

1.142

0.0022592

9

0.78

1.005

0.0022272

10

0.8

0.933

0.0020969

11

0.82

0.925

0.0020408

7.6.4 Simulation Analysis for Spread of Infectious Diseases with Fluctuation in Level of Hygiene of the Dorm Figure 7.10 shows the scenario that illustrates the fluctuation of the consolidity index and the infection rate of the spread of infectious diseases due to the influence of hygiene of the dorm. The results show that if the percentage of hygiene of the dorm is low, the infectious rate will be low and the consolidity will be inferior. However, if the percentage of hygiene of the dorm increased, the infectious rate will be high and the system will be superior consolidated.

125

Figure 7.10 Simulation Analysis for the Fluctuation in Infection Rate and the Consolidity index of Spread of Infectious Diseases due to Different Levels of Hygiene of the Dorm (Scenario III) Table 7.9 shows the variation of the consolidity of the system and the infection rate if there is increase the percentage of hygiene of the dorm. The table also shows the decrease in the consolidity index (which means the system is consolidated at each event step) as the percentage of hygiene of the dorm increases and therefore the infection rate of the spread of infectious diseases will decreases. Table 7.9 The Influence of the Fluctuation of Hygiene of the Dorm on the Consolidity Index and Infection Rate of Spread of Infectious Diseases (Scenario III) Event State ()


Consolidity Index


1

0.534

1.3

0.0023434

2

0.67

1.800

0.0035003

3

0.58

1.23

0.0026857

4

0.82

2

0.0037498

126

Event State ()


Consolidity Index


5

0.78

1.005

0.00198

6

0.735

1.215

0.0020037

7

0.67

1.381

0.0021116

8

0.7

1.521

0.002348

9

0.735

1.315

0.0023549

10

0.75

1.142

0.0022592

11

0.78

1.005

0.0022272

12

0.6

1.665

0.0028234

13

0.78

0.800

0.0018495

14

0.58

1.734

0.0029576

7.7 Concluding Remarks In this chapter, we have developed a simulation analysis for the paradigm if there are events occurrences on the parameter. In this simulation, we have calculated the change and then calculate the consolidity, after choosing relationship between the incremental parameters changes versus affected varying environment and events of various values, In this context, we can calculate the change and then calculate the consolidity in each step. At each step, if there is an event, we will calculate the change P through the relation between the parameter changes and the consolidity index multiplying by the event, and consequently we will have the new parameter after changing its state. In this respect, the three case studies have been modeled and simulated through the suggested paradigm after choosing relationship between the incremental parameters changes versus affected varying environment or events of various values and the consolidity of the system. These case studies are prey predator, HIV/AIDs infectious disease with and without media campaigns and the spread of infectious diseases.

127

Chapter 8 Conclusions and Recommendations for Further Research

8.1 Conclusions The thesis has presented a new updated conceptual change pathway graph of natural and manmade built-as-usual physical systems following a joint two-level event-driven (activity-driven) and time-driven configuration framework, supported by the consolidity- based theory. The conceptual graph was then mathematically expressed as successive event-driven memory less or memory-based formulations as analogous to the original systems change behavior regarding lags ad types. The main structure of the “time driven – event driven – parameters changes” paradigm is mainly based on the assumption that any event affecting at the high level layer lead to changing the system parameters at the physical system level scaled by consolidity. It is therefore the main objective of the thesis to confirm through real life applications the validity of this important assumption. The event driven in this thesis is related to Consolidity which is a new word that describe the internal changes in the parameters on the systems. The property of consolidity is necessary for the system if the system operates with event that affect on its parameters. This theory is operating in a full fuzzy environmental position. The mechanism of the conceptual change pathway development progress was logically conceived that the system behavior changes rate has not accidently happened but is relatively influenced at the point of progress with the associated system consolidity index corresponding to the acting varying environments or events of different types and strength ‘‘on and above’’ their normal stands or original set points. In the investigation, the events, activities, and varying environments operations were recognized in the form of high level influencing layer(s) (referred to as the ‘‘event-driven or upper system layer(s)’’) acting over the ordinary systems state equations designated as the ‘‘time-driven or basic system layer(s).’’ consolidity effect is prevailing at the event-driven or upper system layer(s).

128

In this thesis, a comparison between the time driven and event driven has been discussed in the literature and also graphs for positive, negative and mixed changes and charts have been explained. In addition, consolidity for hypothetical two and three order state space system which describe the influence of changing one parameter of the system have been showed in the thesis. The structure for hypothetical two and three order state space system has been modified to the system and the tuning has been done by the Pole placement to choose all the parameters of the system and then the factor of consolidity increases lead to increase Consolidity. In addition, system change pathway has been discussed depending on the time driven- event driven systemparameter changes, in this thesis. The parameter of the system can be changed due to the influence of event and consequently the consolidity also will be changed. These events can be positive or negative that can increase or decrease the parameter of the system and consolidity of the system, therefore entity will be changed to be built more or destroyed. This thesis has concentrated on developing explicit relationships between the incremental parameters changes versus affected varying environment and events of various value or strengths scaled at each event state by the system self-consolidity; following the general relationship at any event step  as:

