Fractional-Order Design: Devices, Circuits, and Systems introduces applications from the design perspective so that the

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*Table of contents : Front CoverFractional-Order Design: Devices, Circuits, and SystemsCopyrightContentsList of contributors1 MOS realizations of fractional-order elements 1.1 Introduction 1.2 CPE/FI emulation techniques 1.2.1 CPE/FI emulation using electronically controlled RC networks 1.2.2 CPE/FI emulation using fractional-order integrators/differentiators 1.3 Practical aspects 1.3.1 Time constants and scaling factors spread reduction 1.3.2 Reduction of the control terminals of the system 1.3.3 Enhancement of the order range of the emulator 1.4 Conclusions and discussion Acknowledgment References2 A chaotic system with equilibria located on a line and its fractional-order form 2.1 Introduction 2.2 Model of the proposed flow and its dynamics 2.3 Fractional-order form 2.4 Circuit implementation 2.5 FPGA implementation of the chaotic system 2.6 Conclusion References3 Approximation of fractional-order elements for sinusoidal oscillators 3.1 Introduction 3.2 R-C network-based FDs 3.3 FDs for sinusoidal oscillators 3.3.1 Impedance equalization-based FDs 3.3.2 Admittance equalization-based FDs 3.4 Performance analysis 3.4.1 Stability analysis 3.4.2 Sensitivity analysis using Monte Carlo simulation 3.4.3 PSpice simulation and FoM calculation 3.5 Conclusion and scope of future research References4 Synchronization between fractional chaotic maps with different dimensions 4.1 Introduction 4.2 Preliminaries 4.3 Combined synchronization of 2D fractional maps 4.3.1 Master system and slave systems 4.3.2 Combined scheme 4.4 Combined synchronization of 3D fractional maps 4.4.1 Master system and slave systems 4.4.2 Combined scheme 4.5 Concluding remarks and future works Acknowledgments References5 Stabilization of different dimensional fractional chaotic maps 5.1 Introduction 5.2 Basic tools 5.2.1 Caputo delta difference operator and stability 5.2.2 Caputo h-difference operator and stability 5.3 Stabilization of 2D fractional maps 5.4 Stabilization of 3D fractional maps 5.5 Summary and future works Acknowledgments References6 Observability of speed DC motor with self-tuning fuzzy-fractional-order controller 6.1 Introduction 6.2 Mathematical model of DC motor 6.3 Stability of speed estimation 6.4 Proposed speed controller 6.4.1 Literature review 6.4.1.1 Riemann–Liouville fractional difference 6.4.1.2 Caputo fractional difference 6.4.1.3 Grunwald–Letnikov fractional difference 6.4.2 Fractional PID controller 6.4.3 Fractional-order PI controller 6.4.4 Self-tuning PI fractional-order controller with fuzzy logic 6.5 Results and discussion 6.5.1 Test 1 6.5.2 Test 2 6.5.3 Test 3 6.5.4 Test 4 6.6 Conclusions References7 Chaos control and fractional inverse matrix projective difference synchronization on parallel chaotic systems with application 7.1 Introduction 7.2 Preliminaries 7.2.1 Definition 7.2.2 Stability criterion 7.3 The fractional inverse matrix projective difference synchronization 7.3.1 Problem formulation 7.3.2 System description 7.3.3 Simulations and discussions 7.3.4 Comparison with published literature 7.3.5 Chaos control about the stagnation points in the presence of uncertainties and disturbances 7.4 Illustration in secure communication 7.5 Conclusions References8 Aggregation of chaotic signal with proportional fractional derivative execution in communication and circuit simulation 8.1 Introduction 8.2 Fractional-order chaotic systems and their properties 8.2.1 Lyapunov spectrum and Kaplan–Yorke dimension 8.2.2 Dissipativity 8.3 Analog circuit imitation 8.4 Security analysis 8.5 Conclusion References9 CNT-based fractors in all four quadrants: design, simulation, and practical applications 9.1 Introduction 9.2 Fractor: definitions and state-of-the-art 9.2.1 FOE realization: a brief survey 9.3 A wide-CPZ, long-life, packaged CNT fractor 9.3.1 Description of the CNT fractor 9.3.2 Process of fabrication 9.3.3 Electrical characterization 9.3.4 Variation of FO parameters with time 9.3.5 Origin of the wide CP nature in CNT fractors 9.4 Fractors with desired specifications 9.4.1 An RC ladder network with Foster-I topology 9.4.2 Simulation of FO immittances with RC ladder 9.4.3 Change in FO parameters in CNT fractor 9.4.4 Comparison between two different fractor design techniques 9.5 Four-quadrant FO immittances using CNT fractors 9.5.1 Design of Type I fractors 9.5.2 Design of Type II fractors 9.5.3 Design of Type III fractors 9.5.4 Tunability of fractors 9.6 Application of four-quadrant CNT fractors 9.6.1 Design of a high-Q factor FO resonator 9.6.2 Hardware realization and practical tuning 9.7 Conclusion 9.A MATLAB program to determine RC ladder parameters for five FO specifications Acknowledgments References10 Fractional-order systems in biological applications: estimating causal relations in a system with inner connectivity using fractional moments 10.1 Introduction 10.2 Related work 10.3 Fractional moments and fractional cumulants 10.4 Hindmarsh–Rose model 10.5 Estimating causal relations 10.5.1 Complex cumulants 10.5.2 Granger causality 10.6 Causal direction pattern recognition 10.6.1 Clustering 10.6.2 Convolutional neural network 10.7 Discussion 10.8 Conclusion References11 Unitary fractional-order derivative operators for quantum computation 11.1 Introduction 11.2 A brief survey on geometric phase concepts in quantum computation 11.3 Methodology 11.3.1 Fractional calculus preliminaries 11.3.2 Unitary fractional-order derivatives and phasor descriptions 11.3.3 Control of multiqubit quantum interference circuits by unitary fractional-order derivatives 11.4 Some quantum computation implications for unitary fractional-order derivative operators 11.4.1 Modeling of quantum interference computation modes 11.4.2 Design of a measurement probability distribution via a genetic algorithm 11.5 Discussion and conclusions 11.A References12 Analysis and realization of fractional step filters of order (1+α) 12.1 Introduction 12.2 Analysis of fractional step filters 12.2.1 First method 12.2.1.1 Fractional step low-pass filter 12.2.1.2 Fractional step high-pass filter 12.2.1.3 Fractional step band-pass filter 12.2.1.4 Fractional step all-pass filter 12.2.1.5 Fractional step band-stop filter 12.2.2 Second method 12.2.2.1 Fractional step low-pass filter 12.2.2.2 Fractional step high-pass filter 12.2.2.3 Fractional step band-pass filter 12.2.2.4 Fractional step all-pass filter 12.2.2.5 Fractional step band-stop filter 12.3 Numerical analysis and simulations of FSFs of order (1+α) 12.3.1 Circuit simulations based on Method I 12.3.2 Circuit simulations based on Method II 12.4 Stability 12.5 Sensitivity analysis 12.5.1 Sensitivity analysis of Method I 12.5.2 Sensitivity analysis of Method II 12.5.3 Monte Carlo simulations 12.6 Conclusion References13 Fractional-order identification and synthesis of equivalent circuit for electrochemical system based on pulse voltammetry 13.1 Introduction 13.2 Experimental setup 13.3 Fractional-order models 13.3.1 Fractional-order transfer function 13.3.2 Fractional-order circuit elements 13.4 Identification of fractional-order transfer function 13.4.1 Structure of the proposed fractional-order transfer function 13.4.2 Parameter estimation 13.4.3 Results: performance evaluation of the identified FOTF 13.5 Proposed circuit with fractional-order elements 13.5.1 Network synthesis for fractional-order circuit 13.5.2 Analysis with fractional circuit parameters 13.6 Principal component analysis: towards electronic tongue application 13.7 Conclusions References14 Higher-order fractional elements: realizations and applications 14.1 Introduction 14.2 Realization of FOEs with fractional order < 1 14.2.1 CFE approximation-based FOC emulation 14.2.2 FI emulation 14.2.3 Functional block diagram-based emulation 14.3 Realization of fractional-order element with 1 < fractional order< n 14.3.1 IIMC-based realization 14.3.2 GIC-based realization 14.3.3 FBD-based realization 14.4 Application 14.4.1 Stability analysis 14.4.2 Simulation and experimental results 14.4.2.1 Functional verification of FI and FOC 14.4.2.2 Functional verification of FOF 14.5 Conclusion References15 Fabrication of polymer nanocomposite-based fractional-order capacitor: a guide 15.1 Introduction 15.1.1 History 15.1.2 Present trends in polymer NCs 15.1.2.1 Porous polymer-based 15.1.2.2 Ferroelectric polymer-based 15.1.2.3 Epoxy resin-based 15.2 Polymers 15.2.1 Polymer NCs 15.2.2 Polymer NC as FOC dielectric 15.3 Ferroelectric polymers 15.3.1 PVDF 15.3.1.1 Dielectric properties of PVDF 15.3.1.2 Inducing β-phase PVDF 15.3.1.3 Ferroelectric effect 15.3.2 Porous polymers 15.3.2.1 Dielectric properties of PMMA 15.4 Conductive fillers 15.5 Methods of synthesis 15.5.1 Intercalation 15.5.1.1 Chemical intercalation 15.5.1.2 Mechanical intercalation 15.5.1.3 Melt intercalation 15.5.2 Sol-gel method 15.5.3 Direct mixing 15.5.4 Melt compounding 15.5.5 Solution blending 15.5.6 In situ polymerization 15.6 Percolation threshold 15.7 Factors affecting properties of polymer NCs 15.7.1 Alignment of the filler 15.7.2 Dispersion of the filler 15.7.3 Interfacial bonding between filler and the polymer matrix 15.8 A GNS/PVDF FOC 15.8.1 Materials and methods 15.8.2 Results and discussion 15.9 Conclusion Acknowledgments References16 Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor 16.1 Introduction 16.2 Solid-state fractional capacitors 16.2.1 Structure of the fractional capacitor 16.2.2 Fabrication procedure 16.3 Batch analysis of the solid-state fractional capacitors for defining the guidelines 16.3.1 Characterization 16.3.2 Yield rate 16.3.3 Effect of thickness of the nanocomposite and the middle plate 16.4 Validation of the defined guidelines 16.5 Material characterization 16.5.1 Details of the analysis 16.5.2 Results from material characterization 16.5.2.1 FTIR spectra 16.5.2.2 SEM and TEM images 16.6 Correlating the material characterization with the CPA of a solid-state fractional capacitor 16.7 Conclusion Acknowledgments ReferencesIndexBack Cover*

FRACTIONAL-ORDER DESIGN: DEVICES, CIRCUITS AND SYSTEMS

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FRACTIONAL-ORDER DESIGN: DEVICES, CIRCUITS, AND SYSTEMS Edited by

AHMED G. RADWAN Engineering Mathematics and Physics Department Cairo University, Giza, Egypt School of Engineering and Applied Sciences Nile University, Giza, Egypt

FAROOQ AHMAD KHANDAY Department of Electronics and Instrumentation Technology University of Kashmir, Srinagar, India

LOBNA A. SAID Nanoelectronics Integrated Systems Center (NISC) Nile University, Giza, Egypt Series editor

QUAN MIN ZHU

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2022 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this ﬁeld are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-90090-4 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Sonnini R. Yura Editorial Project Manager: Charlotte Rowley Production Project Manager: Kamesh Ramajogi Designer: Matthew Limbert Typeset by VTeX

Contents

List of contributors

xi

1. MOS realizations of fractional-order elements

1

Stavroula Kapoulea, Panagiotis Bertsias, Costas Psychalinos, and Ahmed S. Elwakil 1.1. Introduction 1.2. CPE/FI emulation techniques 1.3. Practical aspects 1.4. Conclusions and discussion Acknowledgment References

2. A chaotic system with equilibria located on a line and its fractional-order form

1 7 16 29 30 30

35

Karthikeyan Rajagopal, Fahimeh Nazarimehr, Alireza Bahramian, and Sajad Jafari 2.1. Introduction 2.2. Model of the proposed ﬂow and its dynamics 2.3. Fractional-order form 2.4. Circuit implementation 2.5. FPGA implementation of the chaotic system 2.6. Conclusion References

3. Approximation of fractional-order elements for sinusoidal oscillators

35 39 42 50 50 52 55

63

Shalabh K. Mishra, Dharmendra K. Upadhyay, and Maneesha Gupta 3.1. Introduction 3.2. R-C network-based FDs 3.3. FDs for sinusoidal oscillators 3.4. Performance analysis 3.5. Conclusion and scope of future research References

4. Synchronization between fractional chaotic maps with different dimensions

63 67 71 75 83 84

89

Adel Ouannas, Amina-Aicha Khennaoui, Iqbal M. Batiha, and Viet-Thanh Pham v

vi

Contents

4.1. Introduction 4.2. Preliminaries 4.3. Combined synchronization of 2D fractional maps 4.4. Combined synchronization of 3D fractional maps 4.5. Concluding remarks and future works Acknowledgments References

5. Stabilization of different dimensional fractional chaotic maps

89 91 93 103 111 111 115

123

Adel Ouannas, Amina-Aicha Khennaoui, Iqbal M. Batiha, and Viet-Thanh Pham 5.1. Introduction 5.2. Basic tools 5.3. Stabilization of 2D fractional maps 5.4. Stabilization of 3D fractional maps 5.5. Summary and future works Acknowledgments References

6. Observability of speed DC motor with self-tuning fuzzyfractional-order controller

123 125 129 134 145 146 149

157

Arezki Fekik, Mohamed Lamine Hamida, Hamza Houassine, Hakim Denoun, Sundarapandian Vaidyanathan, Nacera Yassa, Ahmed G. Radwan, and Lobna A. Said 6.1. Introduction 6.2. Mathematical model of DC motor 6.3. Stability of speed estimation 6.4. Proposed speed controller 6.5. Results and discussion 6.6. Conclusions References

7. Chaos control and fractional inverse matrix projective difference synchronization on parallel chaotic systems with application

157 159 162 162 170 175 176

181

Pushali Trikha, Lone Seth Jahanzaib, and Ayub Khan 7.1. Introduction 7.2. Preliminaries 7.3. The fractional inverse matrix projective diﬀerence synchronization 7.4. Illustration in secure communication 7.5. Conclusions References

181 184 185 200 203 204

Contents

8. Aggregation of chaotic signal with proportional fractional derivative execution in communication and circuit simulation

vii

207

Najeeb Alam Khan, Saeed Akbar, Muhammad Ali Qureshi, and Tooba Hameed 8.1. Introduction 8.2. Fractional-order chaotic systems and their properties 8.3. Analog circuit imitation 8.4. Security analysis 8.5. Conclusion References

9. CNT-based fractors in all four quadrants: design, simulation, and practical applications

207 209 216 219 231 232

235

Avishek Adhikary 9.1. Introduction 9.2. Fractor: deﬁnitions and state-of-the-art 9.3. A wide-CPZ, long-life, packaged CNT fractor 9.4. Fractors with desired speciﬁcations 9.5. Four-quadrant FO immittances using CNT fractors 9.6. Application of four-quadrant CNT fractors 9.7. Conclusion Appendix 9.A MATLAB program to determine RC ladder parameters for ﬁve FO speciﬁcations Acknowledgments References

10. Fractional-order systems in biological applications: estimating causal relations in a system with inner connectivity using fractional moments

235 238 241 252 259 265 269 269 271 271

275

Zahra Tabanfar, Farnaz Ghassemi, Alireza Bahramian, Ali Nouri, Ensieh Ghaffari Shad, and Sajad Jafari 10.1. Introduction 10.2. Related work 10.3. Fractional moments and fractional cumulants 10.4. Hindmarsh–Rose model 10.5. Estimating causal relations 10.6. Causal direction pattern recognition 10.7. Discussion 10.8. Conclusion References

275 277 281 286 287 290 293 296 296

viii

Contents

11. Unitary fractional-order derivative operators for quantum computation

301

Baris Baykant Alagoz and Serkan Alagoz Introduction A brief survey on geometric phase concepts in quantum computation Methodology Some quantum computation implications for unitary fractional-order derivative operators 11.5. Discussion and conclusions Appendix 11.A References 11.1. 11.2. 11.3. 11.4.

12. Analysis and realization of fractional step ﬁlters of order (1 + α)

301 303 306 317 329 330 334

337

Gagandeep Kaur, A.Q. Ansari, and M.S. Hashmi 12.1. Introduction 12.2. Analysis of fractional step ﬁlters 12.3. Numerical analysis and simulations of FSFs of order (1 + α) 12.4. Stability 12.5. Sensitivity analysis 12.6. Conclusion References

13. Fractional-order identiﬁcation and synthesis of equivalent circuit for electrochemical system based on pulse voltammetry

337 340 353 363 365 369 369

373

Sanjeev Kumar and Arunangshu Ghosh 13.1. Introduction 13.2. Experimental setup 13.3. Fractional-order models 13.4. Identiﬁcation of fractional-order transfer function 13.5. Proposed circuit with fractional-order elements 13.6. Principal component analysis: towards electronic tongue application 13.7. Conclusions References

373 376 378 381 391 396 399 399

14. Higher-order fractional elements: realizations and applications

403

Neeta Pandey, Rajeshwari Pandey, and Rakesh Verma 14.1. Introduction 14.2. Realization of FOEs with fractional order < 1 14.3. Realization of fractional-order element with 1 < fractional order < n 14.4. Application 14.5. Conclusion References

403 404 412 416 429 432

Contents

15. Fabrication of polymer nanocomposite-based fractional-order capacitor: a guide

ix

437

Zaid Mohammad Shah, Farooq Ahmad Khanday, Gul Faroz Ahmad Malik, and Zahoor Ahmad Jhat 15.1. Introduction 15.2. Polymers 15.3. Ferroelectric polymers 15.4. Conductive ﬁllers 15.5. Methods of synthesis 15.6. Percolation threshold 15.7. Factors aﬀecting properties of polymer NCs 15.8. A GNS/PVDF FOC 15.9. Conclusion Acknowledgments References

16. Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor

437 445 447 457 461 466 467 470 477 478 478

485

Dina Anna John and Karabi Biswas 16.1. Introduction 16.2. Solid-state fractional capacitors 16.3. Batch analysis of the solid-state fractional capacitors for deﬁning the guidelines 16.4. Validation of the deﬁned guidelines 16.5. Material characterization 16.6. Correlating the material characterization with the CPA of a solid-state fractional capacitor 16.7. Conclusion Acknowledgments References Index

