Synchronization of Nonlinear Drive and Response Systems Through Robust-Adaptive Feedback Control Techniques

Synchronization of Nonlinear Drive and Response Systems through Robust-adaptive Feedback Control Techniques

Muhammad Siddique

2018

Department of Electrical Engineering Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan i Thesis Examinars

Name: Muhammad Siddique Department: Electrical Engineering

Registration:02-7-1-030-2013 Date of Registration: 9 April 2013

Foreign Reviewers (Names and Aﬃliations)

1. Prof. Shen Yin, Harbin Institute of Technology, China.

2. Prof. Tariq Samad, Technological Leadership Institute USA.

3. Dr. Liangrui Peng, Tsinghua University, China.

Thesis Defense Examiners (Names and Aﬃliations)

1. Prof. Dr. Nisar Ahmad, GIK Institute of Engineering Science and Technology.

2. Prof. Dr. Shafqat Karim, Physics Division, PINSTECH, Islamabad.

3. Prof. Dr. Imtiaz Ahmad Taj, Capital University Islamabad.

Head of Department(Name): Dr. Muhammad Arif

Signatures/Date:

ii Thesis Submission Approval

This is to certify that the work contained in this thesis entitled Synchronization of Nonlinear Drive and Response Systems through Robust-adaptive

Feedback Control Techniques, was carried out by Muhammad Siddique, and in my opinion, it is fully adequate, in scope and quality, for the degree of

Ph.D.

Supervisor:

Name: Dr. Muhammad Rehan

Date: July 19, 2018

Place: PIEAS, Islamabad.

Head, Department of Electrical Engineering:

Name: Dr. Muhammad Arif

Date: July 19, 2018

Place: PIEAS, Islamabad.

iii Synchronization of Nonlinear Drive and Response Systems through Robust-adaptive Feedback Control Techniques

Muhammad Siddique

Submitted in partial fulﬁllment of the requirements for the degree of Ph.D. July, 2018

Department of Electrical Engineering Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan Dedications

To my parents

ii Acknowledgements

All glory to Almighty Allah, the creator of this universe, The Gracious and the compassionate whose bounteous blessings gave me the potential thoughts, talented teachers, helping friends, loving parents, sisters and brothers and an opportunity to make this humble contribution and all the respect and Darood-O-Salam are due to His Holy Prophet Muhammad (P.B.U.H.) whose immaculate teachings

ﬂourished my thoughts and thrived my ambition all along the way to have the cherished fruit of modest eﬀort in form of this write-up.

I express my most sincere gratitude, hearty sentiments and thanks to my project supervisor Dr. Muhammad Rehan for his exlcellent supervision, encour- agement, knowledge delivery and guidance. His sweet behavior, keen interest, personal involvement and criticism for the betterment were all the real sources of courage, inspiration and strength during the completion of this thesis.

I would express appreciation to my class fellows at the university especially

Muhammad Awais, Najam Saqib, Muntazir Hussain and Sohaira for their continuous support in learning the tough conceptions. I also express my appreciation to the senior colleagues including the faculty of Electrical Engineering Department and management at NFC, IET Multan for their continuous support.

Finally, I would like to express my deepest gratitude to my mother, father, brother, sisters and all other friends and relatives, for their emotional and moral support throughout my academic career and also for their love, patience, encour- agement and prayers.

(Muhammad Siddique)

PIEAS, Islamabad

iii Declaration of Originality

I hereby declare that the work contained in this thesis and the intellectual content of this thesis are the product of my own work. This thesis has not been previously published in any form nor does it contain any verbatim of the published resources which could be treated as infringement of the international copyright law. I also declare that I do understand the terms ‘copyright’ and ‘plagiarism’, and that in case of any copyright violation or plagiarism found in this work, I will be held fully responsible of the consequences of any such violation.

(Muhammad Siddique)

Date: July 19, 2018

PIEAS, Islamabad

iv Copyrights Statement

The entire contents of this thesis titled Synchronization of Nonlinear Drive and Response Systems through Robust-adaptive Feedback Control

Techniques by Mr. Muhammad Siddique, are an intellectual property of

Pakistan Institute of Engineering & Applied Sciences (PIEAS). No portion of the thesis should be reproduced without obtaining explicit permission from PIEAS.

v Contents

Dedications ii

Acknowledgements iii

Declaration of Originality iv

Copyrights Statement v

Contents vi

Abstract xvii

List of Publications and Patents xviii

List of Abbreviations and Symbols xx

1 Introduction 1

1.1 Introduction ...... 1

1.2 Motivations ...... 3

1.2.1 High Power LASERS Applications ...... 3

1.2.2 Secure Information Transmission ...... 4

1.2.3 Multiple Power Generators Synchronization ...... 4

1.2.4 Synchronization of Two Gyros ...... 5

1.2.5 Synchronization of Electro-Mechanical Systems ...... 6

1.2.6 Application in Master and Slave clocks ...... 6

vi 1.2.7 Synchronization of Robots, Ships, Submarines and Aircrafts7

1.2.8 Synchronization of Biological Oscillations ...... 7

1.3 Control Techniques ...... 8

1.3.1 Observer Based Control ...... 10

1.3.2 Sliding Mode Control ...... 10

1.3.3 Synchronization by Huygen’s Coupling ...... 11

1.3.4 Synchronizing Control by Back Stepping Techniques . . . . . 11

1.3.5 Synchronization Using Adaptive Control ...... 12

1.3.6 Robust Synchronizing Control ...... 13

1.3.7 Robust-adaptive Control for Synchronization ...... 14

1.4 Literature Survey ...... 14

1.5 Contribution of Dissertation ...... 18

1.6 Structure of the Dissertation ...... 21

2 Preludes of the Available Data 23

2.1 Nonlinear Dynamical Systems ...... 23

2.2 Chaos and Chaotic Systems ...... 25

2.3 Time-delay Systems ...... 28

2.4 Nonlinear Drive and Response systems ...... 29

2.4.1 Application of Drive and Response Systems in Real World . 30

2.4.2 Nonlinear Drive and Response Systems with Bounded Time

Delay ...... 31

2.4.3 Nonlinear Drive and Response Systems with Uncertain De-

lay Rate ...... 32

2.5 Error Dynamics ...... 33

2.6 Synchronization of Drive and Response Systems ...... 34

2.6.1 Types of synchronization ...... 35 vii 2.7 Lipschitz Nonlinearities ...... 36

2.8 Lyapunov Theory for Stability and Control ...... 37

2.8.1 Types of stability ...... 39

2.8.2 Lyapunov function and adaptation laws ...... 40

2.9 Robust Control ...... 42

2.10 Robust-adaptive Control ...... 43

2.11 Cone Complementary Linearization Algorithm ...... 44

2.12 Simulation Software ...... 45

2.12.1 MATLAB as a Simulation Software ...... 47

3 Adaptive Feedback Synchronizing Control for Nonlinear Drive

and Response Systems 48

3.1 Introduction ...... 48

3.2 Statement of Systems and The Problem Formulation ...... 50

3.3 Synchronization Control ...... 53

3.4 Case1: Synchronization with Known Parameters ...... 54

3.4.1 Observer Based Feedback Control ...... 55

3.4.2 Coupled Chaotic Synchronous Observers ...... 56

3.4.3 Synchronization Error Dynamics for Known Parameters . . . 58

3.4.4 Feedback Control for Synchronization ...... 63

3.4.5 Simulations and Results ...... 72

3.5 Case2: Synchronization of Nonlinear Systems with Unknown Pa-

rameters ...... 81

3.5.1 Coupled Chaotic Adaptive Synchronous Observers ...... 84

3.5.2 Synchronization Error Dynamics for Unknown Parameters . 86

3.5.3 Feedback Control for Adaptive Synchronization ...... 88

3.5.4 Amended Lyapunov Function ...... 89 viii 3.5.5 Simulation and Results ...... 91

3.6 Conclusion ...... 99

4 Robust-adaptive feedback synchronizing control for nonlinear

drive and response systems 100

4.1 Introduction ...... 100

4.2 Problem Formulation for the Robust-Adaptive Feedback Synchro-

nizing Control ...... 102

4.3 Coupled Chaotic Adaptive Synchronous Observers ...... 104

4.4 CCS Observer Based Control Methodology for Synchronization . . . 105

4.5 Sigma Modiﬁcation for Robustness ...... 107

4.6 Filtering Techniques for Noise Rejection ...... 108

4.7 Robust-Adaptive Feedback Control ...... 108

4.8 Simulation and Results ...... 112

4.9 Conclusion ...... 120

5 Convex Routines for Solution of Matrix Inequalities 121

5.1 Introduction ...... 121

5.2 Problem Statement for The Solution of NMIs using Convex Routines123

5.3 Problem Statement for Alternate Method for The Solution of NMIs

Using Convex Routines ...... 126

5.4 Simulation and Results ...... 131

5.5 Conclusion ...... 137

6 Delay-range-dependent adaptive control for time-delay chaotic

systems 139

6.1 Introduction ...... 139

6.2 Generalized Model of The Delay Incorporated Systems ...... 142 ix 6.3 Synchronizing Error Dynamics for The Delayed Drive and The Re-

sponse Systems ...... 145

6.4 Adaptive Synchronizing Controller Design for Delayed Containing

Drive and Response Systems ...... 146

6.5 A Controller Design Condition for Finding the Controller Gain Ma-

trix ...... 152

6.6 Simulation and Results ...... 158

6.7 Conclusion ...... 163

7 Robust-Adaptive Synchronization of Drive and Response Sys-

tems with Varying Time Delays 165

7.1 Introduction ...... 165

7.2 Statement of the Generalized Model of the Nonlinear Drive and

Response Systems ...... 167

7.3 Synchronizing Error Dynamics for the Delay Comprising Drive and

The Response Systems ...... 169

7.3.1 Problem Formulation ...... 170

7.4 Synchronizing Control for Drive-Response Architecture with Un-

known Parameters and Varying State Delays ...... 170

7.4.1 Robust-adaptive feedback control for synchronization . . . . 171

7.4.2 Adaptation Law ...... 171

7.4.3 Performance index for robust control ...... 171

7.5 Simulation and Results ...... 178

7.6 Conclusion ...... 182

8 Summary and Future Work Directions 183

8.1 Summary ...... 183

x 8.2 Future Work Directions ...... 186

Appendix A 189

Appendix B 194

References 204

xi List of Figures

1.1 Secure information transmission by synchronization of chaotic systems4

1.2 Generic architecture of feedback control scheme ...... 9

2.1 The pendulum system ...... 24

2.2 Synchronization scheme of drive and response systems with time

delay ...... 32

2.3 Robust-adaptive control scheme general architecture ...... 44

3.1 Observer based control scheme block diagram ...... 56

3.2 Coupling architecture for the observer based non-adaptive control

scheme ...... 57

3.3 Phase portraits of the master and the slave FHN systems : (a)

phase portrait of the master system, (b) phase portrait of the slave

system...... 74

3.4 Time evolutions of the normalized membrane potentials ...... 74

3.5 Time evolutions of the recovery variables ...... 75

3.6 Time evolution of errors between corresponding states of the master

and the slave systems ...... 76

xii 3.7 Phase portraits of the master and the slave FHN systems, and phase

portraits of their corresponding observers using the approach pro-

vided in Theorem 3.1 : (a) phase portrait of the master system, (b)

phase portrait of the slave system, (c) phase portrait of the master

observer, (d) phase portrait of the slave observer...... 78

3.8 The time evolution of membrane potentials of the master and slave

systems and their corresponding observers ...... 79

3.9 The time evolution of recovery variables of the master and slave

systems and their corresponding observer ...... 80

3.10 Convergence of the synchronization errors ...... 80

3.11 Error between normalized potentials of master system and its ob-

server for diﬀerent values of observer gains and ﬁxed value of con-

troller gain ...... 82

3.12 Synchronization error eo1 for diﬀerent values of controller gains and

ﬁxed values of observer gains ...... 82

3.13 Architecture for CCAS observers based synchronization scheme . . 83

3.14 Phase portraits of the master and the slave FHN systems, phase

portraits of their corresponding observers : (a) Phase portrait of

the master system, (b) phase portrait of the slave system, (c) phase

portrait of the master observer, (d) phase portrait of the slave ob-

server...... 93

3.15 Temporal evolution of membrane potentials of the master and slave

systems and their corresponding observers ...... 94

3.16 Temporal evolution of recovery variables of the master and slave

systems and their corresponding observers ...... 94

3.17 Error dynamics of the master and the slave systems ...... 95

xiii 3.18 Adaptation of unknown parameters θm and θs ...... 96

3.19 Eﬀect of observer gain matrices Lm and Ls on the error dynamics

em1 ...... 97

3.20 Eﬀect of controller gain matrix F on the error dynamics eo1 .... 98

4.1 Architecture of the robust-adaptive control scheme for the synchro-

nization of the master and the slave systems ...... 106

4.2 Chaotic behavior of the drive and the response FitzHugh-Nagumo

networks: (a) state diagram of the leading system, (b) state diagram

of the following system...... 114

4.3 Time progressions of the membrane potential states of the drive

and the response systems ...... 114

4.4 Time progressions of the membrane potential states of the drive

and the response systems ...... 115

4.5 Errors e1 and e2 between corresponding states of the drive and the

response systems ...... 115

4.6 Chaotic behavior of the drive and the response FitzHugh-Nagumo

systems and their corresponding CCAS observers : (a) state di-

agram of the leading system, (b) state diagram of the following

system, (c) state diagram of the master observer, (d) state diagram

of the slave observer...... 116

4.7 Time progressions of the states xm1 and xs1 of the drive and re-

sponse system ...... 117

4.8 Time progressions of the states xm2 and xs2 of the drive and the

response systems ...... 118

4.9 Errors between corresponding states of leading and following systems119

4.10 Adaptation of unknown parameters after sigma modiﬁcation. . . . . 119 xiv 5.1 Phase portraits of the master and the slave FHN systems, phase

portraits of their corresponding observers using the approach pro-

vided in Theorem 3.2 : (a) Phase portrait of the master system, (b)

phase portrait of the slave system, (c) phase portrait of the master

observer, (d) phase portrait of the slave observer...... 133

5.2 Temporal evolution of membrane potentials of the master and slave

systems and their corresponding observers ...... 134

5.3 Temporal evolution of recovery variables of the master and slave

systems and their corresponding observers ...... 134

5.4 Error dynamics of the master and the slave systems ...... 135

5.5 Adaptation of unknown parameters i.e. θm and θs ...... 136

5.6 Eﬀect of observer gain matrices Lm,Ls on the error dynamics em1 . 137

5.7 Eﬀect of controller gain matrix F on the error dynamics eo1 .... 138

6.1 Phase portraits of the drive and the response Hopﬁeld neural net-

works: (a) state diagram of the leading system, (b) state diagram

of the following system ...... 159

6.2 Time evolution of the states xm1, xm2, xs1, and xs2 of the master

and the slave systems ...... 159

6.3 Time evolution of the errors e1, and e2 between the corresponding

states of the master and the slave Hopﬁeld neural networks . . . . 160

6.4 Synchronized phase portraits of the drive and the response Hopﬁeld

neural networks: (a) state diagram of the leading system, (b) state

diagram of the following system ...... 161

6.5 Temporal evolution of the synchronized states xm1, xm2, xs1, and xs2

of the master and the slave systems ...... 162

xv 6.6 Temporal evolution of the synchronization errors e1, and e2 between

the corresponding states of the master and the slave systems . . . . 162

6.7 Plot for the adaptation of unknown parameters θ1 and θ2 associated

with the nonlinearities present in the dynamics of master and the

slave systems ...... 163

7.1 Behavior of the master-slave Hopﬁeld neural networks with the pro-

posed robust adaptive controller: (a) phase portrait of the master

system, (b) phase portrait of the slave system ...... 179

7.2 Time evolution of the states xm1, xm2, xs1, and xs2 of the drive and

the response systems ...... 180

7.3 Time evolution of the synchronization error e1, and e2 between the

corresponding states of the master and the slave systems ...... 181

7.4 Adaptation of unknown parameters...... 181

xvi Abstract

Synchronization of the nonlinear drive (master) and the response (slave) systems is

explored by utilization of adaptive and robust-adaptive feedback control method-

ologies. The nonlinear drive and response systems with time delays, in their

dynamics, under the inﬂuence of disturbances are synchronized. A step by step

approach is carried out to achieve the solution of the nonlinear synchronization

problem. Initially, coupled chaotic synchronous observer based synchronization

scheme is developed. Afterwards, this scheme is improved and a coupled chaotic

adaptive synchronous observer based synchronization is formulated. Subsequently,

a delay-range-dependent robust-adaptive controller along with adaptation laws for

unknown parameters for attaining the synchronization of nonlinear time delay sys-

tems is proposed. The main contribution of this research work is three fold: First,

a novel approach for the design of coupled chaotic adaptive synchronous observers

for the synchronization of nonlinear systems is presented to address the output

feedback control. Second, a robust-adaptive control, catering the eﬀect of distur-

bances and parametric variations, is explored. Third, a delay-range-dependent

robust-adaptive synchronization control for delayed nonlinear drive and response

systems is studied. It is worth mentioning that the output feedback based and

delay-range-dependent robust-adaptive synchronization methods are scarcely ad-

dressed in the literature, to the best of our knowledge. Further, the use of low pass

filters along with σ-modification for attaining robustness against disturbances is investigated. The effect of external disturbance is also catered by applying specially designed performance index, based on L2 gain reduction. Computer simulations to verify the effectiveness of the proposed methodologies are also provided.

xvii List of Publications

Journal Publications

• Muhammad Siddique and Muhammad Rehan “A concept of coupled chaotic

synchronous observers for nonlinear and adaptive observers-based chaos syn-

chronization”, Nonlinear Dynamics, (2016) volume 84(4) pp 2251-2272 doi:

10.1007/s11071-016-2643-2, (impact factor: 4.339)

• Muhammad Siddique, Muhammad Rehan, M. K. L. Bhatti and Shakeel

Ahmed “Delay-range-dependent local adaptive and robust adaptive synchro-

nization approaches for time-delay chaotic systems.”, Nonlinear Dynamics,

(2017) volume 88(4) pp 2671-2691 doi: 10.1007/s11071-017-3402-8, (impact

factor: 4.339)

• Muhammad Awais Raﬁque, Muhammad Rehan and Muhammad Siddique

“Adaptive mechanism for synchronization of chaotic oscillators with interval

time-delays.”, Nonlinear Dynamics, (2015) volume 81(1) pp 495-509 doi:

10.1007/s11071-015-2007-3, (impact factor: 4.339)

Conference Publications

• Muhammad Siddique, Muntazir Hussain, Muhammad Rehan, and Muham-

mad Majid Hussain “Robust-Adaptive synchronization of drive and response

systems using coupled chaotic adaptive synchronous observers. ”, In Sys-

tems, Process and Control (ICSPC), 2017 IEEE Conference on, pp. 24-29.

IEEE, 2017, Melaca, Malaysia, December 15 - 17, 2017.

• Muntazir Hussain, Najam us Saqib, Muhammad Rehan, and Muhammad

Siddique “Anti-windup Compensator Synthesis for Cascaded Linear Control xviii System. ”, In International Conference on Control Robotics Society (2017):

1634-1639., October, 2017.

• Muhammad Awais Raﬁque, Muhammad Siddique and Muhammad Rehan

“Adaptive delay-dependent synchronization of nonlinear time delay systems

”, In 12th International Conference on Applied Sciences and Technology

(IBCAST), Bhurban, Pakistan, January 13 - 17, 2015.

xix List of Abbreviations and Symbols

Symbol Nomenclature

xm Master system state vector

xs Slave system state vector

A Non-delayed system state coeﬃcient matrix

Ad System Matrix with delayed states

B Coeﬃcient Matrix with non-delayed nonlinearity

Bd Coeﬃcient Matrix with delayed nonlinearity

θ, φ Unknown parameters

τ Time-delay

κ Upper limit on time-delay derivativeτ ˙

AT Transpose of the matrix A

A > 0 Positive deﬁnite matrix A

Lm Observer gain matrix for master system

Ls Observer gain matrix for slave system

F Controller gain matrix

xx Chapter 1 Introduction

This chapter describes the Introduction, motivation, synchronizing control techniques, literature survey, contribution and structure of this dissertation.

1.1 Introduction

The word synchronization is extracted form the ”syn” means the same and

”chronos” means the time [1]. Synchronization means regulation of dynamics of periodic oscillators due to their coupling by some means to manifest the same behavior and the error between the corresponding states converges to zero [2].

This concept of synchronization is often applied to chaotic systems and rotators.

In chaos synchronization, the phenomena of synchronization can be of following types: complete synchronization, phase synchronization, generalized synchronization etc. Synchronization phenomena primarily groups of coupled systems which frequently present themselves as cooperative coherent systems because of synchronization control scheme.

The synchronization was ﬁrst mentioned in 17th century when the famous

Dutch scientist Christian Huygens testiﬁed his observation about synchronization of double pendulum clocks which was invented by him sometime earlier [3]. The motion of double pendulum so assembled that the bobs are attached with the beam made of wood and the two suspended pendulums even with opposite swings

1 Chapter 1: Introduction dynamics, synchronize so much that the motions of both pendulums become same.

Moreover, if this synchronous behavior was perturbed by some intervention, it comes back into its original dynamics by itself in a small interval of time. By having the analysis of dynamics for a long interval, he become astonished at that surprising result, that synchronization was because of the motion of the beam, although that motion was hardly perceptible.

A science perspective about the theory of chaos is somewhat different than the routine meaning as a state of confusion and disorder. Chaos theory is concerned with the complexity of phenomena and unveils purposeful patterns in these phenomena. Significance of chaos can be elaborated by the fact that many natural and synthetic systems, affecting the human beings, possess chaos. The global weather changes, stock market, population of various species, brain states, heart beat, double pendulum dynamics and the butterfly effect are some well-known examples of the chaotic systems. One of the earliest physicist known in the field of chaos, Edward Ott. [4], who made marvelous contribution in developing chaos theory which can be applied to many natural phenomenon and synthetic systems.

Chaos is an interesting aperiodic long-time oscillatory behavior of dynamical nonlinear systems that can demonstrate a sensitive dependence on the initial conditions [5]. Since the development of chaos theory, control and synchronization of chaotic systems flourished as an emerging topic of research. Synchronization is a dynamic progression during which the driven system becomes in line with the driving system so that the synchronized or slave system, in a certain manner, tracks the trajectory of the synchronizing or master system [6, 7]. Carroll and Pecora made first successful attempt to present an experimental set-up for synchronization of chaotic systems with different initial conditions and published a seminal paper [8] that acted as a catalyst in the field of chaos synchronization.

2 Chapter 1: Introduction

Since then, many chaos synchronization techniques have been witnessed by the researchers.

1.2 Motivations

Synchronization of two or more systems in the form of drive and response architecture is an important subject in science and engineering. This synchronization phenomenon plays a vital role in variety of real world situations and have wide applications associated with these situations. Some of these applications are discussed as follows.

1.2.1 High Power LASERS Applications

The extremely synchronized beam of a LASER is used for welding and cutting.

This synchronized and collimated enormously powerful LASER beam is focused to a microscopic dot, and it act as a scalpel for welding and cutting metals. In automobile industry, the use of carbon dioxide LASER is very extensive [9], it is because of the power of carbon dioxide LASER which is up to several kilowatts for automatic and controlled welding on auto-mobiles assembly.

The LASER cutters are also used in garments industry in the advanced countries. The garment industry of those countries become inexpensive and competitive among the market share holders. The LASER garment cutters, which are computer controlled, can be programmed to cut out hundreds of thicknesses of cloth and they can also cut out every single piece of the garment in one go [10].

Hardening and annealing heat treatments have been experienced in metallurgi- cal engineering for a long time. However, LASERS oﬀer several new options for selective heat treatments of metal parts, improving the precision of manufacturing.

3 Chapter 1: Introduction

1.2.2 Secure Information Transmission

Chaos and its applications in the ﬁeld of secure communication have attracted

a lot of attention in various domains of science and engineering during the last

two decades [11]. Drive-response synchronization techniques for synchronizing

Figure 1.1: Secure information transmission by synchronization of chaotic systems

the chaotic networks are found to have capability to be used in designing secure

communication systems as shown in Figure 1.1. The state of the master chaotic

system, i.e., xm(t) is added with the message signal S(t). The resulted signal, i.e.,

xm(t)+S(t) is sent by the transmitter module on the communication channel. On

the receiver side, a slave chaotic system similar to master system is synchronized

with the master chaotic system by utilization of synchronization control signal

u(t). After synchronization, as all the states of master and slave system become

same i.e. xm(t) = xs(t). The message signal S(t) is retrieved by subtracting the synchronized state xm(t) = xs(t).

1.2.3 Multiple Power Generators Synchronization

For multiple power generators to operate on a common grid, the synchronization of these generators is mandatory [12]. Micro-grid is a new and attractive concept of a power supply system, which consists of various types of distributed gener-

4 Chapter 1: Introduction

ators, energy storage devices and customer loads with advanced control system

responsible for stable operations both in the normal utility-connected mode and

incidental stand-alone mode. The controllable distributed generators operated

in the microgrid are required to have proper capability to respond to frequency

changes and voltage of the micro-grid, to contribute a stable operation with their

local controllers under the comprehensive control by the micro-grid energy man-

agement system (EMS). In the stand-alone operation caused by some faults in the

utility grid, the controllable distributed generators must absorb the initial gap in

demand and supply as much as possible depending on the generator capability.

Therefore, study of load-following performance of each type of distributed genera-

tor is needed to develop a cooperative control scheme for the distributed generators

and power storage devices in the micro-grid, and to plan strategic operations of

the micro-grid especially in autonomous stand-alone condition.

1.2.4 Synchronization of Two Gyros

The process of synchronization of two gyros [13] is helpful in determining the gy-

ratory motion in aircrafts, ships, automobiles, and various electronic system such

as cell phone, ﬂying robots etc. Synchronization of gyros, works in similar fashion

to the two accelerometers in cell phone. The application of gyro is categorized

into three diﬀerent classes.

Angular velocity sensing: Measure the value of angular speed produced. Gy- ros can work as sensor for measuring the amount of motion itself.

Angle sensing: Measured angular velocity produced by the sensor’s own move- ment is further utilized to calculate angles. Angles are detected via integration operations by a computing device. These angle measurements are helpful in car navigation systems, game controllers, cellular system, control mechanisms etc.

5 Chapter 1: Introduction

Sensing vibration: Synchronization of gyros is helpful in sensing vibration produced by external factors, and transmits vibration data as electrical signals to a computing device. It is used in correcting the orientation or balance of an object, camera-shake correction, vehicle control, ship balancing etc.

1.2.5 Synchronization of Electro-Mechanical Systems

The process of synchronization is very important in industrial applications [14].

In fact, the manufacturing processes should be synchronized with each other to increase the eﬃciency of industrial plants, to enhance the quality of product and to optimize the cost economics. Beside this, multi-motor applications have become very attractive ﬁeld in industrial units replacing the traditional mechanical coupling. In paper industry, wrapping motors should be synchronized. Similarly, many textile applications are utilizing synchronized speed motors e.g. motors for wrapping of clothes should be synchronized with the speed of weaving spindle to avoid damage.

1.2.6 Application in Master and Slave clocks

Synchronization science has been very effective in the field of clock synchronization [15]. Real mechanical clocks show different time after passing a plenty of time due to clock divergence because of many reasons. The important of those is that the clocks count the time interval at slightly different rates. Many problems occur in the process of synchronization of two or more clocks because of clock rate differences. This problem also complicates the synchronizing control for the last many decades to synchronize independent clocks [16]. For instance, television and radio broadcasting problems can occur because of different timing of clocks at different broadcasting stations. Airport or railway clocks installed at different places

6 Chapter 1: Introduction should be synchronized. If these clocks are not synchronized then it will create trouble for passengers. In communication industry, the clock synchronization play a vital role. Also in large industries and entrepreneurs, clocks should show same time as a necessary requirement to perform their jobs.

1.2.7 Synchronization of Robots, Ships, Submarines and Aircrafts

A physical system that is composed of sensors, actuators and controller equipped with artificial intelligence is called a robot [17]. Scientist in the field of robotics have been working for several years to synchronize the activities of robots working as group. For example, synchronization of dancing robots. This effort is not just for the sake of developing social machines that can amuse us. Notwithstanding, it is an attempt to execute synchronized activities at a small scale for finding the methodologies to perform synchronization activities at large scale, so that the future group of robots can perform coordinated activities in synchronized manner to independently perform desired tasks defined by their masters. Similarly, synchronization of aircrafts [18], submarines [19] and ships [20] is the topic of interest for many researchers.

1.2.8 Synchronization of Biological Oscillations

Biological oscillators are useful for qualitative investigation though already they are elaborated exhaustively in quantitative terms. With the help of qualitative analysis, we can recognize the constituents essential for generating the oscillations and can enhance our understanding of fundamental mechanisms of the oscillator.

Biological oscillations present in cells have generally variable periods, varying in the range of milliseconds to months, and expanse from sub-cellular components to big organisms. The examination of the patterns of collaborations and time

7 Chapter 1: Introduction lags witnessed in biological oscillators is made understandable by the conversion of interactions, variables and time lags into schematic representations.

The biological cells of an organ usually show synchronization, it is because of the reason that the biological cells are coupled through positive feedback. If the coupling strength is more the time of synchronization of two independent coupled oscillators in cell decreases. This positive feedback loop between the oscillators can consist of one of the two types of interactions, one the double positive (PP) interaction and second is the double negative (NN) interaction.

The synchronization of biological models is a hot topic of research in the present era [21].

1.3 Control Techniques

The control technique to control SISO (single input, single output) control systems and MIMO (multiple input, multiple output) control systems in state space is treated in the literature. The diﬀerential equation solution is studied in time domain as well as in the frequency domain and new concepts such as controllability and observability are introduced. With these general insights in mind, the design of

MIMO control systems with state feedback is an important topic for contemporary researchers. The major types of control techniques can be divided into two classes,

Classical control and Modern control. The classical control theory is applicable to only single-input and single-output (SISO) system [22] but in some cases it is also used for analyzing disturbance rejection where disturbance is treated as second input. The analysis and design of control scheme in classical theory is carried using diﬀerential equations. Many higher order systems are over simpliﬁed to a second order systems in time domain. A controller designed using classical theory often requires on-site tuning due to incorrect design approximations. In

8 Chapter 1: Introduction

modern control theory, control schemes can be developed for multiple-input and

multiple-output (MIMO) systems and are designed in state space. The modern

control theory is capable of more sophisticated design problems in comparison to

the limitations of classical control theory.

Feedback control is a technique to control the systems whose output is modiﬁed

using a feedback signal and this signal can be a measurement of the output using

sensor or it can be a state feedback signal from which the output can be calculated.

This diﬀerence of the feedback signal with a reference signal, known as an error

Reference Control Measured Value input Output + Controller System _

Feedback

Figure 1.2: Generic architecture of feedback control scheme signal is fed to the controller for generating the control input to the system. The following block diagram as shown in Figure 1.2 [23] depicts a general feedback system. The desired output value is speciﬁed with a reference input. The controller should adjust the setting of the control input to get the measured output to equal the reference input.The advantages include more accuracy of control than open loop control and can take care of non-linearities and also stabilizes the response of unstable system. But the feedback control ensures stability if the feedback control is designed properly. Without proper stability analysis, the feedback mechanism can induce oscillatory or unstable response of system and reduction of the overall system gain. The feedback control scheme implementation is more costly as well.

9 Chapter 1: Introduction

Under the class of modern control theory, following are some of the major types of control techniques that exist in literature.

1.3.1 Observer Based Control

Observer based control scheme are one of the major types of modern control methodologies that can be effectively implemented for many physical systems. An observer in this control scheme is actually a software program that can provide an estimate of the state of system by using the measurements of the control input and output of that specific system. This observer work as an alternative to the sensors which reduces the cost of control system and increases the accuracy and efficiency of control. It is because the sensors can induce unwanted noise in their measurements. For the estimation of state from observer it is necessary that the system is observable.

1.3.2 Sliding Mode Control

A very well-known nonlinear control technique that is sliding mode control (SMC) possess outstanding features for controlling nonlinear systems. These features include simple tuning rules, accuracy, implementation and robustness. The name sliding mode control is because of the sliding surface, where sliding surface is the upper and lower bounds between which the states of system are maneuvered. This means that the sliding mode control retains the states on the near neighborhood of the sliding surface. The design of sliding mode controller consist of two parts.

One part involves the planning for the sliding surface and second part contains the design of control law for establishing the sliding surface as the final destination to the system dynamics [24]. The main benefits of SMC are that the performance of the system maneuvered by the specific selection of the sliding function and also

10 Chapter 1: Introduction the feedback response get insensitive uncertainties. The sliding mode control can be extended for the systems with disturbance and nonlinearities.

1.3.3 Synchronization by Huygen’s Coupling

The vertically suspended two pendulums, such that the two hooks of these pendulums are embedded in the same wooden beam, if these pendulums are set into motion with diﬀerent initial condition then they always show complete synchronization in their dynamics after some time. This was discovered by Christian

Huygens in 1665 [25]. Moreover, if this synchronization is perturbed by some external force, even then the two pendulums develop the synchronization in their motions. It is because of the reason that, the beam to which the pendulums are suspended is also set into oscillatory motion, which in turn becomes the source of synchronization. The motion of beam necessarily eﬀects on the motion of two pendulums which acts as coupling strength for the phenomenon of synchronization.

The process of synchronization by huygen’s coupling is the process in which two non-identical coupled oscillating structures start to vibrate with equal frequencies because of that coupling. It is important to say that the phase and frequency locking because of synchronization does not means that the amplitudes of oscillating structures are equal, in fact these oscillators can possess dissimilar amplitudes and also waveforms.

1.3.4 Synchronizing Control by Back Stepping Techniques

Back stepping technique in control theory, developed by Circa et. al. [26, 27], for developing stabilizing controls methodologies for speciﬁc nonlinear systems. The systems that are obtained from complex subsystem can be stabilized using this recursive scheme. The control scheme developer initialize this control methodology

11 Chapter 1: Introduction with a stable state of the concerned system and recursively back out to other controllers that one by one stabilize outer sub-system. The control scheme ends or exit when the ﬁnal external control methodology is reached.

1.3.5 Synchronization Using Adaptive Control

Adaptive control theory deals with the development of controllers which have to adapt itself to a system being controlled with varying parameters or the unknown parameter [28]. A good example of an adaptive control is aircraft ﬂying control system, as the adaptive law take care of the decrease of mass because of the fuel combustion and adapts according to the changing conditions. Adaptive control scheme are somewhat dissimilar from robust methodologies for control, as robust control does not need initial knowledge of these parameters which are changing with time. While robust control is capable of controlling the system whose parameter variations are within known limits, and the applied robust control scheme is need not to be altered.

In literature, many synchronizing control schemes to synchronize the two systems that may be similar or different systems are proposed. Most of these schemes are designed and developed under the condition that all the parameters of the drive-response architecture are known. It means that the parameters are either known constants or changing according to known criteria. On the other hand, in real systems, the uncertainties like parameter differences and external disturbances may sabotage the synchronization of nonlinear systems, e.g., chaotic systems. So, for such system with unknown parameters, it is indispensable to propose an adaptive control scheme for parameter adaptation according to the situation for synchronization of chaotic systems to nullify the effect of internal and external noises. Some example of the adaptive control schemes to synchronize a group of

12 Chapter 1: Introduction chaotic systems is as follows. The case of two diﬀerent complex chaotic systems with varying unknown parameters for adaptive synchronization is discussed in [29].

Similarly, for two dissimilar chaotic systems, an adaptive law for synchronization was established by Wu et. al. [30]. In an adaptive scheme for synchronization of chaotic systems with parametric uncertainties was presented by Wang et. al. [31].

1.3.6 Robust Synchronizing Control

Many times, conventional feedback control scheme does not work eﬃciently because of model uncertainties or external disturbances also noise in the sensor measurement. To deal with these type of problems, modern control theories are used. For example, robust control, adaptive control and robust-adaptive control.