 Parameter change (  ) = Function [consolidity (  ) , varying environment or event (  ) ]. After choosing relationship between the incremental parameters changes versus affected varying environment and events of various value. The paradigm has developed to show if there is event occurrence on the parameter, we can calculate the change and then calculate the consolidity. The thesis has demonstrated the concept using three case studies. The first can study in the investigation of prey-predator model operating under external environmental influences. The second can study in the analysis of HIV/AIDS Epidemic problem model subject to variability of awareness and educational programs. The third can study in the analysis of spread of infectious disease problem subject to variability of level of hygiene. In these case studies, a paradigm that shows the relationship between the parameter changes versus (the Event multiplied be the Consolidity Index) have been developed and the parameter changes can be calculated through the least squares approximations. Therefore, the developed paradigms for any system depend on the application as it can be linear, parabolic, exponential, logarithmic …etc. The results of these problems confirm the suitability of the suggested paradigm to handle real life system under external influences “on and above” their normal conditions.

129

After choosing relationship between the incremental parameters changes versus affected varying environment and events of various values, we will develop a simulation analysis for the paradigm if there are events occurrences on the parameter. In this simulation, we can calculate the change and then calculate the consolidity. In this respect, the three case studies have been modeled. These case studies are prey predator, HIV/AIDs infectious disease and the spread of infectious diseases. Finally we can simulate consolidity model for more problems such as engineering, historical, social, environmental and religion by the suggested theory in real life applications not in hypothetical examples. In conclusion, we must be all cautioned of the high complexity and versatility of our real life systems when dealing with the implementation of the above targets of such global theory. We should also keep in mind while pursuing further such targets that the universe in this respect will always be our best inspirer and educator.

8.2 Suggestions for Further Research

1. Exploration the relationship between (the consolidity index and events) and the parameters changes for further real life problems. 2. Studying of occurrence of other events on the other parameters of the system. 3. Exploration the relationship between robustness and the consolidity. 4. Studying the mathematical relationship between the controllability and consolidity. 5. Performing an in-depth survey of the types and classifications of events affecting various real life systems. 6. Searching the relation between forecasting and consolidity. 7. Simulation analysis and modeling for consolidity and parameter changes for real life problems. 8. Studying the use of the concept for Time Driven - Event Driven - Parameter Changes Paradigm with other techniques as Petri net, automata …etc. 9. Using Event Driven in the smart homes to save the amount of energy consumed through number of energy efficiency measures in power system utilization. 10. Applying the paradigm and simulation analysis to choose the best parameters of smart grids after complete researches of smart grids within the framework of consolidity. 11. Searching for additional formulas relating the events with the consolidity and parameters changes. 12. Searching for a new approach the consolidity index calculations different than using fuzzy logic such as by rough sets.

130

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Appendix A Consolidity Index for the 2nd Order of State Space Using Stability Coefficients A.1 Derivation of the Output for the 2nd Order of State Space Problems The state space can be derived as follow:

1,2

 b  b 2  4ac .  2a

(A.1)

1, 2 are the crisp value for the roots of the 2nd order state space problem, and a, b, c are the crisp value for the parameters of the system and then,

( x,  x )  (b,  b ) 2  4(a,  a )(c,  c )  (b 2 ,2 b )  4(ac,  a   c ) Where

(A.2)

 a ,  b ,  c represents the fuzzy level for the parameters of the system a, b, c

( x,  x )  (b  4ac, 2

2 b b 2  4( a   c )ac b  4ac 2

(A.3)

)

( y,  y )  ( x,  x )1/ 2  ( x1/ 2 ,  x / 2)

(A.4)

( z1 ,  z1 )  (b,  b )  ( y,  y )  (b  y,

( z 2 ,  z )  (b,  b )  ( y,  y )  (b  y,

  bb   y y b y   bb   y y b y

138

)

(A.5)

)

(A.6)

(1 ,  1 )  ( z1 ,  z ) / 2(a,  a )  ( z / 2a,  z1   a )

(A.7)

(2 ,  2 )  ( z 2 ,  z2 ) / 2(a,  a )  ( z 2 / 2a,  z2   a )

(A.8)

1 ,  2 are the crisp value of the roots of the system and  1 ,  2 are the fuzzy level for the roots of the system.

y 

 1 1   2  2

(A.9)

1   2

A.2 Derivation of the Input for the 2nd Order of State Space Problems

x 

 a a   bb   c c abc

(A.10)

A.3 Consolidity Index for the 2nd Order of State Space Problems

conslidty index 

y

(A.11)

x

A.4 Special Case: Consolidity Index for Equal Roots Case 1: Equal  & b  4ac  0 2

b  2a, c  a ( ,   )  (b / 2a,  b   a )

(A.12)

conslidty index  b / 2a

(A.13)

We can notice that the results are fixed; therefore, we can make tuning to get the best results

139

Appendix B Consolidity Index for the 3rd Order of State Space Using Stability Coefficients B.1 Derivation of the Output for the 3rd Order of State Space Problems The state space can be derived as follows: 3 2 3 2 2 2 3 b 1 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac) 1    3a 3a 2



3

3

2

2

2

1 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac) 3a 2

2   

2

3 2 3 2 2 2 3 b 1  i 3 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac)  3a 6a 2 3

2

3

2

2

2

1  i 3 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac) 6a 2



2

3

2

2

2

1  i 3 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac) 6a 2

(B.2)

3

3 2 3 2 2 2 3 b 1  i 3 3 2b  9abc  27a d  (2b  9abc  27a d )  4(b  3ac) 3    3a 6a 2 3

(B.1)

3

(B.3)

3

1, 2,3 are the crisp value for the roots of the 3rd order state space problem, and a, b, c, d are the crisp value for the parameters of the system.