485 490 494 500 502 508 512 516 516 523

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List of contributors

Avishek Adhikary Indian Institute of Technology, Bhilai, Department of Electrical Engineering and Computer Science, Raipur, India Saeed Akbar Department of Mathematics, University of Karachi, Karachi, Pakistan Baris Baykant Alagoz Inonu University, Department of Computer Engineering, Malatya, Turkey Serkan Alagoz Inonu University, Department of Physics, Malatya, Turkey A.Q. Ansari Department of Electrical Engineering, Jamia Millia Islamia, New Delhi, India Alireza Bahramian Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Iqbal M. Batiha Department of Mathematics, Faculty of Science, The University of Jordan, Amman, Jordan Panagiotis Bertsias University of Patras, Department of Physics, Electronics Laboratory, Patras, Greece Karabi Biswas Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal, India Hakim Denoun Laboratory of Advanced Technologies of Electrical Engineering (LATAGE), Mouloud Mammeri University Tizi-Ouzou, Tizi-Ouzou, Algeria Ahmed S. Elwakil University of Sharjah, Department of Electrical and Computer Engineering, Sharjah, United Arab Emirates Nile University, Nanoelectronics Integrated Systems Center (NISC), Giza, Egypt University of Calgary, Department of Electrical and Computer Engineering, Calgary, AB, Canada Arezki Fekik Electrical Engineering Department, Akli Mohand Oulhadj University-Bouira, Bouira, Algeria Laboratory of Advanced Technologies of Electrical Engineering (LATAGE), Mouloud Mammeri University Tizi-Ouzou, Tizi-Ouzou, Algeria Farnaz Ghassemi Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran xi

xii

List of contributors

Arunangshu Ghosh Department of Electrical Engineering, National Institute of Technology Patna, Patna, Bihar, India Maneesha Gupta Department of Electronics and Communication Engineering, Netaji Subhas University of Technology, New Delhi, India Tooba Hameed Department of Mathematics, University of Karachi, Karachi, Pakistan Mohamed Lamine Hamida Laboratory of Advanced Technologies of Electrical Engineering (LATAGE), Mouloud Mammeri University Tizi-Ouzou, Tizi-Ouzou, Algeria M.S. Hashmi Deptartment of Electronics and Communication Engineering, IIIT Delhi, New Delhi, India Hamza Houassine Electrical Engineering Department, Akli Mohand Oulhadj University-Bouira, Bouira, Algeria Sajad Jafari Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Health Technology Research Institute, Amirkabir University of Technology, Tehran, Iran Lone Seth Jahanzaib Jamia Millia Islamia, Department of Mathematics, New Delhi, Delhi, India Zahoor Ahmad Jhat Islamia College of Science and Commerce, Hawal Srinagar, JK, India Dina Anna John Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal, India Stavroula Kapoulea University of Patras, Department of Physics, Electronics Laboratory, Patras, Greece Gagandeep Kaur Department of Electrical Engineering, Jamia Millia Islamia, New Delhi, India Ayub Khan Jamia Millia Islamia, Department of Mathematics, New Delhi, Delhi, India Najeeb Alam Khan Department of Mathematics, University of Karachi, Karachi, Pakistan Farooq Ahmad Khanday Department of Electronics and Instrumentation Technology, University of Kashmir, Srinagar, JK, India

List of contributors

xiii

Amina-Aicha Khennaoui Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria Sanjeev Kumar Department of Electrical Engineering, School of Engineering & Technology, Sandip University, Madhubani, Bihar, India Gul Faroz Ahmad Malik Department of Electronics and Instrumentation Technology, University of Kashmir, Srinagar, JK, India Shalabh K. Mishra Department of Electronics and Communication Engineering, ABES Engineering College, Ghaziabad, Uttar Pradesh, India Fahimeh Nazarimehr Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Ali Nouri Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Adel Ouannas Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria Neeta Pandey Department of Electronics and Communication Engineering, Delhi Technological University, Delhi, India Rajeshwari Pandey Department of Electronics and Communication Engineering, Delhi Technological University, Delhi, India Viet-Thanh Pham Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Costas Psychalinos University of Patras, Department of Physics, Electronics Laboratory, Patras, Greece Muhammad Ali Qureshi Department of Physics, University of Karachi, Karachi, Pakistan Ahmed G. Radwan Engineering Mathematics and Physics Department, Cairo University, Cairo, Egypt Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt Karthikeyan Rajagopal Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai, India Lobna A. Said Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt

xiv

List of contributors

Ensieh Ghaffari Shad Ayatollah Kashani Hospital, Tehran University of Medical Sciences, Tehran, Iran Zaid Mohammad Shah Department of Electronics and Instrumentation Technology, University of Kashmir, Srinagar, JK, India Zahra Tabanfar Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Pushali Trikha Jamia Millia Islamia, Department of Mathematics, New Delhi, Delhi, India Dharmendra K. Upadhyay Department of Electronics and Communication Engineering, Netaji Subhas University of Technology, New Delhi, India Sundarapandian Vaidyanathan Research and Development Centre, Vel Tech University, Chennai, India Rakesh Verma Department of Electronics and Communication Engineering, Delhi Technological University, Delhi, India Nacera Yassa Electrical Engineering Department, Akli Mohand Oulhadj University-Bouira, Bouira, Algeria

CHAPTER ONE

MOS realizations of fractional-order elements Stavroula Kapouleaa , Panagiotis Bertsiasa , Costas Psychalinosa , and Ahmed S. Elwakilb,c,d a University

of Patras, Department of Physics, Electronics Laboratory, Patras, Greece of Sharjah, Department of Electrical and Computer Engineering, Sharjah, United Arab Emirates c Nile University, Nanoelectronics Integrated Systems Center (NISC), Giza, Egypt d University of Calgary, Department of Electrical and Computer Engineering, Calgary, AB, Canada b University

1.1. Introduction Fractional-order (FO) elements are fundamental building blocks in the application of fractional calculus on circuit-level realizations. Including FO capacitors, known also in the literature as constant phase elements (CPEs), and FO inductors (FIs), these elements are very useful tools in numerous scientific fields, such as filtering, oscillator design, bio-medicine, and control theory [31,33,34]. An integer-order (IO) system can be directly converted into an FO system by just substituting the IO capacitors/inductors with the corresponding FO counterparts. The parameters of the CPEs and the FIs can offer a more precise control of the system characteristics, due to the extra degree of freedom offered by the (fractional) order. This originates from the fact that the identity of a CPE is determined by two parameters, the order and the pseudocapacitance {α, Cα }. The characteristic frequency-domain impedance of the element is described by ZCPE (s) =

1 , C α · sα

(1.1)

where the order α ∈ (0, 1), Cα has units of Farad/sec 1−α , and s is the Laplacian operator. The relationship between the pseudocapacitance in Farad/sec 1−α and the conventional capacitance C in Farad, at a specific frequency ω, is given by C=

Cα ω1−α

.

Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00006-8 All rights reserved.

(1.2) 1

2

Stavroula Kapoulea et al.

Similarly, the corresponding equations in the case of an FI are given by ZFI (s) = Lβ · sβ , L=

Lβ ω1−β

,

(1.3)

(1.4)

with the order β ∈ (0, 1), the pseudoinductance Lβ in Henry/sec 1−β , and the conventional inductance L in Henry. Therefore, the capacitance/inductance of the elements is a frequency-dependent parameter. The implementation of FO elements is of particular scientific interest, but it is still an ongoing project [2,15,26]. As there is no commercial production of such type of elements, the development of emulators that efficiently approximate their behavior is a research field with practical utility in a wide variety of applications [4,23–25,28,36,40]. The realization of CPEs/FIs is performed using proper approximation methods (i.e., continued fraction expansion [CFE], partial fraction expansion [PFE], Oustaloup, Matsuda, Carlson, Valsa, etc.) and appropriately configured structures (i.e., RC networks, multifeedback configurations, current-mode topologies, etc.) [5,10,11,27,30,44]. The utilized elements, in order to construct the emulator, can be exclusively passive elements (i.e., resistors, capacitors) or a combination of passive and active elements (e.g., operational amplifiers [Op-Amps], second-generation current conveyors [CCIIs], current feedback operational amplifiers [CFOAs], operational transconductance amplifiers [OTAs]), depending on the desired characteristics that must be achieved [12,13,39,41–43]. The approximation methods, mentioned above, focus on the approximation of the operator (τ s)α,β around a center frequency ω0 , leading to a rational nth-order function of the form (τ s)α,β

An sn + An−1 sn−1 + . . . + A1 s + A0 , sn + Bn−1 sn−1 + . . . + B1 s + B0

(1.5)

with τ being a time constant related to the center frequency as τ = 1/ω0 , n being the order of approximation, and Ai (i = 0, 1, ...n), Bi (i = 0, 1, ...n − 1) being real, positive coefficients. Expressing the impedance functions in (1.1) and (1.3) in the form of (1.6a) and (1.6c), respectively, and then substituting the operator (τ s)α,β with the expression in (1.5), the derived approximated impedance functions

3

MOS realizations of fractional-order elements

of a CPE and an FI are given by (1.6b) and (1.6d). We have

ZCPE,approx (s) =

τα

·

Cα

(1.6a)

sn + Bn−1 sn−1 + . . . + B1 s + B0 , An sn + An−1 sn−1 + . . . + A1 s + A0

(1.6b)

ZFI ,approx (s) =

τβ

·

Lβ

·

· (τ s)β ,

(1.6c)

An sn + An−1 sn−1 + . . . + A1 s + A0 . sn + Bn−1 sn−1 + . . . + B1 s + B0

(1.6d)

ZFI (s) = Lβ

Cα

τα

1 , (τ s)α

ZCPE (s) =

τβ

A simple and direct way to implement the impedance function in (1.6b) is to use Foster or Cauer RC networks, constructed by conventional passive resistors and capacitors [10,14]. The configurations of Type-I and Type-II Foster networks are demonstrated in Fig. 1.1, while the corresponding structures of Type-I and Type-II Cauer networks are demonstrated in Fig. 1.2 [19,37].

Figure 1.1 Foster types of RC networks for CPE emulation.

Figure 1.2 Cauer types of RC networks for CPE emulation.

For the Foster networks the PFE tool is applied on (1.6b), leading to the form of (1.7a) for the impedance expression and to the form of (1.7b)

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for the admittance expression. We have ZCPE,PFE (s) = k +

n i=1

ri , s − pi

(1.7a)

ri · s YCPE,PFE (s) k ri = + ⇒ YF −II (s) = k + . s s i=1 s − pi s − pi i=1 n

n

(1.7b)

The coefficients ri and pi (i = 1, ...n) are the residues and poles of (1.6b) and k is a constant term. Considering these functions and the configurations in Fig. 1.1, the expressions for the impedance of Type-I Foster and the admittance of Type-II Foster networks are given by (1.8a) and (1.8b), respectively: ZF −I (s) = R0 +

n

1 Ci

i=1

s + Ri1Ci

,

(1.8a)

1 n 1 Ri · s YF −II (s) = + . R0 i=1 s + Ri1Ci

(1.8b)

In the case of Cauer networks, the CFE tool is used to decompose the approximated impedance function in (1.6b). One option is to arrange the powers of the variable s in the numerator and the denominator of (1.6b) starting from the highest to the lowest power. The derived, decomposed impedance function in this case has the following form: 1

ZCPE,CFE (s) = q0 +

(1.9)

1

q1 s +

1

q2 + q3 s +

1 .................

1 1 q2n−1 s + q2n

,

where qi (i = 0, ...2n) are the coefficients of the CFE. Another form can be obtained by rewriting the polynomials in (1.6b) in the form of the lowest to the highest power of the variable s and, then, divide the impedance

5

MOS realizations of fractional-order elements

function in (1.6b) by s. The derived function is expressed as 1

ZCPE,CFE (s) = s

1

q0 s +

1

q1 +

1

q2 s +

.................

1 q2n−1 + q2n s ⇓

(1.10)

1

ZCPE,CFE (s) =

1

q0 + q1 s

1

+

q2 +

1 .................

1 q2n−1 s

+ q2n

.

Considering the form in (1.9) for the case of the Type-I Cauer network and the form in (1.10) for the case of the Type-II Cauer network, the derived expressions for the impedance of the configurations in Fig. 1.2 are given by 1

ZC−I (s) = R0 +

(1.11)

1

C1 s +

1

R2 + C3 s +

1 .................

1 1 C2n−1 s + R2n

,

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Stavroula Kapoulea et al.

1

ZC−II (s) = 1 R0

(1.12)

1

+ 1 C1 s

1

+ 1 R2

+

1 .................

1 1 C2n−1 s

+ R2n

.

The design equations for calculating the values of resistors and capacitors for both types of each network are summarized in Table 1.1. Table 1.1 Design equations for calculating resistor and capacitor values of the Foster and Cauer networks in Fig. 1.1 and Fig. 1.2. Foster Cauer Element Type I Type II Element Type I Type II

R0 Ri (i = 1, 2, ...n)

k

Ci (i = 1, 2, ...n)

1 ri

ri pi

1 k 1 ri ri pi

R0 Ri (i = 0, 2, ...2n)

Cj j = 1, 3, ...2n − 1

q0 qi qj

1 q0 1 qi 1 qj

An important remark that should be made concerns the behavior of the networks on the frequency limits, i.e., at very low (s → 0) and very high (s → ∞) frequencies. Specifically, the Type-I Foster and Cauer RC networks follow the same conditions, as their impedance at very low frequencies is equal to the series connection of the network resistors ( ZF −I |s→0 = R0 + R1 ... + Rn and ZC−I |s→0 = R0 + R2 ... + R2n ) and at very high frequencies equal to the resistor R0 ( ZF −I |s→∞ = ZC−I |s→∞ = R0 ). A similar behavior is also observed for the Type-II Foster and Cauer RC networks with the impedance at very low frequencies being equal to the resistor R0 ( ZF −II |s→∞ = ZC−II |s→∞ = R0 ) and at very high frequencies equal to the parallel connection of the network resistors ( ZF −II |s→∞ = R0 //R1 ...//Rn and ZC−II |s→∞ = R0 //R2 ...//R2n ). For the FI emulation the above networks are also used, combined with a generalized impedance converter [GIC] in order to achieve inductive behavior [1,32,35]. This technique is the best option in the case of a CPE or FI with a priori specified characteristics {α, Cα }, {β, Lβ }, in the sense that there is no requirement for tunability. But a general CPE/FI emulator, capable of implementing different cases of order and pseudocapacitance/inductance,

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MOS realizations of fractional-order elements

cannot be realized using RC networks that have fixed values of passive elements and, consequently, there is no ability of specifications tuning. This results from the fact that the whole network must be redesigned in order to meet the new specifications.

1.2. CPE/FI emulation techniques 1.2.1 CPE/FI emulation using electronically controlled RC networks The RC networks in Fig. 1.1 and Fig. 1.2 could acquire the advantage of electronic control of their characteristics just by substituting the passive resistors with active elements that express resistance behavior [18]. Considering, for instance, the Type-I Foster network and the design equations given in Table 1.1, and assuming that the values of resistance are variable by a factor x (i.e., R0 → x · R0 and Ri → x · Ri (i = 1, 2, ...n)), the obtained impedance function is given by

ZF −I ,var (s) = x · R0 +

n i=1

Ri

1 Ri Ci

x · s + Ri1Ci

= x1−α · ZF −I (s) ,

(1.13)

with the intermediate pole frequencies described as ω0,i = 1/(Ri Ci ). The corresponding expressions for the impedance/admittance of the other types of networks can also be formed following the same procedure. Using this concept, the values of pseudocapacitance and center frequency of the CPE emulator are adjusted as described in (1.14a) and (1.14b), respectively: Cα,var =

Cα , x1−α

ω0,var =

ω0

(1.14a)

. (1.14b) x So, the adjustment of the resistances of the network by a factor x induces adjustment of the impedance, pseudocapacitance, and center frequency of the emulator. The multiple-output OTA is an efficient option for the implementation of a floating resistor with variable resistance, due to the transconductance parameter (gm ), which is related to the resistance as gm = 1/R and is controlled by a DC voltage or current. Connecting the inputs and outputs of

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the OTA as demonstrated in Fig. 1.3, this configuration can directly replace the passive resistors of the networks in Fig. 1.1 and Fig. 1.2 and form electronically controlled OTA-C networks.

Figure 1.3 Circuit implementation and symbol representation of a multiple-output OTA-based ﬂoating resistor.

A possible implementation of the OTA is given by the circuit in Fig. 1.3, which offers the advantage of enhanced linearity. For operation in the subthreshold region, the expression of the transconductance for this OTA is given by 5 IB gm = · , (1.15) 9 n · VT with IB being the bias current of the OTA, n ∈ (1, 2) the MOS transistor subthreshold slope factor, and VT = 26 mV @ 27°C the thermal voltage. This expression showcases the linear relationship between the transconductance gm and the bias current IB , leading to the fact that the value of gm , and, as a result, the value of the resistance R = 1/gm , can be adjusted through IB , offering in this way the advantage of electronic control of the emulator characteristics [18]. Using the Cadence IC design suite and the Design Kit AMS 0.35 µm process, indicative aspect ratios (W /L ) for the multiple-output OTA in

9

MOS realizations of fractional-order elements

Fig. 1.3 are given in Table 1.2, with the power supply voltages being equal to VDD = −VSS = 0.75 V . Considering the case of a CPE with {α, Cα } = {0.5, 300 pF /sec 0.5 } approximated applying the second-order CFE tool on a Type-II Foster network around f0 = 100 Hz, the values of resistances, along with the corresponding values of bias currents, and also the capacitances are summarized in Table 1.3. The layout design of the circuit is demonstrated in Fig. 1.4, with the active core of the OTA-based resistors being framed by the red rectangle (mid gray in print version) and the passive capacitors being represented by the yellow squares (light gray in print version). Table 1.2 Aspect ratios of the MOS transistors in Fig. 1.3 for emulating a CPE using the electronically controlled OTA-C Type-II Foster network. Transistors (W/L) (μm/μm)

Mb1–Mb3 Mb4–Mb7 Mn1 & Mn4 Mn2 & Mn3 Mp1–Mp6

20/10 1/10 25/1 5/1 0.5/12

Table 1.3 Values of resistances, bias currents, and capacitances for emulating CPE with {α, Cα } = {0.5, 300 pF /s0.5 } around f0 = 100 Hz using second-order CFE approximation on an OTA-C Type-II Foster network. Element Element value R0 / I0 664.9 M / 84.5 pA 31.7 M / 1.8 nA R1 / I1 217.6 M / 258.1 pA R2 / I2 5.3 pF C1 13.9 pF C2

The obtained impedance magnitude and phase frequency responses within a range f = [1, 10 k] Hz, for different values of factor x, are presented in Fig. 1.5. The x = 1 case corresponds to the initial condition, while for x = 0.1 the capacitance of the emulator is scaled to 948.7 pF /sec 0.5 and the center frequency to 1 kHz, while for x = 10 the corresponding values are 94.9 pF /sec 0.5 and 10 Hz.

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Figure 1.4 Layout design of the Type-II Foster impedance emulator (dimensions: 252.55 μm × 102.6 μm). The area included within the red frame (mid gray in print version) is occupied by OTAs conﬁgured as resistors, while the remaining area corresponds to the capacitors [18].

Figure 1.5 Postlayout simulation results of the impedance magnitude and phase frequency responses in the case of CPE {α, Cα } = {0.5, 300 pF /sec0.5 } for x = 0.1, 1, 10.

1.2.2 CPE/FI emulation using fractional-order integrators/differentiators Full electronic control of the specifications of a CPE/FI emulator can be achieved exploiting the advantages of an FO integrator/differentiator

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MOS realizations of fractional-order elements

[6,9,38]. The transfer function that describes the behavior of this element is given by HFO (s) = (τ s)q ,

(1.16)

with τ being a time constant related to the unity-gain frequency as τ = 1/ω0 and q ∈ (−1, 1) being the order of the FO integrator/differentiator. When q spans the range of (−1, 0), the function in (1.16) represents an FO integrator, while in the case that the value of q is within the range of (0, 1), it corresponds to an FO differentiator. The cascade connection of this FO stage with a voltage-to-current (V/I) converter, as demonstrated in the functional block diagram (FBD) of Fig. 1.6, forms an emulator with controllable type (CPE or FI), order (α or β ), and center frequency (ω0 ).

Figure 1.6 FBD of CPE/FI emulator using FO integrator/differentiator.

The impedance of the emulator is described by ZCPE/FI (s) =

υ1 − υ2

i

=

1 gm,V /I · (τ s)q

,

(1.17)

with gm,V /I being the transconductance of the V/I converter. An important point here is that the impedance at the unity-gain frequency ω0 is equal to Z (ω0 ) = 1/gm,V /I and, thus, depends only on the V/I converter transconductance. The desired emulator type (i.e., CPE or FI) is obtained through the appropriate selection of the type of the FO stage. In particular, if the emulator is intended to operate as CPE of order α ∈ (0, 1), then the FO part has to be a differentiator with order q = α . Correspondingly, for the emulation of an FI of order β ∈ (0, 1), the FO part has to be an integrator of order q = −β . Considering the functions in (1.1), (1.3), and (1.17), the expressions for the pseudocapacitance/inductance of the emulator are given

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Stavroula Kapoulea et al.

by (1.18a) and (1.18b), where the dependence of both parameters on the transconductance gm,V /I is obvious. We have Cα = gm,V /I · τ α , Lβ =

τβ

gm,V /I

(1.18a) (1.18b)

.