Horowitz in 1963 developed what is seemingly the ﬁrst detailed mathematical representation of a robust-feedback control methodology [32]. But he did not used the word robust in his paper. Unluckily, the Horowitzs contribution did not get the recognition among the researcher of control at that time. In 1978,Harvey and

Stein made fruitful eﬀort to make modern-control theoretical work [33]. Scientist has been started the reevaluation of classical frequency response schemes and got better understanding at Horowitzs literature about robust feedback control on uncertainty tolerant single loop feedback methodology. With the passage of time many researcher took part in developing the robust control theory.

H∞ control scheme is also a part of robust control theory. Duncan McFar- lane and Keith Glover of Cambridge University added a due share in H∞ control literature [34] . H∞ robust control scheme minimizes the sensitivity of a system, over a range of frequency domain, and make it sure that no signiﬁcant deviation will occur from the desired dynamics of the system under the eﬀect of disturbances. One more example of robust control is the loop-transfer-recovery

13 Chapter 1: Introduction

(LQG/LTR) [35], which was rendered to deal the robustness issues of LQG control. Some more example of robust control methodologies include passivity based robust control schemes, Lyapunov based robust controllers, Quantitative Feed- back Theory (QFT) etc. Most of these above mentioned robust control schemes, deal with the linear time invariant (LTI) systems. The robust control schemes for nonlinear systems are also further discussed in Chapter 2.

1.3.7 Robust-adaptive Control for Synchronization

The issue of instability due to the difference between the actual plant and its model and was flagged in late seventies. With the help of examples, it was presented that an adaptive control scheme for plant model may become ineffective in the existence of minor disturbances [36]. This non-robust behaviour of adaptive control methodologies became a provocative topic in 1980’s. Some additional cases of ineffectiveness of the adaptive control scheme were published which raised the issue of robustness in the presence of disturbances [37, 38]. Many control scientists rushed towards the investigation of the causes of instabilities to develop the schemes to tackle this problem. Several novel methodologies and amendments were proposed in 1980s which introduce a new theory which latermon became a robust adaptive control.

1.4 Literature Survey

Synchronization is a dynamic progression during which the driven system becomes in line with the driving system so that the synchronized or slave system, in a certain manner, tracks the trajectory of the synchronizing or master system [7, 39].

Synchronization of chaotic and nonlinear systems plays a vital role in plenty of scientiﬁc and engineering applications such as secure communication [40], brain

14 Chapter 1: Introduction informatics [41,42], lasers and optics [43], chemical reactions [44] , electromagnetic systems [45,46], image processing [47], physiology, fluid mechanics [48], heartbeat regulation [49,50] and material science [51]. Since the development of chaos theory, the control and synchronization of chaotic systems flourished as an emerging topic of research. Many chaos synchronization techniques have been witnessed by the researchers. Literature review reveals that different approaches like linear feedback control [52], full-order and reduced-order output-affine observers [11], Runge-

Kutta model based nonlinear observer [53], synchronization with Huygens coupling [54], delay-range-dependent methodologies [55], [56], adaptive schemas using fuzzy disturbance observers [57], back-stepping techniques [58], robust adaptive methodologies [59, 60], step-by-step sliding mode observer-based techniques [40],

Chaos synchronization of unknown inputs Takagi-Sugeno fuzzy [39], adaptive generalized projective synchronization (GPS) [61], and evolutionary algorithms [62] are exercised for synchronization of the chaotic systems. All of these techniques for synchronization of the chaotic systems exhibit their strengths in variety of applications such as secure communication [63, 64], chemical reactions [65], neural networks [66], optics and lasers [67], biological systems [68], robotics [69] and information science [70].

The observer-based synchronization techniques are more relevant to the situation where all the states of the master as well as the slave systems are unknown [71]. Researchers are continuously exploring such techniques with diﬀerent types of observers for diﬀerent applications. For instance, synchronous chaos in coupled systems [72], chaos-based secure communications by employing reduced- order and step-by-step sliding mode observers [73], and generalized projective synchronization technique based on state estimation of hyperchaotic systems without calculating Lyapunov exponent [74] are presented. Active sliding mode observer-

15 Chapter 1: Introduction based synchronization, where an active observer increased the attraction strength of the sliding surface, is elaborated in [75]. Similarly, adaptive observer-based synchronization of two non-identical chaotic systems with unknown parameters is described recently in [76]. Further, a robust adaptive control approach for synchronization of the uncertain chaotic networks in the presence of mixed time-delays is described in [57]. In addition, output-aﬃne observers to estimate the system states and to identify the message signal simultaneously, based on synchronization of the uncertain chaotic systems, for establishing secure communication modules are developed in [46]. Some other observer-based synchronization techniques include observers for unknown inputs in Takagi-Sugeno fuzzy models with application to the secure communication [39] and observer-based synchronization methodology in a cascade connection of hyper chaotic systems [77].

Synchronization of the chaotic systems, especially for the delayed drive and response architectures is also an exigent ﬁeld of research. The response of chaotic networks because of inherently present time-delays becomes more complex, therefore, controller design for attaining coherent behaviour of the chaotic systems is a demanding research problem. For this grail, researchers have propounded numerous methodologies, capable of synchronizing the chaotic networks with input or output delay, varying time-delays, and disturbances. For instance, delay- dependent exponential synchronization and an exponential synchronization approach of stochastic chaotic delayed neural network systems is conferred by Sun et al. in [78]. Zhu et. al. accomplished a technique for the adaptive synchronization control of neural entities having varying time-lags in [79] and, further, Lin et. al. realized an adaptive sliding mode control technique for the chaos synchronization in uncertain fractional-order structures under time-delays [80]. Robust sliding mode control scheme for the discrete-time stochastic plants in the pres-

16 Chapter 1: Introduction ence of mixed time-delays has been developed in [81]. Li et. al. characterized an adaptive control method for the synchronization of nonlinear teleoperator systems by considering stochastic time-varying lags due to communication topologies [19].

A recent work in [82] illustrated adaptive mechanisms for the synchronization of chaotic oscillators with interval time-delays, while another work in [21] con- structed robust decentralized adaptive synchronization controllers for a generic form of complex systems with coupling time-delays and uncertain components.

It is imperative to mention that the conventional adaptive control schemes do not guarantee synchronization of the chaotic networks having time-lags because of the divergence of adaptive parameters. Time lags introduce instability in the error dynamics, resulting into non-coherent behaviours of the chaotic entities [59, 79].

Therefore, coherence of chaotic systems under time-delays by means of adaptive feedback control technique has received substantial research attraction in the recent past years. As a result, some remarkable feedback control approaches have been evolved, ensuring asymptotic convergence of the synchronization error. An adaptive control method is developed in [83] for the uncertain and complicated chaotic networks under perturbations. Adaptive synchronization of neural networks having mixed time-delays is investigated in [84]. Synchronization of Lure networks with rate-independent bounded delays using adaptive control methodology is proposed in [85]. However, still synchronization problem for the delayed chaotic systems requires signiﬁcant attention of researches owing to the technical hitches of various types of delays, delayed nonlinear dynamics, unknown parameters, and disturbances. The literature review reveals the fact that adaptive and robust adaptive control methodologies are capable to synchronize the drive and the response networks with bounded interval time-delays with fast delay variations has not been explored in the existing literature.

17 Chapter 1: Introduction

1.5 Contribution of Dissertation

The dissertation of my research is composed of new methodologies for synchronization of nonlinear drive and response system. The worth of contribution of the proposed scheme with respect to the existing techniques is elaborated in the following paragraphs turn by turn. If start with coupled chaotic synchronous

(CCS) and coupled chaotic adaptive synchronous (CCAS) observers-based control methodologies. The observer-based synchronization techniques as discussed in literature survey did not considered CCS and CCAS observers-based control methodologies. The main drawback of the aforementioned techniques, in contrast to the CCS and CCAS observers-based control methods, is their applicability to a lesser extent to synchronize two chaotic systems with unavailable state vectors.

In this thesis, a novel technique for synchronization of the master and the slave chaotic systems based on two observers for estimating states of both of the systems is presented, through which complete synchronization of the master-slave networks is achieved via utilizing their outputs rather than the exact states. In the recent work [86], an error convergent observer-based synchronization technique was proposed by employing estimation of the synchronization error. However, the approach only deals with the chaotic systems for which the overall error system is transformable into a linear combination of various error dynamics. In this thesis a more generic technique based on CCS observers and control input using estimated states is accomplished that can deal with the nonlinear error dynamics for complete synchronization of the master-slave oscillators. Development of the proposed CCS observers-based control method is a non-trivial problem as compared to the existing observer-based techniques for synchronization because the present approach simultaneously estimates the states of both the master and the slave systems using CCS observers and controls the dynamics of the error system 18 Chapter 1: Introduction using a control input. Hence, the proposed synchronization technique is capable for two automations, that is, estimation of the states of the chaotic systems and synchronization of the chaotic systems.

One more contribution of the work presented in this thesis is the adaptation of uncertain parameters present in the nonlinear dynamics by suggesting simple adaptation laws which are employed along with the proposed control signal based on CCAS observers for complete synchronization of the master-slave systems. The

CCS observers and CCAS observers-based synchronization schemes with the static and adaptive controllers to simultaneously estimate and synchronize states of two chaotic systems are not fully elaborated in the literature. Additionally, a two-step approach and a decoupling methodology to determine the controller and the observer gain matrices using linear matrix inequalities (LMIs) are provided herein.

The scope of the proposed observers-based synchronization methodologies diﬀer from the conventional synchronization schemes like [52,53,55,56,62] and [72,76,87].

The conventional observer-based approaches are used to estimate the state vector of a chaotic system and can be employed to specific scenarios like secure communication. The proposed methodologies are useful for monitoring through state- estimation as well as controlled synchronization of the two master-slave systems and have versatile applications. It is worth mentioning that the proposed observers are different from the conventional Luenberger-type and adaptive observers owing to the presence of coupling terms employed to achieve chaos synchronization. It should be noted that these CCS and CCAS observers are specifically designed for synchronization of chaos. Therefore, coupling terms are introduced to aid the synchronization. The application area of the proposed methodologies is broader than the conventional chaos synchronization approaches, which require exact states of the master-slave systems for feedback control. The proposed chaos synchro-

19 Chapter 1: Introduction nization approaches can be applied to the chaotic systems using information of their outputs, when states are not available, and synchronization is achieved using feedback of the estimated states (as well as the parametric estimates in the adaptive synchronization case). Numerical simulation results are demonstrated for synchronization of FitzHugh-Nagumo neurons by estimating states of the neurons and taking feedback of the estimated states under both the known and the unknown parametric information cases.

Another main contribution of this thesis is the derivation of suﬃcient conditions for the delay-range-dependent adaptive and robust adaptive synchronization control of the master and the slave chaotic systems with interval time-delays in the states having bounded delay rate. By considering the initial conditions belonging to a bounded region, matrix inequality based local adaptive control conditions along with simple local adaptation laws are provided with the help of a novel amended Lyapunov-Krasovskii (LK) functional, Jensens inequality, range of the delay, delay-rate bound and Lipschitz condition for attaining asymptotic synchronization. A novel treatment of the adaptive delay-range-dependent stability is explored in this thesis, which can be employed for adaptation of unknown parameters in the presence of interval time-delays. Additionally, a local robust adaptive control scheme is developed to achieve synchronization of the uncertain time-delay nonlinear systems with interval delays under disturbances. The proposed robust adaptive synchronization control methodology guarantees convergence of the synchronization error into a bounded region, which can be reduced to minimize the disturbances eﬀects. Further, a method is provided for computation of the controller gain matrix for chaos synchronization by employing the cone complementary linearization algorithm and linear matrix inequalities. Through this method,

20 Chapter 1: Introduction the controller gain of the resultant methodologies can be computed without any diﬃculty through a recursive computation approach.

In the recent delay-range-dependent control approaches [88], double-integral terms have been eﬀectively employed for incorporating the delay range with lower bound, not necessarily zero and for considering the delay-rate not inevitably less than unity. The current research work provides a solution for incorporation of these useful terms for the adaptive synchronization, which can be utilized to the control problems like stabilization. Note that treatment of the double-integral terms for adaptive counterpart is non-trivial because the state or error dynamics contains highly nonlinear adaptive terms. It is important to mention that the handling of the double-integral terms for ensuring stability of the adaptive error system is attained via rigorous local stability analysis and Lyapunov redesign. The proposed delay-range-dependent approach can be employed to the synchronization of chaotic systems with fast delay variations in contrast to the previously published global adaptive control results in [82]. A numerical simulation example of the proposed methodology for the synchronization of delayed master-slave Hopﬁeld neural networks is provided and the resultant delay-derivative bound is compared with the existing methods.

1.6 Structure of the Dissertation

This document is prepared in the form of a thesis containing eight chapters. A step by step approach is used to address the problems. Also, solutions are proposed with each problem. It is important to mention that the complexity of problems is gradually increased in each chapter and the last problem (if any) of each chapter is more complex then the earlier problems of that chapter. The complete structure

21 Chapter 1: Introduction of this thesis is composed of eight chapters. The brief structure of each chapter is described as follows.

Chapter 1 presents the importance of the synchronization phenomenon and its applications under the heading of motivation. The existing control techniques for nonlinear systems are considered, followed by the literature survey and the analysis about shortcomings of previous work. Then the importance of this contribution in the ﬁeld is analyzed. Chapter 2 presents some basic deﬁnitions and basic concepts as a prerequisite knowledge for control and synchronization of nonlinear systems.

Chapter 3 elaborates the adaptive feedback synchronizing control for nonlinear drive and response systems containing known and unknown parameters. Cor- responding simulation and the results are shown. Chapter 4 extends the the adaptive feedback synchronizing control presented in Chapter 3 and presents a robust-adaptive feedback synchronizing control for nonlinear drive and response networks under the eﬀect of disturbances and noises. The simulation and the results are also depicted in the last section of this chapter. Chapter 5 manifest the convex routines for solution of matrix inequalities which were evolved in Theorems for synchronization of chapter 3 and chapter 4. Corresponding simulation and the results are shown.

Chapter 6 describes delay-range-dependent adaptive synchronizing scheme for time delay chaotic networks synchronization. Simulation and the results are also presented. Chapter 7 renders robust-adaptive synchronization of drive and response system with varying time delays. For validity of proposed control scheme, simulation and the results are shown at the end of chapter 7. Chapter 8 gives the conclusion and future work possibilities in the ﬁeld of synchronization of nonlinear systems.

22 Chapter 2 Preludes of the Available Data

This chapter introduces some basic deﬁnitions and provides the foundation for

understanding of advanced synchronization schemes.

2.1 Nonlinear Dynamical Systems

A set of nonlinear diﬀerential equations in the form of

x˙(t) = g(x, t) (2.1) are used to describe a nonlinear dynamical system [89]. Where g is a n × 1 nonlinear vector function, and x(t) is the n × 1 state vector. The solution of diﬀerential equation (2.1) will result into a time dependent n-dimensional point in state space. The order of the system can be described by the number of states.

So, the order of nonlinear dynamical system described by (2.1) is n. The nonlinear dynamical systems may have control input u and can be represented by

x˙(t) = g(x, u, t). (2.2)

But, usually in state feedback control, the control input depends upon the state x(t). So, in this case the representation of nonlinear dynamical systems is done by (2.1). The term of nonlinear system may either apply to any physical system

23 Chapter 2: Ppreludes of available data

or it can be any mathematical description of some dynamics. The linear system

can be regarded as a special case of the continuous nonlinear system [89]. The

following equations (2.3) which represents the the dynamics of the pendulum is an

example of nonlinear dynamical systems. Where M is the mass, R is the length

of pendulum, and b is the coeﬃcient of viscous friction.

MR2θ¨(t) + bθ˙(t) + MgRsin(θ) = 0. (2.3)

For the stability and control of nonlinear dynamical systems, an important term

Figure 2.1: The pendulum system

is the equilibrium point. A state x∗ is equilibrium point of the system if once x(t) is equal to x∗ , it remains equal to x∗ for all future time. Equilibrium points can be found by solving the nonlinear algebraic equations (2.1) i.e.,

g(x, t) = 0.

A nonlinear system in contrast to linear system can have many isolated equilibrium points. By converting (2.3) into state space form we can get the equilibrium point for the dynamics of pendulum as (0, 0), (π, 0) [89]. Moreover, the nonlinear 24 Chapter 2: Ppreludes of available data dynamical systems diﬀer from the linear system essentially by following features

[90]:

1. Finite time escape

2. Limit cycles

3. quasi periodic oscillations

4. Chaos

5. Multiple modes of behaviour

2.2 Chaos and Chaotic Systems

Chaos and chaotic systems are as old as the nature exist. The start of theory of chaos with its exact deﬁnition is unknown. But, in literature the initial version of modern chaos theory is found in ﬁst half of 20th century [91]. In 1938 Norbert

Wiener discussed the types of homogeneous chaos detail. The present version of chaos theory originated during the study of the turbulence phenomenon in 1971, an account that claimed this scenario of complexity. The Ruelle Takens Newhouse

(RTN) model [92] was proposed as the result of these studies which denied the claim that chaotic behaviour must be modeled by the group of incomparable frequencies. Rather it claimed that a simple set of dynamical equations can be used to model the chaotic systems. The state space of an attractor was used to elaborate the change of laminar state to turbulence that depicts a chaotic behaviour.

The state space attractor is described by group of simple equations. This attractor has many useful properties: First is that it is an attractor which attract all nearby trajectories in state space towards itself. Secondly, it manifest the shape of fractal.

Third is that it exhibits sensitive dependence on initial conditions (SDIC), which

25 Chapter 2: Ppreludes of available data is discussed in the following section. Last it that a simple set of equations can form a strange attractor.

A very well known example of chaotic behaviour is the skeleton of a weather system proposed by Edward Lorenz in 1961 [93], by means of the set of equations as follows. Edward Lorenz found out all of that the hard way ﬁnding behaviour using diﬀerential equations. Lorenz created a set of dynamical equations as a model of weather system. Many physical systems exhibits the phenomenon of Chaos.

In ﬂuid dynamics, turbulence represents the phenomenon of chaos. Dynamics of atmosphere are also chaotic in nature because of this a long time weather prediction becomes diﬃcult. Many electrical, mechanical and physical systems depicts chaotic vibrations. For example buckled elastic systems, play or backlash containing systems and feedback-control systems [89].

The phrase-sensitive dependence on initial conditions-was used by Ruelle [94]

The systems that perform some dynamics and show high dependence of the behavior of dynamics on their initial conditions, are the main field of research in modern chaos theory. Edward N. Lorenz in 1964 elaborated the sensitive dependence on initial conditions (SDIC) by a claim that ”SDIC is shown by the dynamical system if it yields noticeably different dynamics for the two slightly different initial states”

. Lorenz elaborated the sensitive dependence on initial conditions by the simulation of the Lorenz system with different initial conditions. The outputs were the two different trajectories. Initially the two trajectories were only separated one percent which can be confirmed from the initial conditions. The systems behavior or the separation after some time became very large. For small time scale he was not able to notice the effect of the (initially) small difference in initial conditions.

But the diﬀerence in solution exponentially grown and eventually it became a huge diﬀerence. The dependence of separation of the states of similar chaotic system

26 Chapter 2: Ppreludes of available data

with silightly diﬀerent initial conditions is quantitatively measured by Lypunov

exponent [95]. The sensitive dependence of a system on initial conditions also

relates with the very well known metaphor in the ﬁeld of chaos i.e. butterﬂy ef-

fect [96]. It is a phenomenon in which a small perturbation in the initial condition

of a dynamical system produces large changes in ﬁnal results. These phenomena

are often in chaotic dynamical systems.

An orbit that appears in the phase portrait of a dynamical system and it re-

peats itself after certain time interval is called a periodic orbit [97]. An oscillator

always illustrate periodic orbit in the phase portrait. An orbit which does not

show exact periodic behavior but the dynamical system always keeps its dynam-

ics in the vicinity of this orbit is called aperiodic orbit [98]. These orbits come randomly near to the periodic orbit, but does not converge to a periodic orbit.

Usually chaotic systems produce aperiodic orbits. The sensitive dependence on initial conditions is the inherent feature of chaotic systems, and it means a minor variances in the initial value will produce a large change in the aperiodic orbit or in the dynamics of the chaotic systems. This intrinsic property of aperiodicity diﬀerentiates the chaotic systems form other nonlinear systems. The nonlinear systems which possess non periodic behavior do not belong to the class of chaotic systems.

A dynamical system whose future response is predictable and can be mathematically modeled with no uncertainty in the development of impending dynamics of the system [99]. A deterministic system will always yield the identical output using the same initial conditions . Chaotic systems are deterministic it means that if the initial conditions are exactly known, then the upcoming state of chaotic systems can be anticipated. Though, in practical situation, the precision level of measurement of the initial states will determine the accuracy of prediction of the

27 Chapter 2: Ppreludes of available data

future state. Also, it is established fact that chaotic systems are regarded as the

systems which show a strong dependency on the initial states [100].

2.3 Time-delay Systems

Time-delays are natural component of the dynamic processes present in science

and engineering. Time-delay systems are also called systems with aftereﬀect or

dead time systems [101]. Time-delays appear in many engineering systems for

example aircraft, chemical control systems, in lasers models, in internet, biology,

medicine and many more. There can be transport, communication or measurement

delay. Time delay system are mathematically modeled with delay diﬀerential

equations (DDEs). Systems with delays have been studied by many researchers

and continue to be an important area of research in systems and control. Following

is the example of the time-delay equation of general dynamic process with real

positive constant delay τ is

(2.4) x˙(t) = A0x(t) + Ajx(t − τ) + B0u(t)

n Here, x(t) ∈ R is the state vector, A1,A2,B are the state space matrices of matching dimensions. u(t) is the control input

The problem of delay in master-slave synchronization control schemes was reported in [102], after that the chaotic synchronization was thought to be applied to optical communication. Chen and Liu [102] named this phenomenon as a phase sensitivity because of the remoteness of the systems and claimed that the presence of a time delay destroys the synchronization between these systems. Moreover, the synchronization of two coupled chaotic circuits having a time-delay in their coupling, had been explored [103]. In control theory time delay systems often

28 Chapter 2: Ppreludes of available data result in instabilities [104, 105]. The time-delay systems are categorized into two main classes: delay independent [106] and delay dependent [105].

2.4 Nonlinear Drive and Response systems

Any system is nonlinear system if its output does not change in proportion to the input [90]. Drive-response architecture of nonlinear systems is a model of system architecture where one system has unidirectional control over more other systems. This drive and the response configuration is excessively witnessed in the electronic as well as in the mechanical systems where one device acts as the controller or the master system, while the other systems are coupled with it as slave systems. In electronics, communication between the components of computers, such as communication of data in more than disk-drives, instead of communicating with each disk-drive, we can communicate with one drive serving as the master, and the control command for communication of data is sent to the slaves from the master disk-drive [107]. Similarly, mechanical systems can also be configured in drive and response architecture e.g. two motors for driving the same load but energized by two different drives. The drives for controlling the motors are defined in master/slave configuration, the master drive manage the speed of the load, and the slave drive incorporates the required torque. Pneumatic and hydraulic systems can also be coupled in drive and response architecture where master cylinders apply the pressure and maneuver the slave cylinders. For the sake of understanding using mathematical model of general nonlinear system, Let us assume two simple nonlinear systems in the form of a master-slave pair, the equation for the master system is given by

x˙ m(t) = f(xm)(t) + F (xm)(t)α, (2.5)

29 Chapter 2: Ppreludes of available data

n m where xm ∈ R is master state vector. The parameter α ∈ R is the unknown

parameter vector of the system, The function f(xm) is an n × 1 matrix, F (xm) is

an n × m matrix. Similarly for the slave system, the equation may be written as

x˙ s(t) = g(xs)(t) + G(xs)(t)β + u(t), (2.6)

n q where xs(t) ∈ R is slave state vector. The parameter β ∈ R is the unknown

parameter vector of the system, Function g(xs)(t) is an n × 1 matrix, G(xs)(t) is an n × q matrix. u(t) ∈ Rn is a control input vector.

The control of nonlinear systems is an area of research interest to engineers, physicists [108] and mathematicians and many other scientists because most systems are inherently nonlinear in nature.

2.4.1 Application of Drive and Response Systems in Real World

Many practical scenarios are facing drive and response architecture and some of them are given as follow.

1. Master cylinders in pneumatic or hydraulic systems control one or many

slave cylinders.

2. In database management system, slave databases are synchronized to the

master database.

3. In general purpose computers, peripherals attached to a bus may be conﬁg-

ured in master-slave conﬁguration.

4. The master and slave conﬁguration is also witnessed in railway locomotives

systems when multiple locomotives are used for carrying heavy loads. In

this conﬁguration one locomotive is made master and the remaining all are

slaved to the master locomotive. 30 Chapter 2: Ppreludes of available data

5. In large industrial units, where distributed control system operate, PLC

master/slave network is organized for eﬀective control.

6. For the processes of copying data on cassettes and discs, several cassette

tape or CDs recorders are coupled together in master-slave structure

7. In a clock network, a master clock provides time signal, which is utilized to

synchronize one or more slave clocks.

2.4.2 Nonlinear Drive and Response Systems with Bounded Time Delay

The drive and response mechanism of nonlinear systems is often inherently coupled

with the time delays. The constant or bounded time-delays appear in many sys-

tems such as in secure communication mechanism using master and slave chaotic

systems, in synchronization phenomenon of master and slave LASERS, in synchro-

nizing control of chemical systems, in data communication on Internet etc. These

delay can be transport, communication or measurement delay. The transport de-

lay is because of the delaying factors involved in the dynamics of the process.

The communication delays are because of delays involved in the communication

process and the measurement delays are because of the delays contributed by the

measurement process. Moreover, in drive and response architecture, the values of

the variables involved in the dynamics the drive system are received at the con-

trol input of the slave system with some delay (τ), as shown in Figure 2.2. The state vector of master system which the reference value for synchronizing scheme reaches to the controller block after a delay τm. Similarly, the state vector of the

slave system is the output value is fed-back to controller and it reaches to the

controller after a delay τs.

So, this Figure 2.2 demonstrate the generic synchronization scheme for the

drive and response architecture with incorporated delays. The states variables of 31 Chapter 2: Ppreludes of available data

Figure 2.2: Synchronization scheme of drive and response systems with time delay master system and the slave system are being fed to the controller after delays of

τm and τs respectively.

2.4.3 Nonlinear Drive and Response Systems with Uncertain Delay Rate

The time delay parameter are crucial elements in the control and stability of nonlinear drive and response systems. But many a time, the value of delay parameter is uncertain in practical systems. It is because of many known and unknown reasons. Following mathematical models of generalized nonlinear drive and response system provide an insight that how the uncertain delay parameter i.e. τ can be augmented in nonlinear drive and response systems. Consider the master-slave

(drive-response) systems (2.7)-(2.8) with inherent unknown time delay present in the dynamics of these systems. These delays in the dynamics can be of diﬀerent types. Delays may be in the states of the master or the slave system and/or in the the input or output of the master system or the slave system.

p X x˙ m(t) =f1(xm(t)) + f2(xm(t − τ)) + gi(xm, t, τ)θi i=1 (2.7) xm(t) =φ1(t), t ∈ − max(τ1, τ2) 0 ,

32 Chapter 2: Ppreludes of available data

p X x˙ s(t) =f1(xs(t)) + f2(xs(t − τ)) + gi(xs, t, τ)θi + Bu i=1 (2.8) xs(t) =φ2(t), t ∈ − max(τ1, τ2) 0 .

n n Here xm ∈ R and xs ∈ R are the state for the master and the slave systems,

n×n n×n n×m respectively. The symbols A ∈ R , Ad ∈ R and B ∈ R are the constant matrices establishing the linear dynamics of the systems and u(t) ∈ Rm denotes

n n the control input. The vector-functions f1 (x(t)) ∈ R and f2(x(t − τ)) ∈ R symbolize, respectively, the nonlinearities without and with time-delay in the dynamics of the nonlinear systems. These nonlinearities are not associated with

n any unknown parameter. But the vector-function gi(x, t, τ) ∈ R represents the nonlinearity associated with unknown scalar parameter θi for i = 1 ··· p, τ(t) is the time dependent delay, involved in the dynamics of the drive and the response systems.

2.5 Error Dynamics

For the above mentioned generalized nonlinear systems, (2.7) and (2.8), the error or the diﬀerence between the corresponding state variables of the drive and response systems, deﬁned as e(t) = xm − xs is actually the synchronization error.

The synchronization error dynamics between the corresponding states of both master and slave systems may linear or nonlinear depending upon the dynamics of the drive system and the response systems. These error dynamics may also chaotic in nature. This chaotic behavior of error dynamics is usually depicted in the synchronization phenomenon of chaotic drive and response systems. The error dynamics between the corresponding states of the generalized master and the slave systems (2.7) and (2.8) may be represented as:

33 Chapter 2: Ppreludes of available data

e˙(t) =f1(xm(t)) − f1(xs(t)) + f2(xm(t − τ)) − f2(xs(t − τ)) p (2.9) X + B [f3i(xm, t, τ) − f3i(xs, t, τ)] θi − Bu. i=1

For the complete synchronization of nonlinear drive and response systems, this error dynamics show asymptotic convergence i.e. convergence to the origin. Many control scheme as already discussed in chapter 1 are used for synchronization of the nonlinear drive and response systems. Some these techniques which converge the synchronization to zero include unidirectional linear error feedback coupling.

Synchronization is also done by means of coupling the systems physically. Similarly synchronization error is set to zero for coupled nonlinear systems by an external driving agent as a control input.

2.6 Synchronization of Drive and Response Systems

Synchronization is most signiﬁcant phenomena to study the collective behavior of coupled drive and response systems. Synchronization is said to occur if phase locking and consensus among corresponding states of coupled dynamical systems i.e. drive and response systems is achieved. The coupled dynamics of oscillators is described by ordinary diﬀerential equations which comprise of state of dynamics of oscillator with an additional weak coupling term. In [109] single state dynamical model has been considered and all to all coupling between oscillators is via single state. Where as in [110], multi state dynamical model of oscillators has been considered but the coupling is achieved via single state despite that the oscillators possessed the multiple states . The former approach helps to understand basic synchronization process in a simple dynamics, while the later one gives practical approach for dealing with problems of coupling in multi-state oscillators. Synchro-

34 Chapter 2: Ppreludes of available data nization phenomenon is the process of coherence of nonlinear systems that may happen when two or more nonlinear systems are coupled. Though the synchronization of coupled chaotic oscillators is a complex phenomenon however it is well understood theoretically and it is experimentally proven phenomenon.

2.6.1 Types of synchronization

The tuning of oscillations of vacillating objects because of the weak interaction

(coupling) between them is known as synchronization [111]. The coherence or the synchronization phenomenon between the master and the slave systems is achieved by various methodologies like sliding mode control, fuzzy logic, adaptive control, robust control, state-feedback control, neural networks and robust adaptive control with advantages over each other. All of these schemes for synchronization perform diﬀerent types of synchronization. Synchronization phenomena is categorized into various types or categories.

Complete synchronization : If the error dynamics of the two systems exactly go to the origin, then the synchronization between the this type of synchronization between two networks is called complete synchronization. The coupled systems having identical nonlinear parameters often show complete synchronization by applying some control methodology [112]. This type of synchronization is also achievable without control input. This type of synchronization is an indica- tions to the similar dynamics development of the interacting systems [113].

Generalized synchronization: A functional relation based synchronization between the states of coupled systems either in the form of master slave architecture is known as generalized synchronization [114]. Generalized synchronization results in to complete synchronization by taking the static parameter as unity. By changing the value of static parameter, amount of synchronization may change.

35 Chapter 2: Ppreludes of available data

Lag synchronization: The phenomenon of synchronization with time lag τ among the respective variables or the states of the drive system and the response systems is known as lag synchronization [115]. This lag synchronization is most often phenomenon in real word sit

Anticipatory synchronization: Anticipatory synchronization is synchronization process where the states of the leading system is forestalled by the following system with a time delay τ [116]. The anticipation of the behavior of the leading system possess a vital role in anticipatory synchronization.

Phase synchronization: A synchronization in which the corresponding phases of the master and the slave systems congregates to the same values, regardless of their amplitude value is called phase synchronization [117].

2.7 Lipschitz Nonlinearities

If a real valued nonlinear function f(x(t)) : Rn → Rn obey or fulﬁll the following condition

n kf(x(t) − f(¯x(t))k ≤ Lf kx(t) − x¯(t)k, ∀ x(t), x¯(t) ∈ R , (2.10)

then this types of function is called Lipschitz nonlinear function. Where Lf ∈

R+ is a known as Lipschitz constant [118]. Lipschitz functions appear nearly everywhere in mathematics of nonlinear systems. Lipschitz functions appear as examples of functions of bounded variation and it is proved that a real-valued

Lipschitz function on an open interval is almost everywhere diﬀerentiable. The idea of Lipschitz nonlinear continuous functions was presented by Rudolf Lipschitz.

Actually, this idea present some bounds on the rate of change of that continuous function. It is important to mention that all linear continuous functions are also

Lipschitz functions. A Lipschitz continuous function is bounded by some limit for 36 Chapter 2: Ppreludes of available data

the rate of change of that continuous function. A deﬁnite real number always exist

for such continuous function such that, for any two points the curve of function,

the absolute value of the rate of change i.e the slope of the line segment connecting

those two points is not more than constant value. This constant value is named as

a Lipschitz constant of the function. So it is clear that any function with bounded

ﬁrst derivative is a Lipschitz function. These types of nonlinear functions with

bounded ﬁrst derivatives are also called Lipschitz nonlinearities.

2.8 Lyapunov Theory for Stability and Control

The stability of dynamical system are of various types and this stability is ﬁrmed

for that dynamical system by the results achieved from the solution of diﬀerential

equations representing the dynamical systems. A signiﬁcant type of stability is

that relates to the stability near to a equilibrium point [119]. This type of stability

for the dynamical system is conﬁrmed by the Lyapunov theory. Lyapunov stability

is deﬁned as, if the solutions that begin from the neighborhood of an equilibrium

point xe always rest near xe forever, then the point xe is Lyapunov stable. The technique of Lyapunov stability is named after the name of Russian scientist,

Aleksandr Lyapunov, after his publication of book in 1892, ”The General Problem of Stability of Motion” [120]. According to this Lyapunovs seminal idea, the amendments are vital in the linear theory of stability make it implementable to the nonlinear systems. After some time, Lyapunov idea was translated into many languages. During the Cold War, this idea got a new dimension with the invention of the ”Second Method of Lyapunov” which was applied to the control system of aerospace guidance systems comprising high amount of nonlinearities which cannot be dealt by other methods. After this a numerous publications were rendered by

37 Chapter 2: Ppreludes of available data the researcher of control science and mathematicians [121–124]. Recently the notion of the Lyapunov-exponent has become familiar linking with chaos theory.

Stability theory presented by Lyapunov is an instrument for the analysis of nonlinear systems. This theory oﬀer a stratagem for synthesizing a stabilizing feedback controllers. The total energy dissipation of dynamical systems is the basic theme of Lyapunov theory. If the system follows this theme, then the system stabilizes itself. The advantage of this scheme stability conﬁrmation only by analyzing that how an energy-like function , known as Lyapunov function, changes with time.

Lyapunov theory consist of two methods, indirect method and the direct method as presented by Lyapunov [89]. The indirect method ensures that the stability properties of a nonlinear system near an equilibrium point are same as of linearized approximation. The direct method is a strong instrument for the analysis of nonlinear system, which is also termed as Lyapunov analysis or direct method. The direct method is a extension of the energy concepts linked with a mechanical systems: dynamics of the system is stable if the time rate of change of its total mechanical energy is negative . For utilizing direct method, to pursue the stability of a nonlinear system, the main theme is to construct a energy-like scalar function which is known as Lyapunov function for the system, and check whether its value decreases with time. It is very powerful tool and it is because of its generality. It is applicable to all kinds of control systems whether they are ﬁnite dimensional, time-invariant, time-varying or inﬁnite dimensional. Conversely, the main problem in this method to construct a Lyapunov function for a system.

By using this theory, one can claim the system is stable or asymptotically stable, instead of solving the diﬀerential equation. For the elaboration of this scheme,

38 Chapter 2: Ppreludes of available data see the following example. Consider the system with the following dynamics

x˙(t) =y(t)x3(t) (2.11) y˙(t) = − x(t)y3(t).