140

and then, ( x,  x )  2(b,  b )3  9(a,  a )(b,  b )(c,  c )  27(a,  a ) 2 (d ,  d )

where

(3.4)

 a ,  b ,  c ,  d represents the fuzzy level for the parameters of the system a, b, c, d.

( x,  x )  2(b3 ,3 b )  9(a,  a )(b,  b )(c,  c )  27(a 2 ,2 a )(d ,  d )

(3.5)

x  2b3  9abc  27a 2 d

(3.6)

x 

2  3 bb3  9  ( a   b   c )abc  27(2 a   d )a 2 d ) 2b3  9abc  27a 2 d

(3.7)

( y,  y )  (b,  b ) 2  3(a,  a )(c,  c )

(B.8)

( y,  y )  (b 2 ,2 b )  3(ac,  a   c )

(B.9)

( y,  y )  (b 2  3ac,

2 bb 2  3( a   c )ac ) (b 2  3ac)

(B.10)

(n,  n )  ( y 3 ,3 y )

2

(B.11)

2

( z,  z )  ( x,  x )  4(n,  n )  ( x  4n,

2 x x 2  4 n n x 2  4n

)

(B.12)

( w,  w )  ( z,  z )1 / 2  ( z1 / 2 ,  z / 2)

(B.13)

 x   ww (u1,  u1 )  (( x,  x )  ( w,  w )) / 2  (( x  w) / 2, x ) xw

(B.14)

(v1,  v1 )  (u1,  u1 )1 / 3  (u11 / 3 ,  u1 / 3)

(B.15)

141

 x   ww (u2 ,  u 2 )  (( x,  x )  ( w,  w )) / 2  (( x  w) / 2, x ) xw

(B.16)

(v2 ,  v2 )  (u2 ,  u 2 )1 / 3  (u21 / 3 ,  u 2 / 3)

(B.17)

(1,  1 )  

(b,  b ) (v1,  v1 ) (v2 ,  v2 )   3(a,  a ) 3(a,  a ) 3(a,  a )

(B.18)

v v b ,  b   a )  ( 1 ,  v1   a )  ( 2 ,  v2   a ) 3a 3a 3a

(B.19)

(1,  1 )  (

1  

 1 

b v1 v2   3a 3a 3a

 ( b   a )

(2 ,  2 )  

( 2 ,   2 )  (

(B.20)

v v b  ( v1   a ) 1  ( v2   a ) 2 3a 3a 3a b v1 v2    3a 3a 3a

(b,  b ) (1  i 3 )(v1,  v1 ) (1  i 3 )(v2 ,  v2 )   3(a,  a ) 6(a,  a ) 6(a,  a )

(1  i 3 )v1 b ,  b  3 a )  ( , ((1  i 3 ) v1  6 a )) 3a 6a

(1  i 3 )v2 ( , ((1  i 3 ) v 2  6 a )) 6a

2  

 2 

b (1  i 3 )v1 (1  i 3 )v2   3a 6a 6a

 ( b   a )

(B.21)

(B.22)

(B.23)

(B.24)

(1  i 3 )v1 (1  i 3 )v2 b  ( v1   a )  ( v 2   a ) 3a 6a 6a b (1  i 3 )v1 (1  i 3 )v2    3a 6a 6a 142

(B.25)

(3 ,  3 )  

(3 ,  3 )  (

3  

 3 

(b,  b ) (1  i 3 )(v1,  v1 ) (1  i 3 )(v2 ,  v2 )   3(a,  a ) 6(a,  a ) 6(a,  a )

(1  i 3 )v1 (1  i 3 )v2 b ,b   a )  ( , ( v1   a ))  ( , ( v 2   a )) 3a 6a 6a

b (1  i 3 )v1 (1  i 3 )v2   3a 6a 6a

 ( b   a )

(B.26)

(B.27)

(B.28)

(1  i 3 )v1 (1  i 3 )v2 b  ( v1   a )  ( v1   a ) 3a 6a 6a b (1  i 3 )v1 (1  i 3 )v2    3a 6a 6a

(B.29)

1 ,  2 , 3 are the crisp value of the roots of the system and  1 ,  2 ,  3 are the fuzzy level of the roots of the system. y 

 1 1    2 2   3 3

(B.30)

1  2  3

B.2 Derivation of the Input for the 3rd Order of State Space Problems  a   bb   c c x  a abc

(B.31)