The main task in the realization of the emulator in Fig. 1.6 is to implement the FO stage using a common structure capable of realizing both integrators and differentiators. Starting from the approximation of the transfer function in (1.16) and applying one of the approximation tools mentioned in the introduction, the obtained nth-order approximated function has the form of (1.5). The construction of this function can be performed using multifeedback structures (i.e., follow-the-leader-feedback [FLF] and inversefollow-the-leader-feedback [IFLF]) presented in FBD form in Fig. 1.7 and Fig. 1.8, respectively.

Figure 1.7 Functional block diagram of an FLF-based structure.

Figure 1.8 Functional block diagram of an IFLF-based structure.

The expression that describes the FLF and IFLF diagrams is given by H(I )FLF (s) =

Kn s n + sn +

K n −1 τ1

1

τ1

sn−1 + .. +

sn−1 + .. +

K0

τ1 ·τ2 ...τn

1

τ1 ·τ2 ...τn

,

(1.19)

MOS realizations of fractional-order elements

13

with τi (i = 1, 2, ...n) being the time constants and Kj (j = 0, 1, ...n) the scaling factors. The calculation of these parameters is performed through the equation between the corresponding coefficients of the functions in (1.5) and (1.19), leading to the following design equations: τi = Bn−(i−1) /Bn−i and Kj = Aj /Bj , with Bn = 1. The multifeedback structure is an efficient option in the case that the utilized active elements have differential input. Such elements are the single-output OTAs in Fig. 1.9, which can be implemented using the MOS transistor-based circuit demonstrated in the same figure. Considering operation in the subthreshold region and using (1.15), the realized time constants are calculated as τi =

Ci 9nVT Ci = (i = 1, 2, ...n) , gmi 5IBi

(1.20)

while the scaling factors Kj (j = 0, 1, ...n) are formed through the appropriate selection of the transconductances gmj , which are controlled by the bias currents IBj as described by (1.15). The V/I converter of the emulator can be implemented using the multiple-output OTA in Fig. 1.3 with transconductance gm,V /I controlled by the corresponding bias current IB,V /I as described in (1.15).

Figure 1.9 Circuit implementation and symbol representation of the single-output OTA utilized to construct the FBD of the FLF and IFLF structures in Fig. 1.7 and Fig. 1.8.

Another option is to use a PFE-based structure, whose FBD is demonstrated in Fig. 1.10 [8]. The description of this configuration type is based on the decomposition of the expression in (1.5) through PFE as a sum of first-order low-pass filter transfer functions and a constant term. The

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Stavroula Kapoulea et al.

derived expression is given by HPFE (s) = K0 +

K1 K2 Kn + + ... + . τ1 s + 1 τ2 s + 1 τn s + 1

(1.21)

The design equations here are the following: K0 = An , τi = 1/ pi , and Ki = ri / pi , with ri , pi being the residues and poles of the function in (1.5).

Figure 1.10 Functional block diagram of the partial fraction expansion-based structure.

Considering the transfer function in (1.21), the main operation in the PFE concept is the addition. Therefore, current-mode topologies are efficient tools in the implementation of the FBD in Fig. 1.10 [7]. Each of the first-order transfer functions can be implemented using the simple currentmirror-based circuitry in Fig. 1.11, which offers low complexity and a small number of transistors, but suffers from electronic control of the scaling factor K. In order to overcome this difficulty, the circuit based on the log-domain technique, also presented in Fig. 1.11, can be utilized. In this case the scaling factor K is controlled through the bias current IB , while the same also holds for the time constant, which, for operation in the subthreshold region, is calculated by τ=

nVT C . IB

(1.22)

The realization of the V/I converter can be performed using the circuit in Fig. 1.12, with the realized transconductance calculated as gm,V /I = IB,V /I /(nVT ).

MOS realizations of fractional-order elements

15

Figure 1.11 Current-mirror and log-domain technique-based MOS realizations for the implementation of the ﬁrst-order transfer functions (lossy integrators) of the FBD in Fig. 1.10.

Figure 1.12 Current-mode circuit implementation of a V/I converter stage.

The main derivations from all the above are the following: 1. The type (CPE or FI) and the order (α or β ) of the emulator are determined by the type and the order (differentiator q = α or integrator q = −β ) of the FO stage. 2. The center frequency ω0 = 2π f0 is controlled by the FO stage’s bias currents. 3. The pseudocapacitance/inductance (Cα or Lβ ) of the emulator can be controlled through the transconductance gm,V /I of the V/I converter stage. As a result, the type, the order, the pseudocapacitance/inductance, and the center frequency are electronically controlled through appropriate tuning of the bias currents of the system.

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1.3. Practical aspects The FO integrator/differentiator-based CPE/FI emulation technique, discussed in the previous section, is a general concept that offers full electronic control and, thus, can configure a universal emulator usable in a variety of applications. Though, there are some practical aspects, the addressing of which can achieve further improvements.

1.3.1 Time constants and scaling factors spread reduction The spread of a variable is defined as the ratio of the maximum value to its minimum value. In the case of an FO integrator/differentiator, spread of the time constants and scaling factors means spread of the values of bias currents and capacitors. In the case that this spread reaches high values, it can lead to nonpractical requirements for the circuit implementation [17]. In particular, as the order q of the FO integrator/differentiator increases, an increase of the spread of both time constants and scaling factors is observed. The same also holds for the approximation order n. Therefore, in the case of highorder approximation applied on an FO integrator/differentiator with order q > 0.5, the derived spread leads to nonpractical values. A closer view of the problem is obtained through the graphs in Fig. 1.13 and Fig. 1.14, where the spread of time constants and scaling factors as a function of the order q is presented for various orders of CFE and Oustaloup approximation methods. A solution to this problem can be achieved following the concept described in Fig. 1.15. The main idea is to use only the lower orders of the FO stage, where the spread is kept at low values. Connecting the FO stage of order q < 0.5 with an IO integrator/differentiator of order r = ±1, the result is an FO integrator/differentiator of order q + r > 0.5. The mathematical description of the concept is given by the following transfer function: Hs (s) = (τ s)q+r ,

(1.23)

with the different realized cases presented analytically in Table 1.4. Exploiting this concept, the FBD of the obtained CPE/FI emulator is demonstrated in Fig. 1.16, with all the realized cases referred. Indicatively, selecting the IFLF multifeedback structure of Fig. 1.8 constructed from OTAs, the whole CPE/FI OTA-C realization is presented

MOS realizations of fractional-order elements

17

Figure 1.13 Spread of time constants as a function of the order q for various orders of CFE and Oustaloup approximation methods [17].

Figure 1.14 Spread of scaling factors as a function of the order q for various orders of CFE and Oustaloup approximation methods [17].

in Fig. 1.17. Implementing the OTAs of the FO stage and also of the IO stages using the circuit in Fig. 1.9 and the V/I converter using the circuit in Fig. 1.3 and considering the equation in (1.15), the complete emulator is controlled through the bias currents of the OTAs.

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Figure 1.15 FBD for implementing an FO integrator/differentiator using the spread reduction concept. Table 1.4 Realized cases using the concept in Fig. 1.15 for reducing the spread of time constants and scaling factors. FO stage IO stage Result

differentiator q ∈ (0, 0.5) integrator q ∈ (−0.5, 0)

integrator r = −1 differentiator r = +1

integrator q + r ∈ (−1, −0.5) differentiator q + r ∈ (0.5, 1)

Figure 1.16 FBD of the enhanced CPE/FI emulator using the proposed spread reduction technique.

1.3.2 Reduction of the control terminals of the system The tuning of a total number of bias currents that increases as the order of approximation n increases is a difficult aspect that requires a simultaneous control of many different terminals. The exploitation of a mathematical correlation between the bias currents of the emulator reduces the number of bias currents, required to control the CPE/FI emulator, to three: Imain ,

19

MOS realizations of fractional-order elements

Figure 1.17 OTA-C realization of the FBD in Fig. 1.16.

K · Imain , and K0 · I0main [21]. As a result, control of the input current Imain automatically means control of the whole emulator. Considering a second-order CFE approximation, in order to explain the procedure, the required currents of the FO integrator/differentiator are

I01 = K1 · I01 =

I02 =

9nC1 VT 8 − 2q2 · 2 , 5 q − 3q + 2

9nC2 VT q2 + 3q + 2 · , 5 8 − 2q2

K0 I02 =

9nC2 VT q2 − 3q + 2 · , 5 8 − 2q2

(1.24a)

(1.24b)

(1.24c)

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Stavroula Kapoulea et al.

K2 I03 =

q2 + 3q + 2 · I03 . q2 − 3q + 2

(1.24d)

Setting I02 as the main current Imain and I03 equal to K0 I02 , the other currents are formed as follows:

2

8 − 2q2 C1 · Imain ≡ K · Imain , (1.25a) · 2 I01 = K1 I01 = C2 q − 3q + 2 · q2 + 3q + 2

I03 =

q2 − 3q + 2 · Imain = K0 Imain , 8 − 2q2 K2 I03 = Imain .

(1.25b) (1.25c)

The input terminal of the emulator is the Imain current, so the two scaled versions K · Imain and K0 Imain will be produced by an approximation block, in order to feed the main CPE/FI core with the required currents. Performing a second-order polynomial curve-fitting approximation using the MATLAB® inbuilt function polyfit, the scaled currents are described as K · Imain ∼ = mn,1 ·

n−1 n Imain Imain + m · + .... + m1,1 · Imain + m0,1 · Iref , (1.26) n − 1 , 1 Irefn−1 Irefn−2

K0 · Imain ∼ = mn,2 ·

n−1 n Imain Imain + m · + ... + m1,2 · Imain + m0,2 · Iref , (1.27) n−1,2 Irefn−1 Irefn−2

where Iref = 95 nC2 VT ω0 is a reference dc current internally produced and mi,j (i = 0, 1, ...n and j = 1, 2) are real coefficients with values {m2,1 , m1,1 , m0,1 } = {2.752, 0.286, 0.153} and {m2,2 , m1,2 , m0,2 } = {2.399, −2.347, 0.69}. The FBD that describes the equations in (1.26) and (1.27) is presented in Fig. 1.18. The current-scaling stages and the current squarer that construct this diagram can be implemented using the nMOS and pMOS transistor-based circuits in Fig. 1.19. Using as CPE/FI core the emulator in Fig. 1.17 and feeding it with the currents of the approximation block, defined in (1.25a)–(1.25c), the result is a one-terminal controlled CPE/FI emulator based on the control of the Imain current.

MOS realizations of fractional-order elements

21

Figure 1.18 FBD of the second-order polynomial curve-ﬁtting approximation block.

Figure 1.19 MOS transistor-based implementations of the scaling and squaring operations.

The performance of the emulator is evaluated through postlayout simulation results, derived using the Cadence IC design suite and the Design Kit

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Stavroula Kapoulea et al.

AMS 0.35 µm process. Indicative aspect ratios for the transistors of the circuits in Fig. 1.19, for implementing the approximation block in Fig. 1.18, are presented in Table 1.5 [21]. Table 1.5 Aspect ratios of the MOS transistors of the current squarer and scaling circuits shown in Fig. 1.19. Current squarer Current scaling Current scaling Transistor

(W/L) (μm/μm)

Transistor

(W/L) (μm/μm)

Transistor

(W/L) (μm/μm)

Mp1–Mp3 Mp4 Mp5

80/15 m21 ·(80/15) m22 ·(80/15)

Mp1 Mp2 Mp3

120/1.2 m01 ·(120/1.2) m02 ·(120/1.2)

Mp1 Mp2 Mn1 & Mn2 Mn3

120/1 m11 ·(120/1) 0.5/15 m12 ·(120/1)

Correspondingly, aspect ratios for the transistors of the circuits in Fig. 1.3 and Fig. 1.9, which are used to implement the CPE/FI core in Fig. 1.17, are presented in Table 1.6 [21], with the power supply voltages being VDD = −VSS = 0.75V . A second-order CFE approximation is applied on the CPE/FI core around f0 = 100 Hz, with the capacitor values of the FO stage being set as C1 = 5 pF and C2 = 50 pF, while those for the IO stages are set as Cdiff = 10 pF and Cint = 5 pF. Considering these specifications, the layout design of the whole system, including the approximation block (blue frame; dark gray in print version), the CPE/FI core (red frame; mid gray in print version), and also their capacitors (yellow squares; light gray in print version), is demonstrated in Fig. 1.20. Table 1.6 Aspect ratios of the MOS transistors of the OTA in Fig. 1.9 that implements the FO and IO stages and of the OTA in Fig. 1.3 that implements the V/I converter. Transistors OTA in Fig. 1.9 OTA in Fig. 1.3 FO Int/Diff

IO Diff

0.5/10 – 10/2 2/2 10/15 –

1/12 – 5/5 1/5 0.9/18 –

IO Int

V/I converter

(W/L) (μm/μm)

Mb1–Mb3 Mb4–Mb7 Mn1 & Mn4 Mn2 & Mn3 Mp1–Mp3 Mp4–Mp6

25/5 – 5/5 1/5 0.5/5 –

0.5/10 1/2 10/1 2/1 1/2.5 1/2.5

The currents of the IO stages and the V/I converter are internally produced using Iref as reference current and the scaling stages in Fig. 1.19, appropriately configured so that the obtained current values are equal to

MOS realizations of fractional-order elements

23

Figure 1.20 Layout design of the one-terminal controlled CPE/FI emulator (dimensions: 403 μm × 445.7 μm). The blue rectangle frames (dark gray in print version) the approximation block and the red rectangle frames (mid gray in print version) the CPE/FI core, while the remaining area is occupied by the capacitors [21].

IB,diff = 353.2 pA, IB,int = 176.6 pA, and IB,V /I = 500 pA. The values of the main current, required to emulate a CPE of order α ∈ (0, 1), along with the obtained values of pseudocapacitance are presented in Table 1.7 [21], while the corresponding values in the case of an FI are also given in the same table. An important point here is that the concept of the spread reduction, described in the previous subsection, is utilized for the implementation of the higher orders q > 0.5. The efficient operation of the system, in terms of frequency domain, is evaluated through the impedance magnitude and phase responses that are demonstrated in Fig. 1.21 for the CPE case and in Fig. 1.22 for the FI case within a frequency range f = [10, 1 k] Hz. The solid lines correspond to the postlayout simulation results, while the dashed lines represent the theoretically predicted plots.

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Table 1.7 Values of the main bias current for emulating CPE and FI of various orders using the emulator in Fig. 1.20. CPE FI α

Cα

Imain

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

4.67 nF /sec 0.9 2.45 nF /sec 0.8 1.29 nF /sec 0.7 676 pF /sec 0.6 355 pF /sec 0.5 186 pF /sec 0.4 97.8 pF /sec 0.3 51.4 pF /sec 0.2 27 pF /sec 0.1

510.7 pA 588.1 pA 674.6 pA 771.9 pA 882.2 pA 220.5 pA 268.5 pA 320.8 pA 378.1 pA

Iref

β

Lβ

Imain

Iref

1.76 nA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

59 MH /sec 0.9 31 MH /sec 0.8 16 MH /sec 0.7 8.5 MH /sec 0.6 4.5 MH /sec 0.5 2.4 MH /sec 0.4 1.2 MH /sec 0.3 648 kH /sec 0.2 340 kH /sec 0.1

378.1 pA 320.8 pA 268.5 pA 220.5 pA 176.4 pA 771.9 pA 674.6 pA 588.1 pA 510.7 pA

1.76 nA

Figure 1.21 Impedance magnitude and phase frequency responses of the CPE emulator in Fig. 1.20.

Figure 1.22 Impedance magnitude and phase frequency responses of the FI emulator in Fig. 1.20.

The time-domain analysis of the system is performed, indicatively, for the case of the CPE of order α = 0.3. Using as input signal a sinusoid

MOS realizations of fractional-order elements

25

with amplitude V0 = 10 mV and center frequency f0 = 100 Hz, the derived waveform of the output current, along with the input voltage waveform, is presented in Fig. 1.23. The obtained phase difference between the two signals is equal to 25°, which is very close to the theoretical value of 27° and, therefore, verifies the efficient performance of the emulator also in the time domain. The correct operation of the spread reduction concept, discussed in the previous subsection, is also confirmed through these results, as the CPE/FI core of the emulator is based on this concept.

Figure 1.23 Input voltage and output current waveforms in the case that the emulator realizes a CPE of order α = 0.3. The sinusoidal input signal has amplitude V0 = 10 mV and center frequency f0 = 100 Hz.

1.3.3 Enhancement of the order range of the emulator The above realizations of CPEs/FIs are referred to an order range of α, β ∈ (0, 1). Though, there are applications where orders greater than 1 are required [3,16,29]. An extension of the order range of the realized CPE/FI to α, β ∈ (0, 2) can be performed through the exploitation of the concept described by the FBD in Fig. 1.15 [22]. In this case, the FO and IO stages must be of the same type (i.e., integrators or differentiators), in order to realize an element of extended order range. More specifically, if the FO stage with order q ∈ (0, 1) and the IO stage with order r = ±1 are both differentiators or integrators, then the realized FO differentiator or integrator, respectively, will have an order equal to q + r and, as a result, the realized CPE/FI will be of order α, β ∈ (0, 2). The realized cases are explained in detail in Table 1.8.

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Table 1.8 Realized cases using the concept of the order range extension. FO stage IO stage Result Emulator type

differentiator q ∈ (0, 1) integrator q ∈ (−1, 0)

differentiator r = +1 integrator r = −1

differentiator q + r ∈ (0, 2) integrator q + r ∈ (−2, 0)

CPE α ∈ (0, 2)

FI β ∈ (0, 2)

Table 1.9 Values of bias currents for emulating CPE of various orders within the range α ∈ (0, 2) using the emulator in Fig. 1.17. Cα IB2 IB1 IB0 K0 IB2 K1 IB1 K2 IB0 α 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

2.45 nF /sec 0.8 676.6 pF /sec 0.6 186.5 pF /sec 0.4 51.42 pF /sec 0.2 3.91 pF /sec −0.2 1.08 pF /sec −0.4 296.9 fF /sec −0.6 81.86 fF /sec −0.8

470.5 pA 617.5 pA 806.5 pA 1.06 nA 1.06 nA 806.5 pA 617.5 pA 470.5 pA

1.94 nA 2.82 nA 4.59 nA 9.88 nA 9.88 nA 4.59 nA 2.82 nA 1.94 nA

5 nA 1 nA 0.5 nA 0.5 nA 0.5 nA 0.5 nA 1 nA 5 nA

256.6 pA 176.4 pA 108.6 pA 50.41 pA 50.41 pA 108.6 pA 176.4 pA 256.6 pA

1.94 nA 2.82 nA 4.59 nA 9.88 nA 9.88 nA 4.59 nA 2.82 nA 1.94 nA

9.2 nA 3.5 nA 3.7 nA 10.5 nA 10.5 nA 3.7 nA 3.5 nA 9.2 nA

The simulation verification in this case has been performed using the Cadence IC design suite and the Design Kit AMS 0.35 µm process. The OTA-C structure in Fig. 1.17 is utilized and is implemented using the OTA circuits of Fig. 1.3 and Fig. 1.9 with power supply voltages equal to VDD = −VSS = 0.75V . A second-order CFE approximation is applied to the FO stage around a center frequency f0 = 100 Hz with the derived values for the passive capacitors being equal to C1 = 10 pF and C2 = 40 pF. The required bias currents, in order to realize different orders within the range α, β ∈ (0, 2), are tabulated in Table 1.9 for CPE emulation and in Table 1.10 for FI emulation [22]. The corresponding realized pseudocapacitances/inductances are included in the same tables. The parameters of the IO differentiator are Cdiff = 20 pF, IB,diff = 706.4 pA and those of the IO integrator are Cint = 5 pF, IB,int = 176.6 pA, while the bias current of the V/I converter is set equal to IB,V /I = 500 pA. According to these specifications the emulator has been designed at the layout level, as shown in Fig. 1.24, containing the FO stage (red frame; mid gray in print version), the IO differentiator (green frame; light gray in print version), the IO integrator (yellow frame; white in print version), the V/I converter (blue frame; dark gray in print version), and also the passive capacitors (yellow squares; light gray in print version). The impedance

27

MOS realizations of fractional-order elements

Table 1.10 Values of bias currents for emulating FI of various orders within the range β ∈ (0, 2) using the emulator in Fig. 1.17. Lβ IB2 IB1 IB0 K0 IB2 K1 IB1 K2 IB0 β 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

1.55 MH /sec 0.8 426.8 kH /sec 0.6 117.7 kH /sec 0.4 32.44 kH /sec 0.2 2.47 kH /sec −0.2 679.9 H /sec −0.4 187.2 H /sec −0.6 51.67 H /sec −0.8

256.6 pA 176.4 pA 108.6 pA 50.41 pA 50.41 pA 108.6 pA 176.4 pA 256.6 pA

1.06 nA 806.5 pA 617.5 pA 470.5 pA 470.5 pA 617.5 pA 806.5 pA 1.06 nA

10 nA 1 nA 10 nA 500 nA 500 nA 1 nA 10 nA 5 nA

470.5 pA 617.5 pA 806.5 pA 1.06 nA 1.06 nA 806.5 pA 617.5 pA 470.5 pA

1.06 nA 806.5 pA 617.5 pA 470.5 pA 470.5 pA 617.5 pA 806.5 pA 1.06 nA

5.4 nA 1.4 nA 1.3 nA 23.8 nA 23.8 nA 1.3 nA 1.4 nA 5.4 nA

Figure 1.24 Layout design of the CPE/FI emulator of order in the range α, β ∈ (0, 2) (dimensions: 356.6 μm × 358.4 μm). The red rectangle frames (mid gray in print version) the FO stage, the green rectangle frames (light gray in print version) the IO differentiator, the yellow rectangle frames (white in print version) the IO integrator, and the blue rectangle frames (dark gray in print version) the V/I converter, while the remaining space is occupied by capacitors [22].

magnitude and phase frequency responses for the case of Table 1.9 within a frequency range f = [10, 1 k] Hz are presented in Fig. 1.25, while the corresponding frequency responses for the case of Table 1.10 are presented in Fig. 1.26.