Here, x(t) and y(t) are the state variables. The equilibrium point of the above mention dynamics is (0, 0). To consider a Lyapunov candidate function from many possible energy functions for the above mentioned dynamics, take V (t) as the Lyapunov energy function for the above mentioned dynamics (2.11):

V (t) = x2(t) + y2(t). (2.12)

Taking the time derivative of Lyapunov function V we get

V˙ = 2x(t)x ˙(t) + 2yy˙(t). (2.13)

After putting the values ofx ˙(t) andy ˙(t), We get the following result.

V˙ (t) = −x4(t) − y4(t) (2.14)

Which is negative everywhere except at the equilibrium. So the value of V˙ is decreasing along the solutions, meaning that solutions on any level curve continue into the region bounded by the level curve and thus they tend to the equilibrium, which thereby is not only locally stable but also globally stable with whole plane

R2 as basin of attraction.

2.8.1 Types of stability

A scalar function V (t) is a Lyapunov candidate function, with a prerequisite that it should be continuous and positive deﬁnite i.e. V (t) > 0 where t 6= 0 [125]. These

39 Chapter 2: Ppreludes of available data

function also help in determining the type of stability of dynamical systems.

Locally asymptotically stable: If the locally positive deﬁnite Lyapunov candi-

date function is V (x) and its rate of change with respect to time is locally negative semideﬁnite i.e. V˙ (x) < 0 in the neighborhood of origin , then this type of equilibrium is called as locally asymptotically stable [126].

Globally asymptotically stable equilibrium: If the Lyapunov candidate func-

tion V (x) , is globally positive and unbounded radially and the rate of change with respect to time of the this scalar function is globally negative deﬁnite i.e.

V˙ (x) < 0 ∀x ∈ Rn, then this type of equilibrium is called as globally asymptotically stable [127]. The radially unbounded Lyapunov candidate function means if kxk → ∞ ⇒ V (x) → ∞.

2.8.2 Lyapunov function and adaptation laws

Lyapunov function can play a vital role for the derivation of adaptation laws. The adaptation laws are employed for the adaptation of unknown parameters involved in the system dynamics [128]. Following example can elaborate this fact.

x˙ m(t) = − 5xm(t), (2.15)

x˙ s(t) = − 5xs(t) + u(t). (2.16)

Let the controller function is deﬁned as u(t) = K(t)e(t). From equation 2.15 and

2.16, The resulting error dynamics are as follows:

e˙(t) =(−5 − K)e(t). (2.17)

40 Chapter 2: Ppreludes of available data

If the controller gain parameter K(t) is unknown then it has to be adapted for the

asymptotic convergence of error dynamics. The adaptation law for the controller

parameter can be derived using Lyapunov function. Considering the Lyapunov

candidate function.

1 1 V (t) = e2(t) + K2(t). (2.18) 2 2

Taking the time derivative,

V˙ (t) =e ˙T (t)e(t) + K˙ T (t)K(t). (2.19)

Putting the value ofe ˙(t) from equation, and assuming K and e(t)to be scalar quantities,

V˙ (t) = K˙ (t)K(t) − K(t)e2(t) − 5e2(t). (2.20)

From above equation it can be concluded that for the asymptotic convergence

of error e(t), i.e. for the negative time rate of change of the Lyapunov energy

function, the ﬁrst two terms of the adaptation law should cancel each other. For

the sake of cancellation of last two terms, the unknown parameter should be

adapted according to the equation (2.21)

K˙ (t) = e2(t). (2.21)

The above mentioned example is helpful to demonstrate the procedure for

deriving the adaptation law using the Lyapunov function. With the help of this

example, it is clear that for making the time rate of change of Lyapunov function

negative, it is necessary to adopt the unknown controller gain parameters K(t).

41 Chapter 2: Ppreludes of available data

2.9 Robust Control

The control of unknown plants with uncertain dynamics under the effect of disturbances is known as robust control [129]. The main objective of the robust control systems is to deal with uncertainties and disturbances. Over the past three decades, because of the the development of high speed computers, robust control theory has gain a overwhelming success. Robust control technique helps in synthesizing controllers using simplified mathematical models of real system but they still useful for the working of the real plants often difficult to be accurately manifested by a group of linear equations involving differential of variables. The concept of robustness describes that how much a system is sensitive to the internal or external disturbances. The modern approaches to design robust controllers for the linear system which are robust against model uncertainties include Linear fractional fransformation (LFT), µ and µ synthesis [130], Algebraic Riccati equations, H2 optimal control, H∞ control [130]. Moreover L2-gain is a very important characteristic of an input/output system for H2 optimal control design related to linear and nonlinear systems.

Robust control schemes also elaborated for the nonlinear systems by diﬀer- ent researchers in literature. Robust control of nonlinear system for uncertainties compensation is presented by Feng Lin et. al. [131]. For the analyzing the robust performance with linear matrix inequality (LMI) based methods via unknown parameter based energy functions i.e. Lyapunov function is presented by Dimitri et. al. [132]. Robust control for nonlinear systems by parameter dependent Lyapunov functions is presented by Yue Zhi et. al. [133]. A popular approach to solving robust stability analysis and synthesis of systems with uncertainties is the Lya- punov approach [133]. For example, to ﬁnd the feedback control law u = K(x) for robust stabilization of the nonlinear system (2.22) 42 Chapter 2: Ppreludes of available data

x˙((t)) =A(x) + B(x)f(x) + B(x)K(x), (2.22)

Here x(t) ∈ Rn and A(x) and B(x) known functions and f(x) is unknown nonlinear

function of x, one can consider the following quadratic function as the Lypunove function or the performance index for robust control.

2 T T (2.23) V (x) = fmax(x) + x (t)x(t) + u u,

According to Lyapunov theory, if the time rate of change of this Lyapunov candi-

date function become negative,

V˙ (x) < 0

by applying the control signal u = K(x) then the system dynamics becomes

asymptotically stable even under the eﬀect of uncertainties f(x) with norm

bounded by fmax(x). It is, however, well known that the use of the simple quadratic Lyapunov functions usually leads to conservative results. Also, some methodologies developed by some modiﬁcation in the adaptation laws, for example

σ-modiﬁcation [134,135] are used to increase the robustness of the system.

2.10 Robust-adaptive Control

Robust control may be considered as an optimized methodology of controlling the dynamics of systems with bounded perturbations. By utilizing state or the output feedback, a control signal is designed having the strength to robustly maneuver the dynamics of systems to the required path even in the presence of uncertainties. It is sure and an established fact that all familiar and formal control methods

43 Chapter 2: Ppreludes of available data

are model based. The model of the system based on mathematics which is to be

controlled is the foundation of the control architecture. This model is often an

approximation of the physical systems and may contain errors with respect to the

real system due to modeling. This is a signiﬁcant problem that may arises within

approximation based adaptive control literature is the so called loss of controlla-

bility problem [136]. Although the actual system is assumed to be controllable,

the identiﬁcation model may lose its controllability at some points in time, owing

to parameter adaptation. Several solutions have been proposed for linear sys-

tems based on switching strategies to overcome this problem. By utilitarian of

the σ-modiﬁcation and backstepping technique Polycarpou and Ioannou (1993), presented a robust adaptive control scheme for nonlinear systems in the presence of uncertain parameters and nonlinear functions.

Disturbance

Control States of Input Reference Error Robust-adaptive input State of system model System model controller Adaptation parameter Adaptation Error laws

Figure 2.3: Robust-adaptive control scheme general architecture

2.11 Cone Complementary Linearization Algorithm

Cone complementarity linearization algorithm is a technique for solution of non-

linear systems by simple steps [137] as discussed here. For example, consider a

system with state space representation [A, B, C, D] such that the mathematics

44 Chapter 2: Ppreludes of available data

involved in its nonlinear dynamics is of the form as follows:

B⊥(AX + XAT + 2βX)B⊥T ≤ 0,

C⊥(AT S + SA + 2βS)C⊥T ≤ 0.

From above inequalities, we can consider a matrix inequality constraint for the stability analysis as follows:

  XI   M(X,S) =   < 0 (2.24) IS

For solving the problem above, we need to saturate the constraint M(X; S) ≥ 0 in (2.24). The idea is to minimize the constraint by using minT r(XS) subject to (2.24).To solve such a problem, a linearization algorithm (proposed by Franke and Wolfe and described in [138] can be used.

φlin(X; S) = constant + T r(S0X + X0S).

Algorithm:

1) Find a feasible point X0; S0. If no solution, exit. Set t = 0.

2) Set Vt = St; Wt = Xt; and ﬁnd Xt+1; St+1 that solve the LMI problem

Pt : minimizeT r(VtX + WtS), subject to (3).

3) Check the stopping criterion, if it is satisﬁed then exit, else put t = t + 1 and

move to Step 2.

2.12 Simulation Software

Simulation software is a program that permits someone to witness a process

through simulation without performing that process [139]. Cutting-edge com-

puter simulation softwares can simulate electronic circuits, power system behavior,

45 Chapter 2: Ppreludes of available data chemical reactions, weather conditions, feedback control systems, atomic reactions, even complex biological processes. Any process which can be transformed to mathematical representation can be replicated on a computer through simulation software. Additionally, other than reproducing the processes for analyzing the behavior of process under diverse circumstances, simulation softwares are mean of testing the validity of new ideas presented by scientists. Overall simulation software fall into continuous simulation and discrete event. Continuous simulation softwares are utilized to simulate a broad class of physical processes like electric motor response, steam turbine power generation, ballistic trajectories, data communication through radio frequency, human respiration, etc. While discrete event simulators are used to for the statistical models.

Advantages of simulation:

• Computer simulation is often an analysis tool when mathematical analysis

approaches are not available, Similarly in case of complex mathematical

analysis schemes, simulation are the easiest solution for analysis.

• With the help of simulation, it become possible to compare the diﬀerent

control scheme for the process.

• An important useful feature of simulation software is that it can compression

or dilate time of simulation.

• By simulation software we can emulate the actual process with diﬀerent

alteration in its parameters.

• Size of systems to be simulated can be scaled up and down easily

• Simulation of system is usually economical for the researchers

46 Chapter 2: Ppreludes of available data

2.12.1 MATLAB as a Simulation Software

The acronym MATLAB stands for matrix laboratory. MATLAB was developed to facilitate matrix computation developed by the LINPACK and EISPACK projects. MATLAB is a high level language software for methodological computation. Matlab provides a user friendly environment which integrates programming, computation and visualization. The basic data element of Matlab is an array which does not need dimensioning. Because of this the matrix computation and vector formulation is the specialty of MATLAB.

47 Chapter 3 Adaptive Feedback Synchronizing Control for Nonlinear Drive and Response Systems

This chapter elaborates the adaptive feedback control schemes for the synchronization of drive and response architectures.

3.1 Introduction

Adaptive control refers to the methodologies for adaptation of controllers in real time, in order to realize a desired performance of control system in the situation of changing parameters of the plant dynamic model or with unknown parameters.

If the parameters of the mathematical model of the system being controlled are constant but unknown then the precise modiﬁcation of the controller parameters is impossible without knowing their exact values. The methodologies of adaptive control in closed loop provide an automatic tuning process for the controller parameters. With constant unknown parameters, the adaptation process ceased to exist as time goes on and if the variations in the conditions occur, resumption of the adaptation control takes place. For the case when the parameters of mathematical model of the system alter whimsically because of the physical changes of system, the appropriate control is adaptive control to be considered. In these situations, the adaptation laws will continuously function without any stopover

48 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems and this is also known as continuous adaptation. The adaptive control for synchronization of nonlinear systems is rendered and applied by many researcher at diﬀerent times. A modiﬁed Chua’s circuit system’s synchronization by adaptive control is presented in [140] by Yaseen. Distributed-adaptive-control is proposed by Das et al. [141] for synchronization of unknown nonlinear networked systems .

Distributed-adaptive-control for synchronization of complex networks is rendered by Wenwu et al. [142]. Feedback and adaptive scheme for synchronization of a group of hyperchaotic systems is elaborated by Tao et al. [143]. So, the adaptive control methodology for synchronization is a topic of interest for researcher in the last decade. But, still a lot has to do. Especially, for the case of adaptive synchronization where all the states of the drive and the response systems are not available or immeasurable. In these scenarios, observers are used for the estimation of the unknown states. One these scenario of the adaptive feedback synchronizing control for complex systems is discussed in this chapter 3.

This chapter manifest new control schemes for synchronization the drive and the response chaotic systems which is established by means of novel coupled chaotic synchronous (CCS) observers and coupled chaotic adaptive synchronous (CCAS) observers [42]. The simultaneous estimation of the master and the slave systems states is accomplished, by means of the proposed observers for each of the master and the slave systems, to produce error signals between these estimated states.

This estimated synchronization error signal and the state-estimation errors converge to the origin by means of a speciﬁc observers-based feedback control signal to ensure synchronization as well as state-estimation. Using Lyapunov stability theory, non-adaptive and adaptive control laws and properties of nonlinearities, a convergence condition for the state-estimation errors and the estimated synchronization error is developed in the form of nonlinear matrix inequalities. In this

49 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems chapter, mainly two types of control methodologies are presented. Firstly, synchronization methodology for the coupled chaotic systems with known parameters is presented. In this scheme of synchronization, coupled chaotic synchronization

(CCS) observers are designed for synchronizing the chaotic networks using feedback control scheme. Secondly, a synchronization control scheme for the coupled chaotic networks with unknown parameters is elaborated. Coupled chaotic adaptive synchronization (CCAS) observers are designed for such systems to synchronize them using adaptive feedback control. It is worth mentioning that several observer-based synchronization approaches with diﬀerent advantages with respect to each other e.g. [39,40,53,71,75,144] are presented in literature. However, these synchronization approaches are inapplicable to two chaotic oscillators. The proposed observers-based synchronization methodologies in this chapter addresses a relatively diﬀerent synchronization phenomenon, that is, synchronization of the two chaotic oscillators, and has broad applications for two systems’ point of view.

The conventional approaches are limited to a single system only.

The block diagrams for depicting the coupling architecture of chaotic networks

(to be synchronized) are also presented in this chapter. Numerical simulation of the proposed synchronization technique for FitzHugh-Nagumo neuronal systems is also illustrated in this chapter to elaborate the eﬀectiveness of the proposed observers-based control methodologies.

3.2 Statement of Systems and The Problem Formulation

The nonlinear systems synchronization is the subject matter. It means that the dynamics of those systems are to be synchronized which contain homogeneous nonlinearities in their dynamics. These nonlinearities can be classiﬁed into two categories. The nonlinearities with known parameters and the nonlinearities with

50 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

unknown parameters. The mathematical representation of the nonlinear systems

to be synchronized would contain both types of nonlinearities. After above men-

tioned discussion, consider the generalized model of the nonlinear master and the

slave chaotic (nonlinear) systems (3.1)-(3.2), deﬁned by the state-space represen-

tation as,

x˙ m(t) =Axm(t) + f(xm(t)) + Bg(xm(t))θm, (3.1)

ym(t) =Cxm(t),

x˙ s(t) =Axs(t) + f(xs(t)) + Bg(xs(t))θs + Bu(t), (3.2)

ys(t) =Cxs(t),

n n here xm(t) ∈ R and xs(t) ∈ R are the state vectors for the master and the

m m slave systems, respectively. Similarly ym(t) ∈ R and ys(t) ∈ R are the output vectors. A ∈ Rn×n, B ∈ Rn×l and C ∈ Rm×n are the real constant matrices.

The vector functions f(x(t)) ∈ Rn and g(x(t)) ∈ Rl×p are the nonlinear functions,

p p θm ∈ R and θs ∈ R are the unknown parameters in the dynamics of the chaotic oscillators and u(t) ∈ Rl is the control input.

Assumption 3.1: The nonlinearities f(x(t)) and g(x(t)) present in the dynamics of the drive and the response systems i.e. (3.1) and (3.2) are assumed as

Lipschitz nonlinearities. The mathematical representation of nonlinearity to be

Lipschitz is elaborated in (3.3), (3.4), and (3.5). So, the nonlinearities of both the drive and the response systems i.e. f(x(t)) and g(x(t)) are assumed to bounded

51 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

nonlinearities

n kf(x(t) − f(¯x(t))k ≤ Lf kx(t) − x¯(t)k, ∀x(t), x¯(t) ∈ R , (3.3)

n kgm(xm(t)) − gm(¯xm(t))k ≤ Lgmkxm(t) − x¯m(t)k, ∀xm(t), x¯m(t) ∈ R , (3.4)

n kgs(xs(t)) − gs(¯xs(t))k ≤ Lgskxs(t) − x¯s(t)k, ∀xs(t), x¯s(t) ∈ R . (3.5)

Here gm(xm(t)) = Bg(xm(t))θm and gs(xs(t)) = Bg(xs(t))θs. Moreover Lf > 0,

Lgm > 0 and Lgs > 0 are the constant matrices of appropriate dimensions.

Most of the real world practical systems are nonlinear. The mathematical models of these systems for simul ation and control purpose usually involve linearization for making the model simpler. The simple linear assumption in modeling makes the models more erroneous for the practical real world systems. It is because of the reason that in simple linearization, linearization is made in the neighborhood of operating point. But the models behave diﬀerently when the system is operating in a long range of operation and the behaviour of the model becomes diﬀerent from actual systems. Lipschitz nonlinearity approach is the better option with help of which one can make models having more closer behaviour to practical systems.The nonlinearities which are continuous with bounded derivative are known as Lipschitz nonlinearities.

In previous years the interest in the studies of neuronal dynamics initiated the research process . Two approaches, widely used to investigate the neuron response to an external stimulus, are the Hodgkin-Huxley model [145] and its simpliﬁed version, the FitzHugh-Nagumo (FHN) model [146]. The Lipschitz nonlinearity is more common nonlinearity, present in most of the practical systems

52 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems such as single link manipulator with ﬂexible joints [147], synchronous generator connected to an inﬁnite bus [148], inverted pendulum FHN systems [149], transla- tional oscillator with rotating actuator (TORA) [150] and a direct current motor

DC motor [151]. Similarly, Hopeﬁeld neural network [152], FHN system [153] also contain Lipschitz nonlinearities. So the above-mentioned practical systems have

Lipschitz nonlinearities in the mathematical models. Therefore, for the synthesis of the synchronizing controller for nonlinear drive and response systems, we make the utilization of Assumption 3.1 and similarly the same assumption is accounted in the subsequent chapters.

A lot of work has been devoted to the synchronization of chaotic systems using state-feedback control approach. However, in many cases the exact states are not available. Therefore, estimated states of both the drive and the response systems can only be used for synchronization. The purpose of the present study is to device static and adaptive feedback control strategies for synchronization of the master and the slave systems (3.1)-(3.2) by employing the estimated states. For this reason, appropriate approaches for the state-estimation of the master and the slave systems using nonlinear and adaptive nonlinear observers are explored for an eﬃcient synchronization remedy.

3.3 Synchronization Control

The generalized model of nonlinear drive and response systems’ dynamics (3.1)-

p p (3.2) are augmented with the parameters θm ∈ R and θs ∈ R . The values of these augmented parameters may be known in some cases or unknown otherwise.

So, the problem of adaptive synchronization is divided in two cases. That is case 1: Adaptive feedback synchronizing of nonlinear drive and response systems with known parameters,

53 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

case 2: Adaptive feedback synchronizing of nonlinear drive and response systems

with unknown parameters.

As far as the case 1 is concerned, it deals the synchronizing problem of the

master-slave dynamics which contain known nonlinearities i.e. the nonlinearities

with constant parameters or predictable changing parameters. But case 2 refers

to such phenomenon where the parameters of the nonlinearities are unknown or

these nonlinearities are associated with unmodeled changing parameters. In both

cases of synchronization, state-feedback control approach is applied. The designed

control signal is of the form

u(t) = Ψ(ˆxm(t), xˆs(t)), (3.6)

n n herex ˆm(t) ∈ R andx ˆs(t) ∈ R are the estimates of xm(t) and xs(t), respectively.

The following text of this chapter elaborates the static and adaptive feedback control strategies using the control signal (3.6) for synchronization of the master and the slave systems (3.1)-(3.2) by employing their corresponding estimated states.

In addition, appropriate approaches for the state-estimation of the master and the slave systems using nonlinear and adaptive nonlinear observers are explored for an eﬃcient synchronization remedy.

3.4 Case1: Synchronization with Known Parameters

The master and the slave systems with known nonlinearities in their dynamics are to be synchronized. It means that the only possible unknown parameters in the state dynamics of the master and the slave system (3.1) and (3.2) i.e.

p p the augmented parameters θm ∈ R and θs ∈ R respectively, are known. The adaptation for these known constant valued augmented parameters is absurd and the adaptation laws are not needed. The synchronization of the master and the 54 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems response systems is striven through only applying state feedback control containing state estimation using observers.

3.4.1 Observer Based Feedback Control

The state feedback control can only be realized in a situation where the information about the fed-back states of the system is available. But in many practical scenarios especially in nonlinear chaotic systems where the state dynamics are very complex. It becomes very diﬃcult that all states of the system can be measured by sensors. In such situations, observers are used to estimate the unknown states of the system.

Observers are algorithms or the software code that use the available information of some of the states of the system (measured by sensors) and control input to produce the estimation of all or some of the states of the systems. Observers can be used to augment or replace sensors in a control system. The principle of an observer is that by combining a measured feedback signal with knowledge of the control-system components (primarily the plant and feedback system), the behavior of the plant can be known with greater precision then by using the feedback signal alone. The observer output signals can be more accurate, less expensive and more reliable than sensed signals. The observer can be used to enhance system performance. It can be more accurate than sensors or can reduce the phase lag inherent in the sensor. Observers can reduce system cost by augmenting the performance of a low-cost sensor so that both low cost sensor and observer together can provide the performance equivalent to a higher cost sensor.

A generic representation of observer based control scheme is shown in Figure

3.1. For the state estimation, the observer block requires two inputs i.e. control signal and plant output and in turn it produces the estimated state. The estimated

55 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Reference Error Control Control input Plant/ Output + _ Scheme Process

Estimated Control state input

Observer Output

Output

Figure 3.1: Observer based control scheme block diagram states are utilized in controller for synthesizing the control signal. This control signal is synthesized to minimize the error value eﬃciently. So, observer play a vital role in observer based control schemes as an alternative to the physical sensor.

For the sake of synchronization of the master and the slave systems (3.1)-(3.2), with known parameters as mentioned in case 1, coupled chaotic synchronous (CCS) observers are formulated. The detail of coupling architecture of the respective observers for the master and the slave systems and mathematical modeling of their dynamics is given in the following section.

3.4.2 Coupled Chaotic Synchronous Observers

The observers-based control for synchronization of the chaotic systems (3.1)-(3.2) under known dynamics, that is, by assuming θm = θs = 0 is made possible by the specially designed observers coupled with each of the master and slave system.

These observer which are coupled with the nonlinear chaotic systems as shown in

Figure 3.2 are named as coupled chaotic synchronous (CCS) observers.

Two blocks of this ﬁgure i.e. Master system with its CCS observer and Slave system with its CCS observer are interrelated through the estimated states of their respective observers. Beside this, the output of the master network is coupled

56 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

with the synthesized output of the observer for the master network. Similarly,

the output of the slave network is coupled with the synthesized output of the

observer for the slave network. Because of these couplings the whole scheme

is referred as coupled chaotic synchronization observer based control scheme for

synchronization. The mathematical model of the proposed observers (3.7), (3.8)

also elaborated the coupling terms. The term containing the diﬀerence of the

outputs of the master system and its respective observer associated with observer

gain matrix i.e. Lm(ym(t)−yˆm(t)) and Lm(ys(t)−yˆs(t)) show the output coupling.

1 1 Similarly, the terms − 2 BF (ˆxm(t) − xˆs(t)) and − 2 BF (ˆxs(t) − xˆm(t)) show the coupling of estimated states for the synthesizing the feedback control.

Figure 3.2: Coupling architecture for the observer based non-adaptive control scheme

57 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

˙ xˆm(t) =Axˆm(t) + f(ˆxm(t)) + Lm(ym(t) − yˆm(t)) (3.7) 1 − BF (ˆx (t) − xˆ (t)), 2 m s

˙ xˆs(t) =Axˆs(t) + f(ˆxs(t)) + Lm(ys(t) − yˆs(t)) (3.8) 1 − BF (ˆx (t) − xˆ (t)), 2 s m

n×m n×m where Lm ∈ R and Ls ∈ R are the gain matrices of the observers. The

output or the estimated states of these observers i.e.x ˆm(t) andx ˆs(t) are used in

the feedback control vector function Ψ(ˆxm(t), xˆs(t)).

Remark 3.1: The observers proposed in this thesis are different from the traditional observers. The traditional observers do not contain the coupling terms in their structures. The absence of coupling terms is disadvantageous for the synchronization purpose. The proposed observers (3.7)-(3.8), deliberately designed with the specific structures, are different from the traditional Luneberger- oriented observers owing to the specific coupling terms −0.5BF (ˆxm(t)−xˆs(t)) and

−0.5BF (ˆxs(t) − xˆm(t)). For a speciﬁc selection of F , the coupling strength can be increased enough so that the coupled observers achieve synchronization. Conse- quently the proposed observers are explicitly called as coupled chaotic synchronous observers and also has beneﬁt of the coupling terms.

3.4.3 Synchronization Error Dynamics for Known Parameters

The phenomenon of complete synchronization of the drive and response systems

(3.1)-(3.2) occurs if the all the states of both system match. Otherwise the leading and the following systems do not synchronize absolutely. The matching and mismatching of the states of the drive and the response systems can be quanti-

58 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

ﬁed by measuring the diﬀerence in the corresponding states of the both systems.

This diﬀerence between the corresponding states of the master-slave architecture

is known as error. So, the error between the corresponding states of the master

system and its observer is named as em(t). Similarly, the error between the cor-

responding states of the slave system and its observer is named as es(t). And the

error between the corresponding states of the observer for the master system and

observer for the slave system is named as eo(t). In short, the errors between different states of drive and response systems and their coupled observer are deﬁned as

em(t) = xm(t) − xˆm(t), (3.9)

es(t) = xs(t) − xˆs(t), (3.10)

eo(t) =x ˆm(t) − xˆs(t). (3.11)

Remark 3.2: The master and the slave chaotic systems can be made coherent by application of their respective observers. These observers for the master and the slave system produce the estimates of the states of the respective systems. Both estimates of the master and the slave systems are enforced to catch up the same behavior. This is done by applying the proposed control containing the estimated states of the drive and the response systems. The proposed control matches the corresponding estimated states of both systems such that the estimated synchronization error i.e. eo(t) =x ˆm(t)−xˆs(t) approaches to the zero. Once it happens, it is obvious that the states of the master and the slave observers are synchronized, consequently these observers are called as synchronous observers.

59 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Now, taking the derivative of both sides of (3.9), we get

˙ e˙m(t) =x ˙ m(t) − xˆm(t), (3.12)

and putting the values of time rate of change of state vector i.e.x ˙ m(t) and time rate ˙ of change of the estimate of the state vector i.e. xˆm(t), we get the synchronization error dynamicse ˙m(t) as

e˙m(t) =[Axm(t) + f(xm(t))] − [Axˆm(t) + f(ˆxm(t)) (3.13) 1 + L (y (t) − yˆ (t)) − BF (ˆx (t) − xˆ (t))]. m m m 2 m s

Further, after simpliﬁcation of the above equation for the error dynamics by putting the values of ym(t) = Cxm(t) andy ˆm(t) = Cxˆm(t), and rearranging the terms, the error dynamicse ˙m(t) equation is reformed as

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t). (3.14) m m m m m 2 o

Similarly, for calculation of the error dynamics between the corresponding states

of the slave system with its respective observer, process as follows. By taking the

derivative of both sides of (3.10) it results,

˙ e˙s(t) =x ˙ s(t) − xˆs(t), (3.15)

˙ and putting the values ofx ˙ s(t) and xˆs(t), the synchronization error dynamicse ˙s(t) is resulted as

e˙s(t) =[Axs(t) + f(xs(t))] − [Axˆs(t) + f(ˆxs(t)) (3.16) 1 + L (y (t) − yˆ (t)) + BF (ˆx (t) − xˆ (t))]. s s s 2 m s

60 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

After simplifying the above equation for the error dynamics by putting the val-

ues of ys(t) = Cxs(t) andy ˆs(t) = Cxˆs(t), and rearranging the terms, the error dynamicse ˙s(t) equation is reformed as

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t). s s s s s 2 o

Similarly, by taking the time derivative of both sides of (3.11) and putting the ˙ ˙ values of xˆm(t) and xˆs(t), we get the synchronization error dynamicse ˙o(t) as

e˙o(t) =[Axˆm(t) + f(ˆxm(t)) + Lm(ym(t) − yˆm(t)) 1 − BF (ˆx (t) − xˆ (t))] − [Axˆ (t) + f(ˆx (t)) 2 m s s s 1 + L (y (t) − yˆ (t)) + BF (ˆx (t) − xˆ (t))]. s s s 2 m s

Morover, after further simplifying, the error dynamicse ˙o(t) is resulted as

e˙o(t) =(A − BF )eo(t) + f(ˆxm(t)) − f(ˆxs(t)) + LmCem(t) − LsCes(t).

So, it is concluded that by using the systems (3.1)-(3.2), employing the observers (3.7)-(3.8), and incorporating the error equations (3.9)-(3.11), the resulting synchronization error dynamics can be grouped into a set of follwoing three equation (3.17), (3.18) and (3.19). These equations reveals that synchronization of the drive and the slave systems (3.1)-(3.2) can be established by means of convergence of the estimated synchronization error eo(t) to the origin which consequently converges the state-estimation errors em(t) and es(t) to the origin.

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t), (3.17) m m m m m 2 o

61 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t), (3.18) s s s s s 2 o

e˙o(t) =(A − BF )eo(t) + f(ˆxm(t)) − f(ˆxs(t)) + LmCem(t) − LsCes(t). (3.19)

While convergence of state-estimation errors em(t) and es(t) to the origin ensures that the estimated statesx ˆm(t) andx ˆs(t) approach to the actual states respectively. For complete synchronization of the master and the slave systems, these three error dynamics, ultimately converge to origin. Although, the rate of convergence will depend upon the control methodology. This rate of convergence is explored in the following sections of this chapter. The convergence of this error dynamics to the origin for absolute synchronization of drive and response systems is the main objective the proposed control methodology for synchronizing the nonlinear drive and response systems with known dynamincs is presented in the next section.

Now, proceed further for the derivation of the synchronization condition for the drive and the response systems using coupled chaotic synchronous (CCS) observers. The derived condition in mathematical form will be matrix inequality.

This condition is derived with an ultimate intention that the solution of matrix inequality will provide the suitable controller gain matrix F and the observer gain matrices Lm and Ls. The method for solving the matrix inequalities by using convex routines is discussed in Chapter 5.

Remark 3.3: A novel technique for synchronization of the master and the slave chaotic systems based on two observers for estimating states of both of the systems is presented, through which complete synchronization of the master-slave networks is achieved via utilizing their outputs rather than the exact states. In the recent

62 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems work [86], an error convergent observer-based synchronization technique has been proposed by employing estimation of the synchronization error. However, the main disadvantage or weakness of this approach is that it only deals with the chaotic systems for which the overall error system is transformable into a linear combination of various error dynamics. If linear transformation is unattainable, the approach may not lead to a feasible synchronization control.

In the following section, a more generic technique based on CCS observers and control input using estimated states is accomplished that can deal with the nonlinear error dynamics for complete synchronization of the master-slave oscillators.

Development of the proposed CCS observers-based control method is a non-trivial problem as compared to the existing observer-based techniques [39], [53], [40], [144] and [71]- [75] for synchronization because the present approach simultaneously estimates the states of both the master and the slave systems using CCS observers and controls the dynamics of the error system using a control input. Hence, the proposed synchronization technique is capable for two automations, that is, estimation of the states of the chaotic systems and synchronization of the chaotic systems.

3.4.4 Feedback Control for Synchronization

The proposed control signal is aimed so that the master and the slave chaotic networks demonstrate the identical behavior by employing information of the estimated states available from the respective coupled observers of the drive and response networks. For the case of known dynamics of the drive and response systems, the proposed control function is as follows:

Ψ(ˆxm(t), xˆs(t)) = F (ˆxm(t) − xˆs(t)). (3.20)

63 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

From above, it is clear that in addition to the estimates of coupled chaotic syn-

chronous (CCS) observers, controller gain matrix F ∈ Rl×n also plays a critical

role for synchronization of drive and response nonlinear systems. The procedure

for the determination of the value of controller gain matrix F ∈ Rl×n is very

critical and is discussed in Chapter 5.

Lyapunov Function: The Lyapunov energy function is the pivot point of the

Lyapunov theory and negative value of the time rate of change of this function

ensures convergence of the dynamics of the corresponding system to the origin.

In case of known values of parameters associated with the drive and the response

system, the following Lyapunov function is considered for derivation of synchro-

nization condition as presented in the following Theorem.

T T T (3.21) V1(t) = em(t)Pmem(t) + es (t)Pses(t) + eo (t)Poeo(t),

n×n here Pm, Ps, and Po ∈ R are the positive-deﬁnite symmetric matrices.

Theorem 3.1: For the given controller and observer gain matrices F ∈ Rl×n,

n×m n×m Lm ∈ R and Ls ∈ R , a suﬃcient condition for the synchronization of the master-slave networks (3.1)-(3.2), subject to Assumption 3.1, using the control law

(3.20) and CCS observers (3.7)-(3.8) is that there exist positive-deﬁnite symmetric matrices Pm, Ps, and Po of appropriate dimensions and scalars α1 > 0, α2 > 0

64 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

and α3 > 0 such that the matrix inequality

  T T Γ1 0 0.5PmBF + C L Po Pm 0 0  m     T T   ∗ Γ2 0.5PsBF − C Ls Po 0 Ps 0         ∗ ∗ Γ3 0 0 Po  Ω1 =   < 0, (3.22)    ∗ ∗ ∗ −α I 0 0   1 n       ∗ ∗ ∗ ∗ −α2In 0      ∗ ∗ ∗ ∗ ∗ −α3In

is satisﬁed, where

T T T 2 Γ1 = A Pm + PmA − C LmPm − PmLmC + α1Lf In,

T T T 2 Γ2 = A Ps + PsA − C Ls Ps − PsLsC + α2Lf In,

T T T 2 Γ3 = A Po + PoA − F B Po − PoBF + α3Lf In.

Proof: The time rate of change of the candidate Lyapunov energy function

(3.21) is

˙ T T T V1(t) =e ˙m(t)Pmem(t) + em(t)Pme˙m(t) +e ˙s (t)Pses(t) (3.23) T T T + es (t)Pse˙s(t) +e ˙o (t)Poeo(t) + eo (t)Poe˙o(t) .

By putting the value of

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t), m m m m m 2 o

1 e˙ (t) =(A − L C)e (t) + f(x (t)) − f(ˆx (t)) + BF e (t), s s s s s 2 o

e˙o(t) =(A − BF )eo(t) + f(ˆxm(t)) − f(ˆxs(t)) + LmCem(t) − LsCes(t).

65 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

we can write

˙ T T T V1 (t) ≤ [(A − C Lm)em(t)+

T [f(xm(t)) − f(ˆxm(t))]

T T T (3.24) + 0.5eo (t)B F ]Pmem(t)

T + em(t)Pm[(A − LmC)em(t)

+ [f(xm(t)) − f(ˆxm(t))] + 0.5BF eo(t)]

T T T + [(A − C Ls )es(t)

T +[f(xs(t)) − f(ˆxs(t))]

T T T + 0.5eo (t)B F ]Pses(t)

T + es (t)Ps[(A − LsC)es(t)

+ f(xs(t)) − f(ˆxs(t)) + 0.5BF eo(t)]

T T T T (3.25) + eo (t)(A − F B ) Poeo(t)

T + [f(ˆxm(t)) − f(ˆxs(t))] Poeo(t)

T T T T T T + em(t)C Lm − es (t)C Ls Poeo(t)

T + eo (t)Po (A − BF )eo(t)

+ f(ˆxm(t)) − f(ˆxs(t))

T + eo (t)Po [LmCem(t) − LsCes(t)] .