B.3 Consolidity Index for the 3rd Order of State Space Problems C

y

(B.32)

x

143

B.4 Special Cases: Consolidity Index for Equal Roots for 3rd Order Systems B.4.1 Case1: Consolidity Index for Equal Roots Case 1: for equal 1

ax3  bx 2  cx  d  0

(B.33)

x3  3x 2  32 x  3  0

(B.34)

b c d 3  , 32  , 3  a a a

(B.35)

b  3a,

c  32 a, d  3a,

a 1

(B.36)

   ( b   a )

(B.37)

B.4.2 Case2: (b 2  3ac) cancelled

b  a 3, c  a 3

b 2b3  9abc  27a 2 d  3a 3a

(B.38)

2  

b 1 i 3 3 3  2b  9abc  27a 2 d 3a 6a

(B.39)

3  

b 1 i 3 3 3  2b  9abc  27a 2 d 3a 6a

(B.40)

1  

( x,  x )  2(b,  b )3  9(a,  a )(b,  b )(c,  c )  27(a,  a ) 2 (d ,  d )

(B.41)

( x,  x )  2(b3 ,3 b )  9(a,  a )(b,  b )(c,  c )  27(a 2 ,2 a )(d ,  d )

(B.42)

144

( x,  x )  (2b 3  9abc  27a 2 d , 2  3 b b 3  9( a   b   c )abc  27(2 a   d )a 2 d 2b 3  9abc  27a 2 d

(B.43)

)

 x1 / 3 / 3  3 a a ( y1,  y1 )  ( x,  x )1 / 3  3(a,  a )  ( x1 / 3  3a, x ) x1 / 3  3a

( z1,  z1 )  (

b , b   a ) 3a

(1,  1 )  ( y1  z1,

(B.45)

  y1 y1   z1 z1  y1  z1

)

(B.46)

(1  i 3 )( x,  x )1 / 3 (1  i 3 ) x1 / 3 ( y2 ,  y 2 )  ( , x / 3   a ) 6(a,  a ) 6a

(2 ,  2 )  ( y2  z1,

  y 2 y2   z1 z1  y2  z1

)

(B.47)

(B.48)

(1  i 3 )( x,  x )1 / 3 (1  i 3 ) x1 / 3 ( y3 ,  y3 )  ( , x / 3   a ) 6(a,  a ) 6a

(3 ,  3 ) 

(B.44)

  y3 y3   z1 z1

(B.49)

(B.50)

 y3  z1

B.4.3 Case 3: 2b3  9abc  27a 2 d  0 cancelled

2b3  9abc  27a 2 d  0

(B.51)

b  3a , c  a , d  a

(B.52)

145

1  

b 2 i 3a 3

2  3  

(b 2  3ac) b 2 6   i a 3a 3

(B.53)

b 2 6 i 3a 6

(B.54)

 1   2   3   b   a

y 

(B.55)

 1 1   2 2   3 3

(B.56)

1  2  3

 a   bb   c c   d d x  a abcd

C

(B.57)

y

(B.58)

x

146

Appendix C Consolidity Index for the 3rd Order of State Space Using Pole Placement

C.1 Consolidity Index Using Pole Placement Now, we will consider the pole placement of the system

0 1 0 b1   0 1 , B  b2  If A   0 b3 / a   d / a  c / a  b / a 

(C.1)

C.1.1 Ackermann’s Formula Ackermann’s formula: Ackermann’s formula (1972) is a direct evaluation method. It is only applicable to SISO systems and therefore u (t ) and y (t ) are scalar quantities. Let

k  0 0 ... 0 1M 1 ( A) ,

(C.2)

and

 ( A)  An   n 1 An 1  ...  1 A   0 I ,

where A is the system matrix and

(C.3)

 i are the coefficients of the desired closed-loop characteristic

equation.

s  1, s  n ,..., s  n

(C.4)

Then, from (C.4) (s  1)(s  n )... (s  n )  s n   n 1s n 1  ...  1s   0

147

(C.5)

If we consider the 3rd order system

1  0  A  0 0  d / a  c / a 2

0  1   b / a 

1  0  0 0   d / a  c / a

0  1   b / a 

 0 A2    d / a d 2 / a 2

0 c/a  d / a  bc / a 2

1  , b/a   c / a  b 2 / a 2 

 0 A    d / a d 2 / a 2

0 c/a  d / a  bc / a 2

1   b/a   c / a  b 2 / a 2 

3

d /a   A  bd / a 2 dc / a 2  db 2 / a 3 3

(C.6a)

(C.6b)

1  0  0 0   d / a  c / a

c/a  d / a  bc / a 2 d 2 / a 2  c 2 / a 2  cb 2 / a 3

0  1   b / a 

b/a   2 2 c/ab /a  2bc / a 2  d / a  b3 / a 3 

d /a c/a b/a     2 2 2 2  ( A)   bd / a  d / a  bc / a c/ab /a  dc / a 2  db 2 / a 3 d 2 / a 2  c 2 / a 2  cb 2 / a 3 2bc / a 2  d / a  b 3 / a 3  0 1 1 0   0   0    2   d / a c/a b/a 0 1    1  0 d 2 / a 2  d / a  bc / a 2  c / a  b 2 / a 2   d / a  c / a  b / a  1 0 0   0 0 1 0 0 0 1