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Figure 1.25 Impedance magnitude and phase frequency responses of the CPE emulator in Fig. 1.24.

Figure 1.26 Impedance magnitude and phase frequency responses of the FI emulator in Fig. 1.24.

The most important advantage of these OTA-C realizations is the electronic control of the emulator parameters, through the appropriate tuning of the bias currents of the OTAs. If the desired parameter is the pseudocapacitance/inductance of the CPE/FI emulator, then, according to (1.18a)–(1.18b), control can be easily performed through the bias current of the V/I converter. Setting the emulator order as α = β = 1.2 and the center frequency as f0 = 100 Hz, and then tuning the V/I converter bias current as IB,V /I = {1, 2, 5, 10} nA (i.e., gm,V /I = {17.8, 35.6, 89, 178} nS), the obtained impedance magnitude frequency responses are demonstrated in Fig. 1.27. The realized pseudocapacitance value is equal to Cα = {7.82, 15.6, 39.1, 78.2} pF /sec −0.2 and the pseudoinductance value is equal to Lβ = {24.65, 12.33, 4.93, 2.46} kH /sec −0.2 . In a similar way, the center frequency can be accordingly controlled through the appropriate adjustment of the bias currents of the system.

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Figure 1.27 Impedance magnitude frequency responses for various values of realized pseudocapacitance/inductance of the CPE/FI emulator in Fig. 1.24. The control of this parameter is performed through the bias current of the V/I converter.

1.4. Conclusions and discussion The emulation of FO elements (CPEs/FIs) through MOS transistorbased realizations forms enhanced systems, which allow the electronic control of the characteristic features of the element. In this way, the type (CPE or FI), the order (α or β ), the pseudocapacitance/inductance (Cα or Lβ ), and the center frequency (f0 ) of the emulator can be adjusted through the control of the bias currents. The simple approach to construct an RC network with MOS transistorbased resistors offers a controllable CPE emulator, which can also realize an FI emulator with the requirement, though, of an extra GIC. A more general approach proposes an FO integrator/differentiator-based emulator, which composes an advanced system with numerous advantages. First of all, one common structure can, simultaneously, realize both types of FO elements with no requirement for extra stages. An attractive feature is, also, the capability of reducing the spread of the values of the system parameters, such as the capacitances and the bias currents. The application of the proposed concept for spread reduction leads to more compact structures that require reduced silicon area and can even achieve the on-chip integration of the passive capacitors, offering at the same time the benefit of a less consumptive system. The development of enhanced versions of this general emulator can form even more advantageous systems, like the one-terminal electronically controlled emulation system or the emulator with expanded range of

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the order. As a derivation from all these, the MOS transistor-based configurations, discussed in this chapter, are easily modified according to the desired specifications and, thus, are able to fulfill the requirements of a wide range of applications. Further improvement of the presented implementations aims at less complex configurations with extended frequency range of efficient operation, so as to achieve emulators with maximum benefits at the lowest expense. Nowadays research efforts on this subject focus on the development of emulators that approximate the impedance of a whole model and not only the impedance of each FO element individually. For this purpose, alternative approximation tools, like the Padé approximation and curvefitting-based techniques, are required, as conventional methods are able to approach only the Laplacian operator and not a whole function. Exploiting such tools, various biological tissue models, described by complex electrical models, like the well-known Cole–Cole model, can be realized by one emulator, independently of the passive and FO elements it contains [20]. This concept is also very useful in the realization of models where not only the Laplacian operator, but a whole function is raised to the fractional order, as in the case of the Cole–Davidson and Havriliak–Negami models [19]. The obtained, approximated impedance functions have the same form as in the case of the typical approximation methods (i.e., integer rational transfer functions), so their realization can be performed following the concepts presented in this chapter.

Acknowledgment This research is cofinanced by Greece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research-2nd Cycle” (MIS5000432), implemented by the State Scholarships Foundation (IKY). This article is based upon work from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology).

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CHAPTER TWO

A chaotic system with equilibria located on a line and its fractional-order form Karthikeyan Rajagopala , Fahimeh Nazarimehrb , Alireza Bahramianb , and Sajad Jafarib,c a Centre

for Nonlinear Systems, Chennai Institute of Technology, Chennai, India

b Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran c Health Technology Research Institute, Amirkabir University of Technology, Tehran, Iran

Chapter points • • •

3D chaotic ﬂow with inﬁnite equilibria located on a line. The system’s dynamical behavior and its bifurcations. Fractional-order form of the system.

2.1. Introduction Most known chaotic flows such as Lorenz [1] and Chen [2] systems have unstable equilibria, while their chaotic attractors can be found by initial conditions near the saddle points [3]. A category of chaotic systems called hidden attractors has attracted the attention of many researchers [4–6]. Many systems have been proposed in this type, such as systems without any equilibrium points [7–15]. The perpetual point is an interesting tool to investigate chaotic systems with hidden attractors [16–18]. Multistability is an important property of dynamical systems [19]. Multistability in a nonlinear system was studied in [20]. Synchronization of chaotic systems is another interesting topic [21]. Chaotic systems have many applications, such as secure communication [22] and encryption [23–25]. Fractional-order systems can present more complex dynamics than integer-order systems. Being derived from mathematics, they have many applications. The study of fractional-order systems has attracted much attention [26,27]. Various behaviors of a fractional-order map were discussed in [28]. A fractional-order neuron model was investigated in [29]. In [30], dynamical analysis of a fractional-order system was studied. A fractionalorder Shinriki system was studied in [31]. Fractional-order forms of the Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00007-X All rights reserved.

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systems can be applied using various methods such as the Grunwald– Letnikov (GL) method [32,33]. As shown in various literatures, the GL method is more efficient than the others due to its ability to use the maximum history of the variables. Various features of three fractional-order methods, GL, Riemann–Liouville, and Caputo derivatives, were discussed in [34]. A fractional method based on the Caputo derivative was proposed in [35]. The fractional derivative has many applications in physics and mathematics [36,37]. The fractional-order derivatives and some criteria for them were studied in [38]. Synchronization in chaotic systems with fractionalorder derivatives has been a hot topic recently [39,40]. The method for calculating Lyapunov exponents of fractional-order models was studied in [41]. With the use of hardware design, one can ensure the highest probability of success. Designing a reconfigurable emulator has become practical with the emergence of field-programmable gate arrays (FPGAs). FPGA implementation of a chaotic model was studied in [42–44]. When the first chaotic system was discovered by Lorenz, the feasibilities of chaotic systems have become an interesting research topic [45]. Electrical circuits receive much attention to assess the feasibility of chaotic systems [46]. In this way, different circuits have been designed to characterize chaotic oscillators [47]. Besides, these circuits have different applications in different fields of science [48]. Studies that have been done to demonstrate the feasibility of chaotic systems with chaotic circuits are reviewed in the following paragraphs. Besides, some examples of their applications are mentioned. Some simple circuits have been designed to demonstrate autonomous and/or nonautonomous chaotic systems’ feasibilities using few electrical elements. Examples of these researches include Linsay’s circuits, which have an inductor, a resistor, and diodes [49], Lindberg’s system, which has a transistor and pairs of capacitors and resistors [50], and dean’s system that has used a resistor to impliment its circuits [51]. Besides, 3D chaotic behaviors are widely modeled with operational amplifiers (Op-Amps), resistors, and capacitors [52]. Some instances of implicated simple chaotic systems with circuits are a transistor-based circuit which uses a tapped coil [53], a threeelement memristor-based circuit [46], a memristor-based circuit which benefits delays to achieve chaotic behavior [54], Chua’s four-element circuit [55], Op-Amp-based circuits which have time-variable elements [56], and a circuit that uses RLC resonances beside diodes [57]. Jerk systems, as an important category of simple chaotic oscillators, also have been investigated with circuits. For example, a circuit was designed to model a

A chaotic system with equilibria located on a line and its fractional-order form

37

jerk chaotic system with N-scroll attractors [58]. These jerk oscillators have been implicated using Josephson junctions [58] or hyperbolic nonlinearities [59]. A jerk system that has no hyperbolic fixed point was implicated with an Op-Amp-based circuit [60]. Hyperjerk chaotic systems are also attractive for researchers to implement their circuits. Using two diodes to make a hyperbolic function, a chaotic hyperjerk system implementation possibility was shown [61]. Hyperchaotic behaviors of hyperjerk systems are also implemented with circuits using Op-Amps and diodes [62]. Besides these analog circuits, some other digital circuits are used to investigate chaos. For instance, a programmable digital gate was used for observing the effect of different amounts of delay in a chaotic system [63]. Microcontrollers have also been used for the same aim. For instance, Arduino is used for implicating chaotic multiscroll behaviors [64]. FPGA has also been employed for the same purpose. It has been shown that FPGAs prepare a fast platform to identify chaos [65]. FPGAs were used to assess the feasibility of fractal Gaussian maps [66], time-delay chaotic systems [67], and fractional-order systems [68–70]. In this line, circuits are used to investigate the physical realization of chaotic systems which have some special features. Having no equilibrium in the basin of attraction of the attractor is one of these features. For instance, analog circuits are used to assess the hidden attractor of a jerk system which uses the absolute value of a variable as its nonlinearity [71]. A memristivebased circuit has been proposed to model neural network-based equations with no equilibria [72]. A nonequilibrium system that one of its variables has offset boosting freedom is designed using just resistors, capacitors, and Op-Amps [73]. Chaos control of a hidden attractor has been achieved with a circuit that contains affine transformations as its elements [74]. Besides, other circuits have been designed to assess attractors of systems that have a circle [75], a rounded square [76], or a curve [77] of equilibria. Another category of chaotic systems that is interesting for researchers to realize their feasibility is fractional-order systems. As an example of this class, a circuit can be mentioned which has been designed to regenerate responses of an elegant fractional-order system which has just five terms [78]. The circuit of a fractional-order version of the Van der Pol system has been implemented in [79]. In another research, a fractional-order system with no equilibrium was modeled using a chaotic circuit [80]. A 5D system with fractal orders has also been implemented with Op-Amps and multiplier circuits [81]. Using transconductors, a fractional version of Russler system’s feasibility has been assessed [82]. Graphene capacitors have also been used to regenerate

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a 1D fractional system time series [83]. Besides the mentioned circuits, the implementation of fractional systems has been done using FPGA [67]. In addition, these fractional systems are also implicated using active elements and pinched points [84]. Megastability can be mentioned as another property that has been discovered with circuits. A circuit showed how a 2D oscillator when it is forced can demonstrate chaotic behaviors [85]. Systems with delays usually are very hard to analyze analytically. Therefore, researchers are interested to find out their feasibility with circuits. A 3D Wang system which has no equilibria and has some different amounts of delay in its equations [86] has been implemented with circuits. In another work, using circuits the chaos of an equation with one variable which has time delay was shown [87]. Circuits are used not only to represent chaotic behaviors but also to assess the feasibility of chaos control. For instance, an adaptive control method for a chaotic system, which had just one quadratic nonlinearity, was implicated [88]. Chaos control was also implicated for a four-wing attractor which can find application in random number generation [89]. Besides, synchronization control of chaotic oscillators is also implicated using circuits that have Op-Amps and multiplier modules [90]. The possibility of controlling fractal-order systems whose attractors are multiscroll has been assessed using FPGA [91]. Recently, a chaotic system was observed using a real memristor [92]. Multistability is another property that is observed in chaotic systems. This phenomenon has been modeled with memristor-based circuits [93]. Circuits also discover the feasibility of multistability in jerk systems [94]. Systems with this feature were also implicated using Josephson junctions [95]. Hyperchaos is also investigated with circuits. These hyperchaotic systems are implicated with both analog and digital systems [96]. One of the important applications of chaotic circuits in biological engineering is the investigation of biological chaotic models. For instance, bursting behaviors of biological models have been assessed with these circuits [97]. Besides, coupling among neurons is investigated via memristorbased designed circuits [98]. Neuromorphic circuits are used to implement FitzHugh–Nagumo models of neurons [99]. The fractional order of this neural model is implicated in another circuit that uses C-MOS transistors [100]. The period doubling route to chaos in these circuits was also shown [101]. Companding techniques in circuits were used for implicating Hindmarsh–Rose equations as another neuronal dynamical model [102]. The effect of the initial condition on the final response of these systems has also been investigated using circuits [103]. Besides memristive circuits,

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coupling among neurons has also been shown with circuits using coupling capacitors [104]. Using terminal feedback, synchronization among neurons is implicated [105]. Employing compact silicon, bursting behaviors of the neurons have been implemented with circuits [106]. Some other neural behaviors such as propagation [107] have been investigated using circuits. Circuits that have been used for modeling neurons’ behaviors are not limited to analog ones. Microcontrollers [108] and FPGA circuits [109] are also used to model neurons. Chaotic circuits can have other applications in other fields of engineering. For instance, they can be used in robot path planning [110] and in vacuum cleaners [111]. These circuits can have applications in digital image processing for identification purposes [112]. Using regulators in circuits, the use of chaotic behavior in speech encryptions has been demonstrated [113]. They can be used for color encryption [114] to decrease needed space for data and safer transformation. Some of these circuits for these purposes have been implemented with FPGA using chaotic maps [115]. Texts also can be encrypted using these circuits [116]. Random number generation is another application of these chaotic circuits [117]. One of the other functions of these circuits is in the optimization of the needed charge in speech circuits [118]. These circuits also have applications in filtering to tackle noise [119]. In this chapter, a chaotic flow with a line of equilibria and with a different structure to the previous works [14,15] is discussed. In the following, the proposed system is described, and its behavioral properties from the viewpoint of bifurcation analysis are investigated. Then, the fractional-order form of the proposed system is discussed. The circuit of the system is implemented using MATLAB® /Simulink. In this study a chaotic system with equilibria located on a line is implemented with FPGA [120–124] using a forward Euler numerical method. A phase portrait and register-transfer level (RTL) of the chaotic system prove that the system is suitable for hardware realization. The last section gives conclusions of the chapter.

2.2. Model of the proposed ﬂow and its dynamics Consider the following general form: x˙ = y, y˙ = 0.4xz, z˙ = 0.3y − 0.1z − 1.4y2 + Sxy.

(2.1)

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The system in Eq. (2.1) is a new system which is proposed in this chapter. The system is found using an exhaustive computer search in a structure with a line of equilibria. For calculating equilibrium points, the right-hand side of Eq. (2.1) should be zero. From the first equation, y = 0 is obtained, and xz = 0 is obtained from the second one. From the third equation we have z = 0. So x can have any value. So, we find that the structure of Eq. (2.1) has a line of equilibria located on E∗ : y = 0 & z = 0. To explore the stability of the line, the characteristic equation is λ × (λ2 + 0.1λ − 0.12x − 0.4Sx2 ) = 0.

(2.2)

So, three eigenvalues of the line are λ = 0,

−0.1 ±

(−0.1)2 + 0.48x + 1.6Sx2

2

.

(2.3)

Eigenvalues show that the sign of the real part of the two nonzero eigenvalues is dependent on the position of fixed points on the line E∗ . Considering system (2.1) by setting the parameter S = −0.2 shows the coexistence of a chaotic hidden attractor with the line of equilibria. Fig. 2.1 presents the proposed system’s chaotic dynamics with initial values (−1.53, 0.33, 0.39). The equilibria in each projection are shown in black diamonds. In the chaotic attractor, the system has Lyapunov exponents λ1 = 0.004, λ2 = 0, and λ3 = −0.104 and Kaplan–Yorke dimension DKY = 2.0385. Lyapunov exponents of the system are calculated using the numerical Wolf method [125] and with run-time = 20,000. If the system has a positive Lyapunov exponent, the attractor is chaotic. The basin of attraction in plane z = 0 is presented in Fig. 2.2. The proposed system’s behaviors are investigated by changing S. In Fig. 2.3(a), a bifurcation diagram by increasing S is presented, and Fig. 2.3(b) investigates behaviors of the system via Lyapunov exponents. As the parameter S increases, the strange attractor emerged in a perioddoubling route. Between −0.8 < S < −0.4045, the system has a fixed point behavior. Then, in S = −0.4044, the system changes and creates a limit cycle. By increasing S, a period-doubling route to chaos is seen. In S = −0.268, chaotic attractor is destroyed, and a period three attractor emerges. Fig. 2.4 displays the period three limit cycles in S = −0.26. Then another period-doubling route to chaos is seen. Fig. 2.3 reveals that the system has a rare chaotic behavior in −0.25 < S < −0.05 and walks in two different regions.

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Figure 2.1 The chaotic attractor with parameter S = −0.2 and initial conditions (−1.53, 0.33, 0.39). Black diamonds show the line of equilibria.

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Figure 2.2 Basin of attraction with parameter S = −0.2 in z = 0. Initial values in light blue (light gray in print version) show the attraction of chaotic attractors, and red circles (mid gray in print version) show the attraction of equilibria.

2.3. Fractional-order form Various fractional-order calculators are used, such as GL and Caputo [32,33]. As shown in various literatures, the Grunwald–Letnikov method is more efficient than the others due to its ability to use the maximum history of the variables. The GL method is applied in this chapter. In this subsection, the GL fractional-order method is applied to study the fractional form as follows: D q x = y, Dq y = 0.4xz, Dq z = 0.3y − 0.1z − 1.4y2 + Sxy.