From Assumption 3.1, we can write the following inequalities for the Lipschitz nonlinearities as an integral part of nonlinear dynamics of the master system, the

66 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

slave system and their respective observers.

T 2 T − α1[f(xm(t)) − f(ˆxm(t))] [f(xm(t)) − f(ˆxm(t))] + α1Lf em(t)em(t) > 0,

T 2 T − α2[f(xs(t)) − f(ˆxs(t))] [f(xs(t)) − f(xs(t))] + α2Lf es (t)es(t) > 0,

T 2 T − α3[f(ˆxm(t)) − f(ˆxs(t))] [f(ˆxm(t)) − f(ˆxs(t))] + α3Lf eo (t)eo(t) > 0.

It is imperative to mention that the abovementioned inequalities, derived from the Lipschitz condition for f(x(t)), contain scalars α1 > 0, α2 > 0 and α3 >

0 as free variables that can be useful for feasibility of the design constraints.

By utilizing (3.4.4)-(3.4.4) into (3.25) and, subsequently, incorporating the above mentioned inequalities, one can obtain

˙ T T T T T T T V1 (t) ≤ [(A − C Lm)em(t) + [f(xm(t)) − f(ˆxm(t))] + 0.5eo (t)B F ]Pmem(t)

T + em(t)Pm[(A − LmC)em(t) + [f(xm(t)) − f(ˆxm(t))] + 0.5BF eo(t)]

T T T T + [(A − C Ls )es(t) + [f(xs(t)) − f(ˆxs(t))]

T T T T + 0.5eo (t)B F ]Pses(t) + es (t)Ps[(A − LsC)es(t)

+ f(xs(t)) − f(ˆxs(t)) + 0.5BF eo(t)]

h T T T T T i + eo (t)(A − F B ) + [f(ˆxm(t)) − f(ˆxs(t))] Poeo(t) (3.26) T T T T T T + em(t)C Lm − es (t)C Ls Poeo(t)

T + eo (t)Po [(A − BF )eo(t) + f(ˆxm(t)) − f(ˆxs(t))]

T + eo (t)Po [LmCem(t) − LsCes(t)]

T 2 T − α1[f(xm(t)) − f(ˆxm(t))] [f(xm(t)) − f(ˆxm(t))] + α1Lf em(t)em(t)

T 2 T − α2[f(xs(t)) − f(ˆxs(t))] [f(xs(t)) − f(xs(t))] + α2Lf es (t)es(t)

T 2 T − α3[f(ˆxm(t)) − f(ˆxs(t))] [f(ˆxm(t)) − f(ˆxs(t))] + α3Lf eo (t)eo(t).

67 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

After rearranging the terms on the right hand side of the last inequality, it can be

written as :

˙ T T T T T V1 (t) ≤ em(t)(A − C Lm)Pmem(t) + [f(xm(t)) − f(ˆxm(t))] Pmem(t)

T T T 2 T + 0.5eo (t)F B Pmem(t) + α1Lf Pmem(t) + em(t)Pm(A − LmC)em(t)

T T + em(t)Pm[f(xm(t)) − f(ˆxm(t))] + em(t)Pm0.5BF eo(t)

T T T T + es (t)(A − C Ls )Pses(t) (3.27) T T T T + [f(xs(t)) − f(ˆxs(t))] Pses(t) + 0.5eo (t)F B Pses(t)

T T + es (t)Ps(A − LsC)es(t) + es (t)Ps[f(xs(t)) − f(ˆxs(t))]

T T + 0.5es (t)PsBF eo(t) + eo (t)Po(A − BF )eo(t)

T T + eo (t)Po [f(ˆxm(t)) − f(ˆxs(t))] + eo (t)Po [LmCem(t) − LsCes(t)]

T − α1[f(xm(t)) − f(ˆxm(t))] [f(xm(t)) − f(ˆxm(t))]

T − α2[f(xs(t)) − f(ˆxs(t))] [f(xs(t)) − f(xs(t))]

2 T + α2Lf es (t)es(t)

T − α3[f(ˆxm(t)) − f(ˆxs(t))] [f(ˆxm(t)) − f(ˆxs(t))]

2 T + α3Lf eo (t)eo(t).

Furthermore, the simpliﬁcation of the inequality because of the cancellation of some terms on the right hand side of the with each other results into the following

68 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

inequality,

˙ T T T T V1 (t) ≤em(t)(A − C Lm)Pmem(t)

T + [f(xm(t)) − f(ˆxm(t))] Pmem(t)

T T T 2 + 0.5eo (t)F B Pmem(t) + α1Lf Pmem(t)

T + em(t)Pm(A − LmC)em(t)

T + em(t)Pm[f(xm(t)) − f(ˆxm(t))]

T T T T T + em(t)Pm0.5BF eo(t) + es (t)(A − C Ls )Pses(t)

T + [f(xs(t)) − f(ˆxs(t))] Pses(t)

T T T T (3.28) + 0.5eo (t)F B Pses(t) + es (t)Ps(A − LsC)es(t)

T + es (t)Ps[f(xs(t)) − f(ˆxs(t))]

T T + 0.5es (t)PsBF eo(t) + eo (t)Po(A − BF )eo(t)

T + eo (t)Po [f(ˆxm(t)) − f(ˆxs(t))]

T T + eo (t)PoLmCem(t) − eo (t)PoLsCes(t)

T − α1[f(xm(t)) − f(ˆxm(t))] [f(xm(t)) − f(ˆxm(t))]

T − α2[f(xs(t)) − f(ˆxs(t))] [f(xs(t)) − f(xs(t))]

2 T + α2Lf es (t)es(t)

T − α3[f(ˆxm(t)) − f(ˆxs(t))] [f(ˆxm(t)) − f(ˆxs(t))]

2 T + α3Lf eo (t)eo(t).

The above inequality can be written in the form of (3.29), which is the matrix form of above inequality . This type of inequlaity is also known as matrix inequality. Here, on the right hand side of (3.29) the product of matrices results into a scalar value. No doubt, the said matrix inequality manifest that the resultant

69 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

scalar value is always greater than the time rate of change of Lyapunov energy

function.

˙ T (3.29) V (t) ≤ E1 (t)Ω1E1(t),

where, ET (t) = T T T T T T , 1 em(t) es (t) eo (t) ∆fmmˆ (t) ∆fssˆ(t) ∆fmˆ sˆ(t)

∆fmmˆ (t) = f(xm(t)) − f(ˆxm(t)),

∆fssˆ(t) = f(xs(t)) − f(ˆxs(t)),

∆fmˆ sˆ(t) = f(xm(t)) − f(ˆxs(t)),   T T Γ1 0 0.5PmBF + C L Po Pm 0 0  m     T T   ∗ Γ2 0.5PsBF − C Ls Po 0 Ps 0         ∗ ∗ Γ3 0 0 Po  Ω1 =   ,    ∗ ∗ ∗ −α I 0 0   1 n       ∗ ∗ ∗ ∗ −α2In 0      ∗ ∗ ∗ ∗ ∗ −α3In

T T T 2 Γ1 = A Pm + PmA − C LmPm − PmLmC + α1Lf In,,

T T T 2 Γ2 = A Ps + PsA − C Ls Ps − PsLsC + α2Lf In,

T T T 2 Γ3 = A Po + PoA − F B Po − PoBF + α3Lf In.

˙ From (3.29), it is obvious that V (t) < 0 is ensured if Ω1 < 0 is satisﬁed. Hence

the error signals em(t), es(t) and eo(t) are asymptotically stable. Subsequently, the master and the slave systems (3.1)-(3.2) are synchronized, which completes the proof. The limitation of the above mentioned control scheme presented in Theorem

3.1 is that it is eﬀective for synchronizing the nonlinear drive and response systems, having nonlinearities, which are associated with the known parameter. But this 70 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems scheme becomes ineﬀective for the case where nonlinearities are associated with unknown parameters. Theorem 3.2 deals with such type of drive and response architectures having nonlinearities with unknown parameters.

Remark 3.4: Several observer-based synchronization approaches are available in the literature, in which an observer is used to estimate states of a single chaotic oscillator (see, for instance, [39], [40], [53], [144] and [71]- [75]). These so-called observer-based synchronization methodologies are very helpful for secure communication and image processing. However, the main disadvantages of these synchronization approaches is that these methods are inapplicable to two chaotic oscillators because there theme is to estimate states of a single oscillator, rather than to synchronize two independent oscillators. The proposed observers-based synchronization methodology, contrastingly, addresses a relatively diﬀerent synchronization perplexity, that is, synchronization of the two chaotic oscillators, and has broad applications.

Remark 3.5: The traditional observer-based synchronization schemes have a disadvantage of absence of the coupling terms in their traditional observers. Com- pared to the traditional observer-based synchronization approaches [39,40,53,71,

75,144], based on estimation of the unknown state vector of a single chaotic entity, the proposed observers in our work can be applied to synchronize the master-slave networks with unkonwn states using output feedback. The estimated states using CCS observers are utilitarian to synchronize the states of the master and the slave oscillators using a control signal (3.20). By virtue of the proposed CCS observers-based chaos synchronization scheme, exact state vectors are not required in contrast to the conventional methods (such as [52,55,56,87]) and output measurements can be employed for a static feedback control. It is notable that such

71 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems an observers-based chaos synchronization approach for two independent chaotic entities (3.1)-(3.2) is lacking in the literature. So, the CCS observer based synchronization is an addition to the existing control techniques for synchronization.

Remark 3.6: Recently, an observer-based synchronization approach is developed in [76] for synchronization of two chaotic oscillators by employing estimation of error between the master and the slave states by means of a linear state estimation error dynamics. This conventional approach is developed for a speciﬁc class of memristive systems, which is a disadvantage or weakness of this scheme, while the proposed observers-based synchronization method is applicable to a broad class of nonlinear systems. In addition, observer design for estimation of state errors becomes unmanageable in the present case because the estimation error dynamics is nonlinear. In contrast to [76] and similar approaches such as [154], [155] and [156], the proposed methodology demonstrates that synchronization of a broad class of chaotic systems resulting nonlinear state error dynamics can be achieved by application of CCS observers-based control strategy employing estimation of states of both oscillators. For elaborating matching of the dynamics of state variables of the both the master and slave systems, a simulation is performed for the FitzHugh-

Nagumo network.

3.4.5 Simulations and Results

The validity of the proposed techniques for the synchronization of the master and the slave systems, proposed in Theorems 3.1, is illustrated by a corrobo- rating simulation study for FitzHugh-Nagumo (FHN) master-slave architectures.

The FHN systems are utilized to understand behavior of multiple neurons under external electrical stimulation current, such as deep brain stimulation therapy

(see [55], [157] and [158]). Such therapies are used to overcome the symptoms like 72 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

tremor caused by neuronal disorders in the brain (such as Parkinsons disease and

Huntington disorder) because of malfunctioning of diﬀerent parts of the brain.

It is important to mention that FHN systems possess the impulsive behaviour

in their dynamics [159]. Therefore, error between two FHN systems can posses

impulsive behaviour. The FHN systems are described as follows:

x˙ m1 = xm1(xm1 − 1)(1 − r1xm1) − xm2 + Io,

x˙ m2 = bxm2,

x˙ s1 = xs1(xs1 − 1)(1 − r2xs1) − xs2 + Io,

x˙ s2 = bxs2.

Let the stimulation current is Io = (m/ω) cos(ωt) and the parameters are selected as r1 = 10.1, r2 = 9.9, b = 1 ,m = 0.1, ω = 2πf and f = 0.129. The initial conditions of both states, that is, normalized membrane potentials for the master and the slave systems are assumed to be xm,1(0) = 0.2 and xs,1(0) = 0.5, while recovery variables has initial state as xm,1(0) = 0.4 and xs,1(0) = 0.1. Phase portraits of both the master and slave systems are shown in Figure 3.3(a) and Figure

3.3(b), respectively, to endorse the chaotic behavior of FHN drive and response systems. The depicted phase portraits witness the fact that both the master and the slave systems demonstrate the aperiodic orbits in their dynamics which is the characteristics of chaotic systems. The above mentioned initial conditions for both systems show that at start, there is a diﬀerence between the corresponding states

(membrane potentials) of the master and the slave FHN system. The value of

Lipschitz constant is for all nonlinearities is assumed to be unity in accordance with Lipschitz assumption.

The time evolution for normalized potential i.e. xm1(t) and xs1(t) of the

FitzHugh-Nagumo systems drive and response systems are plotted in Figure 3.4.

73 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.3: Phase portraits of the master and the slave FHN systems : (a) phase portrait of the master system, (b) phase portrait of the slave system.

Figure 3.4: Time evolutions of the normalized membrane potentials

The state xm1(t) is the normalized potential and it is one of the states that are

constituting the dynamics of the master system. Similarly xs1(t) is the normalized potential for the slave system. This time evolution of normalized potentials

74 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems of both the drive and response systems exhibits the diﬀerence between these normalized potentials of the drive and the response systems throughout the time of simulation. Moreover the diﬀerence between the normalized potentials of the master and the slave systems are not uniform and nonlinear in nature. Similarly, the

Figure 3.5: Time evolutions of the recovery variables time evolution for recovery variables of the FHN system are plotted in Figure 3.5 which also witness the nonuniform diﬀerence in corresponding states i.e. xm2 and xs2 throughout the time of simulation. Where xm2 is the recovery variable for the master system and xs2 is the recovery variable for the slave system. For complete synchronization of the drive and response systems, these recovery variable have to be synchronized by applying the developed control techniques as presented in the later sections of this thesis. The time series for the errors i.e. e1(t) and e2(t) between the corresponding states of the master and the slave systems are plotted in Figure 3.6. From this ﬁgure, it is clear that with no control applied, errors e1(t) and e2(t) between the corresponding states of the master and response FHN chaotic systems continue to prevail and did not vanish for some interval of time. 75 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.6: Time evolution of errors between corresponding states of the master and the slave systems

A temporal evolution of the values of the states of master and the slave systems and the corresponding errors that is e1(t) and e2(t) manifest that both chaotic systems persistently contains some error during the course of dynamics and never synchronize or tend to synchronize in the absence of an external control input.

Although, these errors always remain bounded but the presence of even bounded errors causes non-synchronization. In other words, these bounded error dynamics is desired to go to origin for complete synchronization of the master and the slave system

Simulation and Results If the nonlinear chaotic master and slave systems have all the parameters known e.g. FHN system dynamics, in accordance with

(3.1)-(3.2), are given written as

    −1 −1 1     A =   ,B =   ,C = 1 0 , 1 0 0

76 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

  11x2 − 10x3 + m cos(ωt)  1 1 ω  f(x(t)) =   . 0

The synchronization problem can be dealt by employing non-adaptive approach propounded in Theorem 3.1. The observer gain matrices Lm and Ls and the

controller gain matrix F can be selected by using a similar two-step procedure

discussed in Theorem 5.1. By selecting

    1.3 1.3     Lm =   ,Ls =   ,F = 1 0 0 0

and applying the control law u = F (ˆxm − xˆs), the simulation results for CCS observer based control technique for synchronization are presented and discussed in the following paragraphs.

The simulation results for the phase portraits of both the drive and the response systems and their corresponding observers are shown in Figure 3.7. The plots aﬃrm the validity of the proposed scheme for synchronization of the master- slave chaotic systems by elaborating complete synchronization. Figures 3.7(a) and

3.7(b) show phase diagram of the FHN master and the slave networks, respectively.

Figure 3.7(c) and Figure 3.7(d) display the phase portraits of observed states of the master and the slave systems, respectively. The simulation results in ﬁgure

3.7 manifest the synchronization of the corresponding states of the master system with its observers and also the corresponding states of the slave system with its observer. The initial conditions as discussed in the beginning of this section of simulation and results were diﬀerent. It means that the corresponding states of the drive system and the response system with the corresponding states of the respective observer were not same at the start. But after applying the designed control scheme for synchronization of the drive and the response systems (3.1)

77 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.7: Phase portraits of the master and the slave FHN systems, and phase portraits of their corresponding observers using the approach provided in Theorem 3.1 : (a) phase portrait of the master system, (b) phase portrait of the slave system, (c) phase portrait of the master observer, (d) phase portrait of the slave observer.

and (3.2) with known parameters, the above mentioned corresponding states of

the drive system and response system converge to the same values.

Similarly, Figure 3.8 and Figure 3.9 represent the temporal evolution of the

normalized potentials of both the master and the slave systems and their respec-

tive observers i.e. xm1,x ˆm1, xs1 andx ˆs1.The temporal evolution for Normalized

potentials of the FHN system are plotted in Figure 3.8. The temporal evolution

of normalized potentials veriﬁes the claim of synchronization by the applied con-

trol. As the plot for all the states of the master and the slave systems and their

respective observers converge to same values.

The temporal evolution of the recovery variables of both the master and the

slave systems and their respective observers i.e. xm2,x ˆm2, xs2 andx ˆs2 are plotted in Figure 3.8 which also witness the complete synchronization of the recovery 78 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.8: The time evolution of membrane potentials of the master and slave systems and their corresponding observers variables of the master and the slave systems and their corresponding observers.

Figure 3.8 presents the fact that during the time of simulation, once the recovery variables are synchronized by applied control, the master and the slave systems’ recovery variables remain synchronized throughout the time of simulation. Ini- tially, the master and the slave system are started with different initial conditions which can be witnessed in Figure 3.8. But as the time goes on, under the effect of applied control, the different states of the master and the slave system and their respective observers converge to same values in the early part of simulation time.

It is worth mentioning the time required to completely synchronize depends upon the controller gain matrix F . In other words, the rate of synchronization can be altered by selecting diﬀerent values of F .

Figure 3.10 witness the fact that by application of the proposed control law, various synchronization errors between the normalized potential of the master system and its observer i.e., em1, recovery variable of the master system and its

79 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.9: The time evolution of recovery variables of the master and slave systems and their corresponding observer

Figure 3.10: Convergence of the synchronization errors

observer i.e., em2, between the normalized potential of the slave system and its observer i.e., es1, recovery variable of the slave system and its observer i.e., es2, between the normalized potential of the both observers i.e.. eo1 and between the

80 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

recovery variables of the both observers i.e., eo2 asymptotically convergence to the zero. From these results of Figure 3.10, one can claim the eﬀective synchronization of both the master (drive) and slave (response) systems. The time required for complete synchronization depend upon the coupling strength between the drive and response systems to their respective observers. This coupling strength can be altered by changing the value of the observer gain matrices i.e. Lm and Ls.

From (3.17) and (3.18) it is clear that the values of observer gain matrices

Lm and Ls have explicit eﬀects on the em and es. Similarly, from (3.19), we

can mention the value of F has an explicit eﬀect on eo. That is why, with the

change of the gains, the trend of synchronization for the nonlinear control scheme

is elaborated in Figure 3.11 and Figure 3.12. It is worth mentioning that the eﬀect

of synchronizing control scheme in case of FHN system only aﬀects em1 and eo1

and it does not aﬀect em2 and eo2. So, only the simulation results for the values

of em1 and eo1 are depicted in the following Figures 3.11 and 3.12. The eﬀect of

variation of Lm = Ls on em1 is elaborated in ﬁrst ﬁgure. From which it is clear

that by increasing the value of Lm = Ls, the rate of error convergence increases

and vice versa. Similarly, Figure 3.12 depicts the eﬀect of change of controller

gain values i.e. F on the eo1, which shows that rate of convergence increases with

the increase in value of F

3.5 Case2: Synchronization of Nonlinear Systems with Unknown Parameters

If the dynamics of the master and the slave systems contain unknown parameters

p θm,θs ∈ R , the synchronization of the both systems cannot be achieved by (3.20)

only, rather we have to use adaptation laws along with the speciﬁc selection of ˆ ˆ control law u = Ψ(ˆxm, xˆs, θm, θs). Here,x ˆm, xˆs are the estimates of the states of

81 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.11: Error between normalized potentials of master system and its observer for diﬀerent values of observer gains and ﬁxed value of controller gain

Figure 3.12: Synchronization error eo1 for diﬀerent values of controller gains and ﬁxed values of observer gains the systems. These estimates are made possible by the coupled chaotic adaptive synchronous (CCAS) observer which are coupled with the respective systems.The

82 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems complete architecture of the synchronizing control scheme using CCAS observers.

These observers along with the adaptation laws are depicted in Figure 3.13.

Figure 3.13: Architecture for CCAS observers based synchronization scheme

This ﬁgure is composed of ﬁve blocks presented with the dashed lines. These include the master system with its coupled adaptive synchronous (CCAS) observer.

Second block is the slave system with its (CCAS) observer. Third and fourth blocks are the adaptation law blocks and last is the control law block. First block, the master system with CCAS observer also contains low pass ﬁlter at the output of drive system and same is the case with second block, i.e., the slave system with its CCAS observer. It is important to mention that the coupling strength of both the master and the slave systems with their respective observers depends upon the observer gain matrices, i.e., Lm and Ls. So, the value of the observer gain matrices 83 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems should be optimized which is possible by the LMI constraint proposed in Theorem

3.2. The coupling strength effects the rate of synchronization of coupled systems to be synchronized. If the coupling strength increases then the master and the slave systems get synchronized early in time and vice versa. The adaptation law blocks are fed by the first two blocks with the output difference, i.e., ym(t)−yˆm(t).

The adaptation law blocks provide the adapted values of unknown parametersx ˆm andx ˆs to control block in the architecture of synchronizing methodology. The control gain matrix F plays a vital role and its value can be achieved by using the linear matrix inequality constraints presented in Theorem 3.2.

3.5.1 Coupled Chaotic Adaptive Synchronous Observers

The main contribution of Chapter 3 is the adaptation of uncertain parameters present in the nonlinear dynamics by suggesting simple adaptation laws. These adaptation laws are employed along with the proposed control signal based on

CCAS observers for complete synchronization of the master-slave systems. The

CCAS observers-based synchronization schemes with the adaptive controllers to simultaneously estimate and synchronize states of two chaotic systems are not fully elaborated in the literature. Coupled chaotic adaptive synchronous observers are aimed for their state estimation under unknown parameters. The proposed nonlinear adaptive observers for the master-slave systems are structured with the ﬂexible coupling strength. The strength of coupling can be varied by changing the value of observer observer gain matrices. The eﬀect of the variations of the observer gain matrices Lm and Ls increases or decreases the coupling strength of the observers with their respective master or drive system. The complete architectures of the

CCAS observers for both the master and the slave systems are

84 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

˙ h ˆ i xˆm =Axˆm(t) + f(ˆxm(t)) + Bg (ˆxm(t)) θm + Lm (ym(t) − yˆm(t)) (3.30) 1 − BF (ˆx (t) − xˆ (t)), 2 m s

˙ h ˆ i xˆs =Axˆs(t) + f(ˆxs(t)) + Bg(ˆxs(t))θs + Ls(ys(t) − yˆs(t)) (3.31) 1 − BF (ˆx (t) − xˆ (t)) + Bu . 2 s m g

n×m n×m where Lm ∈ R and Ls ∈ R are the gain matrices of the observers ˆ ˆ and ug is the nonlinear component of u, that is, ug = g(ˆxm(t))θm − g (ˆxs(t)) θs. ˆ ˆ The values of the parameters of proposed observers i.e. θm and θm are unknown and their values are adapted by means of adaptation laws. For this reason, the architecture of these observers is given the name of coupled chaotic adaptive synchronous (CCAS) observers. It is important to mention here that for the sake of synchronization the convergence of eo(t), em(t) and es(t) to the origin is ensured which means that all the corresponding states of the master and the slave systems are matched. And also, the proposed coupled observers (3.30)-(3.31) are capable of handling unknown parameters during estimations of the state vectors.

Remark 3.7: Unequivocally, it is worth noting that CCAS observers are more generic than the CCS observers developed in the previous section because these

CCAS observers can deal with the nonlinearities of two types, that is, nonlinearities with the known parameters and nonlinearities with the unknown parameters.

However, the main snag of the CCAS observers is their slow response because of ˆ ˆ adaptation of parameters θm(t) and θs(t). If all the parameters are known, it is better to use CCS observers for a fast convergence. However, CCAS observers

85 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems are utilitarian for adaptive synchronization of chaos due to practical limitations in parametric measurements.

3.5.2 Synchronization Error Dynamics for Unknown Parameters

The objective of synchronization can be interpreted in terms of convergence of error dynamics between the corresponding states of the drive-response architecture to the origin. Alternately, the errors between the states of coupled observers can help for the synchronization of the two nonlinear systems in case of unavailability of measurement of the states of the drive and the response systems. The synchronization error dynamics can be formulated using equations (3.9)-(3.11). After taking the time rate of change of (3.9)-(3.11) and the putting the values of state dynamics of the drive and the response systems (3.1)-(3.2) and states of their respective observers i.e. (3.7)-(3.8), the following error dynamics are revealed.

These error dynamics include the time derivative of the error between the states of the master system and its observer, the time derivative of the error between the states of slave system and its observer and the time derivative of the error between the states of the master observer and the slave observer. The novelty of this work for synchronization of nonlinear drive and response system lies in the novel architecture of these error dynamics.

e˙m(t) =Aem(t) + [f(xm(t)) − f(ˆxm(t))]

ˆ + Bg(xm(t))θm − Bg(ˆxm(t))θm (3.32) 1 − L Ce (t) + BF e (t). m m 2 o

The time rate of change of the error between the corresponding states of the slave system and its observer is deﬁned by es(t). This error dynamics formulated

86 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

as,

e˙s(t) =Aes(t) + [f(xs(t)) − f(ˆxs(t))]

ˆ + Bg(xs(t))θs − Bg(ˆxs(t))θs (3.33) 1 − L Ce (t) + BF e (t). s s 2 o

Similarly, the time rate of change of the error between the corresponding states of

the master observer and the slave observer can be written as:

e˙o(t) =Aeo(t) + [(ˆxm(t)) − f(ˆxs(t))]

ˆ ˆ + Bg(ˆxm(t))θm − Bg(ˆxs(t))θs (3.34)

+ LmCem(t) − LsCes(t)

− BF eo(t) − Bug.

˜ ˆ Applying θm = θm − θm and gm(xm) = Bg(xm(t))θm and, further, employ- ˆ ing the mathematical fact Bg(xm(t))θm − Bg(ˆxm(t))θm = gm(xm) − gm(ˆxm) + ˜ Bg(ˆxm(t))θm, we obtain

e˙m(t) =(A − LmC)em(t) + [f(xm(t)) − f(ˆxm(t))] 1 + BF e (t) + g (x ) − g (ˆx ) (3.35) 2 o m m m m

+ Bg(ˆxm(t))θem.

Similarly, it is implicit to achieve

e˙s(t) =(A − LsC)es(t) + [f(xs(t)) − f(ˆxs(t))] 1 + BF e (t) + g(x (t)) − g(ˆx (t)) (3.36) 2 o s s

+ Bg(ˆxs(t))θes.

87 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

ˆ ˆ Using ug = g(ˆxm(t))θm − g (ˆxs(t)) θs, we obtain

e˙o(t) =(A − BF )eo(t) + [f(ˆxm(t)) − f(ˆxs(t))] (3.37)

+ LmCem(t) − LsCes(t).

3.5.3 Feedback Control for Adaptive Synchronization

As the dynamics of the master and the slave chaotic systems contain unknown

p parameters (that is, θm, θs ∈ R ), the synchronization of both systems cannot be achieved by the control law (3.20). Rather we have to use adaptation laws

(as presented in Theorem 3.2), along with the speciﬁc selection of control law u(t) = Ψ(ˆxm(t), xˆs(t)) as,

ˆ Ψ(ˆxm(t), xˆs(t)) =F (ˆxm(t) − xˆs(t)) + g(ˆxm(t))θm(t) (3.38) ˆ − g (ˆxs(t)) θs(t),

ˆ p ˆ p p θm ∈ R and θs ∈ R are the estimates of the unknown parameters of θm ∈ R and

p θs ∈ R respectively. For the realization of the proposed CCAS observer based synchronization scheme, let us introduce the following assumption for derivation of condition as presented in Theorem 3.2

T ⊥ T ⊥ ⊥ Assumption 3.2: Let B PmC = 0 and B PsC = 0, where C denotes the orthogonal projection on to the null of C.

t If Assumption 3.2 holds, matrices Rm and Rs can be selected by solving B Pm−

t RmC = 0 and B Ps − RsC = 0. Now we provide an adaptive controller design condition using CCAS observers.

88 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

3.5.4 Amended Lyapunov Function

The previously deﬁned Lyapunov function (3.21) is unable to develop a synchro-

p p nizing condition in a situation of unknown parameters θm ∈ R and θs ∈ R , so an amended Lyapunov energy function (3.39) is used for this purpose.

T T T V2(t) =em(t)Pmem(t) + es (t)Pses(t) + eo (t)Poeo(t) (3.39) ˜T −1 ˜ ˜T −1 ˜ + θm(t)Θm θm(t) + θs (t)Θs θs(t).

The synchronization error convergence condition, for the adaptive coherence (synchronization) of drive and response systems using CCAS observer (3.30)-(3.31) and adaptive control (3.38) is presented in Theorem 3.2.

Theorem 3.2: For the given controller and observer gain matrices F ∈ Rl×n,

n×m n×m Lm ∈ R and Ls ∈ R , a suﬃcient condition for the synchronization of the

p p master-slave networks (3.1)-(3.2) with unknown parameters θm ∈ R and θs ∈ R , subject to Assumptions 1-2, using the control law in (3.38), and CCAS observers

(3.30)-(3.31) along with the adaptation laws, given by

ˆ˙ T θm(t) = −Θmg (ˆxm(t))Rm(ym(t) − Cxˆm(t)), (3.40)

ˆ˙ T θs(t) = −Θsg (ˆxs(t))Rs(ys(t) − Cxˆs(t)), (3.41)

where Θm and Θs are the adaptation rates of appropriate dimensions, is that there exist positive-deﬁnite matrices Pm, Ps, and Po and scalars α1, α2, α3, β1 and β2, such that matrix inequality

  Φ11 Φ12   Φ =   < 0, (3.42) T Φ12 −diag{α1In, α2In, α3In, β1In, β2In}

is satisﬁed, where 89 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

  Υ 0 0.5P BF + CT LT P  1 m m o    Φ =  T T  , (3.43) 11  ∗ Υ2 0.5PsBF − C Ls Po      ∗ ∗ Υ3

  P 0 0 P 0  m m    Φ =   , (3.44) 12  0 Ps 0 0 Ps      0 0 Po 0 0

T T T 2 2 Υ1 = A Pm + PmA − C LmPm − PmLmC + α1Lf In + β1LgmI,

T T T 2 2 Υ2 = A Ps + PsA − C Ls Ps − PsLsC + α2Lf In + β2LgsI,

T T T 2 Υ3 = A Po + PoA − F B Po − PoBF + α3Lf In.

Proof: The proof of Theorem 3.2 is presented in Appendix A.

It is important to note that above mentioned control methodology has limitations and it is valuable only for synchronization of the drive and response systems which have no disturbances in their dynamics and these systems contain the nonlinearities with unknown parameters. The unknown parameters values are adapted by the proposed adaptation laws presented in Theorem 3.2. But the eﬀect of disturbances in the process of synchronization is not catered in this scheme and this issue is addressed in the next chapter

Remark 3.8: The scope of the proposed observers-based synchronization methodologies diﬀer from the conventional synchronization schemes like [6, 40, 53, 55, 56, 87, 140, 158] and [19, 57, 77, 157, 158]. The conventional observer-based approaches are used to estimate the state vector of a chaotic system and can be employed to speciﬁc scenarios like secure communication. The

90 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems proposed methodologies are useful for monitoring through state-estimation as well as controlled synchronization of the two master-slave systems and have versatile applications. It is worth mentioning that the proposed observers are diﬀerent from the conventional Luenberger-type and adaptive observers owing to the presence of coupling terms employed to achieve chaos synchronization. It should be noted that these CCS observers (discussed in previous section) and CCAS observers are speciﬁcally designed by aiding the coupling terms. The application area of the proposed methodologies is broader than the conventional chaos synchronization approaches, which require exact states of the master-slave systems for feedback control. The proposed chaos synchronization approaches can be applied to the chaotic systems using information of their outputs, when states are not available, and synchronization is achieved using feedback of the estimated states (as well as the parametric estimates in the adaptive synchronization case).

3.5.5 Simulation and Results

The eﬃcacy of the proposed techniques for the synchronization of the master and the slave systems with unknown parameters as proposed in Theorems 2, is illustrated in the following simulation results for FitzHugh-Nagumo (FHN) master- slave architectures. If the value of the parameters r1 and r2 are unknown, one can assign θm = r1 and θs = r2. The FHN systems can be compared with generalized model of the nonlinear master and the slave chaotic systems (3.1)-(3.2). From this comparison, one can deduce the stat-space matrices for the FHN system as follows:     −1 −1 1     A =  , B =  , C = 1 0 , 1 0 0

91 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

  x2 + m cos(ωt)  1 ω  f(x(t)) =  . 0

It is important to mention that to increase the feasibility of the design constraints, the matrix A and the nonlinearity matrix f(x(t)) of the FHN system are modiﬁed as:

    −1 −1 x2 + m cos(ωt)    1 ω  A =  , f(x(t)) =  , 1 −0.3 0.3x2

without loss of generality. By application of Theorem 3, we obtain

    3.286 3.286     Lm =  , Ls =  , F = 1.409 0.7 . 0 0

By applying the control and adaptation laws in (3.38) and (3.40)-(3.41), simulation results are elaborated as follows. Figure 3.14(a) and 3.14(b) show phase portraits of the master and the slave FHN systems. These phase portraits help in understanding the complete synchronization phenomenon of the drive and the response systems (3.1)-(3.2) under the effect of applied control. These phase portraits establish the fact that at initial conditions, both the drive and the response starts their dynamics with different values of the corresponding states. But as the time passes, the control action compels both the systems to similar corresponding states. Similarly, Figure 3.14(c) and 3.14(d) characterize the phase portraits of observed states of the master and the slave systems. The corresponding observed states of the respective observer for the master and the slave systems are also synchronized by the applied control. This applied control has a direct effect on the observed states of the purposefully designed observers (3.30)-(3.31) for the drive

92 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems and the response systems. The structure of these observers witness that controller gain matrix F is also the part of each observer along with the observer gain matrix Lm or Ls.The value of Lipschitz constant is for all nonlinearities is assumed to be unity in accordance with Lipschitz assumption. The temporal evolution of

Figure 3.14: Phase portraits of the master and the slave FHN systems, phase portraits of their corresponding observers : (a) Phase portrait of the master system, (b) phase portrait of the slave system, (c) phase portrait of the master observer, (d) phase portrait of the slave observer.

Normalized potential of the master-slave systems and their respective observers is demonstrated in Figure 3.15. This temporal evolution strengthens the claim of eﬀectiveness of the applied control. The normalized potentials of the master system, the slave system and their respective observers converge gradually to the same values after applying control at time t = 0

Similarly, Figure 3.16 demonstrate the temporal temporal evolution of recovery variable of the master-slave systems and their respective observers. This ﬁgure also proves the eﬃcacy of the applied control by depicting the convergence of recovery

93 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.15: Temporal evolution of membrane potentials of the master and slave systems and their corresponding observers

Figure 3.16: Temporal evolution of recovery variables of the master and slave systems and their corresponding observers variables of the master system, the slave system and their respective observers to the same values.

94 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

The complete synchronization can be aﬃrmed by plotting the time evolution

of the errors (3.9), (3.10) and (3.11) as shown in Fgure 3.17. This ﬁgure demon-

strates that by application of the proposed control (3.38) and adaptation laws

(3.40), (3.41), complete synchronization of the master system and the slave sys-

tem is attained. Actually, the applied control explicitly converges the observer

of the master system and the slave system to the same states. This explicit syn-

chronization of the observers implicitly synchronizes the drive and the response

system. Fgure 3.17 elaborates the whole phenomenon by showing the errors em1,

em2, es1, es2, eo1, eo2 convergence to the origin. The convergence of errors eo1, eo2 to the origin by applied control (3.38) means the normalized potential state and the recovery variable state of the respective observers of the master and the slave system synchronizes with each other. Also, convergence of errors em1, em2

Figure 3.17: Error dynamics of the master and the slave systems

to the origin means the synchronization of the both states of the master system

to the respective states of its observer. Similarly, convergence of errors es1, es2 to

95 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

the origin means the synchronization of the both states of the slave system to the

respective states of its observer.