148

(C.7a)

(C.7b)

(C.8)

 c / a  1   d / a  0  2  ( A)  bd / a   2 d / a  d / a  bc / a 2   2c / a   0  31 32

11  d / a   0 ,  11 

12  c / a  1 ,  12 

13  b / a   2 ,  13 

 b / a  2

  c / a  b / a   2b / a  1   33 2

2

 ( d   a ) d / a

(C.10)

11

 ( c   a ) c / a

(C.11)

12

 ( b  a )b / a

21  bd / a 2   2 d / a ,  21 

(C.9)

(C.12)

13

 ( b   d  2 a )bd / a 2  ( d  a ) 2 d / a

21

(C.13)

22  d / a  bc / a 2   2c / a   0 ,  22 

 ( d  a )d / a  ( b   c  2 a )bc / a 2  ( c  a ) 2c / a

(C.14)

22

23  c / a  b 2 / a 2   2b / a  1 ,  ( c  a )c / a  (2 b  2 a )b 2 / a 2  ( b  a ) 2b / a  23 

23

149

(C.15)

31  dc / a 2  db 2 / a3   2 d 2 / a 2  1d / a ,   31  

( d   c  2 a )dc / a 2  ( d  2 b  3 a )db 2 / a 3

(C.16)

31

(2 d  2 a ) 2 d 2 / a 2  ( d   a )1d / a

31

32  d / a  c / a 2  cb 2 / a3   2 (d / a  bc / a 2 )  1c / a , 2

 32  

2

2

(2 d  2 a )d 2 / a 2  (2 c  2 a )c 2 / a 2  ( c  2 b  3 a )cb 2 / a 3

32

(C.17)

 ( d  a ) 2 d / a  ( b   c  2 a ) 2bc / a 2  ( c  a )1c / a

32

33  2bc / a 2  d / a  b3 / a3   2 (c / a  b 2 / a 2 )  1b / a   0 ,

 33  

( b   c  2 a )2bc / a 2  ( d  a )d / a  (3 b  3 a )b3 / a 3

33

(C.18)

 ( c  a ) 2 c / a  (2 b  2 a ) 2b 2 / a 2  ( b  a )1b / a

33

0 1 0   A 0 0 1  ,  d / a  c / a  b / a 



M B

AB

A2 B

b1  B  b2  b3 / a 

(C.19)



0 1 0   AB   0 0 1   d / a  c / a  b / a 

(C.20)

b2 b1    b   b3 / a  2    b3 / a   b1d / a  b2 c / a  b3b / a 2  150

(C.21)

b2     b3 / a    b1d / a  b2 c / a  b3b / a 2  b3 / a     2 2 A B  b1d / a  b2 c / a  b3b / a   b2 d / a  b3c / a 2  (b1d / a  b2 c / a  b3b / a 2 )b / a  0 1 0   A B 0 0 1   d / a  c / a  b / a  2

(C.22)

Now, we withdraw the matrix to be separate equations as follows.

m11  b1 ,  m11  0

(C.23)

m12  b2 ,  m12  0

(C.24)

m13  b3 / a ,  m13  0

(C.25)

m21  b2 ,  m21  0

(C. 26)

m22  b3 / a ,  m22  0

(C. 27)

m23  b1d / a  b2c / a  b3b / a 2 ,

 ( d   a )b1d / a  ( c   a )b2c / a  ( b   a )b3b / a 2  m23  m23

(C. 28)

m31  b3 ,  m31  0

(C.29)

151

m32  b1d / a  b2c / a  b3b / a 2 ,

 ( d   a )b1d / a  ( c   a )b2c / a  ( b  2 a )b3b / a 2  m32  m32

(C.30)

m33  b2 d / a  b3c / a 2  (b1d / a  b2c / a  b3b / a 2 )b / a ,

 m33  

 ( d   a )b2 d / a  ( c  2 a )b3c / a 2  ( d   b  2 a )b1bd / a 2 m33

(C.31)

( c   b  2 a )b2 cb / a 2  (2 b  3 a )b3b 2 / a 3 m33

b2  b1  M   b2 b3 b3 / a  b1d / a  b2 c / a  b3b / a 2

 m11 m12 1  M  m21 m22  m31 m32

m13  m23  m33 

    b2 d / a  b3c / a 2  (b1d / a  b2 c / a  b3b / a 2 )b / a  b3 / a  b1d / a  b2 c / a  b3b / a

(C.32)

1

(C.33)

11  m22 m33 m23 m32

(C.34)

 21  m12 m33 m13 m32

(C.35)

31  m12 m23 m13 m22

(C.36)

  31 

( m12   m23 )m12 m23  ( m13   m22 )m13 m22  31

(C.37)

12  m21m33 m23 m31

(C.38)

 22  m11m33 m13 m31

(C.39)

152

32  m11m23 m13 m21

  32 

(C.40)

( m11   m23 )m11m23  ( m13   m21 )m13 m21  32

(C.41)

13  m21m32 m22 m31

(C.42)

 23  m11m32 m12 m31

(C.43)

33  m11m22 m12 m21

(C.44)