(2.4)

The parameter is kept constant as S = −0.2 and initial conditions (−1.53, 0.33, 0.39). Here the commensurate order is considered for the study. The system shows various dynamics by changing the fractional order parameter q. For the numerical analysis, we derive the fractional-order discrete form as follows:

q xk = (yk−1 )hq − N i=1 Wi xk−i , q yk = (0.4xk−1 zk−1 )hq − N i=1 Wi yk−i , q zk = (0.3yk−1 − 0.1zk−1 − 1.4y2k−1 + Sxk−1 yk−1 )hq − N i=1 Wi zk−i .

(2.5)

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Figure 2.3 Bifurcation diagram and Lyapunov spectrum.

Using Eqs. (2.5) for a step size of h = 0.01, we can numerically simulate the system. The previous section shows that the system has a line of equilibrium in x and y = 0 and z = 0. Considering that system (2.5) is commensurate, we can derive the condition for the stability of equilibrium points using the relation q

π

2

< arg(eig(λ)).

(2.6)

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Figure 2.4 Projections of a period of three limit cycles in system (1) with parameter S = −0.26 and initial conditions (−1.53, 0.33, 0.39). Black diamonds show the line of equilibria.

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If Eq. (2.6) is satisfied, the system has stable equilibrium points. Using relation (2.6), we have plotted the various regions in the x-S plane for the stability of equilibrium points as shown in Fig. 2.5, where x denotes the various points in the line equilibrium. This plot is derived by fixing the fractional orders to q = 0.8, q = 0.9, q = 0.98. For all three values of q, when the equilibrium points of x are around the origin, they are stable, as could be seen from Fig. 2.5. We could also show that the stable region gets narrower when the fractional order is decreased. To study various dynamics of system (2.4) by changing the fractional order parameter q, the bifurcation diagram is investigated. Fig. 2.6 presents bifurcation diagrams by changing the fractional order parameter q under constant initial conditions. Fig. 2.6(a), (b), and (c) presents bifurcations that are plotted using peaks of x, y, and z variables, respectively. As system (2.4) has infinite equilibrium points, we are interested in studying the dynamical behaviors of system (2.4) for various combinations of commensurate fractional order q and initial condition of state x while keeping the other two initial conditions as y = 0, z = 0 (Fig. 2.7). The magenta region (mid gray in print version) presents the first type of chaotic attractor, which has a very narrow region of attraction for x < 0 and has a wider range for positive values of x. The regions shown in blue (dark gray in print version), green (light gray in print version), and red (mid gray in print version) correspond to the various attractors, and between 3 < x < 10, there are such coexisting attractors. Specifically, the system has a broad range of chaotic regions in q when the initial value of x is between 7 < x < 10. The cyan region (light gray in print version) shows the periodic oscillations. As shown in previous sections, the system has a line of equilibria and shows hidden oscillations. Hence we are interested in studying the basin of attraction of system (2.4). By keeping the fractional order q = 0.98, we derived the basin of attraction plot as shown in Fig. 2.8. The yellow line (white in print version) shows the equilibrium points of the system with the basin in plane z = 0. The red regions (dark gray in print version) show the attractor’s basin while cyan (light gray in print version) shows the unbounded oscillations. Window size has a great effect on the fractional-order dynamics [91, 126,127]. To clearly understand the impact of the window size N shown in Eqs. (2.5), we have plotted the phase portraits of the system for different

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Figure 2.5 Stability of the equilibrium in the x-S plane for various values of the commensurate fractional order.

A chaotic system with equilibria located on a line and its fractional-order form

Figure 2.6 Bifurcation diagram of system (2.4) by changing parameter q.

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Figure 2.7 Different chaotic attractors for various combinations of the initial value of state x and the commensurate fractional order q. The initial conditions of the other two states are kept as y = 0, z = 0.

Figure 2.8 The basin of attraction of system (2.4) at z = 0. The yellow line (white in print version) shows the equilibrium points of the system. The red regions (dark gray in print version) show the attractor’s basin while cyan (light gray in print version) shows the unbounded oscillations.

values of N as in Fig. 2.9. We could see that for the window size N < 500, the system shows only multiperiod limit cycle-like attractors, but when the window size is increased, the chaotic attractors are formed. Also, we

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Figure 2.9 The phase portraits of the system for different window size N.

wish to show that for the window size 1000 < N < 1500, the system shows chaotic attractors similar to the one for the full window size of N = 2000. By fixing the optimum value of N, we could reduce the memory required by the systems and easily implement it in digital platforms.

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Figure 2.10 The circuit schematic for S = −0.2.

2.4. Circuit implementation The chaotic behavior of the introduced 3D system is also assessed with the simulation of its circuit. The circuit is designed using simple elements such as Op-Amps, resistors, and capacitors. The circuit equations are as follows: x˙ = y˙ = z˙ =

R2 R1R3C1 y, R5 R4R6C2 xz, 1 1 R12 2 R7C3 y + R8C3 z + R11R9C3 y

(2.7) +

R14 R13R10C3 xy,

where x, y, and z represent Op-Amp outputs’ voltages O2, O4, and O5, respectively (Fig. 2.10). The other elements’ values are as follows: R1 = 100K, R2 = 100K, R3 = 100K, R4 = 100K, R5 = 40K, R6 = 100K, R7 = 100K, R8 = 100K, R9 = 333K, R10 = 1000K, R11 = 100K, R12 = 140K, R13 = 100K, R14 = 20K, C1 = C2 = C3 = 10nF. The results are shown in Fig. 2.11. The circuit is implemented using MATLAB/Simulink.

2.5. FPGA implementation of the chaotic system In this section, we discuss the implementation of a proposed chaotic system using a digital implementation method such as FPGA design. FPGA implementation provides a variety of applications, including chaotic random number generation for image cryptography [128], memristor-based

A chaotic system with equilibria located on a line and its fractional-order form

Figure 2.11 The chaotic attractor of the implemented circuit for S = −0.2.

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chaotic systems [121], and higher-dimensional digital chaotic systems [129]. The proposed chaotic system is designed using the Xilinx System Generator. Different arithmetic operators of a system are selected from the built-in blocks of the system generator toolbox, such as Add/Sub and multiplier blocks with zero latency, and configured using 32/16-bit fixed point settings according to the IEEE 754 standard. An integrator which is not readily available in the Xilinx System Generator Toolbox is designed using the following mathematical equations: dpi [pi (n + 1) − pi (n)] = lim . dt h→0 h

(2.8)

To get an accuracy, we select h = 0.001, and the initial conditions are fed into the forward register of the integrator block designed using the forward Euler method. A set of discretized system equations is given as follows: xk+1 = xk + h(yk−1 ), yk+1 = yk + h(0.4xk−1 zk−1 ), zk+1 = zk + h(0.3yk−1 − 0.1zk−1 − 1.4y2k−1 + Sxk−1 yk−1 ),

(2.9)

where h is the step size for the discrete numerical solution. These discrete state equations of the proposed chaotic system are implemented on the FPGA in discrete-time [130]. After designing the Simulink design, the system is interfaced with Kintex 7 through the Vivado synthesis tool. An RTL of the proposed system is generated after the simulation and synthesis of the system using Vivado software. Phase portrait, time series, and the RTL schematic of the proposed chaotic system (2.9) are shown in Fig. 2.12, Fig. 2.13, and Fig. 2.14.

2.6. Conclusion A chaotic system whose equilibrium points are located on a line was proposed in this chapter. The system had various dynamics such as equilibrium points, periodic and chaotic attractors. Bifurcation analysis of the proposed system has shown a period-doubling route to chaos and also a jump. Bifurcation analysis of the simple system has presented its ability to have complex behaviors. Also, the fractional-order form of the system was investigated. Bifurcation study of the fractional-order system has revealed that the system has different dynamics just by changing the fractional order parameter. The circuit realization of the system was investigated using

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Figure 2.12 Strange attractor projections on various planes of system (2.9) with S = −0.2 under initial conditions (−1.53, 0.33, 0.39).

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Figure 2.13 Time series of the proposed chaotic system (2.9).

Figure 2.14 The RTL schematic of the proposed chaotic system (2.9) implemented in Kintex 7.

MATLAB/Simulink. The FPGA implementation of the system was done using the Xilinx System Generator, showing the feasibility of the chaotic system.

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CHAPTER THREE

Approximation of fractional-order elements for sinusoidal oscillators Shalabh K. Mishraa , Dharmendra K. Upadhyayb , and Maneesha Guptab a Department of Electronics and Communication Engineering, ABES Engineering College, Ghaziabad, Uttar Pradesh, India b Department of Electronics and Communication Engineering, Netaji Subhas University of Technology, New Delhi, India

3.1. Introduction Electronic oscillators, which generate sinusoidal waveforms without any input signal, are one of the most widely used active devices. They are used in several areas of science and technology; some potential applications of sinusoidal oscillators are wired and wireless communication systems, navigation systems, RADAR, and biomedical equipment. Nowadays the concept of fractional calculus is being utilized in the design and realization of various signal processing systems such as integrators, differentiators, filters, and oscillators [1,32,54,55,59]. This mathematical concept provides some additional degree of freedom and flexibility in the designed system. For example, the phases of the oscillatory waveforms cannot be controlled independently for the conventional designs, whereas it is possible to control the phases by using the fractional calculus approach. Before moving further, it is worthy to briefly discuss the foundation of fractional calculus. Fractional calculus is a mathematical tool dealing with the differentiation and integration of noninteger orders in a generalized sense. It can be defined in various ways; popularly, it is defined by the Grunwald–Letnikov, regularized Liouville, Riemann–Liouville, and Caputo derivatives [5,29–31,50]: ∞ 1 k α 1 f t − kh , Dα f (t) = Lim (− ) α h→0 h k k=0

D f (t ) = α

1 (−α)

∞

τ 0

−α−1

f (t − τ ) −

N −1 0

(−)m f m (t) m τ dτ, m!

Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00008-1 All rights reserved.

(3.1)

(3.2) 63

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D α f (t ) =

dn (n − α) dtn

Dα f (t) =

1

(n − α)

(t − u)n−1−α f (u)du ,

(3.3)

0

1

t

t

(t − u)n−α−1 f n (u) du.

(3.4)

0

In electrical engineering, conventional passive energy storing circuit elements are replaced with fractance devices (FDs) to approximate the system in a fractional sense. FDs are considered to be fractional energy storing elements since their voltage–current relations are expressed by the fractional-order derivative [2,26]. These FDs are further classified as the fractional-order capacitor and fractional-order inductor. The current– voltage relationship and the impedances of the fractional-order capacitor and fractional-order inductor are given in (3.5) and (3.6), respectively: 1 dα v i= α , C dt

(3.5a)

Zc = Cs−α ,

(3.5b)

dα i 1 v= α, L dt

(3.6a)

ZL = Lsα .

(3.6b)

Therefore, the impedance FDs can also be represented as ZFD = Ksδ ,

(3.7)

where K is the magnitude and δ is the behavior defining parameter. For the cases δ < 0 and δ > 0, this impedance function represents a fractional-order capacitor and a fractional-order inductor. The SI units of these devices are Farad sec α−1 and Henry sec α−1 , whereas the dimensional formulas are [M −1 L −2 T (3+α) I 2 ] and [M L 2 T (−3+α) I −2 ], respectively [26]. The electrical systems designed using these FDs are known as fractional-order systems, and such electrical systems are more flexible as-well-as accurate than the conventional integer-order systems. FDs are not yet available commercially as a single circuit component; therefore, R-C tree- or ladder networkbased circuits are often used to approximate them [10–12,23,48,49,53]. Although few attempts have been made to approximate the FDs as a sin-

Approximation of fractional-order elements for sinusoidal oscillators

65

gle lumped element [7,21,22,28], the R-C network-based designs are often preferred for the practical realization of various fractional-order circuits and systems. During the past decade, various fractional-order oscillators have been proposed and designed to employ the advantages of fractional calculus. In this area, initially, the Wien-bridge oscillator was simulated in a fractional sense by replacing the conventional capacitors with the fractional-order capacitors [2]. The authors derived the condition of oscillation (CO) and frequency of oscillation (FO) and illustrated the merits of the fractional-order Wien oscillator over the conventional design. It was also illustrated that the fractional-order oscillators have additional parameters (order of FDs) that can control CO and FO. The orders are also capable of controlling the phase shifts of the waveforms generated across the FDs. In 2007, Radwan– Soliman–Elwakil derived the design equations for a fractional-order negative resistance RC oscillator, a Twin-T oscillator, and a Hartley oscillator [34]. They implemented a fractional-order Wien-bridge oscillator practically using the chemically developed FDs. Further, they derived a generalized theorem for designing the fractional-order sinusoidal oscillator [33]. In this work, the authors simulated the RC, LC, RC phase shift, and Colpitts oscillators in the PSpice environment using the R-C tree and ladder network-based FDs. Furthermore, the fractional-order oscillator was realized using two integrator loops of noninteger order [52]. Besides, the two-port network-based design of fractional-order oscillators is presented [42,43,46,47]. In the next development, modern current-mode active building blocks (ABBs) such as current feedback operational amplifiers (CFOAs), operational transresistance amplifiers (ORTAs), and second-generation current conveyors (CCIIs) are used since they exhibit improved performances over the conventional operational amplifier (Op-Amp) [36–40,44]. The modern current-mode ABBs have several advantages over conventional active devices such as simple circuitry, greater linearity, larger signal bandwidth, wider dynamic range, high input impedance, and low power consumption. Said et al. designed two-port network-based fractional-order sinusoidal oscillators using CCII, and implemented the oscillatory system on an NI ELVIS II kit using an Op-Amp as an active element [41]. In 2016, Kubánek et al. proposed a fractional-order oscillator using a differential voltage current conveyor (DVCC) as an active block [19]. In this work, CO and FO were derived and the oscillatory circuit was implemented using BD-QFG DVCC. Furthermore, Kartci et al. designed MOS-RC-based fractionalorder oscillators [17]. Later on, Hartley oscillators were also realized in a

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generalized fractional sense [8,56]. Besides these works, several attempts have been made to design and realize fractional-order multiphase oscillators. Fouda et al. introduced fractional-order multiphase oscillators for higher-order phase shift keying (PSK) applications [14,15]. Thereafter, Said et al. proposed classes of fractional oscillators consisting of three fractionalorder capacitors [45]. In their work, three different Op-Amp-, CFOA-, and CCII-based oscillatory circuits had been simulated and implemented practically. Also, the DVCC has been used in the realization of a fractional-order multiphase oscillator generating four sinusoidal waveforms of controlled phase shifts [24]. Additionally, few more fractional-order oscillators and relaxation oscillators have been designed using modern ABBs such as operational transconductance amplifier (OTA), voltage differencing inverting buffered amplifier (VDIBA), OTRA, etc. [9,13,16,51]. Although a substantial amount of work has been done in this area, researchers are still exploring the area of fractional oscillators due to their unique features. It has been noticed that the practical implementation of fractional-order oscillators is still a challenging task due to the commercial unavailability of FDs. Mostly R-C tree- or ladder network-based circuits consisting of a large number of passive components are used to approximate the FDs. A large number of resistive and capacitive elements may inject noise and exert parasitic effects in the oscillatory circuit. Consequently, phase noise and harmonic distortion may significantly increase in the fractionalorder oscillators implemented using the tree- and ladder network-based FDs. Besides this, these FDs are also responsible for poor power efficiency, bulky hardware, and high cost. To overcome these limitations, impedance and admittance equalization-based techniques have been proposed that are capable of approximating the FDs using just two passive components. These types of FDs show ideal behavior only at a single frequency, and are therefore best suited for fractional-order sinusoidal oscillators. Initially, fractional-order Wien-bridge and LC oscillators have been realized using the impedance equalization-based FDs [25,26]. Furthermore, a differential difference current conveyor (DDCC)-based fractional oscillator with independent phase and frequency controlling capability is implemented using admittance equalization-based FDs [27]. This chapter aims to compare the performances of the recently developed impedance equalization-based FDs, admittance equalization-based FDs, and the well-known R-C network-based FDs so that a generalized criterion can be deduced for the selection of a suitable FD for a specific application. For this purpose, a well-known fractional-order Wien-bridge

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oscillator is simulated using these three types of FDs, and the obtained responses are compared to each other to analyze the performance of the FDs. This chapter is organized as follows. Section 3.2 discusses the design and development of conventional R-C network-based FDs. The recently developed R-C pair-based FDs are described in Section 3.3. Furthermore, their performances are compared in Section 3.4. Finally, the conclusion and scope of future research are given in Section 3.5.

3.2. R-C network-based FDs It is well known that the ideal FDs do not exist; nevertheless, several attempts have been made to approximate these devices for the realization of fractional-order systems. In 1964, a rational approximation of the function 1 s n was derived using the regular Newton process and applied in the implementation of driving point impedances using the R-C ladder network [6]. The starting point of the method is the statement of the following relationships: (X (s))1/δ − Y (s) = 0;

X (s) = (Y (s))δ .

(3.8)

For the initial value X0 (s) = 1, the rational approximation is obtained as

q − n (Hi−1 (s))2 + q + n Y (s) , X1 (s) = Xi−1 (s) q + n (Hi−1 (s))2 + q − n Y (s)

(3.9)

where δ = 1/q and n = q/2. Similarly, Matsuda’s method is based on the approximation of an irrational function by a rational one, obtained by fitting the original function in a set of logarithmically spaced points. Considering the selected points sk (where k = 1, 2, 3, . . . ), the Matsuda approximation is defined as [57] X (s) = A0 +

s − s0 , A1 + A +s−ss−1 s2 2

(3.10)

A3 +...

where Ai = Wi (si ), W0 (s) = X (s), and Wi+1 (s) = (s − si )/(Wi (s) − Ai ). Also, some approximation scheme is based on curve fitting or identification techniques such as Oustaloup approximation and Chareff’s approximation [57]. Oustaloup’s method gives the rational approximation of a function given in the following form: X (s) = sδ , ∀δ ∈ R+ .

(3.11)

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Chareff ’s method is similar to Oustaloup’s method and gives the rational approximation of a function given in the following form:

X (s) = 1 +

s −δ . k

(3.12)

Continued fraction expansion (CFE) is one of the most famous rational approximation schemes used to approximate the fractional-order impedances as given below [18,58,60]: (1 + x)δ =

1 1 − 1+

.

δx

(3.13)

(1+δ)x (1−δ)x 2+ x 3+ (2(+δ) x 2+ 25−δ) +...

This approximation scheme converges much more rapidly than the several existing schemes, and is hence widely used in the rational approximation of fractional-order circuits and systems. The above rational approximations may be used for the design and realization of FDs. Initially, the impedances of FDs are approximated to an integer-order transfer function. After the rationalization, these impedances can be implemented using R-C networks. The complete process for designing R-C network-based FDs is summarized as follows. Step I: Rationalize the impedance of FDs as an nth-order rational transfer function: n zi si ZFD = Ksδ ∼ (3.14) = in=0 j . j=0 pj s Step II: Realize this rationalized impedance function using Cauer or Foster forms. Now, we will see one example to realize the 0.5th-order fractionalorder capacitor. For simplicity, we have taken the capacitance 1 F sec −0.5 . This fractional impedance can be rationalized using any one of the available techniques; however, we have selected the CFE scheme in our example due to its rapid convergence speed. By the replacement x → s − 1 and δ = −0.5 in Eq. (3.13), we get s−0.5 =

1 1 − 1+

−0.5(s−1)

0.5(s−1) 1.5(s−1) 2+ 1) 3+ 1.25.(5s(−s− 2+ 5+...1)

.

(3.15)

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Table 3.1 Rational approximation of s−0.5 using continued fractional expansion. S. No.

Order of approximation

Rational approximation

1. 2. 3. 4. 5.

1st 2nd 3rd 4th 5th

s+3 3s+1 s2 +10s+5 5s2 +10s+1 s3 +21s2 +35s+7 7s3 +35s2 +21s+1 s4 +36s3 +126s2 +84s+9 9s4 +84s3 +126s2 +36s+1 s5 +55s4 +330s3 +462s2 +165s+11 11s5 +165s4 +462s3 +330s2 +55s+1

Figure 3.1 Magnitude and phase response of CFE-based fractional-order capacitor.