Simulation results for the adaptation of the unknown parameters are plotted

in Figure 3.18, which shows convergence of the adaptive parameters to their true

values, that is θm = r1 = 10.1 and θs = r2 = 9.9. The true value means the actual

Figure 3.18: Adaptation of unknown parameters θm and θs

values for the model FHN system, and these values may vary under diﬀerent phys-

ical conditions, in which the FHN system may operate. By the change of these

true values of parameters of the FHN system, the behavior of the FHN system

changes. Any how, in the present simulation environment, the values for parame-

ters θm and θs is opted as θm = r1 = 10.1 and θs = r2 = 9.9. The adaptation laws

(3.40) and (3.41) play vital role in the determination of these parameters which are unknown to the controller. These adaptation laws gradually adapt the unknown parmeter and send the adapted values to the controller continuously through the course of synchronization. The rate of adaptation of unknown parameters θm and

θs can be varied by changing the values of Θm and Θs present in laws (3.40) and 96 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

(3.41). The rate of adaptation also eﬀects the performance of controller. So, we

have to choose a suitable adaptation rate depending upon the speed of controller

for computation.

It is noteworthy that synchronization behavior varies with the change of the

observer gain matrices Lm and Ls and the controller gain matrix F . The rate

of synchronization or the extent of synchronization in same time duration may

be different for different values of observer gain matrices Lm and Ls or controller gain matrix F . Figure 3.19 shows dependency of extent of synchronization in same time duration on the observer gain matrices Lm and Ls , by keeping the controller gain matrix as fixed. The error em1 is plotted for Lm = T T T Ls = 1 0 , 2 0 , 8 0 by keeping the controller gain matrix F = T 1.409 0.7 as fixed. The increase in the magnitude of observer gain matrix as shown above increases the extent synchronization and decreases the synchronization error in same time interval of simulation as shown in Figure 3.19. Though

Figure 3.19: Eﬀect of observer gain matrices Lm and Ls on the error dynamics

em1

97 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

Figure 3.20: Eﬀect of controller gain matrix F on the error dynamics eo1

matrix on the synchronization phenomenon is shown in Figure 3.20. By keeping T the observer gain matrices Lm = Ls = 13.286 0 as fixed and changing the T T T controller gain matrix as F = 1 0 , 2 0 , 8 0 the synchronization error e01 becomes smaller respectively as shown in the following figure . It is worth mentioning that the effect of synchronizing control scheme in case of FHN system only affects em1 and eo1 and it does not affect em2 and eo2. So,only the

simulation results for the values of em1 and eo1 are depicted in the following Fig-

ures 3.11 and 3.12. The eﬀect of variation of Lm = Ls on em1 is elaborated in ﬁrst

ﬁgure. From which it is clear that by increasing the value of Lm = Ls, the rate

of error convergence increases and vice versa. Similarly, Figure 3.12 depicts the

eﬀect of change of controller gain values i.e. F on the eo1, which shows that rate

of convergence increases with the increase in value of F 98 Chapter 3: Adaptive Feedback Synchronizing Control for Nonlinear Systems

3.6 Conclusion

A new approach for the synchronization of the master-slave chaotic systems by means of CCS observers and CCAS observers-based control schemes is propounded in this chapter. The proposed techniques are applicable to attain multi-objectives, that is, estimation of states of chaotic systems and control of the synchronization errors in the absence and the presence of unknown parameters. The novel CCAS observers presented are more generic than the CCS observers but with a snag of slow response because of adaptation of the unknown parameters. By means of Lya- punov stability theory, a convergence condition for the synchronization errors is developed in the form of nonlinear matrix inequalities. The recommended methodologies for synchronization are dissimilar with the conventional chaos synchronization approaches, requiring exact states of the master-slave systems for feedback control. Numerical simulation for synchronization of FitzHugh-Nagumo neuronal systems is illustrated to demonstrate eﬀectiveness of the proposed observers-based chaos synchronization control methodologies.

99 Chapter 4 Robust-adaptive feedback synchronizing control for nonlinear drive and response systems

In this chapter, the modified adaptation laws are presented. The adaptation laws presented in chapter 3, are modified using σ-modification. The use of low pass

ﬁlters for ﬁltering the high frequency measurement noise at the output is also presented.

4.1 Introduction

These robust controller attempts to maintain system performance in the existence of perturbations. Notwithstanding an adaptive controller stab to make an online guesstimate of the uncertainty augmented with the systems dynamics and then impart a control input to diminish the unwanted deviances from the prescribed systems behavior. But an adaptive scheme is not the eventual control methodology to realize the dynamics of real world systems according to demanded dynamics.

The robust-adaptive synchronization is the phenomenon of concern in many real world problems. The previous couple of decades witness synchronization of chaotic networks in the master and the slave formalism is topic of interest for researchers in nonlinear science, as it possesses a valuable position for applications in

100 Chapter 4: Robust-Adaptive Feedback Synchronizing Control communications, optics and lasers, robotics, secure communication [160], information science [70], electronic circuits [161], neurophysiology [162]. Researchers have rendered many ideas for synchronization of nonlinear chaotic systems e.g. sliding mode observer based techniques, using Huygens coupling for synchronization, utilizing linear feedback scheme for synchronization, back stepping techniques, adaptive generalized projective synchronization [61], delay dependent methodologies [56], evolutionary algorithms [62] and robust adaptive methodologies [60], are applied for synchronization of the chaotic network. So, we may proclaim that many versatile synchronization techniques have been established, tested and implemented, for example feedback control, intermittent control, impulsive control coupling control. The observer based synchronizing control techniques are accounted as an important contributor in larger class of synchronization methodologies. These observer based methodologies include fuzzy disturbance observer [163] methodologies, sliding mode observer based techniques [164], adaptive sliding observer based schemes [165] etc. A robust-adaptive control is a methodology to develop controllers for a category of systems with disturbances and parameter uncertainties to get required performance and this is done by the modiﬁcation in the scheme presented in the previous chapter.

In the Chapter 3, a new idea of synchronous observers, that is, coupled chaotic synchronous (CCS) observers and coupled chaotic adaptive synchronous (CCAS) observers [42] had been introduced for synchronization of chaotic systems. The synchronization using coupled chaotic observers enables the estimation of unavailable states and subsequently the use of those states in the feedback control signal for asymptotic convergence of synchronization error. As the observers are coupled with the drive and the response chaotic systems, so these observers were called coupled chaotic. There are also some deﬁciencies in the synchronizing control

101 Chapter 4: Robust-Adaptive Feedback Synchronizing Control elaborated in Chapter 3. The foremost shortcoming is that this scheme cannot accommodate the eﬀect of disturbances and noises in the dynamics of the systems and in the output measurement respectively. So, the amendments in the control scheme presented in Chapter 3 are done. These amendments become valuable for achieving the robustness feature against the unknown disturbances aﬀecting the dynamics of the drive and the response formalism. Also, with help of these amendments the output noises are purged out.

The important contribution of this chapter 4 is the modification of adaptation laws already presented in Chapter 3, using σ-modification along with the use of low pass filters for filtering the high frequency measurement noise at the output.

This chapter is arranged as follows. First of all, problem statement is presented which describes the systems with some assumptions. Subsequent is the discussion about some available techniques for achieving robustness. After this proposed robust-adaptive scheme is presented. At the end of chapter, simulation section provides the results that verify the authenticity of the presented scheme.

4.2 Problem Formulation for the Robust-Adaptive Feed- back Synchronizing Control

The drive and the response systems, which are to be synchronized are taken as nonlinear in their dynamics. The mathematical model of these homogeneous nonlinear systems are presented in (4.1) and (4.2). These generic models represent equations for the state dynamics and also the output dynamics for the drive and the response systems respectively. Both the state dynamics and output dynamics in each of the master and the slave system contain the disturbances and the noise respectively. Also, two types of nonlinearities are involved in the state dynamics of the drive and the response systems, that is nonlinearities with known parameters

102 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

and nonlinearities with unknown parameter.The unknown quantities augmented

in the dynamics of the master and the slave system demand the robust-adaptive

control synchronization of the master and the slave systems. Mathematical repre-

sentation of the dynamics of both the drive and the response system is provided

here:

x˙ m(t) =Axm(t) + φ(xm(t)) + Bξ(xm(t))Pm + dm, (4.1)

ym(t) =Cxm(t) + ηm,

x˙ s(t) =Axs(t) + φ(xs(t)) + Bξ(xs(t))Ps + ds + Bu(t), (4.2)

ys(t) =Cxs(t) + ηs.

n n The state vectors xm(t) ∈ R and xs(t) ∈ R represent the state vectors of

z the drive and the response systems respectively. The vectors ym(t) ∈ R and

z n×m ys(t) ∈ R represent the output vectors. The real constant matrices A ∈ R ,

B ∈ Rn×m and C ∈ Rz×n nominate the diﬀerent parameters of the systems. The

n n z z nonlinearities φ(x(t)) ∈ R , Bξ(xs(t))P ∈ R , ηs ∈ R , dm ∈ R and ξ(x(t)) ∈

m ×p m×p R are inherited by the systems. Where ξ(xs(t)) ∈ R is the nonlinearity associated with unknown parameters Pm and Ps. The unknown parameters Pm ∈

p p R ,Ps ∈ R are the factors of later nonlinear terms as presented in (4.1)and

z z z z (4.2). Similarly ηm ∈ R , ηs ∈ R output noise vectors and dm ∈ R , ds ∈ R are disturbance vectors for the drive and response networks respectively. u(t) ∈ Rm is the control input.

Assumption 4.1: The nonlinearities φ(x(t)) and ξ(x(t)) augmented with the drive and the response systems are considered as Lipchitz nonlinearities. The

103 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

mathematical representation of nonlinearity to be Lipchitz is elaborated in (4.3),

(4.4), and (4.5). So, the nonlinearities of both the drive and the response systems

i.e. φ(x(t)) and ξ(x(t)) are assumed to be bounded nonlinearities.

^ ^ ^ n φ(y(t)) − φ(y(t)) ≤ L y(t) − y(t) , ∀y(t), y(t) ∈ R (4.3)

^ ^ ^ n (4.4) ξm(ym(t)) − ξm(ym(t)) ≤ Jm ym(t) − ym(t) , ∀ym(t), ym(t) ∈ R

^ ^ ^ n (4.5) ξs(ys(t)) − ξs(ys(t)) ≤ Js ys(t) − ys(t) , ∀ys(t), ys(t) ∈ R

where ξm(ym(t)) = Bξ(ym(t))Pm and ξs(ys(t)) = Bξ(ys(t))Ps. Also L > 0, Jm > 0 and Js > 0 are the matrices of arbitrarily ﬁxed values and of appropriate dimensions for compatibility.

4.3 Coupled Chaotic Adaptive Synchronous Observers

The synchronization of chaotic systems by means of state-feedback control methodology is explored by the researchers as elaborated by the literature review.

But, more often the exact states are not available or cannot be measured exactly; consequently, only estimations of the states of the leading and the following systems (4.1)-(4.2) can be utilized for coherence. As this chapter may be taken as the extension of [42] as presented in previous Chapter 3, which rendered a state feedback synchronization scheme for the master and the slave systems with no disturbance that is dm = ds = 0 and noise i.e. ηm = ηs = 0 in its state dynamics or at the output. That paper recommends coupled chaotic adaptive synchronous

(CCAS) observers approach for complete synchronization of the drive and the response systems. In accordance with that recommendation, the CCAS observers

104 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

for the systems (4.1)and (4.2) can have the following structure.

˙ h ˆ i xˆm(t) =Axˆm(t) + φ(ˆxm(t)) + Bξ(ˆxm(t))Pm (4.6) 1 + L (y (t) − yˆ (t)) − BF (ˆx (t) − xˆ (t)), m m m 2 m s

˙ h ˆ i xˆs(t) =Axˆs(t) + φ(ˆxs(t)) + Bξ(ˆxs(t))Ps (4.7) 1 + L (y (t) − yˆ (t)) + BF (ˆx (t) − xˆ (t)) + Bu . s s s 2 m s g

4.4 CCS Observer Based Control Methodology for Syn- chronization

The synchronization of (4.1)and (4.2) with a prerequisite of zero values of disturbance and noise is made possible using some amendment of the control function for synchronization as presented in Chapter 3. This amended control function which is based on the observed states of the CCAS observers (4.6), (4.7)is given as follows:

u(t) =Φ(ˆxm(t), xˆs(t)), (4.8) ˆ ˆ =F (ˆxm(t) − xˆs(t)) + ξ(ˆxm(t))Pm(t) − ξ (ˆxs(t)) Ps(t),

n n herex ˆm(t) ∈ R andx ˆs(t) ∈ R present the estimation of xm(t) and xs(t) respec-

ˆ p ˆ p tively. Also Pm ∈ R and Pm ∈ R are the respective estimations of the uncertain parameters Pm and Ps. The adaptation laws according to the methodology provided in Chapter 3 for the complete synchronization of drive and response systems are proposed as

ˆ˙ T Pm(t) = −∆mξ (ˆxm(t))Tm[filt1(ym(t)) − Cxˆm(t)], ∆m > 0 (4.9)

105 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

ˆ˙ T Ps(t) = −∆sξ (ˆxs(t))Ts[filt2(ys(t)) − Cxˆs(t)], ∆s > 0 (4.10)

where filt1{ym(t)} and filt2{ys(t)} are the filtered output of the master and the slave i.e. output without noise. The objective of the present work is to develop a robust-adaptive feedback control methodology by utilizing the control signal (4.8) along with adaptation laws (4.9) and (4.10) for synchronization of the drive and the response systems (4.1)and (4.2) under the influence of disturbance i.e. dm, ds in their state dynamics of the master and the slave systems respectively and also has the effect of noise ηm , ηs in the output measurement because of sensors.

The complete architecture of the synchronizing control scheme using CCAS observers these observers along with the modiﬁed adaptation laws is presented in

n×m n×m Figure 4.1. where Lm ∈ R and Ls ∈ R are the observer gain matrices and

Figure 4.1: Architecture of the robust-adaptive control scheme for the synchronization of the master and the slave systems

ˆ ˆ the nonlinear component of u, that is, ξ(ˆxm(t))Pm −ξ (ˆxs(t)) Ps. The values of the ˆ ˆ parameters of proposed observers i.e. Pm and Ps are unknown and their values are

106 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

adapted by means of adaptation laws. The architecture of these observers is given

the name of coupled chaotic adaptive synchronous (CCAS) observers. It is worth

mentioning that the convergence of eo(t), em(t) and es(t) to the origin is ensured for the sake of complete synchronization. For this purpose the coupled observers

(4.6)-(4.7) are capable of handling unknown parameters during estimations of the state vectors.

4.5 Sigma Modiﬁcation for Robustness

The gist of work presented in this chapter is achieving the robustness in the methodology (for synchronization of the drive system and the response systems) elaborated in previous chapter . This is done by utilizing σ-modiﬁcation technique along with use of ﬁlters for removing the noise from the output measurement.

The methodology proposed by Ioannou and Kokotovic known as σ-modiﬁcation

[165, 166] has been established as an effective way to synthesize robust-adaptive controllers. Beside this, no prior information about the disturbance upper bounds is needed in this . The adaptive law with the σ-modification restricts any deviation of the value of parameters from their prior estimate converse to the techniques applied in [82,167]. However, the existence of Limit cycles may develop in case of poor prior estimates or it can lead to the burst phenomenon. So, it is established fact that the σ-modification is performed with σ > 0 , to increase robustness

and to ensure that the adaptive parameters remain bounded. The mathematical

description of sigma modiﬁcation in adaptation laws is represented by (4.11).

θ˙(t) = ψ(t) − σθ(t), (4.11) where ψ(t) describes the adaptation of unknown parameters θ associated with the dynamics of the systems. 107 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

4.6 Filtering Techniques for Noise Rejection

No doubt, σ-modiﬁcation helps in developing the restrictions on the divergence

of adaptation of systems unknown parameters. But, it cannot cater the high fre-

quency noise interference in the output measurement. To deal with this problem,

low pass ﬁlters can help and are adopted as an eﬃcient possible option in the

proposed methodology for coherence of the master and slave system to remove the

eﬀect of noise. The mathematical representation of the low pass ﬁlter

ω Y (s) = filt{y}(s) = o y(s), s + ωo

and in time domain, the above equation can be written as

d (filt{y(t)}) + ω filt{y(t)} − ω y(t) = 0. (4.12) dt o o

Here, ωo is upper bound of frequency for the proposed ﬁlters. Now, by applying

this low pass ﬁlter (4.12), the extraction of noise free output from the noise cor-

rupted output in the leading and the following systems’ dynamics can be made

possible. These noise free outputs, now can be utilized by the CCAS observers

(4.6) and (4.7) for the estimation of unknown states as proposed in Siddique et.al.

Proposed scheme for robust-adaptive synchronizing control

4.7 Robust-Adaptive Feedback Control

From above discussion, it can be conferred that the strength of synchronizing con-

trol scheme proposed in Chapter 3 can be enhanced by augmenting the robust-

adaptive feature in this methodology by means of output noise ﬁlters and σ- modiﬁcation in the adaptation laws propounded by Chapter 3. As the state dynamics of the drive and the response systems hold unknown parameters Pm, Ps ∈

108 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

Rp, the robust-adaptive coherence of the drive-response formalism can only be

achieved by means of adaptation laws (4.9) , (4.10) along with the explicitly pro-

posed control law as

ˆ ˆ u(t) = F (ˆxm(t) − xˆs(t)) + ξ(ˆxm(t))Pm(t) − ξ (ˆxs(t)) Ps(t). (4.13)

The adaptive observers are designed for both the drive and the response systems for

guesstimate of their states augmented with unknown parameters. The suggested

nonlinear adaptive observers using ﬁltered outputs for the drive and the slave

systems respectively are as follows:

˙ h ˆ i xˆm(t) =Axˆm(t) + φ(ˆxm(t)) + Bξ (ˆxm(t)) Pm(t) (4.14) 1 + L (filt (y (t)) − yˆ (t)) − BF (ˆx (t) − xˆ (t)), m 1 m m 2 m s

˙ h ˆ i xˆs(t) =Axˆs(t) + φ(ˆxs(t)) + Bξ(ˆxs(t))Ps(t) (4.15) 1 + L (filt (y (t)) − yˆ (t)) − BF (ˆx (t) − xˆ (t)) + Bu , s 2 s s 2 s m g

ˆ ˆ where ug(t) = ξ(ˆxm(t))Pm(t) − ξ (ˆxs(t)) Ps(t) is the nonlinear component of u(t).

Modiﬁed Adaptation Laws: After σ − modiﬁcation for robust adaptive control, adaptation laws (4.9) and (4.10) with ∆m > 0 and ∆s > 0 are transformed as follows:

ˆ˙ T ˆ Pm(t) = −∆mξ (ˆxm(t))Tm[filt1(ym(t)) − Cxˆm(t)] − σ1 ∗ Pm, (4.16)

ˆ˙ T ˆ Ps(t) = −∆sξ (ˆxs(t))Ts(filt(ys(t)) − Cxˆs(t)) − σ2 ∗ Ps. (4.17)

109 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

The appropriate values of σ1, σ2 in (4.16), (4.17) establish the bounds on the

rate of change of adaptation parameter and inhibits the divergence of adaptation

parameter values. Higher values of σ1, σ2 provides the faster adaptation capability

but higher values can lead to the instability in synchronization. Smaller values of

σ1, σ2 decreases the rate of adaptation and the response of synchronizing control

become less eﬃcient. So, a trade oﬀ is made between the higher and lower values

of σ1, σ2.

The synchronization of the drive and the response system is also eﬀected by

high frequency noise in the output measurement by sensor. As these outputs are

fed back for generating the error signal, the high frequency noise is also fed back

and causes problem for synchronizing control. To get rid of the high frequency

noise in output of the drive and the response system, ﬁlters functions (4.18) and

(4.19)are used. The ﬁltered outputs Ym, Ys of the leading and the following systems after deduction of the high frequency noise from the measured output can be obtained by

Ym(t) = filt1(ym(t)), (4.18)

Ys(t) = filt2(ys(t)). (4.19)

T ⊥ T ⊥ ⊥ Assumption 4.2: Let B SmC = 0 andB SsC = 0 where CC = 0. If

Assumption 4.2 prevails, the selection of matrices Tm and Ts can be done by

T T B Sm − TmC = 0 and B Ss − TsC = 0.

The Assumption 4.2 is important to resolve a problem that occurs in the deriva-

T ⊥ tion of mathematical results presented in Theorem 4.1. By assuming B SmC =

T T 0, it will become possible to write B Sm = TmC = 0. It means that the B Sm can

110 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

be replaced by TmC = 0. After this we can provide a condition for robust-adaptive controller development using CCAS observers (4.14), (4.15).

Theorem 4.1: A suﬃcient condition for the coherence of drive-response archi-

p p tecture (4.1) and (4.2) having unknown parameters Pm ∈ R and,Ps ∈ R along

n×z n×z with the known observer and controller gain matrices Lm ∈ R and Ls ∈ R ,

F ∈ Rm×n and also fulﬁlling Assumptions 4.1-4.2, utilizing the control scheme in

(4.13), CCAS observers (4.14), (4.15) using the low pass ﬁlters (4.18), (4.19) and also the adaptation laws (4.16), (4.17) is that there exist Sm > 0 , Ss > 0, So > 0 and scalars λ1 > 0 ,λ2 > 0, λ3 > 0 , γ1 > 0, γ2 > 0 such that matrix inequality

  H 0 0.5S BF + CT LT S S 0 0 S 0  1 m m o m m     T T   ∗ H2 0.5SsBF − C S So 0 Ss 0 0 Ss   s       ∗ ∗ H3 0 0 So 0 0         ∗ ∗ ∗ −λ1In 0 0 0 0    < 0    ∗ ∗ ∗ ∗ −λ I 0 0 0   2 n       ∗ ∗ ∗ ∗ ∗ −λ3In 0 0         ∗ ∗ ∗ ∗ ∗ ∗ −γ1In 0      ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ2In

(4.20) holds, where

T T T 2 2 H1 = A Sm + SmA − C LmSm − SmLmC + λ1L In + γ1ZmI,

T T T 2 2 H2 = A Ss + SsA − C Ls Ss − SsLsC + λ2L In + γ2ZmIn,

T T T 2 H3 = A So + SoA − F B So − SoBF + λ3L In.

111 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

In Chapter 3, the above mentioned condition was employed for asymptotic stability of synchronization errors between the corresponding states of the master and the slave systems having no or zero disturbances in their dynamics. However, in the presence of disturbances, some modiﬁcations make the synchronization of two nonlinear drive and response systems possible. These include modiﬁed observers

(4.14), (4.15) and redeﬁned adaptation laws (4.16), (4.17) along with (4.18), (4.19), by which we can attain convergence of synchronization errors in a bounded region.

The limitation of the above mentioned synchronizing control scheme is that it is useful for the case where nonlinear drive and response systems contain the nonlinearities with unknown parameters and also the disturbance in their dynamics.

But this scheme is not eﬀective to synchronize the nonlinear drive and response architecture which have time-delays in their dynamics. Time-delays are addressed in the subsequent chapters for synchronization of nonlinear systems.

4.8 Simulation and Results

The robust-adaptive synchronization of control presented in this chapter is an extension of the synchronizing control technique proposed in Chapter 3. The authenticity of this extension is tested through simulation. For this purpose,

FitzHugh-Nagumo (FHN) drive and response systems are synchronized by applying proposed robust-adaptive control technique which is derived from the CCAS observer based control methodology of Chapter 3 and concluded in the above mentioned Theorem 4.1. The synchronization of the FitzHugh-Nagumo systems through simulation is presented as follows. The mathematical model of the master and the slave FitzHugh-Nagumo systems [168] is written as, in which each of the master and the slave systems consist of two states i.e. zm1,zm2, for master system and zs1 ,zs2 for slave system. Where zm1, zs1 represent normalized membrane

112 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

potentials and zm2 , zs2 are the recovery variables for the drive and the response systems respectively.

z˙m1 =zm1(zm1 − 1)(1 − κ1zm1) − zm2 + Io + dm1,

z˙m2 =Gzm2,

z˙s1 =zs1(zs1 − 1)(1 − κ2zs1) − zs2 + Io + ds1,

z˙s2 =Gzs2.

By taking the stimulation current Io = (H/ω) cos(ωt) and the parameters

G = 1, H = 0.1 , κ1 = 10.1, κ2 = 9.9, ω = 2πf, where f = 0.129. The initial conditions for the states of master system are considered as zm,1(0) = 0.2 and zm,2(0) = 0.5, while initial values for the states of the slave system are taken as zs,1(0) = 0.4 and zs,2(0) = 0.1. Furthermore, the drive and response systems are simulated under the disturbances dm1 = 0.1sin(10t) and ds1 = 0.05sin(6t).

The chaotic behavior of the drive and the response architecture under the inﬂuence of disturbances is veriﬁed and manifested by the Phase portraits of both systems as elaborated in Figure 4.2. The phase portrait or state diagram of the drive system and the response system are presented in Figure4.2 (a) and Figure

4.2 (b) respectively. These portrait manifest the chaotic behavior of both the drive and response systems with different initial conditions. These phase portraits establish the fact that both the drive and the response systems start their dynamics with different values of the at initial conditions of the corresponding states. This difference in the corresponding state continue to exist under no control.

As the drive and the response systems with diﬀerent initial conditions continue to show diﬀerence in their corresponding states i.e. normalized potentials xm1(t), xs1(t) and recovery variables xm2(t), xs2(t). This fact can be witnessed in Figures

4.3 and 4.4. The temporal evolution of xm1(t) and xs1(t) is elaborted in Figure

113 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

Figure 4.2: Chaotic behavior of the drive and the response FitzHugh-Nagumo networks: (a) state diagram of the leading system, (b) state diagram of the following system.

4.3 while the the evolution of the xm2(t) and xs2(t) is depicted in Figure 4.4. To

Figure 4.3: Time progressions of the membrane potential states of the drive and the response systems verify the claim of diﬀerence in temporal evolution of the normalized potentials xm1(t), xs1(t) and recovery variables xm2(t), xs2(t), the error between correspond- 114 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

Figure 4.4: Time progressions of the membrane potential states of the drive and the response systems ing normalized potential and also the error in the corresponding recovery variable is plotted in Figure 4.5. Where e1(t) is the error between the xm1(t)and xs1(t),

Figure 4.5: Errors e1 and e2 between corresponding states of the drive and the response systems

115 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

and e2(t)is the error between the xm2(t)and xs2(t). It is important to note that

Figure4.5 manifests that errors e1 and e2 prevails for all the time and did not vanish under no control. The proposed control methodologies can eﬀectively deal this problem for synchronization of chaotic networks, discussed as follows.

Figure 4.6: Chaotic behavior of the drive and the response FitzHugh-Nagumo systems and their corresponding CCAS observers : (a) state diagram of the leading system, (b) state diagram of the following system, (c) state diagram of the master observer, (d) state diagram of the slave observer.

For complete synchronization of the drive and response chaotic networks, the error signals e1(t) and e2 should converge to zero. The effectiveness of the designed control methodology can be witnessed in the following description for simulation performed for synchronization of the drive and the response system. On applying proposed robust-adaptive synchronization scheme by utilization of sigma modification resulted adaptation laws (4.16), (4.17) and output filters (4.12) in the simulation environment both the drive and response systems synchronizes as shown in Figure 4.6 depicts the whole story. Figure 4.6 (a), 4.6 (b) manifest the phase portrait of the chaotic master and slave systems while Figure 4.6 (c), Figure 4.6 116 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

(d) represents phase portraits of their observers respectively. From these Figures,

it is conﬁrmed that on applying control, the state of both the drive and response

system ultimately converges to the equal values. In other words, both the drive

and the response chaotic systems depict same behavior or both master and slave

systems get synchronized.

Figure 4.7: Time progressions of the states xm1 and xs1 of the drive and response system

Figure 4.7 and Figure 4.8 shows temporal evolution of the states of the leading and the following system dynamics respectively. These figures depicts the complete synchroization of the drive and the response systems under effect of disturbances dm, ds and noises ηm, ηs. The effectiveness of the proposed control scheme is also being illustrated by these figures. Both figures depicts that the matching of the corresponding states of the master and the slave system dominates in small interval of simulation time i.e less than 5 seconds.

For authenticating the synchronization of both the drive and response systems, it is mandatory to show the convergence of synchronization errors i.e. e1(t) and

117 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

Figure 4.8: Time progressions of the states xm2 and xs2 of the drive and the response systems

e2(t) to the origin. Figure 4.9 renders the evidence of merging of corresponding

states of the drive-response architecture to the same values. This ﬁgure witness

the fact that under the inﬂuence of the designed control, errors e1 and e2 converges to the asymptotic region. The blue dotted line represent the error value beteween the normalized potentials of the master and the slave systems, similarly the black bold faced line represents the error value between the recovery variables of the master and the slave systems respectively.

The presented control scheme can cater the unknown parameters associated with nonlinearities of the drive and the response systems. The controller function contains the estimates of these unknown parameters, estimated by the modified adaptation laws (4.16), (4.17). Figure 4.10 depicts the adaptation of unknown parameters which show the effect of sigma modification as it bounds the divergence of adaptation. In this figure, the effect of sigma modification technique can be observed as the values of unknown parameters θm and θs is being pushed back

118 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

Figure 4.9: Errors between corresponding states of leading and following systems when these values start exploding from certain limit. By this the phenomenon, robustness is achieved in the synchronization process.

Figure 4.10: Adaptation of unknown parameters after sigma modiﬁcation.

119 Chapter 4: Robust-Adaptive Feedback Synchronizing Control

4.9 Conclusion

An amendment in an already published adaptive procedure for coherence of the drive-response formalism by applying CCAS observers is proposed for making this adaptive scheme robust against disturbance and noises. The utilitarian of modiﬁcation in adaptation laws and ﬁltered output in the CCAS observers for state estimation, coherence is made possible between the corresponding states of the drive and the response systems, even in the presence of disturbances and noises.

The use of σ-modification and filters improves the already published [42] control strategy for achieving the robust-adaptive control for making the synchronous behavior of nonlinear (FitzHugh-Nagumo) drive and response systems as verified in the simulation section.

120 Chapter 5 Convex Routines for Solution of Matrix Inequalities

In this chapter, new convex routines for solution of matrix inequalities derived in chapter 3 are presented. These convex routine are utilitarian to calculate the controller gain matrix F and observer gain matrices Lm , Ls.

5.1 Introduction

Optimization processes are numerous and consistently implicate the process of choosing the outstanding choice from many of the available options. Several illus- trations of optimization phenomena can be seen in sciences and engineering for example chemical engineering, electrical engineering, mechanical engineering, in economics, social sciences, biological and elogical processes. For instance, manufacturing and fabrication methods in industry are market dependent and need an ever growing allowance of manufactured goods specifications change because of purchaser requirement in price,specification quality. The products which are made necessarily under certain specifications, having expanse in variations of quality, compared to competitive prices augmented with the nominal waste of resources and with best profit.This economic benefit is realized by optimization procedures in manufacturing processes.

121 Chapter 5: Convex Routines for Solution of Matrix Inequalities

The optimization problem to minimize the objective function Z over a set of

feasible values S involves various speciﬁc questions such as what is the optimal

value? How to determine an almost optimal solution? Does there exist an optimal

solution? If an optimal solution exists, then, is it also unique? After formulating

the optimization problem from the above mentioned questions, a conditions for

optimal solutions to exist is sought. Consequently, it is obvious to move to a such

an analysis which facilitates these type of requirements i.e.

Convex analysis: For the complete understanding of convex analysis, reader has

to go through literature for the description of some important deﬁnitions and facts

about the convex analysis such as Weierstrass, Convex sets, Convex combinations,

Convex hull, Convex functions. In convex optimization, a linear matrix inequality

(LMI) can be expressed by

  ST   Ω =   < 0 (5.1) 0 Q

The linear matrix inequality Ω describe a convex condition. In other words,

the set of solutions of the LMI Ω < 0 is convex. Similarly a ﬁnite set of LMIs

Ω1 < 0, Ω2 < 0, Ω3 < 0, ..., Ωn < 0, can be written as

  Ω (x) 0 0 ... 0  1       0 Ω2(x) 0 0 0    Ω(x) =   < 0 (5.2)    0 0 Ω3(x) ... 0      0 0 0 ... Ωn(x)

It is obvious to see, for any value of x the inequality (5.2) Ω(x) is Hermitian

or symmetric. Moreover, the combination of eigenvalues of the matrix Ω(x) can

be obtained by taking the union of all eigenvalues of Ω1(x), Ω2(x)...,Ωn(x).

122 Chapter 5: Convex Routines for Solution of Matrix Inequalities

Furthermore, the value of x that solves the inequality (5.2) also satisﬁes tall the

LMIs. It can be concluded that many linear matrix inequalities can be transformed

in one linear matirx inequality condition. Similar to the linear matrix inequalities,

nonlinear matrix inequalities (NMIs) can also exist. The matrix inequalities can

be of many forms, such as of Linear matrix inequalities, matrix norm inequalities,

convex quadratic inequalities, similarly, numerous other conditions from theory

of control e.g. Riccati inequalities and Lyapunov inequalities are also written as

matrix inequalities. These matrix inequalities are the eﬀective instruments to deal

with large range of optimization and control problems. Many of the problems in

control theory are often described in the form of NMIs or BMIs. The process of

solving the bilinear matirx inequalities is more complex than the linear matirx

inequlities. The matrix inequalities presented as the constraints in the Theorems

of chapters 3 and chapter 4 are also the nonlinear matrix inequalities (NMIs).

The convex solution of these NMIs, provide the solutions of the synchronization

problems. The proposed methodologies for the synchronization of the master-

slave chaotic systems is constrained with known controller gain matrix F and the

observer gain matrices Lm and Ls. To get rid of this limitation, some routines

(procedures) are proposed in this chapter to evaluate the possible values of F ,

Lm and Ls. First, a two-step procedure which is LMI-based approach for solving the matrix inequalities. Secondly a decoupling methodology to determine the controller and the observer gain matrices are provided herein.

5.2 Problem Statement for The Solution of NMIs using Convex Routines

The constraint in the form of nonlinear matrix inequalities i.e. (3.42) for the synchronization the drive and the response systems (3.1)-(3.2) can not be solved

123 Chapter 5: Convex Routines for Solution of Matrix Inequalities

by conventional convex routines which are used for the linear matrix inequalities.

The solution of (3.42) to achieve the controller gain matrix F and the observer

gain matrices Lm and Lsis the problem, pursued in the following description.

Assumption 5.1: Let there exist a scalar matrix Z such that Z < 0 and it can be partitioned as

  Z Z Z  11 12 13    Z =   . (5.3)  ∗ Z22 Z23      ∗ ∗ Z33

This Assumption provides a negative deﬁnite matrix Z < 0, which is used in the two step procedure for the solution of matrix inequality pesented in Theroem 5.1.

Theorem 5.1: A solution to the matrix inequality (3.42) presented in Theorem

3.2 is achievable, if and only if there exist positive-deﬁnite matrices Pm,Ps,Po, and scalars α1 > 0, α2 > 0, α3 > 0, and β1 > 0, β2 > 0, and a matrix Z < 0 such that the inequalities

  Z Z Z P 0 0 P 0  11 12 13 m m       ∗ Z22 Z23 0 Ps 0 0 Ps         ∗ ∗ Z33 0 0 Po 0 0         ∗ ∗ ∗ −α1In 0 0 0 0    < 0, (5.4)    ∗ ∗ ∗ ∗ −α I 0 0 0   2 n       ∗ ∗ ∗ ∗ ∗ −α3In 0 0         ∗ ∗ ∗ ∗ ∗ ∗ −β1In 0      ∗ ∗ ∗ ∗ ∗ ∗ ∗ −β2In

124 Chapter 5: Convex Routines for Solution of Matrix Inequalities

  Z − Υ Z Z − 0.5P BF − CT LT P  11 1 12 13 m m o     T T  ≥ 0 (5.5)  ∗ Z22 − Υ2 Z23 − 0.5PsBF + C Ls Po      ∗ ∗ Z33 − Υ3 are satisﬁed.