  33 

( m11   m22 )m11m22  ( m12   m21 )m12 m21  33

  m11(m22m33 m23 m32 )  m12 (m21m33 m23 m31)  m13 (m21m32 m22 m31)

(C.45)

(C.46)

( m11   m22   m33 )m11m 22 m33  ( m11   m23   m32 )m11m 23 m32  ( m12   m21   m33 )m12 m 21m33  ( m12   m23   m31 )m12 m23 m31  ( m13   m21   m32 )m13 m21m32  ( m13   m22   m31 )m13 m 22 m31

  



 11     M 1   21     31  

  12   22   32  

13     23       33   

(C.48)

153

(C.47)

 11     K  0 0 1  21     31  

 K   31  

  32 

  12   22   32  

13       23     33   

11 12 13     21 22 23  31 32 33 

11 12 13   33   21 22 23      31 32 33 

(C.49)

(C.50)

   K1  31 11  32 21  33 31   

(C.51)

   (  31   11    ) 31 11  (  32    21    ) 32 21  (  33   31    ) 33 31     K1  K1

(C.52)

   K 2  31 12  32 22  33 32   

(C.53)

   (  31   12    ) 31 12  (  32    22    ) 32 22  (  33   32    ) 33 32     K2  K2

(C.54)

   K 3  31 13  32 23  33 33   

(C.55)

154

   (  31   13    ) 31 13  (  32    23    ) 32 23  (  33   33    ) 33 33     K3  K3

(C.56)

k1 , k 2 , k3 are the crisp value of the feedback gain of the system and  k1 ,  k 2 ,  k 3 are the

fuzzy level of feedback gain of the system.

y 

 k1 k1   k 2 k 2   k 3 k3

(C.57)

k1  k 2  k3

C.1.2 Derivation of the Input for the 3rd Order of State Space Problems  a   bb   c c x  a abc

(C.58)

C.1.3 Consolidity Index for the 2nd Order of State Space Problems C

y

(C.59)

x

C.2.1 Direct Comparison Method Direct comparison method: If the desired locations of the closed-loop poles (eigen values) are

s  1, s  n ,..., s  n

(C.60)

Then, from (3.19)

sI  A  Bk  ( s  1 )( s   n )... ( s   n )

(C.61)

s n   n 1s n 1  ...  1s   0

155

Solving (C.61) will give the elements of the state feedback matrix.

1 0   b1   s 0 0  0    sI  A  Bk  0 s 0   0 0 1    b2  k1 k 2 0 0 s   d / a  c / a  b / a  b3 / a 

k3 

(C.62)

s 1 0     sI  A  Bk   0 s 1  k1 / a  d / a k 2 / a  c / a s  k3 / a  b / a 

(C.63)

sI  A  Bk  s 3  (k3 / a  b / a)s 2  (k2 / a  c / a)s  k1 / a  d / a

(C.64)

 2  (k3 / a  b / a), 1  (k2 / a  c / a),  0  k1 / a  d / a

(C.65)

k1   2  d / a,  k1   d   a

(C.66)

k2  1  c / a,  k 2   c   a

(C.67)

k3   0  b / a,  k3   b   a

(C.68)

k1 , k 2 , k3 are the crisp value of the roots of the system and  k1 ,  k 2 ,  k 3 are the fuzzy level of the roots of the system.

y 

 k1 k1   k 2 k 2   k 3 k3

(C.69)

k1  k 2  k3

C.2.2 Derivation of the Input for the 3rd Order of State Space Problems  a   bb   c c x  a abc

(C.70)

C.2.3 Consolidity Index for the 3rd Order of State Space Problems C

y

(C.71)

x

156

‫مهنـــــــــدس‪ :‬أحمد ربيع جنيدي جنيدي‬ ‫تاريــخ الميــــالد‪1168 / 1 /1 :‬‬

‫الجنسيـــــــــة‪:‬‬

‫مصري‬

‫تاريخ التســـــجيل‪1111 / 3 /1 :‬‬ ‫تــــاريخ المنـــح‪:‬‬

‫‪/‬‬

‫‪1115/‬‬

‫القســــــــــم‪ :‬هندسة القوي واالالت الكهربية‬ ‫الدرجـــــــــة‪ :‬دكتوراة الفلسفة‬

‫المشـــــــرفون ‪ :‬أ‪.‬د‪ /.‬حسن طاهر دره‬ ‫الممتحنـــــــون ‪:‬‬ ‫أ‪.‬د‪ /.‬حسن طاهر دره‬

‫أ‪.‬د‪ /.‬أسامة يوسف أبو الحجاج‬

‫أ‪.‬د‪ /.‬أحمد محمد الجارحي‬

‫(جامعة حلوان)‬

‫عنـــــوان الرسالـــة ‪ :‬تطوير نمووي لننظمةووعنردتووين و ن"ويفي روعنلنيتو ن–نويفي روعنلندوي ن–ن‬ ‫تغر ل نوداوال نلنمةامن"نوعنلنتطبرقنن‬ ‫الكلمات الدالـة ‪ -:‬مدفوعية الوقت‪ ,‬مدفوعية الحدث‪ ,‬تغيرات معامالت النظام ‪ ,‬التمثيل والنمذجة لتغير معامالت االنظمة‬