This gives a rational approximation of the impedance of a fractionalorder capacitor with α = 0.5. This is an infinite series; however, it can be truncated for higher-order terms. The first-, second-, third-, fourth-, and fifth-order approximations have been done to rationalize this fractionalorder capacitor as given in Table 3.1. The magnitude and phase response of the rationalized fractional impedances are compared with those of the ideal one as illustrated in Fig. 3.1. Since the fourth- and higher-order approximations give a satisfactory response, first-, second-, and third-order expansion is often ignored. Considering this, fourth-order rational approximation is often preferred since fifth-order approximation is complex and requires more complex circuitry. It should also be noted that the center frequency of these approximations is ω = 1 since this approximation has been done at the point s = 1. This center frequency can be easily shifted to ωi by the replacement s → ωsi in

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Figure 3.2 Schematic of CFE-based FDs using Foster form.

Figure 3.3 Schematic of CFE-based FDs using Cauer form.

Figure 3.4 Magnitude and phase response of ideal and R-C network-based FDs.

the approximation formulas. For the time being, we are considering the fourth-order approximation centered at ω = 1 only. In the next step, this fourth-order approximation is realized by the R-C network using Foster and Cauer form as illustrated in Figs. 3.2 and 3.3 respectively. This fourthorder approximation requires four capacitances and five resistances with the numerical values listed in Table 3.2. The magnitude and phase response of these two structures are almost identical as illustrated in Fig. 3.4. However, the structure and numerical values of these two realizations are different. Here we can see that realization of FDs is a complex task, which requires tedious calculations (during the rational approximation and then circuit im-

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Table 3.2 Circuit components used for Cauer and Foster realization. S. No. Circuit Realization scheme component Cauer form Foster form

1. 2. 3. 4. 5. 6. 7. 8. 9.

R0 R1 R2 R3 R4 C1 C2 C3 C4

0.111 0.57 1.16 1.91 5.24 0.34 F 0.84 F 1.58 F 3F

0.111 7.37 0.251 0.38 0.888 4.63 F 0.52 F 1.86 F 3.37 F

plementation). We should not forget that this calculation has been done for the realization of a specific 0.5th-order fractional-order capacitor with unit pseudocapacitance. Any adjustment in the order and/or pseudocapacitance may change the entire circuit configuration and, hence, require a fresh similar calculation. It should be noted that the R-C network-based FDs are bulky and power-inefficient due to the presence of a large number of passive components. These components may also degrade the system performance as they are the key reason for the existence of thermal noise in the system. Noise is a serious concern, and it is undesirable for any electrical circuit. This seriousness intensifies several times in an oscillatory system since noise components may be added up through the positive feedback loop. Hence, a small noise component may lead to a harsh change in the spectrum and steady-state response of the generated waveforms. Hence, by considering the design and performance issues of R-C network-based FDs, a few alternative approaches have been developed to approximate the FDs for fractional-order sinusoidal oscillators. A detailed description and analysis of these recently developed schemes are provided in the next section.

3.3. FDs for sinusoidal oscillators In recent years, a few alternate schemes have been proposed to realize FDs for fractional-order sinusoidal oscillators. Theoretical anticipation of the impedance and admittance equalization-based realization techniques emerged by analyzing the phase angle of the impedance of FDs and conventional passive components as illustrated in Fig. 3.5. These schemes are

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Figure 3.5 Phase angle of passive circuit components.

based on impedance and admittance equalization techniques through which it is possible to approximate single-frequency FDs using R-C/R-L series or parallel pairs. Therefore, this scheme provides fractional impedance through only a pair of passive circuit components. These passive pairs show ideal fractional characteristics at a single frequency, and therefore sinusoidal oscillators can be impeccably realized using these FDs. Now, we will discuss the impedance and admittance equalization-based FDs, one by one.

3.3.1 Impedance equalization-based FDs The phase angles of the fractional-order capacitor and the fractional-order inductor lie in the fourth and first quadrant respectively, as illustrated in Fig. 3.5. The phase angles of conventional passive circuit elements, i.e., resistors, inductors, and capacitors, are 0◦ , 90◦ , and −90◦ , respectively; therefore, the phase angles of R-C series and R-L series pairs will also lie in the fourth and first quadrant respectively. So, it is possible to approximate the FDs using the R-C/R-L passive pairs by using the impedance equalization technique [26], discussed below. Let a fractional-order capacitor (C) and a fractional-order inductor (L) of order α (for both FDs) be designed. In this FD approximation approach, the impedances of R-C and R-L series pairs (named Rf − Cf and Rf − Lf , respectively) are equated with the impedances of an ideal fractional-order capacitor and fractional-order inductor as given in (3.16a) and (3.16b), respectively: 1 1 = Rf + , sα C sCf

(3.16a)

sα L = Rf + sLf ,

(3.16b)

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where s = jω and ω is the angular frequency in radians per second. By using Euler’s formula, the above equations are modified as 1 ωα C

1 cos (απ /2) − j sin (απ /2) = Rf − j , ω Cf

ωα L cos (απ /2) + j sin (απ /2) = Rf + jωLf .

(3.17a)

(3.17b)

By comparing real and imaginary parts in (3.17a) for fractional-order capacitors, the relations for Rf and Cf are obtained as Rf =

cos(απ /2) , ωα C

(3.18a)

Cf =

ω(α−1) C . sin(απ /2)

(3.18b)

Similarly, for fractional-order inductors, the relations of Rf and Lf are obtained from (3.17b) as follows: Rf = L ωα cos (απ /2) ,

(3.19a)

Lf = L ω(α−1) sin (απ /2) .

(3.19b)

The mathematical relations obtained in (3.18) and (3.19) depend upon the angular frequency ω, which should be considered as the oscillation frequency of the sinusoidal oscillators where these FDs are to be used. Therefore, it is possible to approximate FDs using only two passive components, as illustrated in Fig. 3.6. Alternatively, the admittance equalization-based technique can also be used to approximate FDs for sinusoidal oscillators. The admittance equalization-based technique is similar to the impedance equalization-based technique except that admittances of an ideal FD are equalized with the R-C or R-L parallel pair, as discussed in the next subsection.

3.3.2 Admittance equalization-based FDs In this FD approximation approach, admittances of R-C and R-L parallel pairs (named Rf − Cf and Rf − Lf , respectively) are equated with the admittance of fractional-order capacitor and fractional-order inductor as given

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Figure 3.6 Schematic of impedance equalization-based FDs.

in (3.20a) and (3.20b), respectively [25]: sα C = 1 sα L

=

1 + sC , Rf

(3.20a)

1 1 + , Rf sLf

(3.20b)

where s = jω and ω is the angular frequency in radians/second. By using Euler’s formula, the above equations are modified as 1 + j ω Cf , ωα C cos (απ /2) + j sin (απ /2) =

(3.21a)

1 1 1 cos (απ/2) − j sin (απ /2) = −j . α ω L Rf ω Lf

(3.21b)

Rf

After comparing real and imaginary parts in (3.21a) for fractional-order capacitors, the relations for Rf and Cf are obtained as Rf =

ω−α , C cos (απ /2)

Cf = ω(α−1) C sin (απ /2) .

(3.22a) (3.22b)

Similarly, comparing real and imaginary parts in (3.21b) for fractional inductors, the relations for Rf and Lf are obtained as Rf =

ωα L , cos (απ /2)

(3.23a)

Lf =

ω(α−1) L . sin (απ/2)

(3.23b)

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Figure 3.7 Schematic of impedance equalization-based FDs.

Here also the angular frequency ω should be taken as the oscillation frequency of the sinusoidal oscillators. In this case also, it is possible to approximate FDs using only two passive components, as illustrated in Fig. 3.7.

3.4. Performance analysis It is essential to analyze the performance and characteristics of the recently developed impedance (R-C series pair)- and admittance equalization (R-C parallel pair)-based FDs, with respect to the well-known R-C ladder network-based FDs. For this purpose, a well-known fractional-order Wien-bridge oscillator is realized using these three FDs. The schematic of a fractional-order Wien-bridge oscillator is shown in Fig. 3.8. This oscillator consists of two fractional-order capacitors C1 and C2 of order α and β , respectively, four resistors, and one Op-Amp as an active block. The voltages at the fractional-order capacitors can be expressed by the following matrix: ⎛ ⎝

dVCα 1 dtα dVCβ 2 dtβ

⎞ ⎠=

(Rb /Ra )

1 R2 C1 − R1 C1 (Rb /Ra ) R2 C2

−1 R2 C1 −1 R2 C2

VC 1 VC 2

.

(3.24)

Consequently, the characteristic equation of the fractional-order Wienbridge oscillator is

s

α+β

−

( Rb / Ra )

R2 C 1

1 1 α 1 − sβ + s + = 0. R1 C 1 R2 C 2 R1 R2 C 1 C 2

(3.25)

To avoid computational complexity, resistances and pseudocapacitances are taken equal (R1 = R2 = R and C1 = C2 = C), which gives the CO and FO

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Figure 3.8 Schematic of a fractional-order Wien-bridge oscillator.

as

απ Rb , = 2 1 + cos Ra 2 1

ωosc = (1/RC ) α .

(3.26) (3.27)

The phase shift of the voltages at terminals VC2 and VC1 are related α−1 , α = 0.5, and as ∠VC2 − ∠VC1 = απ 4 [33]. For the case C = 1 µF s R = 10 k, the CO and FO are obtained as Rb = 3.141Ra and ωosc = 10000 radians/second, respectively. The fractional-order capacitor is simulated using the three different approaches, namely R-C ladder network (fourth order)-based FDs, R-C series pair-based FDs, and R-C parallel pairbased FDs. The numerical values of resistances and capacitances for R-C series pairbased FDs are Rf ≈ 7071 and Cf ≈ 14.14 nF, whereas for the R-C parallel pair-based FDs these values are Rf ≈ 14.142 k and Cf ≈ 7.07 nF. The circuit components for the ladder network-based fractional-order capacitor are summarized in Table 3.3. The considered fractional oscillator is numerically simulated using all three FDs and their spectra are shown in Fig. 3.9. It is observed that the R-C series and parallel pair-based designs have a nearly 100 dB peak gain at the oscillation frequency that is comparable to the ideal peak gain. However, the R-C ladder network-based design has a gain of 75 dB only. Therefore, the R-C pair-based FDs are expected to perform better than the R-C network-based FDs.

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Table 3.3 Circuit components and their numerical values for the R-C network-based FD. S. No. Circuit component Numerical value 1. R0 1.402 k 2. R1 3.17 k 3. R2 4.78 k 4. R3 11.2 k 5. R4 92.9 k

6. 7. 8. 9.

C1 C2 C3 C4

6.64 nF 23.1 nF 43.3 nF 54.8 nF

Figure 3.9 Spectrum of fractional-order Wien-bridge oscillators using different FDs.

3.4.1 Stability analysis Stability analysis is essential for any oscillatory system since it assesses the ability to produce a sustained oscillation. For the fractional-order systems, this assessment is often done in the n-plane Riemann surface or in the W -plane as it is not so easy to track down their poles and zeros in the conventional s-plane [20,35]. In the W -plane, there are no fixed predefined stable and unstable regions as in the conventional s-plane. So, the stability analysis in the W -plane is followed by an additional step for defining the stable and unstable regions. However, it can be possible to locate the poles of the fractional-order system in the s-plane also if the fractional impedance/admittance (i.e., Csα ) is approximated to an integer-order function. For this purpose, the ideal fractional-order capacitor is replaced with the well-known fourth-order R-C network-based and the newly proposed

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Figure 3.10 Pole plot of a fractional-order Wien oscillator using different FDs.

R-C pair-based approximations. Therefore, the fractional-order function Csα is replaced by integer-order (fourth-order) transfer functions, given in Table 3.1 for CFE-based design, and by Eqs. (3.16a) and (3.20a) for the newly proposed impedance and admittance-based designs. By this replacement, the fractional-order characteristic equation is converted into an integer-order characteristic equation, and therefore, their poles can be easily plotted in the s-plane. After plotting their poles in the s-plane, the stability of the system can be easily analyzed. For a sustained oscillatory system, the circuit must have a pair of poles on the imaginary axis, and the remaining poles, if any, must lie in the left half of the s-plane. The R-C series and R-C parallel pair-based FDs reduce this characteristic equation in a second-order function, whereas the R-C network-based (fourth-order approximation) FDs lead to an eighth-order function. Consequently, the fractional-order oscillator designed using the recently proposed R-C series and R-C parallel pair-based designs have only two poles on the imaginary axis, whereas the same fractional-order oscillator designed with the R-C network-based FDs has six additional poles, i.e., a total of eight poles, as illustrated in Fig. 3.10. Among these eight poles in the R-C network-based design, two lie on the imaginary axis which are responsible for the generation of a sustained oscillatory sinusoid. Although these additional poles are residing in the left half of the s-plane and may not affect the ability to generate the sustained oscillation of desired frequency, they can produce unwanted converging (dying out) oscillatory components that may deteriorate the transient behavior of the system.

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3.4.2 Sensitivity analysis using Monte Carlo simulation The system response depends upon the numerical values of the circuit components used in it. Consequently, the tolerances of resistors and capacitors become very crucial for the reliability of a system containing a significant amount of passive components. Additionally, the tolerances become even more significant for the oscillatory system since the error in the circuit component may affect the oscillation frequency as well as the oscillation condition, and the oscillator might be destabilized. So, the Monte Carlo simulation was performed to analyze the effect of resistance and capacitance tolerances on the oscillator performance. This simulation has been performed with 500 runs while considering 2% tolerance in the numerical values of components. The effects of the resistance and capacitance tolerances on the pole plot and the spectrum for the R-C series pair-, R-C parallel pair-, and R-C ladder network-based designs are shown in Figs. 3.11, 3.12, and 3.13, respectively. This simulation shows that the location of the spectrum peak varies in the neighborhood frequency regions of the actual frequency of oscillation. A significant deviation of poles from their actual locations is also observed. Therefore, resistance and capacitance tolerances significantly affect the oscillation frequency as well as the capability of the oscillator to produce sustained oscillation. The variation in oscillation frequency can be in the ranges of 9810–10190 rad/sec, 9820–10180 rad/sec, and 9710–10300 rad/sec for the considered R-C series pair-, parallel pair-, and ladder network-based FDs, respectively. Hence, it can be concluded that the R-C series pair-based FDs have more tolerance towards the resistance and capacitance values than the well-known R-C network-based FDs. However, the R-C parallel pair-based FDs have the most tolerance among them.

3.4.3 PSpice simulation and FoM calculation The fractional-order Wien oscillator was simulated using R-C series pair-, R-C parallel pair-, and R-C ladder network-based FDs in the PSpice environment. In these simulations, the order of both fractional-order capacitors is fixed at the same numerical value, i.e., α = β = 0.5. The output waveforms obtained from these simulations are shown in Figs. 3.14, 3.15, and 3.16. It can be observed that all three FDs give the expected steady-state response, i.e., they generate two sinusoidal waveforms of angular frequency 10000 rad/sec, while maintaining the phase shift of π/8. The steady-state response as well as the transient response indicates that the peak-to-peak voltage is a little higher in the R-C series pair-based design than in the R-C

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Figure 3.11 Monte Carlo simulation of an R-C series-based design.

Figure 3.12 Monte Carlo simulation of an R-C parallel pair-based design.

Figure 3.13 Monte Carlo simulation of an R-C ladder network-based design.

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Figure 3.14 Time response of R-C series pair-based 0.5th-order fractional oscillator.

Figure 3.15 Response of an R-C parallel pair-based 0.5th-order fractional oscillator.

parallel pair- and R-C ladder network-based designs. In these simulations, total harmonic distortion (THD) was measured as 2.62%, 2.48%, and 3.2% for the R-C series pair-, R-C parallel pair-, and R-C ladder network-based designs, respectively. The THD gives the proportion of powers associated with the harmonics of oscillation. The higher the THD value, the more power is associated with the harmonics and correspondingly higher distortion in the output oscillatory signals. Therefore, it can be concluded that the R-C ladder network-based FDs produce more unwanted harmonics than the R-C pair-based FDs. Moreover, it is also noted that the R-C parallel pair-based FDs produce the least unwanted harmonics among the three considered FDs.

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Figure 3.16 Response of an R-C ladder network-based 0.5th-order fractional oscillator.

Additionally, the fractional-order Wien oscillator (FO = 10000 radians per second) was simulated again with two different orders (α = β = 0.3 and α = β = 0.7) using the newly proposed R-C series and R-C parallel pairbased FDs, as shown in Figs. 3.17 and 3.18. Numerical values of resistances and pseudocapacitances are taken as 10 k and 6.3 µF, respectively, for the 0.3rd-order oscillator, whereas these values are 1 k and 1.58 µF for the 0.7th-order system. These simulations give the expected oscillation frequency and phase shifts. Phase noise and power dissipation are also important parameters from the performance measurement point of view. The phase noise may also depend upon the power consumed by the system. Phase noise can be reduced by feeding more power in the system; therefore, phase noise alone is not self-sufficient to classify the system performance. Hence, another parameter, the figure of merit (FoM), is used for analyzing the system since it takes care of the cumulative effects of phase noise, offset frequency, and power dissipation during the performance measurement [3,4]:

FoM = PN δ f + 20 log10

fo δf

− 10 log10

PT , 1 mW

(3.28)

where PN δ f is phase noise at δ f offset frequency in dBc/Hz, and PT is the total power consumed. The phase noise for the R-C ladder-, R-C series pair-, and R-C parallel pair-based designs are −54.86 dBc/Hz, −64.06 dBc/Hz, and −64.95 dBc/Hz, whereas the FoMs are 169.41, 186.26, and 187.15 for the respective cases. The analysis results presented in this section are summarized

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Figure 3.17 Response of an R-C pair-based 0.3rd-order fractional oscillator.

Figure 3.18 Time response of an R-C pair-based 0.7th-order fractional oscillator.

in Table 3.4, which indicates that the fractional-order sinusoidal oscillators designed using the recently developed R-C pair-based FDs are more compact, more reliable, more stable, and less noisy than similar systems designed through R-C network-based FDs.

3.5. Conclusion and scope of future research In this chapter, a comparative study of the R-C network-based and some recently developed FDs used for sinusoidal oscillators has been undertaken. It has been observed that the fractional-order sinusoidal oscillators can be successfully designed using recently developed impedance

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Table 3.4 Performance of a fractional Wien-bridge oscillator designed using various FDs. S. No. Oscillator R-C network Recently developed parameter R-C series pair R-C parallel pair

1. 2. 3. 4. 5. 6.

THD Phase noise FoM Poles Comp. count Variation in FO

3.2% −54.77 dBc/Hz 169.41 8 22 5.2%

2.62% −64.06 dBc/Hz 186.26 2 8 3.8%

2.48% −64.95 dBc/Hz 187.15 2 8 3.8%

equalization- and admittance equalization-based FDs. The oscillators designed using these FDs are more compact than the traditional design since only two passive components (R-C series or parallel pair) are required to approximate the FDs. It has also been noted that the performances of R-C series and R-C parallel pair-based oscillators are comparable to each other. Besides, the R-C pair-based oscillator design gives improved results compared with the R-C ladder network-based oscillatory system in terms of cost, hardware complexity, compactness, stability, reliability, efficiency, and accuracy. Hence, the R-C pair-based FDs are remarkably promising for several practical applications where portable, lightweight, and efficient oscillatory systems with controlled phase and frequency are desirable, such as mobile communication devices, biomedical devices, health monitoring systems, and tracking systems. The recently developed R-C pair-based design shows ideal fractional behavior only at a single frequency, and therefore, these FDs are suitable for sinusoidal oscillators only. As a future scope of this work, these recently developed designs could be modified or improved in such a way so that the bandwidth of FDs might be widened and the FDs could be used in designing filters and several other electrical systems also, in a fractional sense.