Proof: Suﬃciency: Let Ω2 = Φ < 0 represents the matrix inequality (3.42).

It can be partitioned as

  Φ11 Φ12   Φ =   < 0, (5.6) T Φ12 −diag{α1In, α2In, α3In, β1In, β2In}

  Υ 0 0.5P BF + CT LT P  1 m m o    Φ =  T T  , (5.7) 11  ∗ Υ2 0.5PsBF − C Ls Po      ∗ ∗ Υ3

  P 0 0 P 0  m m    Φ =   (5.8) 12  0 Ps 0 0 Ps      0 0 Po 0 0

Let us introduce a matrix Z < 0 such that Z ≥ Φ11. Note that if

  Z Φ12     < 0 (5.9) T Φ12 −diag{α1In, α2In, α3In, β1In, β2In}

remains valid, Ω2 = Φ < 0 is ensured. One can also witness the inequality (5.4) by substituting (??) and (5.8) into (5.9). The matrix inequality Z ≥ Φ11 can be written as (5.5) by putting the value of Z ≥ Φ11 and Φ12 from (??) and

(5.7), respectively. Therefore, the matrix Z and the inequalities (5.4) and (5.5)

125 Chapter 5: Convex Routines for Solution of Matrix Inequalities in Theorem 5.1 provide a suﬃcient condition to establish a solution of matrix inequality (3.42) in Theorem 3.2.

Necessity: If the inequality (3.42) is verified, there always exists a negative- definite matrix such that the inequalities Z ≥ Φ11 and (5.9) are satisfied. At least, we can always choose a matrix Z validating Z = Φ11 (and thus ensuring Z ≥ Φ11), through which Z can be replaced by Φ11 to obtain (5.9). Z ≥ Φ11 and (5.9) can be written as (??)-(5.4). Hence the condition in Theorem 3 is necessary for the statement in Theorem 2 of chapter 3, which completes the proof.

Remark 5.1: Obtainment of a solution to the matrix inequality (3.42) can be a diﬃcult task. Notwithstanding, we provided a technique in Theorem 3 for the solution of the controller and the observer gain matrices. It is signiﬁcant to note that the whole procedure discussed in Theorem 3 can be arranged into a two-step

LMI-based approach. In the step-1, LMI (5.4) along with Z < 0 can be solved. The solution of this LMI in the form of Z, Pm, Ps, and Po can be utilitarian for solving the step-2. In the step-2, LMI (5.5) can be resolved for obtaining the controller gain matrix F and the observer gain matrices Lm and Ls. These controller and observer gain matrices will ultimately used in formulating the observers dynamics for both the drive and response systems and in the synthesis of controller.

5.3 Problem Statement for Alternate Method for The So- lution of NMIs Using Convex Routines

Analysis of the earlier Theorem 1 reveals that it provides a necessary and suﬃcient condition for the solution of Theorem 2 of chapter 3. The retrieval of the matrices

Lm, Ls, and F using Theorem 3 may be comparatively diﬃcult, as the value of

Z achieved in the step-1 may not be suitable to attain feasible values of Lm, Ls,

126 Chapter 5: Convex Routines for Solution of Matrix Inequalities

and F in the step-2. So a methodology for determining the values of Lm, Ls and F with less computational eﬀorts is main objective. The following theorem, proposed the required alternate methodology for more eﬃcient results.

Theorem 5.2: A suﬃcient condition for the solution of the given constraints in ˜ ˜ ¯ Theorem 3.2 is that there exist positive-deﬁnite matrices Pm, Ps and Po, matrices

H1 , H2 and H3 of appropriate dimensions, and scalarsα ¯1 > 0,α ¯2 > 0,α ¯3 > 0, ¯ ¯ β1 > 0, β2 > 0, η1 > 0 and η2 > 0 such that the following LMIs are satisﬁed:

  Θ P˜ P˜ β¯ L α¯ L  1 m m 1 gm 1 f       ∗ −α¯1In 0 0 0       ¯  < 0, (5.10)  ∗ ∗ −β1In 0 0       ¯   ∗ ∗ ∗ −β1In 0      ∗ ∗ ∗ ∗ −α¯1In

  Θ P˜ P˜ β¯ L α¯ L  2 s s 2 gs 2 f       ∗ −α¯2In 0 0 0       ¯  < 0, (5.11)  ∗ ∗ −β2In 0 0       ¯   ∗ ∗ ∗ −β2In 0      ∗ ∗ ∗ ∗ −α¯2In

  Θ P¯ 0α ¯ L  3 o 3 f     ¯   ∗ −2Po α¯3In 0      < 0. (5.12)    ∗ ∗ −α¯3In 0      ∗ ∗ ∗ −α¯3In

Where, 127 Chapter 5: Convex Routines for Solution of Matrix Inequalities

T ˜ ˜ T T Θ1 = A Pm + PmA − C H1 − H1C,

T ˜ ˜ T T Θ2 = A Ps + PsA − C H2 − H2C,

¯ T ¯ T T Θ3 = PoAo + APo − H3 B − BH3.

The observer and controller gain matrices can be obtained by solving Lm =

˜−1 ˜−1 ¯−1 Pm H1, Ls = Ps H2 and F = H3Po .

Note: To prove this theorem by showing its agreement with the Theorem 3.2, one can regenerate the matrix inequality (3.42) as presented in the following text.

Proof: Applying the congruence transformation, that is, by pre and post multi- ¯ plying (5.10) by diag(In,In,In, β1In, α1In, ) , where α1 = 1/η1α¯1 and β1 = 1 η1β1 for an appropriate number η1, the resultant matrix inequality is

  ˜ ˜ Lgm Lf Θ1 Pm Pm  η1 η1       ∗ −α¯1In 0 0 0       ¯  < 0. (5.13)  ∗ ∗ −β1In 0 0      1  ∗ ∗ ∗ − β1In 0   η1     1  ∗ ∗ ∗ ∗ − α1In η1

Employing Schur complement to the above matrix (5.13) inequality yields the following inequality

  ∆ P˜ P˜  1 m m    ζ =   < 0. (5.14) 1  ∗ −α¯1In 0     ¯  ∗ ∗ −β1In

Here,

T ˜ ˜ T T 2 ¯ 2 ∆1 = A Pm + PmA − C H1 − H1C +α ¯1Lf + β1Lgm.

128 Chapter 5: Convex Routines for Solution of Matrix Inequalities

¯ Similarly, by using α2 = 1/η2α¯2 and β2 = 1 η2β2 for a scalar η2, and following the same procedure as above, the matrix inequality (5.11) can be modiﬁed as

  ∆ P˜ P˜  2 s s    ζ =   < 0. (5.15) 2  ∗ −α¯2In 0     ¯  ∗ ∗ −β2In

Here,

T ˜ ˜ T T 2 ¯ 2 ∆2 = A Ps + PsA − C H2 − H2C +α ¯2Lf + β2Lgm.

By application of congruence transformation to (5.12) using diag(In,In, α3In, α3In, ),

where α3 = 1/¯α3, the resultant inequality is obtained as

  Θ P¯ 0 L  3 o f     ¯   ∗ −2Po In 0      < 0. (5.16)    ∗ ∗ −α3In 0      ∗ ∗ ∗ −α3In

By applying Schur complement, one can achieve

  2 ¯ Θ3 +α ¯3L Po  f    < 0. (5.17) ¯ ∗ −2Po +α ¯3In

Since,

1 ¯ ¯ ¯ ¯ −1 ¯ PoPo − 2Po +α ¯3In = (Po − α¯3In)(¯α3In) (Po − α¯3In) ≥ 0 α¯3 1 ¯ ¯ ¯ − PoPo ≤ −2Po +α ¯3In α¯3

129 Chapter 5: Convex Routines for Solution of Matrix Inequalities

Consequently, the resulted inequality is

  2 ¯ Θ3 +α ¯3L Po  f    < 0.. 1 ¯ ¯ ∗ − PoPo α¯3

¯ By considering H3 = F Po and applying congruence transformation by diag(Po,Po)

  ∆3 Po ζ =   < 0, (5.18) 3   1 ∗ − In α¯3 where,

T T T 2 ∆3 = A oPo + PoA − F B Po − PoBF +α ¯3Lf .

By lumping together the linear matrix inequalities (5.14), (5.15), and (5.18) ˜ ˜ and, further, using H1 = PmLm and H2 = PsLs it produces

  η ζ ζ ζ  1 1 4 5      < 0. (5.19)  ∗ η2ζ2 ζ6      ∗ ∗ ζ3 where,

    0 0 0 0.5P BF + CT LT P 0    m m o      ζ =  , ζ =  , 4  0 0 0  5  0 0          0 0 0 0 0   0.5P BF − CT LT P 0  s s o    ζ =  . 6  0 0      0 0

130 Chapter 5: Convex Routines for Solution of Matrix Inequalities

We can regenerate the matrix inequality (3.42), by pre- and post-multiplying

T T T T T T T T T (5.19) by [k¯ , k¯ , k¯ , k¯ , k¯ , k¯ , k¯ , k¯ ]T , and its transpose, respectively, where k¯ 1 4 7 2 5 8 3 6 i is the matrix generated by replacing the ith 0n×n with In in 0n×8n matrix (for ¯ example k = [0n×n,In , 0n×n , 0n×n , 0n×n, 0n×n , 0n×n , 0n×n]) and substituting 2 ˜ −1 ˜ −1 ˜ −1 Pm = η1 Pm, Ps = η1 Ps, Ps = η1 Ps ,α ¯1 = 1/(η1α1) ,α ¯2 = 1/(η2α2),

−1 ¯ ¯ α¯3 = α3 , β1 = 1/(η1β1), and β2 = 1/(η2β2).

5.4 Simulation and Results

The above mentioned convex routines for the solution nonlinear matrix inequlities made it possible for calculating the controller and observer gain matrices. By using the values of these controller and observer gain matrices, the eﬃcacy of the techniques presented in chapter 3 for the synchronization of the master and the slave systems with unknown parameters as proposed in Theorems 3.2, is illustrated in the following simulation results for FitzHugh-Nagumo (FHN) master-slave architectures [?]. If the value of the parameters r1 and r2 are unknown, one can assign θm = r1 and θs = r2. The FHN systems can be compared with generalized model of the nonlinear master and the slave nonlinear systems (3.1)-(3.2). From this comparison, one can deduce the stat-space matrices for the FHN system as follows:     −1 −1 1     A =  , B =  , C = 1 0 , 1 0 0   x2 + m cos(ωt)  1 ω  f(x(t)) =  . 0

It is important to mention that to increase the feasibility of the design constraints, the matrix A and the nonlinearity matrix f(x(t)) of the FHN system are

131 Chapter 5: Convex Routines for Solution of Matrix Inequalities modiﬁed as:

    −1 −1 x2 + m cos(ωt)    1 ω  A =  , f(x(t)) =  , 1 −0.3 0.3x2

without loss of generality. By application of Theorem 3, we obtain

    3.286 3.286     Lm =  , Ls =  , F = 1.409 0.7 . 0 0

By applying the control and adaptation laws in (3.38) and (3.40)-(3.41), simulation results are elaborated as follows. Figure 5.1(a) and 5.1(b) show phase portraits of the master and the slave FHN systems. These phase portraits help in understanding the complete synchronization phenomenon of the drive and the response systems (3.1)-(3.2) under the effect of applied control. These phase portraits establish the fact that at initial conditions, both the drive and the response starts their dynamics with different values of the corresponding states. But as the time passes, the control action compels both the systems to similar corresponding states. Similarly, Figure 5.1(c) and 5.1(d) characterize the phase portraits of observed states of the master and the slave systems. The corresponding observed states of the respective observer for the master and the slave systems are also synchronized by the applied control. This applied control has a direct effect on the observed states of the purposefully designed observers (3.30)-(3.31) for the drive and the response systems. The structure of these observers witness that controller gain matrix F is also the part of each observer along with the observer gain matrix

Lm or Ls.

The temporal evolution of Normalized potential of the master-slave systems and their respective observers is demonstrated in Figure 5.2. This temporal evolu- 132 Chapter 5: Convex Routines for Solution of Matrix Inequalities

Figure 5.1: Phase portraits of the master and the slave FHN systems, phase portraits of their corresponding observers using the approach provided in Theorem 3.2 : (a) Phase portrait of the master system, (b) phase portrait of the slave system, (c) phase portrait of the master observer, (d) phase portrait of the slave observer. tion strengthens the claim of eﬀectiveness of the applied control. The normalized potentials of the master system, the slave system and their respective observers converge gradually to the same values after applying control at time t = 0

Similarly, Figure 5.3 demonstrate the temporal temporal evolution of recovery variable of the master-slave systems and their respective observers. This ﬁgure also proves the eﬃcacy of the applied control by depicting the convergence of recovery variables of the master system, the slave system and their respective observers to the same values.

The complete synchronization can be aﬃrmed by plotting the time evolution of the errors (3.9), (3.10) and (3.11) as shown in Fgure 5.4. This ﬁgure demonstrates that by application of the proposed control (3.38) and adaptation laws

133 Chapter 5: Convex Routines for Solution of Matrix Inequalities

Figure 5.2: Temporal evolution of membrane potentials of the master and slave systems and their corresponding observers

Figure 5.3: Temporal evolution of recovery variables of the master and slave systems and their corresponding observers

(3.40), (3.41), complete synchronization of the master system and the slave sys-

tem is attained. Actually, the applied control explicitly converges the observer

of the master system and the slave system to the same states. This explicit syn-

chronization of the observers implicitly synchronizes the drive and the response

system. Fgure 5.4 elaborates the whole phenomenon by showing the errors em1,

em2, es1, es2, eo1, eo2 convergence to the origin. The convergence of errors eo1, eo2 to the origin by applied control (3.38) means the normalized potential state

134 Chapter 5: Convex Routines for Solution of Matrix Inequalities

and the recovery variable state of the respective observers of the master and the

slave system synchronizes with each other. Also, convergence of errors em1, em2

to the origin means the synchronization of the both states of the master system

to the respective states of its observer. Similarly, convergence of errors es1, es2 to

the origin means the synchronization of the both states of the slave system to the

respective states of its observer.

Figure 5.4: Error dynamics of the master and the slave systems

Simulation results for the adaptation of the unknown parameters are plotted

in Figure 5.5, which shows convergence of the adaptive parameters to their true

values, that is θm = r1 = 10.1 and θs = r2 = 9.9. The true value means the actual

values for the model FHN system, and these values may vary under diﬀerent phys-

ical conditions, in which the FHN system may operate. By the change of these

true values of parameters of the FHN system, the behavior of the FHN system

changes. Any how, in the present simulation environment, the values for parame-

ters θm and θs is opted as θm = r1 = 10.1 and θs = r2 = 9.9. The adaptation laws

(3.40) and (3.41) play vital role in the determination of these parameters which are 135 Chapter 5: Convex Routines for Solution of Matrix Inequalities

unknown to the controller. These adaptation laws gradually adapt the unknown

parmeter and send the adapted values to the controller continuously through the

course of synchronization. The rate of adaptation of unknown parameters θm and

θs can be varied by changing the values of Θm and Θs present in laws (3.40) and

(3.41). The rate of adaptation also eﬀects the performance of controller. So, we

have to choose a suitable adaptation rate depending upon the speed of controller

for computation.

Figure 5.5: Adaptation of unknown parameters i.e. θm and θs

It is noteworthy that synchronization behavior varies with the change of the

observer gain matrices Lm and Ls and the controller gain matrix F . The rate

of synchronization or the extent of synchronization in same time duration may

be different for different values of observer gain matrices Lm and Ls or controller gain matrix F . Figure 5.6 shows dependency of extent of synchronization in same time duration on the observer gain matrices Lm and Ls , by keeping the controller gain matrix as fixed. The error em1 is plotted for Lm = T T T Ls = 1 0 , 2 0 , 8 0 by keeping the controller gain matrix F = 136 Chapter 5: Convex Routines for Solution of Matrix Inequalities

T 1.409 0.7 as ﬁxed. The increase in the magnitude of observer gain matrix

as shown above increases the extent synchronization and decreases the synchro-

nization error in same time interval of simulation as shown in Figure 5.6. Though

Figure 5.6: Eﬀect of observer gain matrices Lm,Ls on the error dynamics em1

the graph for synchronization error em1 for all the three cases is nonuniform, but the maximum peak values for synchronization error decreases with the increase of maginitude of Lm and Ls. Similarly, the effect of variations in controller gain matrix on the synchronization phenomenon is shown in Figure 5.7. By keeping the T observer gain matrices Lm = Ls = 13.286 0 as fixed and changing the con- T T T troller gain matrix as F = 1 0 , 2 0 , 8 0 the synchronization error e01 becomes smaller respectively as shown in the following figure .

5.5 Conclusion

The evaluation of controller and observer gain matrices from the resulted nonlinear matrix inequalities using a two-step approach is described and, further, a decoupling methodology to attain LMI-based solution is presented. The two-step

137 Chapter 5: Convex Routines for Solution of Matrix Inequalities

Figure 5.7: Eﬀect of controller gain matrix F on the error dynamics eo1 technique is more generic and the decoupling method requires less computational eﬀorts for the design of the controller and the observer.

138 Chapter 6 Delay-range-dependent adaptive control for time-delay chaotic systems

In this chapter synchronization control methodologies for chaos synchronization of the drive and the response systems with ﬁnite time-lags, bounded delay-rates, unknown parameters and perturbations are discussed.

6.1 Introduction

Time-delays are inherently present in the natural and the synthetic systems. These delays can induce oscillation or even instability in the time evolution of synchronization error [43,56,169–171]. Therefore, adaptive synchronization of chaotic networks in the presence of time lags, is an elusive task requiring substantial research attention. For instance, delays in information flow of neuronal systems under both electrical and chemical synapses can either stimulate or destroy synchronization of coupled neurons [172,173]. The effects of time-delays on synchronization between coupled bursting neurons, demonstrating different synchronization outcomes for small and large time-delays, are explored in a recent study [174]. Moreover, it is found that the coupling delays can result into variation in the synchronization patterns in the presence of stochastic noises [175]. It is interesting to mention that in addition to the synapses delays, a neuron receives its own activation potential

139 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization after a delay due to electric autapses [176–178]. These electric autapses can cause oscillations in neurons, can alter their bursting synchronization behavior and can have capability of introducing diversity in the neuronal activity due to variations in closed-loop delays [176–180]. In addition, adaptive parameters, designed for adaptive handling of the unknown dynamics, can diverge on the account of modeled time lags, leading to non-coherent behaviors of the chaotic oscillators.

Adaptive synchronization of chaotic identities under time-delays has gained a signiﬁcant research attention over a few past years, and recently, researchers have developed some prodigious feedback control strategies ensuring the asymptotic convergence of synchronization error or the convergence of the synchronization error in the neighborhood of the origin. In [181, 182], adaptive synchronization and anti-synchronization approaches, based on simple feedback controllers and adaptation rules, by utilizing a special matrix structure and the global Lipschitz condition, by exploiting linear matrix inequalities (LMIs), have been developed to address mixed and constant time-delays. A delay-independent coherence of chaos containing networks with multiple time lags in the state vector is explored in [183], and further, an LMI-based simple adaptive control approach is developed in a recent work [184] for uncertain chaotic networks containing bounded perturbations. Adaptive synchronization of competitive neural networks, Lure networks and Cohen-Crossberg network works with mixed delays. Rate independent bounded delays and generalized mixed delays is investigated in the research papers [184, 185] and [73], respectively. Sliding mode control for attaining synchronous behavior of chaotic oscillators under unknown time-delay by incorporating neuro-fuzzy networks is described in [73].

However, further work is still needed for adaptive and robust adaptive synchronization in delayed chaotic systems owing to the complexities of various types

140 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization of delays, nonlinear and time-delay nonlinear dynamics, unknown parameters, uncertainties and perturbations. As far as the author’s knowledge is concerned, adaptive control mechanism for synchronization of the time-delay chaotic systems with interval delays ranging between a zero or nonzero lower and a ﬁnite upper value has not been studied in the previous studies. It is imperative to mention that the conventional adaptive control schemes do not guarantee synchronization of the chaotic networks having time-lags because of the divergence of adaptive parameters. Time-lags introduce instability in the error dynamics, resulting into non-coherent behaviors of the chaotic entities [186–190]. This instability in the synchronization error dynamics is unwanted in the synchronization phenomenon.

To get rid of the instability in synchronization error dynamics, either synchronization control schemes are modiﬁed or new control scheme made to purged out the hitches in the synchronization of the drive and response system because of time delay.

Synchronization of drive and response chaotic systems under time-delays using adaptive feedback control technique has received substantial research attraction in the recent past years. Many researcher of this field of synchronization have made efforts. As a result, some remarkable feedback control approaches have been evolved, ensuring asymptotic convergence of the synchronization error. control method is developed in [191] for the uncertain and complicated chaotic networks under disturbances. Adaptive coherence of chaotic networks having mixed time-delays is investigated in [192]. Synchronization of Lure networks with rate-independent bounded delays using adaptive control methodology is proposed in [193]. However, still synchronization problem for the delayed chaotic systems requires significant attention of researches owing to the technical hitches

141 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization of various types of delays, delayed nonlinear dynamics, unknown parameters, and disturbances.

This chapter discusses synchronization control methodologies for chaos synchronization of the master and the slave systems with finite time-lags, bounded delay-rates, unknown parameters and perturbations. A coherence control condition for the drive and response systems under bounded initial condition is developed to synthesize a feedback controller capable of executing successful synchronization of the master and the slave systems with unknown parameters and delay belonging to a range (with zero or non-zero lower bound). A sufficient condition for the adaptive control approach is derived using an amendment in the conventional Lyapunov-Krasovskii (LK) functional, which enables to develop the adaptation laws for the unknown parameters associated with the nonlinearity in the dynamics of the master-slave systems. Numerical simulations for the chaotic delayed Hopfield neural networks are presented at end of chapter to show the efficacy of the proposed delay-range-dependent adaptive synchronization control methodologies.

6.2 Generalized Model of The Delay Incorporated Sys- tems

Consider the master-slave (drive-response) systems which are to be synchronized are taken as nonlinear in their dynamics. The mathematical model of these nonlinear systems are presented in (6.1) and (6.2). The nonlinearities presents in both master and slave systems are homogeneous nonlinearities. These generic models represent equations for the state dynamics for the drive and the response systems respectively. The state dynamics and output dynamics in each of the master and the slave system contain the disturbances. Also, two types of nonlinearities with

142 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

known and unknown parameters are involved in the state dynamics of the drive

and the response systems.

x˙ m(t) =Axm(t) + Adxm(t − τ) + f1(xm(t)) + f2(xm(t − τ)) p X + gi(xm, t, τ)θi + ϕ1(t), (6.1) i=1

xm(t) =φ1(t),

x˙ s(t) =Axs(t) + Adxs(t − τ) + f1(xs(t)) + f2(xs(t − τ)) p X + gi(xs, t, τ)θi + ϕ2(t) + Bu, (6.2) i=1

xs(t) =φ2(t).

The range of t in both of the above equations (6.1) and (6.2)is deﬁned by

t ∈ − max(τ1, τ2) 0 ,

n n where xm ∈ R and xs ∈ R are the state vectors for the master and the slave systems, respectively. The symbol u(t) ∈ Rm denotes the control input.A ∈ Rn×n,

n×n n×m Ad ∈ R and B ∈ R are the constant matrices establishing the linear dy-

n n namics of the systems. The vector-functions f1 (x(t)) ∈ R and f2(x(t − τ)) ∈ R symbolize, respectively, the nonlinearities without and with time-delay in the

n dynamics of the nonlinear systems. The vector-function gi(x, t, τ) ∈ R represents the nonlinearity associated with unknown scalar parameter θi for i = 1 ··· p.

n n ϕ1(t) ∈ R and ϕ2(t) ∈ R represent the bounded time-dependent perturbations

aﬀecting the master and the slave systems’ dynamics, respectively. τ(t)is the time

dependent delay, involved in the dynamics of the master system and the slave

system, satisfying the relations

143 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

0 ≤ τ1 ≤ τ(t) ≤ τ2, (6.3) |τ˙(t)| ≤ κ.

Assumption 6.1: Let the nonlinearity gi(x, t, τ) associated with the unknown parameters θi can be reformed as

gi(x, t, τ) = Bf3i(x, t, τ), ∀i = 1 ··· p (6.4) where

m f3i(x, t, τ) ∈ R , for i = 1 ··· p, are continuous nonlinear functions depend upon the time delay τ and value of state vector x(t).

Remark 6.1: Assumption 6.1 restricts the function gi(x, t, τ), for ∀i = 1 ··· p, associated with the unknown parameters, to a matching nonlinearity, which can be adaptively canceled through the control signal u(t). Note that the known non-matching nonlinearities can be included into f1(x(t)) and f2(x(t − τ)), while the non-matching uncertain components can be covered through ϕ1(t) and ϕ2(t).

Consequently, the master-salve systems in (6.1)-(6.2) can be used to represent a wide class of systems with both the matching and the non-matching nonlinearities having both the known and the unknown components by means of ϕ1(t), gi(x, t, τ), f1(x(t)), f2(x(t − τ)), ϕ1(t) and ϕ2(t). But to deal with these to deal with these types of nonlinearities, following assumption is taken for pursuing mathematical solution of the control problem for synchronization.

Assumption 6.2: The nonlinear functions f1(x(t)) , f2(x(t − τ)) and gi(x, t, τ) are the Lipschitz nonlinear function and satisfy the inequalities

144 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

kf1(xm(t)) − f1(xs(t))k ≤ kL1 (xm(t) − xs(t))k , (6.5)

kf2(xm(t − τ)) − f2(xs(t − τ))k ≤ kL2 (xm(t − τ) − xs(t − τ))k , (6.6)

kgi(xm, t, τ) − gi(xs, t, τ)k ≤ kL3i(xm − xs)k , ∀i = 1, 2, ··· , m. (6.7)

where gi(x, t, τ) and L1 , L2, and L3i are the constant matrices of appropriate dimensions and the operator k·k stands for the Euclidean norm.

6.3 Synchronizing Error Dynamics for The Delayed Drive and The Response Systems

Let us deﬁne the synchronization error between the states of the drive and the response systems as e(t) = xm(t)−xs(t). This error dynamics between the master and the slave network (6.1)-(6.2) by incorporating (6.4) can be derived as follows.

By taking the time rate of change of both sides of e(t) = xm(t)−xs(t), we have

e˙(t) =x ˙ m(t) − x˙ s(t)

Putting the values ofx ˙ m(t) andx ˙ s(t) from (6.1) and (6.2) in the above equation yields

e˙(t) =[Axm(t) + Adxm(t − τ) + f1(xm(t)) + f2(xm(t − τ)) p X + gi(xm, t, τ)θi + ϕ1(t)] − [Axs(t) + Adxs(t − τ) + f1(xs(t)) + f2(xs(t − τ))] i=1 p X + gi(xs, t, τ)θi + ϕ2(t) + Bu. i=1

145 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

Furthermore, rearranging the terms in the above equation,

e˙(t) =A(xm(t) − xs(t)) + Ad(xm(t − τ) − xs(t − τ))

+ f1(xm(t)) − f1(xs(t))

+ f2(xm(t − τ)) − f2(xs(t − τ)) p p X X + gi(xm, t, τ)θi − gi(xs, t, τ)θi i=1 i=1

+ ϕ1(t) − ϕ2(t) − Bu.

Putting the value of gi(x, t, τ) = Bf3i(x, t, τ) from assumption 6.1 and also replacing the diﬀerence of states i.e. (xm(t) − xs(t)), and (xm(t − τ) − xs(t − τ)) with e(t) and e(t − τ) the error dynamics are reformed as

e˙(t) =Ae(t) + Ade(t − τ) + f1(xm(t)) − f1(xs(t))

+ f2(xm(t − τ)) − f2(xs(t − τ)) p (6.8) X + B [f3i(xm, t, τ) − f3i(xs, t, τ)] θi i=1

+ ϕ1(t) − ϕ2(t) − Bu.

The objective of the present work discussed in this chapter is to design delay- range-dependent adaptive control methodologies to ensure stability of the synchronization error dynamics (6.10) for attainment of the synchronization of the nonlinear systems (6.1)-(6.2) subject to (6.3) and the assumptions 6.1 and 6.2.

6.4 Adaptive Synchronizing Controller Design for Delayed Containing Drive and Response Systems

The designing of the required synchronizing control function is deliberately composed of two parts. First term of synthesized control function (6.9) is for mini- mizing the synchronization erroril by applying the feedback of the synchronization 146 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

error to The proposed control law. The controller gain matrix F helps in reducing

synchronization error dynamics. The Second term of the control function (6.9) is

for taking care of the nonlinearity and the unknown parameters associated with

this nonlinearity. The proposed control law for adaptive synchronization is given

p X ˆ u = F e(t) − [f3i(xm, t, τ) − f3i(xs, t, τ)]θi, (6.9) i=1

ˆ where F is a constant controller gain matrix of appropriate dimensions and θi is an estimate of θi. By utilizing (6.9), the error dynamic between corresponding states of the master and the slave systems (6.10) takes the form

e˙(t) =Ae(t) + Ade(t − τ) + f1(xm(t)) − f1(xs(t))

+ f2(xm(t − τ)) − f2(xs(t − τ)) p X + B [f3i(xm, t, τ) − f3i(xs, t, τ)] θi (6.10) i=1

+ ϕ1(t) − ϕ2(t) − B[F e(t) p X ˆ − [f3i(xm, t, τ) − f3i(xs, t, τ)]θi]. i=1

In the above equation (6.10), the nonlinear terms associated with the unknown ˆ parameters θi, θi can be diﬀerentiated into two types. First type of nonlinearities are associated with the unknown parameters while the second type of nonlinearities are associated with the estimate of the unknown parameters. The rearrangement of the terms in the above equation yields

e˙(t) =(A − BF )e(t) + Ade(t − τ) + f1(xm(t)) − f1(xs(t))

+ f2(xm(t − τ)) − f2(xs(t − τ)) (6.11) p X ˜ + B [f3i(xm, t, τ) − f3i(xs, t, τ)]θi + ϕ1(t) − ϕ2(t). i=1 147 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

˜ ˆ Here, θi = θi − θi. For the simpler representation of the mathematics of the

synchronization error dynamics, deﬁne the equations (6.12), (6.13) and (6.14) as

follows,

ψ1(xm, xs, t) = f1(xm(t)) − f1(xs(t)), (6.12)

ψ2(xm, xs, t, τ) = f2(xm(t − τ)) − f2(xs(t − τ)), (6.13)

ψ3i(xm, xs, t, τ) = f3i(xm(t − τ)) − f3i(xs(t − τ)). (6.14)

Here, i = 1 ··· p. By means of (6.12)-(6.14), the synchronization error dynamics

(6.11) can be rewritten as

e˙(t) =(A − BF )e(t) + Ade(t − τ) + ψ1(xm, xs, t) p X ˜ + ψ2(xm, xs, t, τ) + Bψ3i(xm, xs, t, τ)θi (6.15) i=1

+ ϕ1 − ϕ2.

For the stability analysis of the synchronization error dynamics to synchronize the master and the slave system using adaptive feedback control, Lyapunov theory of stability is used. As the choice of energy function plays a vital in Lyapunov theory, the following energy function is considered for analyzing the synchronization phenomenon. This type of energy function, which very well in the control literature as the Lyapunov-Krasovskii functional is used in [82] for adaptive synchronization of chaotic oscillators with interval time-delays. But in present case, a modiﬁed version of the Lyapunov-Krasovskii functional is used as presented in the following text. This modiﬁcation allows to adapt the unknown parameters in situation of varying time delays and varying delay rates. 148 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

Lyapunov function: By assigning τ12 = τ2 − τ1, let us construct a novel

Lyapunov-Krasovskii functional as

t t Z 2 Z T T X T V (t) =e (t)P e(t) + e (z)Qe(z)dz + e (z)Mie(z)dz i=1 t−τ t−τi 0 t −τ t Z Z Z 1 Z T T (6.16) + τ1e˙ (z)N1e˙(z)dzdα + τ12e˙ (z)N2e˙(z)dzdα

−τ1 t+α −τ2 t+α p X −1 ˜T ˜ + σi θi θi. i=1

In the above Lyapunov-Krasovskii functional P > 0, Q > 0, M1 > 0, M2 > 0,

N1 > 0 , and N2 > 0 are the positive-deﬁnite matrices. Similarly, σi with i = 1 ··· p are positive scalars and τ12 = τ2 −τ1. It is viable to derive a constraint for adaptive synchronization of the master and the slave systems (6.1)-(6.2) by utilizing a control law (6.9) and by taking ϕ1 = ϕ2 = 0 . The resultant condition is presented in the form of Theorem 6.1 as follows.

Theorem 6.1: A suﬃcient condition for the local synchronization of (6.1)-(6.2) subject to Assumptions 1-2, ϕ1 = ϕ2 = 0, and time-delay bounds presented by

(6.3) by utilization of the proposed feedback control law (6.9) and the proposed adaptation law

˙ T T ˆθi =σiψ 3i(xm, xs, t, τ)B P e

T T + σiψ 3i(xm, xs, t, τ)B (6.17) 2 2 ¯ × [τ1 N1 + τ12N2][Ae(t) + Ade(t − τ)

+ ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)],

for i = 1 ··· p, is that there exist positive deﬁnite matrices P > 0, Q > 0,

M1 > 0, M2 > 0, N1 > 0, N2 > 0 and positive scalars σi for i = 1 ··· p such that 149 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization the inequality

  Θ PA N 0 PP 0 τ A¯T N τ A¯T N LT 0 Θ  11 d 1 1 1 12 2 1 12     T T T   ∗ Θ22 N2 N2 0 0 0 τ1A N1 τ12A N2 0 L 0   d d 2       ∗ ∗ Θ33 0 0 0 0 0 0 0 0 0         ∗ ∗ ∗ Θ44 0 0 0 0 0 0 0 0       ∗ ∗ ∗ ∗ −I 0 0 τ N τ N 0 0 0   1 1 12 2       ∗ ∗ ∗ ∗ ∗ −I 0 τ1N1 τ12N2 0 0 0      < 0    ∗ ∗ ∗ ∗ ∗ ∗ −I τ1N1 τ12N2 0 0 0         ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N1 0 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N 0 0 0   2       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗∗∗∗∗∗∗ ∗ ∗ ∗−I 0      ∗∗∗∗∗∗∗ ∗ ∗ ∗∗−I (6.18)

holds, where

A¯ = A − BF

T T T Θ11 = A P + PA − BF − F B + Q + M1 + M2 − N1, p T P Θ12 = σiL3i i=1

Θ22 = −(1 − κ)Q − 2N2,

Θ33 = −N1 − N2 − M2,

Θ44 = −N2 − M2.

The error trajectory converges to the origin for all initial conditions V (0) < 1 and remains bounded in the region V (t) < 1.

Proof: The proof of Theorem 3.2 is presented in Appendix B.

150 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

The limitation of the above mentioned control methodology is that it is eﬀec-

tive for the synchronization of the control of the nonlinear systems with known

parameters associated with the nonlinearities and also having the time delay in

the dynamics of the master and the slave systems. But it is not capable of dealing

the drive and response formalism with the disturbance in their dynamics.

Remark 6.2: The proposed synchronization scheme in Theorem 6.1 is based

on a novel LK functional (6.16) by extending the LK function in [88] for the p P −1 ˜T ˜ delay-range-dependent adaptive control by incorporating the term σi θi θi. i=1 0 t R R T In contrast to [82], the double-integral terms τ1e˙ (z)N1e˙(z)dzdα and −τ1 t+α −τ1 t R R T τ12e˙ (z)N2e˙(z)dzdα are employed in the present work that can be utilitar- −τ2 t+α ian for attaining large delay bounds such as delay range and limit on delay-rate.