‫ملخص البحـث ‪:‬‬ ‫تناولت الرسالة موضوعاً هاماً وهو مسار تغير النظام معتمداً علي نموذج لمدفوعية الوقت ومدفوعية الحدث وتغيرات‬ ‫معامالت النظام‪ .‬فمدفوعية الوقت أو المستوي االساسي يتم التحكم فيه من خالل معادالت النظام والذي تتغير معامالته‬

‫مع الوقت‪ ,‬بينما المستوي ال علوي هو مدفوعية الحدث الذي يؤثر علي النظام وحالته الطبيعية وهذه االحداث تسبب‬

‫تغيرات لمعامالت النظام في كل حدث يؤثر عليه‪ .‬ففي هذه الرسالة تم استنتاج العالقات الرياضية التي تبين التغيرات‬

‫التي تحدث علي معامالت النظام‪ ,‬كما تم عمل نمذجة ومحاكاه الظهار كيفية تغير التماسكية النظام نتيجة تغير بعض‬ ‫معامالت النظام نتيجة التغيرات الخارجية التي تؤثر علي معامالت النظام‪ .‬وقد نشأت فكرة البحث من تعرض كثير من‬

‫النظم العملية إلي مؤثرات خارجية تغير من قيم المتغيرات الخاصة بالمشكلة المراد حلها‪ .‬وقد تم تطبيق النموذج الجديد‬

‫علي ثال ثة تطبيقات مهمة من التطبيقات في الحياة‪ ,‬التطبيق اوأول هو دراسة المعادالت الرياضية لسلو نموذج الفريسة‬ ‫والمفترس تحت المؤثرات البيئية الخارجية مثل االمطار والتي حينما تتزايد يزيد معدل نمو النباتات ومن ثم يزيد عدد‬ ‫الفريسة والمفترس ويكون النظام أكثر تماسكاً و هي مسألة معروفة في التطبيقات الهندسية‪ ,‬والتطبيق الثاني هو تحليل‬ ‫نموذج االيدز عندما يتم التأثير عليه من خالل تزايد الحمالت التوعوية وبرامج التعليم وتحليل نموذج االيدز عندما‬ ‫يتزايد انتشارة مع قلة الحمالت التوعوية‪ ,‬والتطبيق الثالث هو دراسة تأثير نسبة نظافة المنشأة علي نموذج انتشار‬ ‫االمراض المعدية‪ .‬وقد أثبتت الرسالة فاعلية وكفاءة هذا النموذج المقترح لمعرفة التغيرات التي تحدث لمعامالت النظام‬ ‫عند التأثير عليهم بمؤثرات خارجية أو الظروف الطبيعية للنموذج‪.‬‬

‫تطوير نموذج لألنظمة يعتمد علي "مدفوعية الوقت – مدفوعية الحدث – تغيرات‬ ‫معامالت النظام " مع التطبيق‬

‫إعداد‬ ‫المهندس ‪ /‬أحمد ربيع جنيدي جنيدي‬

‫رسالة مقدمة إلي كلية الهندسة‪ ،‬جامعة القاهرة‬ ‫كجزء من متطلبات الحصول علي درجة‬ ‫دكتوراة الفلسفة‬ ‫في‬ ‫هندسة القوي واآلالت الكهربية‬

‫كلية الهندسة‪ ،‬جامعة القاهرة‪،‬‬ ‫الجيزة‪ ،‬جمهورية مصر العربية‬ ‫‪5102‬‬

‫تطوير نموذج لألنظمة يعتمد علي "مدفوعية الوقت – مدفوعية الحدث‪ -‬تغيرات‬ ‫معامالت النظام " مع التطبيق‬ ‫إعداد‬ ‫المهندس ‪ /‬أحمد ربيع جنيدي جنيدي‬

‫رسالة مقدمة إلي كلية الهندسة‪ ،‬جامعة القاهرة كجزء من متطلبات الحصول علي درجة‬ ‫دكتوراة الفلسفة‬ ‫في‬ ‫هندسة القوي واآلالت الكهربية‬ ‫تحت إشراف‬

‫األستاذ الدكتور ‪ /‬حسن طاهر دره‬ ‫هندسة القوي واآلالت الكهربية‬ ‫كلية الهندسة‪ ،‬جامعة القاهرة‬


‫مهنـــــــــدس‪ :‬أحمد ربيع جنيدي جنيدي‬ ‫تاريــخ الميــــالد‪1168 / 1 /1 :‬‬

‫الجنسيـــــــــة‪:‬‬

‫مصري‬

‫تاريخ التســـــجيل‪1111 / 3 /1 :‬‬ ‫تــــاريخ المنـــح‪:‬‬

‫‪/‬‬

‫‪1115/‬‬

‫القســــــــــم‪ :‬هندسة القوي واالالت الكهربية‬ ‫الدرجـــــــــة‪ :‬دكتوراة الفلسفة‬