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CHAPTER FOUR

Synchronization between fractional chaotic maps with different dimensions Adel Ouannasa , Amina-Aicha Khennaouib , Iqbal M. Batihac , and Viet-Thanh Phamd a Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria b Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria c Department of Mathematics, Faculty of Science, The University of Jordan, Amman, Jordan d Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

Chapter points •

This chapter presents combined synchronization among fractional-order discrete-time chaotic systems.

•

Using a nonlinear control method, the combined synchronization can be achieved.

•

Two nonlinear controllers are constructed to make the synchronization error converge towards zero under the stability theory of linear fractional-order discrete-time systems.

•

Some numerical experiments are considered to verify the eﬀectiveness of the combined synchronization.

4.1. Introduction Ever since the disclosure of the Lorenz system in one of Lorenz’ papers on climatic expectation [1], chaos has become a hot research topic and a large number of chaotic systems were proposed over the last 50 years [2,3]. Basically, chaotic systems are systems of nonlinear equations whose states exhibit high sensitivity to the small change in the initial condition. Until now, several works have been dedicated to analyze the chaotic systems outlined by differential and difference equations [4–6,97]. Synchronization and control of chaos in both continuous-time systems [7–12] and discretetime systems [15–17,50,53] are regarded as the most interesting topics. The objective of synchronization is to force the states of the response chaotic Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00009-3 All rights reserved.

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system towards that of a master chaotic system [4–6,62,63]. In this case, the synchronization error converges towards zero as t → +∞ [64,65]. Referring to discrete-time chaotic systems, a wide number of techniques have been used to design different synchronization schemes [18,20–22,24,25]. In fact, many types of chaos synchronization have been reported, such as hybrid projection synchronization [19,26,45,52,56,59], inverse matrix projection synchronization [23,51], Q-S synchronization [13,54,55], and complete synchronization [14,46–49], among others [50,52,57,58,60]. Among these types of synchronizations the combined synchronization of discretetime chaotic systems is defined by taking one master system and more than one system as slave systems. Recently, the search for synchronization has moved to high-dimensional nonlinear dynamical systems. Besides integer-order systems, attention has recently been focused on systems described by fractional-order differential equations [74,75,108,109]. Studies have shown that chaotic fractional-order systems can also be synchronized [68–73,76–79,81,82]. In recent years, many scientists who are interested in the field of chaos synchronization have struggled to achieve the synchronization between integer-order and fractional-order chaotic systems [66,67,80,83,84]. In recent years attention has been focused on discrete fractional calculus and fractional difference operators [27–30,93,94,100]. Several papers regarding the presence of chaotic phenomena in fractional discrete-time systems have been published [31–33,36,37,102,107]. For example, in [104] the dynamics of the fractional Duffing map have been investigated. In [37] the presence of chaos in fractional rational maps has been analyzed, whereas in [34,35,101] the chaotic behavior of fractional generalized Hénon maps has been illustrated. In [38] the presence of chaos in the fractional sine maps has been analyzed in detail. In references [39,44,61] different fractional maps with no fixed point have been studied, whereas in [40–43,95,99] the chaotic dynamics of different 3D fractional maps with stable and without fixed point have been investigated. Referring to the synchronization of fractional-order discrete-time systems, a number of papers have been proposed [87,90,91]. In [88] the synchronization of integer- and fractionalorder chaotic maps with different dimension is studied. Similarly, in [85] two control schemes for the Q-S synchronization of two different dimensional fractional maps are proposed and analyzed in detail. Reference [89] investigates the synchronization of two fractional chaotic systems using two scaling matrices, whereas a generalized and inverse generalized synchronization was studied in [86,89,96,98].

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Based on previous considerations regarding the synchronization of fractional discrete-time systems, in this chapter we study the combined synchronization between fractional chaotic maps in 2D and 3D. Based on stability of linear fractional-order discrete-time systems and nonlinear control laws, combined synchronization schemes are presented. Moreover, numerical simulations are used to verify the feasibility of the proposed control schemes and the derived synchronization criterion. This chapter is organized as follows. In Section 4.2, some preliminaries related to discretetime fractional calculus and stability are introduced. Section 4.3 proposes a combined synchronization scheme between 2D chaotic fractional maps. Section 4.4 presents synchronization control between 3D chaotic fractional maps using a combined scheme. Finally, Section 4.5 provides a general summary of the main findings of this study and some future works.

4.2. Preliminaries Before we start talking about chaotic combined synchronization for chaotic fractional maps, some preliminaries and basic concepts associated with discrete fractional calculus are presented here for completeness. There are three different definitions for fractional difference operators, namely, Grunwald–Letnikov, Caputo, and Riemann–Liouville operators, which have indeed played a practical tool in engineering applications. In this context, we consider a function X defined from a discrete time scale Na where Na = {a, a + 1, a + 2, . . .} and a ∈ R. The forward difference operator is defined by x(n) = x(n + 1) − x(n).

(4.1)

In order to give a reasonable treatment of the discrete fractional calculus, we begin by recalling Euler’s Gamma function: (μ) =

∞

sμ exp−s .

(4.2)

0

This function is a generalization of a factorial in the following form: (n) = (n − 1)!.

(4.3)

Definition 4.1. The falling factorial function of real order μ is defined as s(μ) =

(s + 1) . (s + 1 − μ)

(4.4)

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Definition 4.2. In [28], the μth fractional sum for mτ X (s) with μ > 0 is defined by −μ a X ( s) =

1 (μ)

s−μ

(s − τ − 1)(μ−1) X (τ ) .

(4.5)

τ =a

Based on the above definition of the μth fractional sum, it is possible to define the μ-Caputo-like difference operator. Definition 4.3. [29] Let X denote any function defined from Na+m−μ . The Caputo difference operator with order μ = N is defined by X (s) : Na → R, which is of the form C

m−μ) m μa X (s) = −( X (s) a

=

s−( m−μ) 1 (s − τ − 1)(m−μ−1) m τ X (τ ) , (m − μ) τ =a

(4.6)

where μ ∈/ N is the fractional order, s ∈ Na+m−μ , and m = μ + 1. Now a theorem is briefly summarized which will allow us to define the discrete formula of the chaotic fractional maps in the following. Theorem 4.1. [30] For the fractional difference equation

C μ X (s) = f (s + μ − 1, X (s + μ − 1)), a b X (a) = Xb , m = μ + 1, b = 0, 1, ..., m − 1,

(4.7)

the unique solution of the initial value problem (4.7) is given by X (s) = X0 (s) +

1

s−μ

(μ) τ =a+m−μ

(s − σ (τ ))(μ−1) f (τ + μ − 1, X (τ + μ − 1)),

s ∈ Nμ+m ,

(4.8)

where

m−1 (s − a) b b X (a). X0 (s) = b+1 b=0

(4.9)

Now a stability result is briefly illustrated, with the aim to identify the stability conditions of the zero equilibrium point for a general linear fractional-order discrete-time system.

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Theorem 4.2. [92] Consider the following linear fractional discrete-time system: C

μa X (s) = AX (s + μ − 1) ,

X (0) = X0 ,

(4.10)

where 0 < μ ≤ 1, X (s) ∈ Rn , and A ∈ Rm×m . The zero equilibrium point of system (4.10) is asymptotically stable ∀s ∈ Na+1−μ if and only if |arg λ| > μ

and

π

2

|arg λ| − π μ |λ| < 2 cos 2−μ

(4.11a)

(4.11b)

are satisfied for all eigenvalues λ of matrix A.

4.3. Combined synchronization of 2D fractional maps 4.3.1 Master system and slave systems In this paragraph a master–slave system based on 2D fractional maps is considered. In particular, we consider the slave system as a combination between the fractional Lorenz map and the fractional flow map, where the fractional Lozi map is considered as the master system. We will indicate the states of the master and slave systems by m and s, respectively. In the following, we consider the fractional Lozi map as the master system. The Lozi map is a 2D fractional map, which has been proposed in [90] as an example of a fractional discrete-time system that can display chaotic behavior. We have ⎧ ⎪ ⎨ ⎪ ⎩

C μ x (s) = −α |x (s − 1 + μ)| + x (s − 1 + μ) + 1 1 1m 2m a 1m

− x1m (s − 1 + μ) , C μ x (s) = α x (s − 1 + μ) − x (s − 1 + μ) , 2m 2 1m 2m a

(4.12)

where x1m , x2m are the states of the master system and α1 , α2 are parameter values. This fractional map was introduced by replacing the integer-order difference operator with the μ-Caputo delta difference operator. The authors in paper [90] showed that the Lozi fractional map (4.12) is more dynamically rich than the classical Lozi map, described by

x1 (n + 1) = −α1 |x1 (n)| + x2 (n) + 1, x2 (n + 1) = α2 x1 (n) − x2 (n) .

(4.13)

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To study the properties of the fractional Lozi map (4.12), the following discrete numerical solution is defined based on Theorem 4.1: ⎧ 1 s−μ (μ−1) x ( s ) = x ( a ) + (−α1 |x1m (τ − 1 + μ)| 1m 1m ⎪ τ =a+1−μ (s − τ − 1) (μ) ⎪ ⎪ ⎨ + x2m (τ − 1 + μ) + 1 − x1m (τ − 1 + μ)) , 1 s−μ (μ−1) ⎪ x2m (s) = x2m (a) + (μ) (α2 x1m (τ − 1 + μ) ⎪ τ =a+1−μ (s − τ − 1) ⎪ ⎩ −x2m (τ − 1 + μ)) .

(4.14) Let n = s − a, a = 0. Since (s − τ − 1)μ−1 / (μ) is equal to (s − τ )/ (s − τ − μ + 1), it follows that

1 x1m (n) = x1m (0) + (μ)

x2m (n) = x2m (0) + (μ) 1

n−1

(n−i+μ) i=0 (n−i+1) (n−i+μ) i=0 (n−i+1)

n−1

(−α1 |x1m (i)| + x2m (i) + 1 − x1m (i)) , (α2 x1m (i) − x2m (i)) ,

(4.15) where x1m (0), x2m (0) are the initial states. According to the discrete equation (4.15), the proposed fractional Lozi map (4.20) has memory effects, which means that the iterated solution xm is determined by all the previous states. Considering the parameter value α2 = 0.5 and varying α1 from 0 to 1.8, the resulting bifurcation diagram and the largest Lyapunov exponents (LEs) are depicted in Fig. 4.1 with fractional order μ = 0.98. Different dynamic behaviors including chaos and periodic orbits are observed in the fractional Lozi map (4.12). It can be seen that the system (4.12) has the largest positive LE when α1 takes the highest values, indicating that the system has indeed a chaotic attractor, as shown in Fig. 4.1(c) for α1 = 1.7. Then, the equations of the Lorenz slave system are given by ⎧ ⎪ ⎨

(s) = β1 β2 x1s1 (s − 1 + μ) − β2 x2s1 (s − 1 + μ) x1s1 (s − 1 + μ) + L1 (s − 1 + μ) , ⎪ ⎩ C μ x (s) = −β x (s − 1 + μ) + β x2 (s − 1 + μ) + L (s − 1 + μ) , 2s 2 2s1 2 1s1 2 1 a C μ x a 1s1

(4.16) where x1s1 and x2x1 are the system states, whereas L1 and L2 denote the synchronization controller. When L1 = L2 = 0, then system (4.16) degenerates into the fractional Caputo Lorenz map. This 2D map has been introduced in [90] using the μ-Caputo difference operator. When β1 = 1.25, β2 = 0.75, and μ = 0.98 the fractional Lorenz map displays chaos. Fig. 4.2(c) shows the chaotic attractor where the initial condition is selected as x1s1 = 0.1,

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Figure 4.1 (a) Bifurcation diagram of x1m and (b) largest LE of the fractional Lozi map (4.12) calculated for α1 ∈ [0, 1.8] and α2 = 0.5 with fractional order value μ = 0.98 using initial condition [x1 (0), x2 (0)] = [0, 0]. Additionally, (c) represents the chaotic attractor for α1 = 1.7.

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Figure 4.2 (a) Bifurcation diagram of x1 s1 and (b) largest Lyapunov exponent of the fractional Lorenz map (4.16) calculated for β1 ∈ [0.4, 1.3] and β2 = 0.75 with fractional order μ = 0.98. Additionally, (c) represents the chaotic attractor for β1 = 1.25.

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x2s1 = 0. For better observation, the bifurcation diagram and the maximum LE are considered by changing the control parameter β1 from 0.4 to 1.3, as depicted in Fig. 4.2(a) and Fig. 4.2(b), respectively. The computation of the largest LE shows that the dynamics of the fractional Lorenz map change from periodic to chaotic when β1 is increased. Similarly, the index s2 is used to denote the states of the fractional flow map, given by

c μ x a 1s2 c μ x a 2s2

(s) = x2s2 (s − 1 + μ) + (γ1 − 1)x1s2 (s − 1 + μ) + L3 (s − 1 + μ) , (s) = γ2 + x21s2 (s − 1 + μ) − x2s2 (s − 1 + μ) + L4 (s − 1 + μ) ,

(4.17) where the parameters L3 (s) and L4 (s) denote the synchronization controllers. The system (4.17) has been studied extensively in the literature and is known to exhibit a chaotic behavior for the bifurcation parameters γ1 = −0.1 and γ2 = −1.7. The resulting attractor for μ = 0.98 is depicted in Fig. 4.3(c). In order to highlight the properties of the fractional flow map, the bifurcation diagrams and maximum LE with varying system parameter γ1 for fractional order value μ = 0.98 are reported in Fig. 4.3(a) and Fig. 4.3(b), respectively. As one can see, the fractional flow map can dynamically display chaos for different values of the system parameter.

4.3.2 Combined scheme In order to make the master system (4.12) and the slave system states (4.16)–(4.17) achieve combination synchronization, we need to define proper controller functions Li ∀i = 1.4. Below we provide the combination synchronization scheme. Definition 4.4. The fractional Lozi map (4.12) and the slave systems, fractional Lorenz map (4.16) and fractional flow map (4.17), are said to be combination synchronized if there exist controllers Li (s) , i = 1, 2, 3, 4, such that

lims→+∞ ||e1 (s)|| = lims→+∞ ||x1s1 (s) + x1s2 (s) − x1m (s) || = 0, lims→+∞ ||e2 (s) || = lims→+∞ ||x2s1 (s) + x2s2 (s) − x2m (s) || = 0,

with s ∈ Nμ+1 .

(4.18)

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Figure 4.3 (a) Bifurcation diagram of x1s2 and (b) largest Lyapunov exponent of the fractional ﬂow map (4.17) calculated for γ1 ∈ [−0.1, 0.1], γ2 = −1.7 with fractional order μ = 0.98 using the initial condition [x1s2 (0), x2s2 ] = [1, 0.2]. Additionally, (c) represents the chaotic attractor for γ1 = −0.1.

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We propose the following control law for this type of synchronization. Theorem 4.3. If the following control laws hold, the master system (4.12) and the combined slave systems (4.16)–(4.17) will achieve combined synchronization for any initial state: ⎧ ⎪ L1 (s) = − (1 + β1 β2 ) x1s1 (s) + x2s1 (s) + β2 x2s1 (s) x1s1 (s) − α1 |x1m (s)| + 1, ⎪ ⎪ ⎨ L s = −9 x s + α + 9 x s + β + 1 x s − β x2 s , 2( ) 2 1m ( ) 2 2s1 ( ) 2 1s1 ( ) 16 1s1 ( ) 16 2 ⎪ = −γ , L s x s ( ) ( ) 3 1 1s2 ⎪ ⎪ ⎩ L (s) = −9 x (s) + 3 x (s) − 3 x (s) − γ − x2 (s) . 4 2 1s2 16 1s2 2 2s2 2 2m

(4.19)

Proof. The synchronization error system (4.18) has the following form: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C μ e (s) = β β x 1 2 1s1 a 1

(s − 1 + μ) − β2 x2s1 (s − 1 + μ) x1s1 (s − 1 + μ) + L1 (s − 1 + μ) + x2s2 (s − 1 + μ) + (γ1 − 1) x1s2 (s − 1 + μ) + L3 (s − 1 + μ) + α1 |x1m (s − 1 + μ)| − x2m (s − 1 + μ) − 1 + x1m (s − 1 + μ) , C μ e (s) = −β x (s − 1 + μ) + β x2 (s − 1 + μ) + L (s − 1 + μ) 2 2s1 2 1s1 2 a 2 + γ2 + x21s2 (s − 1 + μ) − x2s2 (s − 1 + μ) + L4 (s − 1 + μ) − α2 x1m (s − 1 + μ) + x2m (s − 1 + μ) .

(4.20) Substituting Eq. (4.19) into the error system (4.20) yields the following error system: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

C μ e (s) = −x 1s1 a 1

(s − 1 + μ) − x1s2 (s − 1 + μ) + x1m (s − 1 + μ) + x2s1 (s − 1 + μ) + x2s2 (s − 1 + μ) − x2m (s − 1 + μ) , C μ e (s) = 9 x (s − 1 + μ) + x (s − 1 + μ) − x (s − 1 + μ) 1s2 1m a 2 16 1s1 + 12 x2s1 (s − 1 + μ) + x2s2 (s − 1 + μ) − x2m (s − 1 + μ) ,

(4.21)

and then we have

C μ e (s) = −e (s − 1 + μ) − e (s − 1 + μ) , 1 2 a 1 C μ e (s) = 9 e (s − 1 + μ) + 1 e (s − 1 + μ) . a 2 16 1 2 2

(4.22)

Now, the design process reduces to demonstrating if the equilibrium (0, 0, 0) of (4.22) is globally asymptotically stable. The 2D dynamical sys-

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tem (4.22) reduces to C

μa (e1 (s) , e2 (s))T = A × (e1 (s − 1 + μ) , e2 (s − 1 + μ))T ,

where

A=

−1

−1

9 16

1 2

(4.23)

.

(4.24)

1 = 0. 16

(4.25)

The characteristic equation of matrix A is 1 2

λ2 + λ +

It is easy to see that the eigenvalues λi of matrix A are negative real numbers, that is, |arg λi | = π >

and

μπ

2

|arg λi | − π μ |λi | < 2 cos , 2−μ

(4.26a)

,

for i = 1, 2.

(4.26b)

With Theorem 4.2 one can find that the equilibrium point (0, 0) is asymptotically stable. This indicates that the combined slave systems (4.16)–(4.17) are synchronized to the fractional Lozi map. The proof is thus completed. Here, some numerical experiments are considered to verify the effectiveness of the combined synchronization. First, the numerical solution of the 2D error system is given as

e2 (n) = e2 (0) +

n−1

(n−i+1) i=0 (n−i+1) (−e1 (i) − e2 (i)), 1 n−1 (n−i+1) 9 1 i=0 (n−i+1) ( 16 e1 (i) + 2 e2 (i)). (μ)

1 e1 (n) = e1 (0) + (μ)

(4.27)

The bifurcation parameters are considered as above to ensure chaos. The initial value of the fractional Lozi map is set as x1m (0) = 0, x2m (0) = 0, and the initial values of the fractional Lorenz map and flow map are considered as x1s1 = 0.3, x2s1 = 0.4 and x1s2 = 1, x2s2 = 0.2, respectively. Fig. 4.4 illustrates the synchronization error system (4.22) with μ = 0.98. It is very clear that the error states e1 , e2 can converge to zero when the controller functions L1 , L2 are added to the slave systems, which implies that the combination synchronization between the master system (frac-

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Figure 4.4 Combined synchronization error between the fractional Lozi map (4.12) and fractional Lorenz (4.16) and ﬂow maps (4.17) with μ = 0.98. (a) The ﬁrst error state e1 (s). (b) The second error state e2 (s).

tional Lozi map) is realized. Furthermore, we plot the evolution of states of the master and slave systems for the fractional order value μ = 0.98. Fig. 4.5 illustrates the results. Clearly, the state variables of the master and slave systems are synchronized completely. Thus, the numerical results show

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Figure 4.5 The time histories of the master system (4.12) and the slave systems (4.16)–(4.17) with μ = 0.98. (a) The ﬁrst state of the master system and the combined slave system. (b) The second state of the master system and the combined slave system.

very well the effectiveness of the proposed combined synchronization for fractional-order discrete-time systems.