0 t R R T Remark 6.3: It is worth noting that incorporation of τ1e˙ (z)N1e˙(z)dzdα −τ1 t+α −τ1 t R R T and τ12e˙ (z)N2e˙(z)dzdα in the LK functional for a delay-range-dependent −τ2 t+α adaptive control methodology is not an easy task. To deal with these double-

integral terms, a local synchronization control methodology is developed by con-

sidering Lyapunov redesign for determining a region of stability. Further, the adap-

T T 2 2 ¯ tation law is also modiﬁed by introducing ψ3i(xm, xs, t, τ)B [τ1 N1 +τ12N2][Ae(t)+

Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)]. Note that such terms were not em-

ployed in the conventional adaptive control and required in the present delay-

range-dependent adaptive synchronization case.

Remark 6.4: The parameters σi for i = 1 ··· p in (6.17) are employed to attain ˆ reasonable adaptation rates for θi. In the conventional global adaptive control

schemes, large values of such parameters can be used to attain fast convergence

of the adaptive parameters. However, large values of σi for i = 1 ··· p can increase 151 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

sensitivity of adaptive parameters against noise; therefore, careful attention is

required for selection of σi. In the present case, the parameters σi with i =

1 ··· p have more signiﬁcance due to their involvement in the region of stability

V (t) < 1 and in the matrix inequality (6.18). By increasing σi , the region

of stability with respect to parametric estimation error can be enlarged because p ¯ P −1 ˜T ˜ V (t) = V (t)+ σi θi (t)θi(t). On the other hand, large values of the parameters i=1 can result into infeasibility of the constraint (6.18).

6.5 A Controller Design Condition for Finding the Con- troller Gain Matrix

The eﬃcacy of Theorem 6.1 is witnessed only in a situation where the controller

gain matrix F is provided. The provision of the value of controller gain matrix

F to be known is met only under some certain situations practically. It means

the known value of controller gain matrix F is possible under certain conditions.

But more often, it is diﬃcult to ﬁnd the controller gain matrix F . To solve this

problem of unknown value of the controller gain matrix for implementation of the

developed scheme for synchronization of the master and the slave systems, the

following condition presented in Theorem 6.2 helps in this regard. A suﬃcient

condition for the controller design is described in Theorem 6.2 for ﬁnding the

controller gain matrix F . In this theorem, the eﬀect of distrubances i.e. ϕ1

and = ϕ2 is considered as zero. Similarly, to develop this Theorem 6.2, some

assumptions which are mentioned earlier i.e. sssumptions 1-2 are also considered.

Theorem 6.2: A suﬃcient condition for condition for complete synchronization

of (6.1)-(6.2) subject to assumptions 1-2 ϕ1 = ϕ2 = 0, and time-delay bounds

presented by (6.3), by utilization of a feedback control (6.9) and an adaptation law

152 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

(6.17), is that there exist matrices P > 0, Q > 0, M1 > 0, M2 > 0, N1 > 0, and

N2 > 0 and positive scalars σi for i = 1 ··· p such that the constraint

  ζ A X N¯ 0 II 0 τ ζ τ ζ XLT 0 ζ  11 d 1 1 18 12 19 1 12     ¯ ¯ T T T   ∗ ζ22 N2 N2 0 0 0 τ1XA τ12XA 0 XL 0   d d 2       ∗ ∗ ζ33 0 0 0 0 0 0 0 0 0         ∗ ∗ ∗ ζ44 0 0 0 0 0 0 0 0       ∗ ∗ ∗ ∗ −I 0 0 τ I τ I 0 0 0   1 12       ∗ ∗ ∗ ∗ ∗ −I 0 τ1I τ12I 0 0 0      < 0    ∗ ∗ ∗ ∗ ∗ ∗ −I τ1I τ12I 0 0 0       ˜   ∗ ∗ ∗ ∗ ∗ ∗ ∗ N1 0 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ N˜ 0 0 0   2       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗∗∗∗∗∗∗∗ ∗ ∗−I 0      ∗∗∗∗∗∗∗ ∗ ∗∗−I

(6.19) holds, where

˜ ¯ −1 N1 = −XN1 X

˜ ¯ −1 N2 = −XN2 X

T T T ¯ ¯ ¯ ¯ ζ11 = XA + AX − BJ − J B + Q + M1 + M2 − N1,

p T P ¯ ¯ ζ12 = X σiL3i , ζ22 = −(1 − κ)Q − 2N2, i=1

¯ ¯ ¯ ¯ ¯ ζ33 = −N1 − N2 − M2, ζ44 = −N2 − M2,

T T T ζ18 = ζ19 = XA + J B .

153 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

The error trajectory converges to the origin for all initial conditions and re-

mains bounded in the region for all time. The controller gain can be determined

through F = JX−1.

Proof: Applying the congruence transformation using diag(φ1, φ2, φ3, φ4)

−1 −1 −1 −1 to (6.18), where φ1 = diag(P ,P ,P ,P ), φ2 = diag(I,I,I), φ3 =

−1 −1 diag(N1 ,N2 ) and φ4 = diag(I,I,I), we have

  ζ A X N¯ 0 II 0 τ ζ τ ζ XLT 0 ζ  11 d 1 1 18 12 19 1 12     ¯ ¯ T T T   ∗ ζ22 N2 N2 0 0 0 τ1XA τ12XA 0 XL 0   d d 2       ∗ ∗ ζ33 0 0 0 0 0 0 0 0 0         ∗ ∗ ∗ ζ44 0 0 0 0 0 0 0 0       ∗ ∗ ∗ ∗ −I 0 0 τ I τ I 0 0 0   1 12       ∗ ∗ ∗ ∗ ∗ −I 0 τ1I τ12I 0 0 0      < 0.    ∗ ∗ ∗ ∗ ∗ ∗ −I τ1I τ12I 0 0 0       −1   ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N1 0 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N −1 0 0 0   2       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗∗∗∗∗∗∗∗ ∗ ∗−I 0      ∗∗∗∗∗∗∗ ∗ ∗∗−I

(6.20)

−1 ¯ −1 −1 ¯ −1 −1 ¯ Substituting X = P , J = FX, N1 = P N1P , N2 = P N2P , M1 =

−1 −1 ¯ −1 −1 ¯ −1 −1 ¯ P M1P , M2 = P M2P , Q = P QP and A = A − BF , we obtain the

inequality (6.19), and this concludes the proof of Theorem 6.2.

Remark 6.5: The proposed methodologies in Theorems 6.1 and 6.2 can be

employed to the adaptive synchronization of the nonlinear master-slave systems 154 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

with large variations in the delays. In [82], the largest possible value of the delay-

derivative bound κ is restricted to be less than unity due to infeasibility of the controller design constraints. In practice, delay-rate can be either small or large and may not have any restriction. However, the proposed method in Theorem

6.1 can be employed to enlarge κ up to a desired value that can be either less or greater than one. A comparison of the proposed method is provided in the simulation section to elaborate this feature.

Remark 6.6: The condition provided in Theorem 6.2 is necessary and sufficient for the validity of the constraints in Theorem 6.1. The necessity and sufficiency can be easily validated through the congruence transformation by validating that the inequalities (6.18) and (6.19) are equivalent. However, the condition in Theorem 2 offers an extra feature of computation of the controller gain matrix by F = JX−1 for designing an appropriate adaptive controller.

It is important to mention that Theorem 6.2 is more generic with respect to time-delay bounds and delay-rate κ. By substituting τ1 = 0, a speciﬁc delay- dependent adaptive synchronization control case is presented in Corollary 1.

Corollary 6.1: A suﬃcient condition for the local adaptive synchronization of

(6.1)-(6.2) subject to Assumptions 1-2, ϕ1 = ϕ2 = 0, and time-delay bounds τ1 = 0,

τ2 > 0 and τ˙(t) ≤ κ by utilization of the proposed feedback control law (6.9) and the proposed adaptation law (6.17) for i = 1 ··· p is that there exist positive-deﬁnite ¯ ¯ ¯ ¯ matrices X > 0, Q > 0, M1 > 0,M2 > 0, and N2 > 0 and positive scalars σi (for i = 1 ··· p ) such that the inequality

155 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

  ¯ T T Σ11 AdX + N2 0 II 0 τ2ζ19 XL XL Σ12  1 2     ¯   ∗ ζ22 N2 0 0 0 τ2XAd 0 0 0         ∗ ∗ Σ33 0 0 0 0 0 0 0       ∗ ∗ ∗ −I 0 0 τ I 0 0 0   2       ∗ ∗ ∗ ∗ −I 0 τ2I 0 0 0      < 0    ∗ ∗ ∗ ∗ ∗ −I τ2I 0 0 0       ¯   ∗ ∗ ∗ ∗ ∗ ∗ −N2 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0      ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I (6.21)

holds, where

˜ ¯ −1 N2 = −XN2 X

T T T ¯ ¯ ¯ Σ11 = XA + AX − BJ − J B + Q + M2 − N2, ¯ ¯ Σ33 = −N2 − M2.

The error trajectory converges to the origin for all initial conditions V (0) < 1 and

remains bounded in the region V (t) < 1. The controller gain can be determined

through F = JX−1.

If the delay-rate bound is unknown, then by assigning Q = 0 (or Q¯ = 0), we can derive the following corollary from Theorem 2.

Corollary 6.2: A suﬃcient condition for the local adaptive synchronization of

(6.1)-(6.2) subject to Assumptions 1-2, ϕ1 = ϕ2 = 0, and time-delay bound 0 ≤

τ1 ≤ τ(t) ≤ τ2 by utilization of the proposed feedback control law (6.9) and the proposed adaptation law (6.17), for i = 1 ··· p, is that there exist positive-deﬁnite 156 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

¯ ¯ ¯ matrices X > 0 ,M1 > 0 , M2 > 0and N2 > 0 positive scalars σi (for i = 1 ··· p), such that the inequality

  ∆ A X N¯ 0 II 0 τ ζ τ ζ XLT XLT ζ  11 d 1 1 18 12 19 1 2 12     ¯ ¯ ¯ T T   ∗ −2N2 N2 N2 0 0 0 τ1XA τ12XA 0 0 0   d d       ∗ ∗ ζ33 0 0 0 0 0 0 0 0 0         ∗ ∗ ∗ ζ44 0 0 0 0 0 0 0 0       ∗ ∗ ∗ ∗ −I 0 0 τ I τ I 0 0 0   1 12       ∗ ∗ ∗ ∗ ∗ −I 0 τ1I τ12I 0 0 0      < 0    ∗ ∗ ∗ ∗ ∗ ∗ −I τ1I τ12I 0 0 0       ˜   ∗ ∗ ∗ ∗ ∗ ∗ ∗ N1 0 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ N˜ 0 0 0   2       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗∗∗∗∗∗∗∗ ∗ ∗−I 0      ∗∗∗∗∗∗∗ ∗ ∗∗−I (6.22)

holds, where

T T T ¯ ¯ ¯ ∆11 = XA + AX − BJ − J B + M1 + M2 − N1.

Further, the error trajectory converges to the origin for all initial conditions

V (0) < 1 and remains bounded in the region V (t) < 1. The controller gain can be

determined through F = JX−1.

Remark 6.7: It should be noted that the conditions provided in Corollaries 1

and 2 are although novel and useful, however, these conditions are speciﬁc cases

of the proposed approach in Theorem 2 for the delay-dependent and delay-rate-

independent scenarios, respectively. It should be noted that the delay-dependent

157 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization and delay-rate-independent local adaptive control and synchronization approaches as in Corollaries 1-2 are deﬁcient in the literature.

6.6 Simulation and Results

For veriﬁcation of the synchronization methods provided in this chapter, we select the master-slave Hopﬁeld neural networks in [181] (see also [82]) with the following dynamics:       −1 0 0 0 1 0       A =  , Ad =  , B =  , 0 −1 0 0 0 1

  2 tanh(x1)   f1(x(t)) =   , −5 tanh(x1) + 1.5 tanh(x2)

  −1.5 tanh(x1(t − τ)) − 0.1 tanh(x2(t − τ))   f2(x(t − τ)) =   , − tanh(x2(t − τ))

  tanh(x2(t)   g1(x, t, τ)) = Bf31(x, t, τ)) =   , 0

  0   g2(x, t, τ)) = Bf32(x, t, τ)) =   . tanh(x2(t)

For simulation purpose, time-varying delay is taken as τ(t) = 1−0.001sin(0.01t).

The unknown parameters used in the master-slave networks are selected as

θ1 = 0.1 and θ2 = 0.2. Figure 6.1 depicts the behavior of the master and the slave

Hopfield neural networks without applying a control law. Figure 6.1(a) and 6.1(b) show that the phase diagram of the drive and the response networks, respectively, have chaotic behavior. Also these chaotic behaviors are different with different

158 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

Figure 6.1: Phase portraits of the drive and the response Hopﬁeld neural networks: (a) state diagram of the leading system, (b) state diagram of the following system initial conditions. The time evolution of the states of the drive and the response systems is presented in Figure 6.2 as follows. The value of Lipschitz constant is for all nonlinearities is assumed to be unity in accordance with Lipschitz assumption.

States of the master of salve States and master the systems States of the master of salve States and master the systems

Figure 6.2: Time evolution of the states xm1, xm2, xs1, and xs2 of the master and the slave systems

159 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

While Figure 6.3 elaborates the synchronization errors e1(t) and e2(t) between matching states of the drive and the response systems. It can be observed that the master-slave Hopﬁeld neural networks have not coherent behavior due to diﬀerence in initial conditions. Morover the error dynamics do not converge to zero in the

presence of no control input through out the time of simulation. Synchronization errors errors Synchronization

Figure 6.3: Time evolution of the errors e1, and e2 between the corresponding states of the master and the slave Hopﬁeld neural networks

Now for the validity of control scheme proposed in Theorem 2, the controller parameters are obtained by solving the inequality (36) for L1 = L2 = 50, L31 =

L32 = 500, σ1 = σ2 = 1, κ = 10 , τ1 = 0.9, and τ2 = 1.1 . The solution is obtained as   674.8507 0   F =  , 0 674.8507   424.9273 0   P =  , 0 424.9273

160 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

  1.293 0 −6   N1 = 10 ×  , 0 1.293   1.1839 0 7   N2 = 10 ×  . 0 1.1839 By utilizing these values for the proposed adaptation and control laws, the effectiveness of the proposed synchronization control technique is aﬃrmed as shown in following Figures. The phase portraits of the master and the slave networks are manifested in Figures 6.4(a) and 6.4(b), respectively. Note that the behavior of both networks looks similar compared to the results of Figures 6.1(a) and (b).

Figure 6.4: Synchronized phase portraits of the drive and the response Hopﬁeld neural networks: (a) state diagram of the leading system, (b) state diagram of the following system

Figure 6.5 shows the time evolution of all the states of the master and the slave chaotic networks.The complete synchronization of corresponding states of the master and the slave networks can be observed from these plots.

161

Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization States of the master and salve salve of systems and master the States

Figure 6.5: Temporal evolution of the synchronized states xm1, xm2, xs1, and xs2

of the master and the slave systems Synchronization Synchronization errors errors

Figure 6.6: Temporal evolution of the synchronization errors e1, and e2 between the corresponding states of the master and the slave systems

As the drive and the response systems with diﬀerent initial conditions continue

to move i.e. normalized potentials xm1(t), xs1(t) and recovery variables xm2(t), xs2(t), the corresponding states match with each other. Figure 6.6 conﬁrms the 162 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization

validity of the proposed control approach in Theorem 2, by elaborating asymptotic

convergence of the errors e1(t) and e2(t) to the origin. So, the claim of removing

diﬀerence in temporal evolution of the normalized potentials xm1(t), xs1(t) and re-

covery variables xm2(t), xs2(t), that is, the error between corresponding normalized potential recovery variable is veriﬁed.

Adaptive parameters Adaptive Adaptive parameters Adaptive

Figure 6.7: Plot for the adaptation of unknown parameters θ1 and θ2 associated with the nonlinearities present in the dynamics of master and the slave systems

Figure 6.7 conﬁrms the validity of the proposed control approach in Theorem 2, by elaborating convergence of the adaptive parameters to bounded values. Hence, the proposed approach in Theorem 2 can be employed for delay-range-dependent adaptive synchronization of chaotic networks.

6.7 Conclusion

This chapter describes the adaptive synchronization of the master and the slave chaotic systems subjected to time-delays in a range, bounded delay-rate, unknown parameters. Suﬃcient conditions for the chaos synchronization, based on matrix

163 Chapter 6: Delay-Range-Dependent Adaptive Control for Synchronization inequalities, are derived for the adaptive synchronization by means of an amended

Lyapunov-Krasovskii functional. The proposed amendment in the LK functional enables us to develop adaptation laws for the uncertain parameters coupled with the nonlinearity in the error dynamics between the drive and the response systems. Moreover, the proposed control methodology, by employing the local stability theory, is capable of handling the synchronization problem in a situation where delay-rate bounds are larger than unity, compared to the existing results.

A numerical example for the delay-range-dependent adaptive synchronization of the master-slave Hopﬁeld neural networks was illustrated, and synchronization simulation results were detailed.

164 Chapter 7 Robust-Adaptive Synchronization of Drive and Response Systems with Varying Time Delays

In this chapter, a robust-adaptive synchronization control condition for asymptotic convergence of the synchronization error to a bounded region in the presence of uncertainties and disturbances is presented.

7.1 Introduction

Time delay in practical systems i.e. transport lag, transport delay, dead time is a phenomenon because of delayed response in sensors, actuators and in communication of network. These delays posses risk of degradation of performance system.

The control design of delay containing system need special intention by the control engineer to these delays. Time delay many times come across in physical and technical systems, e.g. hydraulic systems, pneumatic systems, electric systems (long transmission lines, robotics), chemical processes etc. The presence of time lag in the control or the state, may result in degradation of system performance, or even instability. That is why, controllability, observability, optimization, robustness, adaptive control, pole placement has been the area of interest of many researcher in time delays systems. Particularly robust and adaptive stabilization for such systems, has been the area of interests for many researchers during the last ﬁve

165 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays decades. Because of complex nature of nonlinear systems, time delay in these type of system has attracted the researchers more to develop some techniques for stabilization of such systems. Especially, the synchronization of time delayed nonlinear system has been explored by the researcher in the last decade.

A delay-independent synchronization of chaotic networks with multiple time lags in the state vector is explored in [183], and further, an LMI-based simple adaptive control approach is developed in a recent work [184] for uncertain chaotic networks containing bounded perturbations. Adaptive synchronization of competitive neural networks, Lure networks and CohenCrossberg network works with mixed delays, rate-independent bounded delays and generalized mixed delays is investigated in the research papers [73,184] and [185] , respectively. Sliding mode control for attaining synchronous behavior of chaotic oscillators under unknown time-delay by incorporating neuro-fuzzy networks is described in [73]. However, still synchronization problem for the delayed chaotic systems requires signiﬁcant attention of researches owing to the technical hitches of various types of delays, delayed nonlinear dynamics, unknown parameters, and disturbances.

This chapter exhibits a robust adaptive synchronization control condition for asymptotic convergence of the synchronization error to a bounded region in the existence of uncertainties and perturbations In contrast to the existing methodologies, delay-range-dependent adaptive synchronization approaches are derived in this chapter and the proposed controller design conditions are applicable to both the slow and the fast variations in time-delays. Numerical simulations for the chaotic delayed Hopﬁeld neural networks are presented to present the eﬃcacy of the presented delay-range-dependent robust adaptive synchronization control methodologies.

166 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

7.2 Statement of the Generalized Model of the Nonlinear Drive and Response Systems

Consider the master-slave (drive-response) systems which are to be synchronized are taken as nonlinear in their dynamics.The nonlinearities presents in both master and slave systems are homogeneous nonlinearities. The mathematical model of these nonlinear systems are presented in (7.1) and (7.2). These generic models represent equations for the state dynamics for the drive and the response systems respectively. The state dynamics and output dynamics in each of the master and the slave system contain the disturbances. Also, two types of nonlinearities with known and unknown parameters are involved in the state dynamics of the drive and the response systems.

x˙ m(t) =Axm(t) + Adxm(t − τ)

+ f1(xm(t)) + f2(xm(t − τ)) p (7.1) X + gi(xm, t, τ)θi + ϕ1(t), i=1

xm(t) =φ1(t),

x˙ s(t) =Axs(t) + Adxs(t − τ)

+ f1(xs(t)) + f2(xs(t − τ)) p (7.2) X + gi(xs, t, τ)θi + ϕ2(t) + Bu, i=1

xs(t) =φ2(t).

The range of t in both of the above equations (7.1) and (7.2)is deﬁned by

t ∈ −τ2 0 .

167 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

n n where xm ∈ R and xs ∈ R are the state vectors for the master and the slave systems, respectively. The symbol u(t) ∈ Rm denotes the input as control signal.

n×n n×n n×m A ∈ R , Ad ∈ R and B ∈ R are the constant matrices establishing

n the linear dynamics of the systems. The vector-functions f1 (x(t)) ∈ R and

n f2(x(t−τ)) ∈ R symbolize, respectively, the nonlinearities without and with time- delay in the dynamics of the nonlinear systems. The vector-function gi(x, t, τ) ∈

n R represents the nonlinearity associated with unknown scalar parameter θi for

n n i = 1 ··· p. ϕ1(t) ∈ R and ϕ2(t) ∈ R represent the bounded time-dependent perturbations aﬀecting the drive and the response systems’ dynamics, respectively.

τ(t)is the time dependent delay, involved in the dynamics of the master and the slave systems, satisfying the relations,

0 ≤ τ1 ≤ τ(t) ≤ τ2, (7.3) |τ˙(t)| ≤ κ.

Assumption 7.1: Let the nonlinearity gi(x, t, τ) associated with the unknown parameters θi are reformed as

gi(x, t, τ) = Bf3i(x, t, τ)∀i = 1 ··· p, (7.4)

m where f3i(x, t, τ) ∈ R , for i = 1 ··· p, are continuous nonlinear functions.

Remark 7.1: Assumption 7.1 restricts the function gi(x, t, τ), for ∀i = 1 ··· p, associated with the unknown parameters, to a matching nonlinearity, which can be adaptively canceled through the control signal u(t). Note that the known non-matching nonlinearities can be included into f1(x(t)) and f2(x(t − τ)), while the non-matching uncertain components can be covered through ϕ1(t) and ϕ2(t).

Consequently, the master-salve systems in (1)-(2) are used to represent a wide

168 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

range of systems with both the matching and the non-matching nonlinearities

having both the known and the unknown components by means of ϕ1(t), gi(x, t, τ),

f1(x(t)) , f2(x(t − τ)) , ϕ1(t) and ϕ2(t).

Assumption 7.2: The nonlinear functions f1(x(t)) , f2(x(t − τ)) and gi(x, t, τ) obey the inequalities

kf1(xm(t)) − f1(xs(t))k ≤ kL1 (xm(t) − xs(t))k (7.5)

kf2(xm(t − τ)) − f2(xs(t − τ))k ≤ kL2 (xm(t − τ) − xs(t − τ))k (7.6)

kgi(xm, t, τ) − gi(xs, t, τ)k ≤ kL3i(xm − xs)k ∀i = 1, 2, ··· , m (7.7)

where gi(x, t, τ) and L1 , L2, and L3i are the constant matrices of appropriate dimensions and the operator k·k stands for the Euclidean norm.

7.3 Synchronizing Error Dynamics for the Delay Compris- ing Drive and The Response Systems

Synchronizing error dynamics can be derived by deﬁning the synchronization error between the states of the drive and the response systems as e(t) = xm(t) − xs(t).

Now this error dynamics between the master and the slave networks (7.1)-(7.2) by incorporating the assumption 7.1 represented by (7.4) are written as

e˙(t) =Ae(t) + Ade(t − τ) + f1(xm(t)) − f1(xs(t)) + f2(xm(t − τ)) p (7.8) X − f2(xs(t − τ)) + B [f3i(xm, t, τ) − f3i(xs, t, τ)] θi + ϕ1(t) − ϕ2(t) − Bu. i=1

169 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

7.3.1 Problem Formulation

Theorems 6.1-6.2, similarly Corollaries 6.1-6.2 presented in Chapter 6 provide adaptive synchronization controller design approaches by ignoring disturbances, that is, by substituting ϕ1 = ϕ2 = 0. However, to take care of the synchronization problem under disturbances, we derive a delay-range-dependent robust adaptive synchronization condition in Theorem 7.1 in the next section. The following assumption is considered for the controller synthesis.

Assumption 7.3: Let the diﬀerence in disturbance vectors ϕ1 = ϕ2 satisﬁes

T (ϕ1 − ϕ2) (ϕ1 − ϕ2) < δ, (7.9) where δ is a positive scalar.

Now we extend the proposed delay-range-dependent adaptive synchronization control methodology of Theorem 2 for robustness against undesirable perturbations.

7.4 Synchronizing Control for Drive-Response Architec- ture with Unknown Parameters and Varying State De- lays

A suﬃcient conditions for the delay-range-dependent adaptive and robust adaptive synchronization control of the master and the slave chaotic systems is derived in this section. The master and the slave systems to be synchronized contain interval time-delays in the states and also exhibit bounded delay-rate . A novel treatment of the adaptive delay-range-dependent stability is explored. By this novel treatment for the objective of stability, a local robust adaptive control scheme is developed to achieve synchronization of the uncertain time-delay nonlinear systems 170 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

with interval delays under disturbances. The proposed robust adaptive synchro-

nization control methodology guarantees convergence of the synchronization error

into a bounded region which can be reduced to minimize the disturbances eﬀects.

The reduced bounded region into which the synchronization error converges guar-

anties more robust stability against disturbances which eﬀect the delays and delays

rates in the synchronization phenomenon of the master and the slave systems.

7.4.1 Robust-adaptive feedback control for synchronization

The proposed control law for adaptive synchronization is given by

p X ˆ u = F e(t) − [f3i(xm, t, τ) − f3i(xs, t, τ)]θi (7.10) i=1

ˆ where F is a constant controller gain matrix of appropriate dimensions and θi is an estimate of θi.

7.4.2 Adaptation Law

The proposed adaptation law

ˆ˙ T T T T θi =σiψ 3i(xm, xs, t, τ)B P e + σiψ 3i(xm, xs, t, τ)B (7.11) 2 2 ¯ × [τ1 N1 + τ12N2][Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)],

and the value of index can vary in the range of i = 1 ··· p.

7.4.3 Performance index for robust control

For the convergence of the synchronization error e(t) to the origin for the robust adaptive synchronization of the master and the slave systems, a Lyapunov function based performance index is designed. This performance index J(t) consist of time rate of the Lyapunov energy function, the quadratic function based on error and

171 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

the quadratic function based on diﬀerence of disturbances i.e. (ϕ1 − ϕ2). This

performance index for the robust-adaptive stability is intuited in two steps. Fisrt

is the L2 gain of error e(t) with respect to diﬀerence of disturbances i.e. (ϕ1 − ϕ2) with ε > 0 is deﬁned as,

eT (t)He(t) T < ε. (ϕ1 − ϕ2) (ϕ1 − ϕ2)

This can also be written as

T T e (t)He(t) − (ϕ1 − ϕ2) ε(ϕ1 − ϕ2) < 0.

Secondly, The addition of time rate of change Lyapunov function with the above

inequality produces the desired performance index

˙ T T J(t) = V (t) + e (t)He(t) − (ϕ1 − ϕ2) ε(ϕ1 − ϕ2). (7.12)

Here H ∈ Rn×n is a positive-deﬁnite matrix. The condition for delay-range-

dependent robust adaptive synchronization of the drive and the response systems

is derived in Theorem 7.1. The robustness oﬀered by this suﬃcient condition is

against undesirable perturbations occurring in the phenomenon of synchroniza-

tion.

Theorem 7.1: A suﬃcient condition for synchronization of the master-slave

systems in (7.1)-(7.2) with time-delay bounds presented by (7.3) subject to As-

sumptions 7.1-7.3, by utilization of a feedback control (7.10) and an adaptation ¯ ¯ ¯ ¯ law (7.11), is that there exist matrices X > 0, Q > 0, M1 > 0, M2 > 0, N1 > 0, ¯ ¯ N2 > 0, H > 0, and scalars σi > 0, for i = 1, 2, ··· p, δ > 0 and λ > 0 such that

172 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays the matrix inequalities

  HX¯     > 0, (7.13) ∗ λ2X

  MAT 11 MAT 12     < 0, (7.14) ∗ MAT 22

  Ω A X N¯ 0 II 0 I  11 d 1     ¯ ¯   ∗ ζ22 N2 N2 0 0 0 0         ∗ ∗ ζ33 0 0 0 0 0         ∗ ∗ ∗ ζ44 0 0 0 0  MAT 11 =   ,    ∗ ∗ ∗ ∗ −I 0 0 0         ∗ ∗ ∗ ∗ ∗ −I 0 0         ∗ ∗ ∗ ∗ ∗ ∗ −I 0      ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI

 p T  τ ζ τ ζ XLT 0 X P σ L  1 19 12 110 1 i 3i   i=1     τ XAT τ XAT 0 XLT 0   1 d 12 d 2       0 0 0 0 0         0 0 0 0 0  MAT 12 =   ,      τ1I τ12I 0 0 0       τ I τ I 0 0 0   1 12       τ1I τ12I 0 0 0      τ1I τ12I 0 0 0

173 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

  −XN¯ −1X 0 0 0 0  1       ∗ −XN¯ −1X 0 0 0   2      MAT 22 =  ∗ ∗ −I 0 0  ,        ∗ ∗ ∗ −I 0      ∗ ∗ ∗ ∗ −I hold, where

T T T ¯ ¯ ¯ ¯ ¯ Ω11 = XA + AX − BJ − J B + Q + M1 + M2 − N1 + H, ¯ ¯ ζ22 = −(1 − κ)Q − 2N2, ¯ ¯ ¯ ζ33 = −N1 − N2 − M2, ¯ ¯ ζ44 = −N2 − M2,

T T T ζ19 = ζ110 = XA + J B

µ = λX.

The error trajectory converges to an ellipsoidal region eT (t)λ2He(t) ≤ 1 for all initial conditions V (0) < 1 and remains bounded in the region V (t) < 1, where

H = X−1HX¯ −1 .

The controller gain can be determined through F = JX−1 and the tolerable bound on disturbance can be estimated as δ = ε−1λ−2 .

Proof: To ensure the asymptotic convergence of the synchronization error e(t) for the robust adaptive synchronization of the master and the slave systems, a Lyapunov function based performance index (7.12) is used. For the robust adaptive synchronization J(t) he should be negative i.e.

˙ T T J(t) = V (t) + e (t)He(t) − (ϕ1 − ϕ2) ε(ϕ1 − ϕ2) < 0,

174 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays where H ∈ Rn×n is a positive-deﬁnite matrix and ε is a positive scalar.

  Ξ PA N 0 PP 0 P τ A¯T N τ A¯T N LT LT S  11 d 1 1 1 12 2 1 2     T T   ∗ Θ22 N2 N2 0 0 0 0 τ1A N1 τ12A N2 0 0 0   d d       ∗ ∗ Θ33 0 0 0 0 0 0 0 0 0 0         ∗ ∗ ∗ Θ44 0 0 0 0 0 0 0 0 0       ∗ ∗ ∗ ∗ −I 0 0 0 τ N τ N 0 0 0   1 1 12 2       ∗ ∗ ∗ ∗ ∗ −I 0 0 τ1N1 τ12N2 0 0 0        < 0  ∗ ∗ ∗ ∗ ∗ ∗ −I 0 τ1N1 τ12N2 0 0 0         ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε τ1N1 τ12N2 0 0 0       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N 0 0 0 0   1       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −N2 0 0 0         ∗∗∗∗∗∗∗∗ ∗ ∗−I 0 0         ∗∗∗∗∗∗∗∗ ∗ ∗ ∗−I 0      ∗∗∗∗∗∗∗∗ ∗ ∗ ∗∗−I (7.15)

This performance index is the sum of time rate of change of the Lyapunov energy function, the synchronization error based quadratic term eT (t)He(t) with

T H > 0 and disturbance diﬀerence based term (ϕ1 − ϕ2) ε(ϕ1 − ϕ2) with ε > 0.

So, if J(t) < 0, the energy of both quadratic functions i.e. eT (t)He(t) and

T (ϕ1 − ϕ2) ε(ϕ1 − ϕ2) converges to origin with time. By performing the similar steps as in the proof of Theorem 6.1, the condition (7.12) can be validated if holds, where p T P S = σiL3i i=1 ¯T ¯ Ξ11 = A P + P A + Q + M1 + M2 − N1 + H,

A¯ = A − BF,

Θ22 = −(1 − κ)Q − 2N2,

175 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

Θ33 = −N1 − N2 − M2,

Θ44 = −N2 − M2.

By applying the congruence transform as presented in the proof of Theorem

2 of chapter 6, to the inequality (7.15), we obtain the matrix inequality (7.15).

Hence, a suﬃcient condition for validity of J(t) < 0 is that the inequality (7.15)

T T must hold. Let us assign Λ = e (t)He(t) − ε(ϕ1 − ϕ2) (ϕ1 − ϕ2). If Λ < 0,

T −2 using (ϕ1 − ϕ2) (ϕ1 − ϕ2) < δ from Assumption 3 and by taking λ = (εδ) , we can conclude eT (t)λ2He(t) < 1. It enables us to infer that the stability prevails for synchronization error dynamics with suitable values of λ and H inside the ellipsoidal region eT (t)λ2He(t) < 1. On the other hand if Λ > 0 , employing

J(t) < 0 implies V˙ (t) < outside the region eT (t)λ2He(t) ≥ 1. Hence, the error asymptotic converges to the ellipsoidal region eT (t)λ2He(t) < 1. Unequivocally, it is well-established that V˙ (t) < 0 ensures V (t) < V (0). Using initial condition

V (0) < 1 , we can write V (t) < 1. From (6.16) and V (0) < 1, we have eT (t)P e(t) <

1. The ellipsoidal region eT (t)λ2He(t) < 1 can be included into the region of stability eT (t)P e(t) < 1 through λ2H > P . By assigning X = P −1, we have

λ2H − X−1 > 0, which by application of the Schur complement produces

  λ2HI     > 0. (7.16) ∗ X

By applying the congruence transformation using diag(λ−1X, λI) to (7.16) and substituting H¯ = XHX result into the matrix inequality (7.13). Consequently, the inequalities (7.13) and (7.15) must be satisﬁed for the robust adaptive synchronization of the master and the slave systems, which completes the proof.

Remark 7.2: Theorem 7.1 provides a condition in (7.13)-(7.15) for the convergence of the synchronization error e(t)into an ellipsoidal region eT (t)λ2He(t) < 1 176 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

in presence of disturbances. The proposed methodology in Theorem 7.1 is worthier

than that of Theorem 6.2 as it propounds the resultant adaptive control scheme

to handle the bounded disturbances. Robustness of the synchronization error

against perturbations can be attained by reducing the size of ellipsoidal region

eT (t)λ2He(t) < 1.

Remark 7.3: In Theorem 7.1, the constraints for robust adaptive synchronization methodologies are advantageous for ﬁnding the controller gain matrix F .

However, these constraints contain nonlinear terms, which cannot be straight- forwardly ﬁxed using the conventional convex routines. To ﬁnd an appropriate controller gain matrix, the feasibility problem of the conditions in Theorems 7.1 and also the Corollaries 1-2 of chapter 6 and Theorem 1 of this chapter can be converted into an optimization problem of a nonlinear objective function with linear constraints, and the resultant problem can be resolved via recursive convex routines, based on cone complementary linearization approach (see, for instance, [194]-

[195] and references therein). For instance, the nonlinear constraints in Theorem

1 can be converted into

  min T race XP + N¯ N˜ + N¯ N˜ + R P N¯ P + R P N¯ P + R λX˜ λ˜  1 1 2 2 1 1 2 2 3    subject to            ¯ ¯ ˜  XI N1 I N2 I λI I             ≥ 0 ,   ≥ 0,   ≥ 0,   ≥ 0,  ˜ ˜  ∗ P ∗ N1 ∗ N2 ∗ λI        N¯ X N¯ X X λI  1 2    ≥ 0,   ≥ 0,   ≥ 0,         ∗ R1 ∗ R2 ∗ R3    ˜  andtheinequalitiesin Theorem 3bysubstituingR1 = XN1X,    ˜ 2  R2 = XN2X, andR3=λ X. (7.17)

177 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

Cone complementary linearization algorithm can be followed to obtain a solu-

tion to the problem (7.17) using convex routines.