‫المشـــــــرفون ‪ :‬أ‪.‬د‪ /.‬حسن طاهر دره‬ ‫الممتحنـــــــون ‪:‬‬ ‫أ‪.‬د‪ /.‬حسن طاهر دره‬

‫أ‪.‬د‪ /.‬أسامة يوسف أبو الحجاج‬

‫أ‪.‬د‪ /.‬أحمد محمد الجارحي‬

‫(جامعة حلوان)‬

‫عنـــــوان الرسالـــة ‪ :‬تطوير نمووي لننظمةووعنردتووين و ن"ويفي روعنلنيتو ن–نويفي روعنلندوي ن–ن‬ ‫تغر ل نوداوال نلنمةامن"نوعنلنتطبرقنن‬ ‫الكلمات الدالـة ‪ -:‬مدفوعية الوقت‪ ,‬مدفوعية الحدث‪ ,‬تغيرات معامالت النظام ‪ ,‬التمثيل والنمذجة لتغير معامالت االنظمة‬

‫ملخص البحـث ‪:‬‬ ‫تناولت الرسالة موضوعاً هاماً وهو مسار تغير النظام معتمداً علي نموذج لمدفوعية الوقت ومدفوعية الحدث وتغيرات‬ ‫معامالت النظام‪ .‬فمدفوعية الوقت أو المستوي االساسي يتم التحكم فيه من خالل معادالت النظام والذي تتغير معامالته‬

‫مع الوقت‪ ,‬بينما المستوي ال علوي هو مدفوعية الحدث الذي يؤثر علي النظام وحالته الطبيعية وهذه االحداث تسبب‬

‫تغيرات لمعامالت النظام في كل حدث يؤثر عليه‪ .‬ففي هذه الرسالة تم استنتاج العالقات الرياضية التي تبين التغيرات‬

‫التي تحدث علي معامالت النظام‪ ,‬كما تم عمل نمذجة ومحاكاه الظهار كيفية تغير التماسكية النظام نتيجة تغير بعض‬ ‫معامالت النظام نتيجة التغيرات الخارجية التي تؤثر علي معامالت النظام‪ .‬وقد نشأت فكرة البحث من تعرض كثير من‬

‫النظم العملية إلي مؤثرات خارجية تغير من قيم المتغيرات الخاصة بالمشكلة المراد حلها‪ .‬وقد تم تطبيق النموذج الجديد‬

‫علي ثال ثة تطبيقات مهمة من التطبيقات في الحياة‪ ,‬التطبيق اوأول هو دراسة المعادالت الرياضية لسلو نموذج الفريسة‬ ‫والمفترس تحت المؤثرات البيئية الخارجية مثل االمطار والتي حينما تتزايد يزيد معدل نمو النباتات ومن ثم يزيد عدد‬ ‫الفريسة والمفترس ويكون النظام أكثر تماسكاً و هي مسألة معروفة في التطبيقات الهندسية‪ ,‬والتطبيق الثاني هو تحليل‬ ‫نموذج االيدز عندما يتم التأثير عليه من خالل تزايد الحمالت التوعوية وبرامج التعليم وتحليل نموذج االيدز عندما‬ ‫يتزايد انتشارة مع قلة الحمالت التوعوية‪ ,‬والتطبيق الثالث هو دراسة تأثير نسبة نظافة المنشأة علي نموذج انتشار‬ ‫االمراض المعدية‪ .‬وقد أثبتت الرسالة فاعلية وكفاءة هذا النموذج المقترح لمعرفة التغيرات التي تحدث لمعامالت النظام‬ ‫عند التأثير عليهم بمؤثرات خارجية أو الظروف الطبيعية للنموذج‪.‬‬

‫تطوير نموذج لألنظمة يعتمد علي "مدفوعية الوقت – مدفوعية الحدث‪ -‬تغيرات‬ ‫معامالت النظام " مع التطبيق‬ ‫إعداد‬ ‫المهندس ‪ /‬أحمد ربيع جنيدي جنيدي‬

‫رسالة مقدمة إلي كلية الهندسة‪ ،‬جامعة القاهرة كجزء من متطلبات الحصول علي درجة‬ ‫دكتوراة الفلسفة‬ ‫في‬ ‫هندسة القوي واآلالت الكهربية‬ ‫تحت إشراف‬

‫األستاذ الدكتور ‪ /‬حسن طاهر دره‬ ‫هندسة القوي واآلالت الكهربية‬ ‫كلية الهندسة‪ ،‬جامعة القاهرة‬


‫تطوير نموذج لألنظمة يعتمد علي "مدفوعية الوقت – مدفوعية الحدث – تغيرات‬ ‫معامالت النظام " مع التطبيق‬

‫إعداد‬ ‫المهندس ‪ /‬أحمد ربيع جنيدي جنيدي‬

‫رسالة مقدمة إلي كلية الهندسة‪ ،‬جامعة القاهرة‬ ‫كجزء من متطلبات الحصول علي درجة‬ ‫دكتوراة الفلسفة‬ ‫في‬ ‫هندسة القوي واآلالت الكهربية‬


âTime Driven-Event Driven-Parameters Changes

âTime Driven-Event Driven-Parameters Changes

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âTime Driven-Event Driven-Parameters Changes