Remark 7.4: It is important to note that the novel results for local delay-

dependent and delay-rate-independent are similar to Corollaries 1-2, for robust

adaptive synchronization of nonlinear systems. These results can be derived as

speciﬁc cases of Theorem 1, which are left for the readers.

7.5 Simulation and Results

For evaluating the authenticity of the proposed robust adaptive control

methodology rendered by Theorem 1, we consider a case where the slave

system’s states are inﬂuenced by disturbances (by taking ϕ1(t) = 0 and T ϕ2(t) = sin 19πt sin 20πt ) for L1 = L2 = 10, L31 = L32 = 0.5,

σ1 = σ2 = 1, κ = 10, τ1 = 0.9, and τ2 = 1.1. The solution of inequality

7.15 subjected to constraints presented in Theorem 7.1 yields.

    2584 0 13.1608 0     F =  , P =  , 0 2584 0 13.1608 0

    4159.8 0 3833.4 0     N1 =  , N2 =  , 0 4159.8 0 3833.4

and ε = 33.7739 and λ = 14.0504.

By using these values, the closed-loop simulation results are shown in the

following Figure which conﬁrms the claim of Theorem 7.1 by depicting the syn-

chronized behavior of the master-slave Hopﬁeld neural networks under the eﬀect

178 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays

disturbances. We can witness from these Figures that both synchronization errors

converge in the neighborhood of the origin with bounded adaptive parameters

in the presences of disturbances by application of the propounded delay-range-

dependent robust-adaptive control methodology. Figure 7.1(a) and 7.1(b) provide

the phase portraits of the master and the slave systems, respectively.

Figure 7.1: Behavior of the master-slave Hopﬁeld neural networks with the proposed robust adaptive controller: (a) phase portrait of the master system, (b) phase portrait of the slave system

By applying the proposed control scheme, the drive and the response systems

with diﬀerent initial conditions continue to move, so as, the normalized correspond-

ing potentials xm1(t), xs1(t) and recovery variables xm2(t), xs2(t) match with each other. The time evolutions of the states of the nonlinear systems, conﬁrming synchronization of the networks, by the proposed robust adaptive control approach, are shown in Figure 7.2.

Convergence of the synchronization errors e1(t)and e2(t) under disturbances is

elaborated in Figure 7.3. Note that convergence of the errors is robust against

perturbations. This ﬁgure conﬁrm the validity of the proposed control approach 179 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time

Delays States of States salvethe master and systems of salve the master and systems

Figure 7.2: Time evolution of the states xm1, xm2, xs1, and xs2 of the drive and the response systems

in Theorem 2, by elaborating asymptotic convergence of the errors e1(t) and e2(t) to the origin. So, the claim of removing diﬀerence in temporal evolution of the normalized potentials xm1(t), xs1(t) and recovery variables xm2(t), xs2(t), that is, the error between corresponding normalized potential recovery variable is veriﬁed.

It is worth mentioning that the eﬀect of synchronizing control scheme in case of FHN system aﬀects e11 and e2. So, the simulation results for the values of e1 and e2 are depicted in the Figure 7.3. ˆ ˆ In Figure 7.4(e), convergence of the adaptation parameters θ1and θ2 is demonstrated. Hence, the proposed synchronization control methodology in Theorem 3 is robust against disturbances and adaptive against unknown parameters in the master-slave systems.

To compare the proposed control methodology with the control scheme developed in [82], feasibility of the constraint (6.19) for diﬀerent values of the lower and the upper delay bounds and maximum allowable delay-rates is tested along with

180 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time

Delays Synchronization Synchronization errors errors

Figure 7.3: Time evolution of the synchronization error e1, and e2 between the corresponding states of the master and the slave systems

Adaptive parametersAdaptive Adaptive parameters Adaptive

Figure 7.4: Adaptation of unknown parameters. the feasibility of the approach developed in [82]. The corresponding results show that the eﬀectiveness (found to be feasible) for extremely larger bounds on the delay-rates (of the order of ≈ 1019), while the control method of [82] can deal with

181 Chapter 7: Robust-Adaptive Synchronization of System with Varying Time Delays the problem of synchronization with the delay-rates less than unity. Consequently, the proposed control methodology for diﬀerent time-delay bounds and various time-delay ranges over the existing methodology on delay-range-dependent adaptive synchronization. The proposed adaptive feedback provided herein is capable to synchronize the master-slave networks of various time-delay bounds, versatile time-delay ranges, and larger delay-rates.

7.6 Conclusion

This chapter presents a suﬃcient conditions for the chaos synchronization, based on matrix inequalities, which are derived for the robust adaptive synchronization of drive and response chaotic systems with bounded delays and under the inﬂuence of nonzero disturbances . An extension to the asymptotic stability results presented in chapter 6 for the synchronization error is provided to deal with the external perturbations. A cone complementary linearization algorithm based scheme for iterative computation of the controller gain matrix is also propounded.

A numerical example for the delay-range-dependent adaptive synchronization of the master-slave Hopﬁeld neural networks is illustrated, and synchronization simulation results are elaborated.

182 Chapter 8 Summary and Future Work Directions

In this chapter, summary and future work are presented.

8.1 Summary

In this thesis report synchronization of the nonlinear drive and response chaotic systems through adaptive and robust-adaptive feedback control is investigated.

Introduction to the nonlinear system and subclass of nonlinear systems i.e. chaotic systems is presented. Synchronization problem and its importance in context to applications is manifested. The work already done in the ﬁeld synchronization is explored through literature survey and presented in introductory chapter.

A detailed discussion for the problem of synchronization of drive and response chaotic (nonlinear) systems containing known and unknown parameters, delayed and non-delayed dynamics and under the effect of zero and nonzero disturbances is elaborated in problem formulation section of each chapter. The synchronizing controller design conditions are derived for different cases including nonadaptive control, adaptive control and robust adaptive control. The necessary and sufficient conditions for deriving the controller and observer gain matrices are deduced using the Lyapunov stability theory. Similarly, adaptation laws are derived by Lyapunov

183 Chapter 8: Summary and Future Work Directions energy function and amended LK functional.The controller and adaptation laws are veriﬁed for derive and response chaotic systems to be synchronize.

New control approaches for synchronization the master and the slave chaotic systems is established by means of novel coupled chaotic synchronous observers and coupled chaotic adaptive synchronous observers.The simultaneous estimation of the master and the slave systems states is accomplished, by means of the proposed observers for each of the master and the slave systems, to produce error signals between these estimated states.

This estimated synchronization error signal and the state-estimation errors converge to the origin by means of a speciﬁc observers-based feedback control signal to ensure synchronization as well as state estimation. Using Lyapunov stability theory, nonadaptive and adaptive control laws and properties of nonlinearities, a convergence condition for the state-estimation errors and the estimated synchronization error is developed in the form of nonlinear matrix inequalities.

Convex routines for the solution of the resulted inequality constraints using a two-step approach is presented, which provides the necessary and suﬃcient condition to obtain values of the observer gain and controller gain matrices. Further, a method requiring less computational eﬀorts for solving the matrix inequalities for obtaining the observer and the controller gain matrices using decoupling technique is also proposed.

A local adaptive and robust adaptive control methodologies for the synchronization of the chaotic drive and the response systems with ﬁnite time lags, bounded delay rates, unknown parameters, and perturbations. A local adaptive coherence control condition under bounded initial condition is developed to synthesize a feedback controller capable of executing successful synchronization of the drive and the response systems with unknown parameters and delay belong-

184 Chapter 8: Summary and Future Work Directions ing to a range (with either zero or nonzero lower bound). A suﬃciency condition for the adaptive control approach is inculcated through an amendment in the conventional LyapunovKrasovskii functional, which enables to develop the adaptation laws for the unknown parameters associated with the nonlinearity in the dynamical behavior of the drive and the response systems.

Further, a robust adaptive synchronization control condition for asymptotic convergent behavior of the synchronization error to a bounded region under perturbations as well as disturbances is provided. Indifferent to the existing works, delayrange-dependent adaptive synchronization methodologies are derived in this paper, and the rendered controller design conditions are valid for both the slow and the fast variations in time delays. Numerical simulations for the chaotic delayed Hopfield neural networks are presented to elaborate the efficacy of the proposed delay-range-dependent adaptive and robust adaptive synchronization control methodologies.

The proposed synchronization schemes by the use of controller and adaptation laws are verified by using FitzHugh-Nagumo (FHN) master-slave architectures and the master-slave Hopfield neural networks for to synchronize them. The simulation for these system shows the asymptotic convergence of synchronization errors and convergence of adaptive parameters to a finite value.

Then the concept of delay range dependent with time varying delays is introduced. Finally an adaptive handling technique is introduced for nonlinearities having time-delays and combined with previous adaptive technique for non delayed nonlinearity and an adaptive delay-range-dependent controller is introduced for synchronization of nonlinear time-delay chaotic systems with varying time-delays in an interval and aﬀected by the disturbance. Simulation results elaborates the eﬀectiveness of the rendered methodologies.

185 Chapter 8: Summary and Future Work Directions

The synchronization of two nonlinear, time-delay, chaotic systems with uncertainties and external disturbance is considered in this research. A delay-rate- dependent controller is designed using robust-adaptive feedback control theory.

The designed controller, along with the adaptation laws for estimation of parameters guarantees the chaos synchronization by converging synchronization error to zero. Computer simulations verify the eﬀectiveness of proposed method. Although the model considered is for complex nonlinear chaotic systems with time-delays with unknown parameters. However the solution is also valid for relatively simple, nonlinear systems with known parameters and constant delays.

8.2 Future Work Directions

The work presented in this report mainly focused on the study of the synchronization of time-delayed chaotic systems with unknown parameters. Methodologies for the control of nonlinear, time-delayed and chaotic systems with unknown parameters are also proposed. There is still a lot of room available for the research in the ﬁeld. Following issues can be considered for the future options for research.

1. Delay-range-dependent robust adaptive output feedback control for synchro-

nization of nonlinear systems is not addressed in literature.

2. Instead of taking Lipschitz condition for linearity, one sided Lipschitz con-

straints can be used.

3. Adaptive and robust adaptive output feedback synchronization conditions

for more than two system can be explored.

4. Asymptotic stability for the delay-range-dependent state feedback synchro-

nization can also be explored for synchronization two nonlinear systems un-

der quantization noise. 186 Chapter 8: Summary and Future Work Directions

5. Proposed schemes should be explored for discrete models of nonlinear sys-

tems and controller.

6. The coeﬃcient matrix of the controller u(t) is assumed to be identity. This

issue may be explored in future for a non-identity matrix.

7. Although a novel Lyapunov Krasovskii functional is used in the proof of

stability theorems for synchronization of two nonlinear systems, however,

the matrix P is taken to be identity which may be explored in future for a

non-identity matrix.

8. The derivative of time-delayτ ˙ is assumed to be bounded by a quantity κ < 1,

this work may be explored in future without this assumption.

9. The applicability of the presented controller for more complex systems such

as systems with larger delay rate bound along with hardware constraints

may also be tested in future.

10. The results presented in this thesis are veriﬁed and simulated for relatively

simple and less complex systems. These results can be considered for veriﬁca-

tion with more complex systems under input saturation for synchronization

187 Appendices

188 Appendix A

Proof of Theorem 3.2

Consider the Lyapunov function

T T T V2(t) =em(t)Pmem(t) + es (t)Pses(t) + eo (t)Poeo(t) (A.1) ˜T −1 ˜ ˜T −1 ˜ + θm(t)Θm θm(t) + θs (t)Θs θs(t).

By taking the time rate of change of the Lyapunov function (A.1) , one can write

˙ T T T V2(t) =e ˙m(t)Pmem(t) + em(t)Pme˙m(t) +e ˙s (t)Pses(t)

T T T (A.2) + es (t)Pse˙s(t) +e ˙o (t)Poeo(t) + eo (t)Poe˙o(t)

˜˙T −1 ˜ ˜T −1 ˜˙ ˜˙T −1 ˜ ˜T −1 ˜˙ + θmΘm θm + θmΘm θm + θs Θs θs + θs Θs θs.

By putting the values of error dynamics (3.32)-(3.34), the time rate of change of

Lyapunov energy function results into

˙ V2(t) =[Aem(t) + [f(xm(t)) − f(ˆxm(t))]

ˆ + Bg(xm(t))θm − Bg(ˆxm(t))θm 1 − L Ce (t) + BF e (t)]T P e (t) (A.3) m m 2 o m m

T + em(t)Pm[Aem(t) + [f(xm(t)) − f(ˆxm(t))]

ˆ + Bg(xm(t))θm − Bg(ˆxm(t))θm − LmCem(t)

189 1 + BF e (t)] + [Ae (t) + [f(x (t)) − f(ˆx (t))] 2 o s s s ˆ + Bg(xs(t))θs − Bg(ˆxs(t))θs − LsCes(t) 1 + BF e (t)]T e (t) + [Ae (t) + [f(x (t)) − f(ˆx (t))] 2 o s s s s ˆ + Bg(xs(t))θs − Bg(ˆxs(t))θs − LsCes(t) 1 + BF eo(t)]e ˙s(t) + [Aeo(t) + [f(ˆxm(t)) − f(ˆxs(t))] 2 (A.4) ˆ ˆ + Bg(ˆxm(t))θm − Bg(ˆxs(t))θs + LmCem(t) − LsCes(t)

T T − BF eo(t) − Bug] Poeo(t) + eo (t)Po[Aeo(t)

ˆ ˆ + [f(ˆxm(t)) − f(ˆxs(t))] + Bg(ˆxm(t))θm − Bg(ˆxs(t))θs

+ LmCem(t) − LsCes(t) − BF eo(t) − Bug]

˜˙T −1 ˜ ˜T −1 ˜˙ ˜˙T −1 ˜ ˜T −1 ˜˙ + θmΘm θm + θmΘm θm + θs Θs θs + θs Θs θs.

T T By employing the assumption 3.2, i.e. B Pm −RmC = 0 and B Ps −RsC = 0, the time rate of change of Lyapunov function is further reformed as follows,

˙ T T T T T V2(t) =em(t)(A − C Lm)Pmem(t) + [f(xm(t)) − f(ˆxm(t))] Pmem(t)

T T T T + 0.5eo (t)F B Pmem(t) + [gm(xm(t)) − gm(ˆxm(t))] Pmem(t)

T T + θem(t)g (ˆxm(t))Rm(ym(t) − Cxˆm(t))

T + em(t)Pm (A − LmC) em(t)

T T + em(t)Pm [f(xm(t)) − f(ˆxm(t))] + 0.5em(t)PmBF eo(t) (A.5) T + em(t)Pm [gm(xm(t)) − gm(ˆxm(t))]

T T ˜ + (ym(t) − Cxˆm(t)) Rmg(ˆxm(t))θm(t)

T T T T T + es (t)(A − C Ls )Pses(t) + [f(xm(t)) − f(ˆxm(t))] Pses(t)

T T T T + 0.5eo (t)F B Pses(t) + [gs(xs(t)) − gs(ˆxs(t))] Pses(t)

˜T T + θs (t)g (ˆxs(t))Rs(ys(t) − Cxˆs(t))

190 T + es (t)Ps(A − LsC)es(t)

T T + es (t)Ps [f(xs(t)) − f(ˆxs(t))] + 0.5es (t)PsBF eo(t)

T + es (t)Ps [gs(xs(t)) − gs(ˆxs(t))]

T T ˜ + (ym(t) − Cxˆm(t)) Rs g(ˆxs(t))θs(t)

T T T T + eo (t)(A − F B )Poeo(t)

T + [f(ˆxm(t)) − f(ˆxs(t))] Poeo(t) (A.6) T T T T T T + em(t)C LmPoeo(t) − es (t)C Ls Poeo(t)

T + eo (t)Po(A − BF )eo(t)

T + eo (t)Po [f(ˆxm(t) − f(ˆxs(t)]

T T + eo (t)PoLmCem(t) − eo (t)PoLsCes(t)

˜˙T −1 ˜ ˜T −1 ˜˙ + θmΘm θm + θmΘm θm

˜˙T −1 ˜ ˜T −1 ˜˙ + θs Θs θs + θs Θs θs.

Now, using Assumption 3.1, which represents the Lipschitz boundedness of the nonlinear function and also scaling the inequality (resulted by Lipschitz assumption) by the factors β1 and β2, we have

T 2 T −β1[gm(xm(t)) − g(ˆxm(t))] [gm(xm(t)) − gm(ˆxm(t))] + β1Lgmem (t) em (t) > 0,

T 2 T −β2[gs(xs(t)) − gs(ˆxs(t))] [gs(xs(t)) − gs(ˆxs(t))] + β2Lgses (t) es (t) > .

Employing the above inequalities, using (A.6), ˜˙ ˆ˙ θm(t) = −θm(t)

and ˜˙ ˆ˙ θs(t) = −θs(t) and incorporating the adaptation laws (3.40)-(3.41) under Assumption 2, the above equality (A.6) is resulted into the following inequality, 191 ˙ T T T T T V2 (t) ≤em(t)(A − C Lm)Pmem(t) + [f(xm(t)) − f(ˆxm(t))] Pmem(t)

T T T T + 0.5eo (t)F B Pmem(t) + [gm(xm(t)) − g(ˆxm(t))] Pmem(t)

T T + em(t)Pm(A − LmC)em(t) + em(t)Pm [f(xm(t)) − f(ˆxm(t))]

T T + 0.5em(t)PmBF eo(t) + em(t)Pm [g(xm(t), θm) − g(ˆxm(t), θm)]

T T T T T + es (t)(A − C Ls )Pses(t) + [f(xm(t)) − f(ˆxm(t))] Pses(t)

T T T T + 0.5eo (t)F B Pses(t) + [g(xs(t), θs) − g(ˆxs(t), θs)] Pses(t)

T T + es (t)Ps(A − LsC)es(t) + es (t)Ps [f(xs(t)) − f(ˆxs(t))]

T T + 0.5es (t)PsBF eo(t) + es (t)Ps [g(xs(t), θs) − g(ˆxs(t), θs)]

T T T T T + eo (t)(A − F B )Poeo(t) + [f(ˆxm(t)) − f(ˆxs(t))] Poeo(t)

T T T T T T + em(t)C LmPoeo(t) − es (t)C Ls Poeo(t)

T T (A.7) + eo (t)Po(A − BF )eo(t) + eo (t)Po [f(ˆxm(t) − f(ˆxs(t)]

T T + eo (t)PoLmCem(t) − eo (t)PoLsCes(t)

T − α1[f(xm(t)) − f(ˆxm(t))] [f(xm(t)) − f(ˆxm(t))]

2 T + α1Lf em(t)em(t)

T 2 T − α2[f(xs(t)) − f(ˆxs(t))] [f(xs(t)) − f(ˆxs(t))] + α2Lf es (t)es(t)

T 2 T − α3[f(ˆxm(t)) − f(ˆxs(t))] [f(ˆxm(t)) − f(ˆxs(t))] + α3Lf eo (t)eo(t)

T − β1[gm(xm(t)) − g(ˆxm(t))] [gm(xm(t)) − gm(ˆxm(t))]

2 T + β1Lgmem (t) em (t)

T − β2[gs(xs(t)) − gs(ˆxs(t))] [gs(xs(t)) − gs(ˆxs(t))]

2 T + β2Lgses (t) es (t) ,

192 which further reveals that the time rate of Lyapunov energy function is less than

product of matrices as written in the following equation (A.8)

˙ T (A.8) V2 (t) ≤ E2 (t)Ω2E2(t).

ET (t) = T T T T T T T T , 2 em(t) es (t) eo (t) ∆fmmˆ (t) ∆fssˆ(t) ∆fmˆ sˆ(t) ∆gmmˆ (t) ∆gssˆ(t)

∆gmmˆ (t) = [gm(xm(t)) − gm(ˆxm(t))] ,

∆gssˆ(t) = [gs(xs(t)) − gs(ˆxs(t))] .

If (3.42) is satisﬁed, then the above inequality (A.8) implies V˙ (t) < 0. Hence, the errors em(t), es(t) and eo(t) converge to the origin, which entails synchronization of the master and the slave chaotic oscillators.

193 Appendix B

Proof of Theorem 6.1

Taking the time derivative of both sides of (6.16) for evaluating the rate of change

of energy of the dynamics with time, which is useful for developing the stability

condition for convergence of synchronization error dynamics to the origin results

into (B.1). Moreover, it is worth mentioning that this synchronizing condition is

developed with a prerequisite that the initial values of the master system and the

slave system are bounded such that V (t) < 1.

V˙ (t) =e ˙T (t)P e(t) + eT (t)P e˙(t) + eT (t)Qe(t)

− (1 − τ˙)eT (t − τ)Qe(t − τ)

T T + e (t)M1e(t) − e (t − τ1)M1e(t − τ1)

T T + e (t)M2e(t) − e (t − τ2)M2e(t − τ2)

t Z (B.1) 2 T T + τ1 e˙ (t)N1e˙(t) − τ1e˙ (t)N1e˙(t)dz

t−τ1 t−τ Z 2 2 T T + τ12e˙ (t)N2e˙(t) − τ12e˙ (t)N2e˙(t)dz

t−τ1 p p X −1 ˜˙T ˜ X −1 ˜T ˜˙ + σi θi θi + σ θi θi. i=1 i=1

The Jensen’s inequality, is very well known concept in the literature as mention in

Chapter 2. By using this Jensen’s inequality concept for diﬀerent values of upper and lower bounds of integration i.e t − τ1 to t and t − τ1 to t − τ2 as given in (B.2) 194 and (B.3), following two inequalities are obtained. First inequality is resulted for the integral limits ranging from t − τ1 to t as mentioned in (B.2)

t Z T T T − τ1e˙ (t)N1e˙(t)dz ≤ − e (t)N1e(t) + e (t)N1e(t − τ1)

t−τ1 (B.2)

T T + e (t − τ1)N1e(t) − e (t − τ1)N1e(t − τ1).

For the integral limits ranging from t − τ1 to t − τ2 the resulted inequality is mentioned in (B.3)

t−τ Z 2 T T − τ12e˙ (t)N1e˙(t)dz ≤ − [e(t − τ) − e(t − τ2)] N2[e(t − τ) − e(t − τ2)]

t−τ1 (B.3)

T − [e(t − τ1) − e(t − τ)] N2[e(t − τ1) − e(t − τ)].

Using (B.2)-(B.3) the integral terms are replaced with the non-integral terms, moreover this replacement reforms the inequality into simpler form involving the error dynamics. Similarly, for further simpliﬁcation of (B.1), incorporateτ ˙(t) ≤ κ and obtain the following result.

V˙ (t) ≤e˙T (t)P e(t) + eT (t)P e˙(t) + eT (t)Qe(t) − (1 − κ)eT (t − τ)Qe(t − τ)

T T T + e (t)M1e(t) − e (t − τ1)M1e(t − τ1) + e (t)M2e(t)

T T 2 2 T − e (t − τ2)M2e(t − τ2) +e ˙ (t)[τ1 N1 + τ12N2]e ˙(t) − e (t)N1e(t)

T T T + e (t)N1e(t − τ1) + e (t − τ1)N1e(t) − e (t − τ1)N1e(t − τ1) (B.4)

T − [e(t − τ) − e(t − τ2)] N2[e(t − τ) − e(t − τ2)]

T − [e(t − τ1) − e(t − τ)] N2[e(t − τ1) − e(t − τ)] p p X −1 ˜˙T ˜ X −1 ˜T ˜˙ + σi θi θi + σ θi θi. i=1 i=1

195 Putting the value of time rate of change of error i.e.e ˙(t) from (6.15) in the above inequality and applying the relationship between the matrices A, B and F which the diﬀerence between matrix A and BF i.e. A¯ = A − BF reveals,

˙ ¯ V (t) ≤[Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ) p X ˜ T + Bψ3i(xm, xs, t, τ)θi + ϕ1 − ϕ2] P e(t) i=1

T ¯ + e (t)P [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ) p X ˜ + Bψ3i(xm, xs, t, τ)θi + ϕ1 − ϕ2] i=1

+ eT (t)Qe(t) − (1 − κ)eT (t − τ)Qe(t − τ)

T T + e (t)M1e(t) − e (t − τ1)M1e(t − τ1)

T T + e (t)M2e(t) − e (t − τ2)M2e(t − τ2)

¯ + [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) p X ˜ T + ψ2(xm, xs, t, τ) + Bψ3i(xm, xs, t, τ)θi + ϕ1 − ϕ2] i=1

2 2 ¯ × [τ1 N1 + τ12N2][Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) p X ˜ + ψ2(xm, xs, t, τ) + Bψ3i(xm, xs, t, τ)θi + ϕ1 − ϕ2] i=1

T T − e (t)N1e(t) + e (t)N1e(t − τ1)

T T + e (t − τ1)N1e(t) − e (t − τ1)N1e(t − τ1)

T − [e(t − τ) − e(t − τ2)] N2[e(t − τ) − e(t − τ2)]

T − [e(t − τ1) − e(t − τ)] N2[e(t − τ1) − e(t − τ)] p p X −1 ˜˙T ˜ X −1 ˜T ˜˙ + σi θi θi + σ θi θi. i=1 i=1

Further simpliﬁcation of the right hand side of the above iequality implies

196 ˙ T ¯T T T T V (t) ≤e (t)A P e(t) + e (t − τ)Ad P e(t) + ψ1 (xm, xs, t)P e(t) p T X ˜T T T + ψ2 (xm, xs, t, τ)P e(t) + θi ψ3i(xm, xs, t, τ)B P e(t) i=1

T T ¯ T + (ϕ1 − ϕ2) P e(t) + e (t)P Ae(t) + e (t)PAde(t − τ)

T T + e (t)P ψ1(xm, xs, t) + e (t)P ψ2(xm, xs, t, τ) p T X ˜ T T + e (t)P Bψ3i(xm, xs, t, τ)θi + e (t)P (ϕ1 − ϕ2) + e (t)Qe(t) i=1

− (1 − κ)eT (t − τ)Qe(t − τ)

T T T + e (t)M1e(t) − e (t − τ1)M1e(t − τ1) + e (t)M2e(t)

T ¯ − e (t − τ2)M2e(t − τ2) + [Ae(t) + Ade(t − τ)

T 2 2 + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)] [τ1 N1 + τ12N2]

¯ × [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)]

¯ T (B.5) + [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)] " p # 2 2 X ˜ × [τ1 N1 + τ12N2] Bψ3i(xm, xs, t, τ)θi + (ϕ1 − ϕ2) i=1 " p # X ˜T T T 2 2 + θi ψ3i(xm, xs, t, τ)B + (ϕ1 − ϕ2) [τ1 N1 + τ12N2] i=1 ¯ × [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)] " p # X ˜T T T + θi ψ3i(xm, xs, t, τ)B + (ϕ1 − ϕ2) i=1 " p # 2 2 X ˜ × [τ1 N1 + τ12N2] Bψ3i(xm, xs, t, τ)θi + (ϕ1 − ϕ2) i=1

T T T − e (t)N1e(t) + e (t)N1e(t − τ1) + e (t − τ1)N1e(t)

T T − e (t − τ1)N1e(t − τ1) − e (t − τ)N2e(t − τ)

T T + e (t − τ)N2e(t − τ2) + e (t − τ2)N2e(t − τ)

T T − e (t − τ2) × N2e(t − τ2) − e (t − τ1)N2e(t − τ1)

197 T T + e (t − τ1)N2e(t − τ) + e (t − τ)N2e(t − τ1) p p T X −1 ˜˙T ˜ X −1 ˜T ˜˙ + e (t − τ)N2e(t − τ) + σi θi θi + σ θi θi. i=1 i=1

By substituting ϕ1 = ϕ2 = 0 and applying the proposed adaptation law in (6.17) ˜ ˆ along with θi = θi − θi, it implies that

˙ T ¯T ¯ V (t) ≤e (t)[A P + P A + Q + M1 + M2 − N1]e(t)

T T T + e (t)PAde(t − τ) + e (t)N1e(t − τ1) + e (t)P ψ1(xm, xs, t)

T T T + e (t)P ψ2(xm, xs, t, τ) + e (t − τ)Ad P e(t)

T T − (1 − κ)e (t − τ)Qe(t − τ) − 2e (t − τ)N2e(t − τ)

T T + e (t − τ)N2e(t − τ1) + e (t − τ)N2e(t − τ2)

T T + e (t − τ1)N1e(t) + e (t − τ1)N2e(t − τ)

T T − e (t − τ1)N2e(t − τ1) − e (t − τ1)N1e(t − τ1) (B.6)

T T − e (t − τ1)M1e(t − τ1) + e (t − τ2)N2e(t − τ)

T T − e (t − τ2)M2e(t − τ2) − e (t − τ2)N2e(t − τ2)

T T + ψ1 (xm, xs, t)P e(t) + ψ2 (xm, xs, t, τ)P e(t) p p X ˜T T T 2 2 X ˜ + θi ψ3i(xm, xs, t, τ)B [τ1 N1 + τ12N2] Bψ3i(xm, xs, t, τ)θi i=1 i=1

T ¯T T T T T + [e (t)A + e (t − τ)Ad + ψ1 (xm, xs, t) + ψ2 (xm, xs, t, τ)]

2 2 ¯ × [τ1 N1 + τ12N2][Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) + ψ2(xm, xs, t, τ)].

The stability of the synchronization phenomenon of the drive and the response nonlinear systems can be established by ensuring the negative values of the time rate of change of LK functional i.e. V˙ (t) < 0, which further implies V (t) < V (0) for all time t > 0. Given the initial condition satisfying the bound V (0) < 1, the error trajectory remains bounded in the region V (t) < 1 for all t > 0. By splitting

198 p ¯ P −1 ˜T ˜ the LK functional V (t) into V (t) + σi θi θi, where i=1

t Z V¯ (t) =eT (t)P e(t) + eT (z)Qe(z)dz

t−τ t 2 Z X T + e (z)Mie(z)dz i=1 t−τ i (B.7) 0 t Z Z T + τ1e˙ (z)N1e˙(z)dzdα

−τ1 t+α −τ t Z 1 Z T + τ12e˙ (z)N2e˙(z)dzdα.

−τ2 t+α

Since V (t) < 1, the Lyapunov function can be a summation of two terms i.e.

p ¯ X −1 ˜T ˜ V (t) + σi θi θi < 1. i=1

As all the terms on right side of (B.7) are positive therefore one can infer

p X −1 ˜T ˜ σi θi θi < 1, i=1

Multiplying both sides of above inequality with a scalar value σi results into (??),

which implies that the product of ith component of the parameter vecotr θi with

its corresponding transposed component is less than a scalar value σi

p X ˜T ˜ θi θi < σi. (B.8) i=1

The above inequality helps to deduce the inequality (B.9)

˜T ˜ (B.9) θi θi < σi.

199 Furthermore, by utilitarian Lipschitz nonlinearity condition as mentioned in As-

sumption 6.2, the following two positive inequalities are achieved corresponding

to the nonlinearities f1(x(t)) , f2(x(t − τ)) of the master and the slave system

dynamics.

T T T (B.10) e (t)L1 L1e(t) − ψ1 (xm, xs, t)ψ1(xm, xs, t) ≥ 0

T T T (B.11) e (t)L2 L2e(t) − ψ2 (xm, xs, t)ψ2(xm, xs, t) ≥ 0

Similarly, for the nonlinearities gi(x, t, τ), where each of of these implies g(x, t, τ) = p P gi(x, t, τ). By application of (6.7), i=1

p p

X X (gi(xm, t, τ) − gi(xs, t, τ)) ≤ L3i(xm − xs) 0 (B.12) i=1 i=1

˜T ˜ By utilizing (??) i.e. θi θi < σi, the above inequality can be modiﬁed to the following inequality

p p X ˜ ˜ X gi(xm, t, τ)θi − gi(xs, t, τ)θi ≤ σiL3i(xm − xs) , (B.13) i=1 i=1 which further helps the to develop the following inequality (B.14)

p !T p ! " p #T T X X X ˜ e (t) σiL3i σiL3i e(t) − Bψ3i(xm, xs, t, τ)θi i=1 i=1 i=1 (B.14) " p # X ˜ × Bψ3i(xm, xs, t, τ)θi ≥ 0. i=1

By employing gi(x, t, τ) = Bf3i(x, t, τ) and (6.14). Incorporating (B.10)-(B.12) into (B.6) produces

200 ˙ T ¯T ¯ V (t) ≤e (t)[A P + P A + Q + M1 + M2 − N1]e(t)

T T + e (t)PAde(t − τ) + e (t)N1e(t − τ1)

T T + e (t)P ψ1(xm, xs, t) + e (t)P ψ2(xm, xs, t, τ)

T T T + e (t − τ)Ad P e(t) − (1 − κ)e (t − τ)Qe(t − τ)

T T − 2e (t − τ)N2e(t − τ) + e (t − τ)N2e(t − τ1)

T T + e (t − τ)N2e(t − τ2) + e (t − τ1)N1e(t)

T T + e (t − τ1)N2e(t − τ) − e (t − τ1)N2e(t − τ1)

T T − e (t − τ1)N1e(t − τ1) − e (t − τ1)M1e(t − τ1)

T T + e (t − τ2)N2e(t − τ) − e (t − τ2)M2e(t − τ2)

T T − e (t − τ2)N2e(t − τ2) + ψ1 (xm, xs, t)P e(t)

T + ψ2 (xm, xs, t, τ)P e(t) (B.15)

T ¯T T T + [e (t)A + e (t − τ)Ad

T T + ψ1 (xm, xs, t) + ψ2 (xm, xs, t, τ) p X ˜T T T 2 2 + θi ψ3i(xm, xs, t, τ)B ][τ1 N1 + τ12N2] i=1 ¯ × [Ae(t) + Ade(t − τ) + ψ1(xm, xs, t) p X ˜ + ψ2(xm, xs, t, τ) + Bψ3i(xm, xs, t, τ)θi] i=1

T 2 T + e (t)L1e(t) − ψ1 (xm, xs, t)ψ1(xm, xs, t)

T 2 + e (t − τ)L2e(t − τ)

T − ψ2 (xm, xs, t, τ)ψ2(xm, xs, t, τ)

p !T p ! T X X + e (t) σiL3i σiL3i e(t) i=1 i=1

201 " p #T X ˜ − Bψ3i(xm, xs, t, τ)θi i=1 " p # X ˜ × Bψ3i(xm, xs, t, τ)θi . i=1

The above inequality (B.15) can be summarized into an inequality containing the product of matrices as presented in (B.16). Here the Φ is a square matrix and

Ω(xm, xs, t, τ) are the vector matrices. The (B.16) inequality reveals that the time rate of Lyapunov energy function is less than product of matrices as written in the following equation

˙ T V (t) ≤ Ω(xm, xs, t, τ) ΦΩ(xm, xs, t, τ), (B.16)

 ^  Θ PA N 0 PP 0  11 d 1     ^   ∗ Θ22 N2 N2 0 0 0     ^     ∗ ∗ Θ33 0 0 0 0     ^  Φ =  ∗ ∗ ∗ Θ 0 0 0   44    (B.17)    ∗ ∗ ∗ ∗ −I 0 0         ∗ ∗ ∗ ∗ ∗ −I 0      ∗ ∗ ∗ ∗ ∗ ∗ −I

+[ ¯T T ]T [τ 2N + τ 2 N ] ¯ A Ad 0 0 III 1 1 12 2 [AAd 0 0 III],

p T T T T T T P ˜T T T T Ω = [e (t) e (t − τ) e (t − τ1) e (t − τ2) ψ1 ψ2 θi ψ3iB ] . i=1

T ^ p p ¯T ¯ T P P Θ11 = A P + P A + Q + M1 + M2 − N1 + L1 L1 + σiL3i σiL3i i=1 i=1 ^ T Θ22 = −(1 − κ)Q − 2N2 + L2 L2 202 ^ Θ33 = −N1 − N2 − M2

^ Θ44 = −N2 − M2

Applying the Schur complement to Φ < 0 under (B.17), the matrix inequality

(6.18) is obtained, which completes the proof.

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