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4.4. Combined synchronization of 3D fractional maps 4.4.1 Master system and slave systems Synchronization is one challenging topic in the field of chaos theory. The field of synchronization of fractional-order discrete-time systems has great potential application in a number of fields including secure communication and data encryption [61,103,105,106]. In this section, we aim to synchronize the dynamics of combination of the fractional Stefanski and Wang systems to the fractional Rössler map, which is considered as master system. We consider as our drive system a 3D fractional map known as the fractional Rössler map given for s ∈ Na+1−μ by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C μ x (s) = a x (s − 1 + μ) (1 − x (s − 1 + μ)) 1 1m 1m a 1m

−a2 (x3m (s − 1 + μ) + a3 ) (1 − 2x2m (s − 1 + μ)) −x1m (s − 1 + μ) ,

C μ x (s) = a x (s − 1 + μ) (1 − x (s − 1 + μ)) 4 2m 2m a 2m

+a5 x3m (s − 1 + μ) − x2m (s − 1 + μ) ,

C μ x (s) = a (1 − a x (s − 1 + μ)) 6 7 1m a 3m

× [(x3m (s − 1 + μ) + a3 ) (1 − 2x2m (s − 1 + μ)) − 1] − x3m (s − 1 + μ) ,

(4.28) where bi , ∀i = 1.7, are bifurcation parameters and x1m , x2m , x3m are the states of the drive system denoted by the subscript m. In [91], this fractional-order map was firstly constructed by adding the μ-Caputo difference operator to an integer-order Rössler map which can be described by ⎧ ⎪ x1m (n) = a1 x1m (n − 1) (1 − x1m (n − 1)) ⎪ ⎪ ⎪ ⎪ − a2 (x3m (n − 1) + a3 ) (1 − 2x2m (n − 1)) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2m (n) = a4 x2m (n − 1) (1 − x2m (n − 1)) + a5 x3m (n − 1) , ⎪ ⎪ ⎪ ⎪ x3m (n) = a6 (1 − a7 x1m (n − 1)) ⎪ ⎪ ⎪ ⎪ ⎪ × [(x3m (n − 1) + a3 ) (1 − 2x2m (n − 1)) − 1] ⎪ ⎪ ⎩ − x (n − 1) . 3m

According to Theorem 4.1, the numerical formula is designed as

(4.29)

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⎧ 1 n−1 (n−i+μ) x1m (n) = x1m (0) + (μ) ⎪ i=0 (n−j+1) (a1 x1m (n − 1) (1 − x1m (i)) ⎪ ⎪ ⎪ ⎪ − a2 (x3m (i) + a3 ) (1 − 2x2m (i)) − x1m (i)), ⎪ ⎪ ⎪ ⎪ ⎨ x (n) = x (0) + 1 n−1 (n−i+μ) (a x (i) (1 − x (i)) 2m

2m

(μ)

i=0 (n−j+1)

4 2m

2m

⎪ + a5 x3m (i) − x2m (i)), ⎪ ⎪ ⎪ ⎪ 1 n−1 (n−i+μ) ⎪ ⎪ x3m (n) = x3m (0) + (μ) ⎪ i=0 (n−j+1) (a6 (1 − a7 x1m (i)) ⎪ ⎩ × [(x3m (i) + a3 ) (1 − 2x2m (i)) − 1] − x3m (i) − x3m (i)),

(4.30) where x1m (0), x2m (0), x3m (0) are the initial states. Fig. 4.6 shows the chaotic attractor of the fractional Rössler system (4.28) with order value μ = 0.97 and a1 = 3.8, a2 = 0.05, a3 = 0.35, a4 = 3.78, a5 = 0.2, a6 = 0.1, a7 = 1.9. In Fig. 4.7 we present the bifurcation diagram of the fractional Rössler map (4.28) when μ = 0.97. These diagrams are obtained by changing a from 1 to 3.8 with step size 0.2. As can be seen in those diagrams, the fractional Rössler system (4.28) shows chaotic behavior. As for the response system, we consider a combination of two 3D fractional maps. Herein, the Stefanski map with fractional order μ is described by [91] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(s) = −x1s1 (s − 1 + μ) + x3s1 (s − 1 + μ) + 1 − b1 x22s1 (s − 1 + μ) + C1 (s − 1 + μ) , C μ x (s) = b − 1 x (s − 1 + μ) + 1 − b x2 (s − 1 + μ) 2 2s1 1 1s1 a 2s1 + C2 (s − 1 + μ) , C μ x a 1s1

C μ x a 3s1

(s) = b2 x1s1 (s − 1 + μ) − x3s1 (s − 1 + μ) + C3 (s − 1 + μ) ,

(4.31) in which C1 , C2 , and C3 are designed fractional system controllers based on the linearization method and s1 denotes the first slave system. Setting b2 = 0.2, when b1 varies from 0.9 to 1.4 the resulting bifurcation diagram is shown in Fig. 4.8. The fractional map (4.31) is chaotic over the interval ]1.18, 1.4] when the fractional order takes the value μ = 0.97. A chaotic attractor is plotted in Fig. 4.9 to identify the chaotic behavior when b1 = 1.4. Next, the expression for the second slave system is ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

C μ x a 1s2 C μ x a 2s2 C μ x a 3s2

(s) = c3 x2s2 (s − 1 + μ) + x4 x1s2 (s − 1 + μ) + C4 (s − 1 + μ) , (s) = c1 x1s2 (s − 1 + μ) + c2 x3s2 (s − 1 + μ) + C5 (s − 1 + μ) , (s) = c7 x3s2 (s − 1 + μ) + c6 x2s2 (s − 1 + μ) x3s2 (s − 1 + μ) + c5 + C6 (s − 1 + μ) ,

(4.32)

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Figure 4.6 The phase portrait of the fractional master system (4.28) with μ = 0.97 and for a1 = 3.8, a2 = 0.05, a3 = 0.35, a4 = 3.78, a5 = 0.2, a6 = 0.1, a7 = 1.9.

in which x1s2 , x2s2 , x3s2 are the states of the slave Wang map (4.32) with fractional order μ ∈]0, 1] and C1 , C2 , C3 are control parameters to be designed. To make the fractional Wang map (4.32) in chaotic motion, the system parameters are set as c1 = −1.9, c2 = 0.2, c3 = 0.5, c4 = −2.3, c5 = 2, c6 = −0.6, c7 = −1.9, and the fractional order μ = 0.97. Fig. 4.10 illustrates

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Figure 4.7 The bifurcation diagram the fractional master system (4.28) versus a1 with μ = 0.97 and for a2 = 0.05, a3 = 0.35, a4 = 3.78, a5 = 0.2, a6 = 0.1, a7 = 1.9.

the chaotic attractor. For better observation the bifurcation diagram versus system parameter c3 is illustrated in Fig. 4.11.

4.4.2 Combined scheme Here, a combination–synchronization scheme is presented in order to synchronize the slave systems (4.31)–(4.32) with the Rössler map (4.28). The objective is achieved if there exist fractional controllers C1 (s), . . . , C6 (t) such that the synchronization errors ei with s ∈ Na+1−μ converge to zero asymptotically, i.e., lim |ei (s)| = 0, i = 1, 2, 3. s→∞ One can define the corresponding error synchronization system as ⎧ ⎪ ⎨ e1 (s) = x1s1 (s) + x2s2 (s) − x1m (t) , e2 (s) = x2s1 (s) + x2s2 (s) − x2m (s) , ⎪ ⎩ e (s) = x (s) + x (s) − x (s) . 3 3s1 3s2 3m

(4.33)

The following theorem proposes suitable adaptive laws for controllers C1 (s), . . . , C6 (s) in order to guarantee that (4.33) holds. Theorem 4.4. If the following control laws hold, the master system (4.28) and the combined slave system (4.31)–(4.32) will achieve combined syn-

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Figure 4.8 The bifurcation diagram of the fractional slave system (4.31) versus b1 with μ = 0.97 and for b2 = 0.2.

chronization for any initial states: ⎧ C1 (s) = −x3s1 (s) − 1 + b1 x22s1 (s) + a1 x1m (s) − a1 x21m (s) − a2 x3m (s) ⎪ ⎪ ⎪ ⎪ ⎪ − x (s) , ⎪ ⎪ 2m ⎪ ⎪ C2 (s) = 1 − b2 x2s1 (s) − 1 + a1 x21s1 (s) − x2s1 (s) − x3s1 (s) + x2m (s) , ⎪ ⎪ ⎪ ⎪ ⎪ C3 (s) = −b2 x1s1 (s) + a6 x3m (s) − a6 a7 x1m (s) − 2a3 a6 x2m (s) , ⎪ ⎪ ⎨ C s = 1 − c x s − 1 + c x s − 2a x s x s ( ( 4( ) 3 ) 2s2 ( ) 4 ) 1s2 ( ) 2 3m ( ) 2m ( ) ⎪ − 2a2 a3 x2m (s) + a2 a3 , ⎪ ⎪ ⎪ ⎪ = − C s c1 x1s2 (s) − (c2 + 1) x3s2 (s) + 2x3m (s) − x2s2 (s) ( ) ⎪ 5 ⎪ ⎪ ⎪ ⎪ a6 (1 − a7 x1m (s)) [(x3m (s) + a3 ) (1 − 2x2m (s)) − 1] , − ⎪ ⎪ ⎪ ⎪ C6 (s) = (−1 − c7 ) x1s2 (s) − c6 x2s2 (s) x3s2 (s) − c5 − 2a6 x3m (s) x2m (s) ⎪ ⎪ ⎩ + 2a6 a7 x1m (s) x3m (s)x2m (s) + 2a3 a6 a7 x1m (s) x2m (s) .

(4.34)

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Figure 4.9 The phase portrait of the fractional slave system (4.31) with μ = 0.97 and for b1 = 1.4, b2 = 0.2.

Proof. Taking the fractional differences of the synchronization errors (4.33) and substituting the proposed controllers (4.34) yields the error dynamics C

μa (e1 (s) , e2 (s) , e3 (s))T = B× (e1 (s − 1 + μ) , e2 (s − 1 + μ) , e3 (s − 1 + μ))T ,

(4.35a)

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Figure 4.10 The phase portrait of the fractional Wang system (4.32) with μ = 0.97 and for c1 = −1.9, c2 = 0.2, c3 = 0.5, c4 = −2.3, c5 = 2, c6 = −0.6, c7 = −1.9.

where ⎛ ⎜

−1

B=⎝ 0 0

1 −1 0

⎞

0 ⎟ −1 ⎠ . −1

(4.35b)

It is easy to see that the matrix B satisfies the stability theorem of fractionalorder discrete-time systems. This means that the synchronization error

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Figure 4.11 The bifurcation diagram of the fractional Wang system (4.32) versus c3 with μ = 0.97.

in (4.35a) converges towards zero asymptotically for all initial conditions. Therefore, master system (4.28) and slave systems (4.31)–(4.32) are combined synchronized. Similarly, to verify the effectiveness of the synchronization method, numerical simulations are considered. For that, we rewrite the error system (4.35a) in the following form: ⎧ C μ ⎪ ⎪ a e1 (s) = −e1 (s − 1 + μ) + e2 (s − 1 + μ)), ⎪ ⎨ C μ e (s) = −e (s − 1 + μ) + e (s − 1 + μ), 2 3 a 2 ⎪ ⎪ ⎪ ⎩C μ a e3 (s) = −e3 (s − 1 + μ),

(4.36)

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in which C μa is the Caputo-like difference operator for s ∈ Na−μ+1 and a is the starting point. When a = 0, the numerical solution of the 3D error system is given by ⎧ ⎪ ⎪ ⎪e1 (n) = e1 (0) + ⎨ ⎪ ⎪ ⎪ ⎩

1 (μ)

1 e2 (n) = e2 (0) + (μ) 1 e3 (n) = e3 (0) + (μ)

n−1

(n−i+μ) i=0 (n−i+1) (−e1 (i) + e2 (i)),

n−1

(n−i+μ) i=0 (n−i+1) (−e2 (i) + e3 (i)),

(4.37)

n−1

(n−i+μ) i=0 (n−i+1) (−e3 (i)),

where e1 (0), e2 (0), and e3 (0) are the initial conditions. Let μ = 0.97 and μ = 0.98. The synchronization error states (4.35a) are shown in Fig. 4.12 and Fig. 4.13, respectively. Let the parameter values of the master system (4.28) and the slave systems (4.31)–(4.32) be {a1 = 3.8, a2 = 0.05, a3 = 0.35, a4 = 3.78, a5 = 0.2, a6 = 0.1, a7 = 1.9}, {b1 = 1.4, b2 = 0.2}, {c1 = −1.9, c2 = 0.2, c3 = 0.5, c4 = −2.3, c5 = 2, c6 = −0.6, c7 = −1.9}. For the initial condition {x1m , x2m , x3m } = {0.01, 0.2, −0.5}, simulation results of the combined synchronization between systems (4.28) and (4.31)–(4.32) are shown in Fig. 4.14. The simulation results demonstrate that combined synchronization between the fractional Rössler system and the fractional Stefanski and Wang systems is achieved.

4.5. Concluding remarks and future works Based on adaptive control, two combination synchronization schemes are obtained in this chapter. The first scheme is considered for 2D chaotic fractional maps, whereas the second scheme is considered for 3D chaotic fractional maps where the fractional Rössler, Stefanski, and Wang maps are considered as examples. The synchronization approaches introduced in this chapter have more advantages than synchronization between one master discrete-time system and one slave discrete-time system. Also, this method can be applied to any chaotic fractional discrete-time system defined in the literature. Numerous studies have demonstrated the feasibility of using chaotic maps in the generation of pseudorandom keys. If the proposed chaos synchronization schemes can be utilized to encrypt data and images, then it may be possible to attain vastly better performance for, for example, encryption schemes in data encryption and the advanced encryption standard.

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Figure 4.12 Combined synchronization error between the fractional Rössler map (4.28) and fractional Stefanski (4.31) and Wang maps (4.32) with μ = 0.97. (a) The ﬁrst error state e1 (s). (b) The second error state e2 (s). (c) The third error state e3 (s).

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Figure 4.13 Combined synchronization error between the fractional Rössler map (4.28) and fractional Stefanski (4.31) and Wang maps (4.32) with μ = 0.98. (a) The ﬁrst error state e1 (s). (b) The second error state e2 (s). (c) The third error state e3 (s).

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Figure 4.14 The time histories of the master system (4.28) and the slave systems (4.31)–(4.32) with μ = 0.97. (a) The ﬁrst state of the master system and the combined slave system. (b) The second state of the master system and the combined slave system. (c) The third state of the master system and the combined slave system.

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Acknowledgments The result of this approach is to produce a top-down view of a framework that standardizes the manner in which organizations can refer to complex data content, thereby reducing the overhead support and providing an incremental approach to improving business applications and operating efficiency. It can be applied as and when needed, depending on the manner in which organizations refer to complex data content, thereby facilitating business applications and the adoption of new systems.

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[35] L. Jouini, A. Ouannas, A.A. Khennaoui, X. Wang, G. Grassi, V.T. Pham, The fractional form of a new three-dimensional generalized Hénon map, Advances in Difference Equations 2019 (1) (2019) 122. [36] A. Ouannas, A.A. Khennaoui, Z. Odibat, V.T. Pham, G. Grassi, On the dynamics, control and synchronization of fractional-order Ikeda map, Chaos, Solitons and Fractals 123 (2019) 108–115. [37] A. Ouannas, A.A. Khennaoui, S. Bendoukha, Z. Wang, V.T. Pham, The dynamics and control of the fractional forms of some rational chaotic maps, Journal of Systems Science and Complexity 30 (2020) 1–20. [38] A. Gasri, A. Ouannas, A.A. Khennaoui, S. Bendoukha, V.T. Pham, On the dynamics and control of fractional chaotic maps with sine terms, International Journal of Nonlinear Sciences and Numerical Simulation 1 (2020). [39] A.A. Khennaoui, A. Ouannas, S. Boulaaras, V.T. Pham, A. Taher Azar, A fractional map with hidden attractors: chaos and control, The European Physical Journal Special Topics 229 (2020) 1083–1093. [40] A. Ouannas, A.A. Khennaoui, S. Momani, G. Grassi, V.T. Pham, Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization, AIP Advances 10 (4) (2020). [41] A. Ouannas, A.A. Khennaoui, S. Momani, V.T. Pham, R. El-Khazali, Hidden attractors in a new fractional-order discrete system: chaos, complexity, entropy, and control, Chinese Physics B 29 (5) (2020). [42] A. Ouannas, A.A. Khennaoui, S. Momani, G. Grassi, V.T. Pham, R. El-Khazali, D. Vo Hoang, A quadratic fractional map without equilibria: bifurcation, 0-1 test, complexity, entropy, and control, Electronics 9 (5) (2020). [43] A. Ouannas, O.A. Almatroud, A.A. Khennaoui, M.M. Al-sawalha, D. Baleanu, V.V. Huynh, V.T. Pham, Bifurcations, hidden chaos and control in fractional maps, Symmetry 12 (6) (2020) 879. [44] F. Hadjabi, A. Ouannas, N. Shawagfeh, A.A. Khennaoui, G. Grassi, On twodimensional fractional chaotic maps with symmetries, Symmetry 12 (5) (2020). [45] A. Ouannas, M.M. Al-sawalha, On inverse full state hybrid projective synchronization of chaotic dynamical systems in discrete-time, International Journal of Dynamics and Control 5 (2) (2017) 252–258. [46] A. Ouannas, Synchronization criterion for a class of N-dimensional discrete chaotic systems, Journal of Advanced Research in Dynamical and Control Systems 7 (1) (2015) 82–89. [47] A. Ouannas, A new synchronization scheme for general 3D quadratic chaotic systems in discrete-time, Nonlinear Dynamics and Systems Theory 15 (2) (2015) 163–170. [48] A. Ouannas, Z. Odibat, Generalized synchronization of different dimensional chaotic dynamical systems in discrete time, Nonlinear Dynamics 81 (1) (2015) 765–771. [49] A. Ouannas, A new generalized-type of synchronization for discrete-time chaotic dynamical systems, Journal of Computational and Nonlinear Dynamics 10 (6) (2015) 061019. [50] A. Ouannas, M.M. Al-Sawalha, A new approach to synchronize different dimensional chaotic maps using two scaling matrices, Nonlinear Dynamics and Systems Theory 15 (4) (2015) 400–408. [51] A. Ouannas, R. Abu-Saris, On matrix projective synchronization and inverse matrix projective synchronization for different and identical dimensional discrete-time chaotic systems, Journal of Chaos (2016). [52] A. Ouannas, G. Grassi, Inverse full state hybrid projective synchronization for chaotic maps with different dimensions, Chinese Physics B 25 (9) (2016) 090503. [53] A. Ouannas, Coexistence of various synchronization-types in hyperchaotic maps, Nonlinear Dynamics and Systems Theory 16 (3) (2016) 312–321.

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[72] A. Ouannas, S. Abdelmalek, S. Bendoukha, Coexistence of some chaos synchronization types in fractional-order differential equations, Electronic Journal of Differential Equations 2017 (128) (2017) 1. [73] A. Ouannas, Z. Odibat, T. Hayat, Fractional analysis of co-existence of some types of chaos synchronization, Chaos, Solitons and Fractals 105 (2017) 215–223. [74] V.T. Pham, A. Ouannas, C. Volos, T. Kapitaniak, A simple fractional-order chaotic system without equilibrium and its synchronization, AEÜ. International Journal of Electronics and Communications 86 (2018) 67–79. [75] X. Wang, A. Ouannas, V.T. Pham, H.R. Abdolmohammadid, A fractional-order form of a system with stable equilibria and its synchronization, Advances in Difference Equations 2018 (1) (2018) 20. [76] A. Ouannas, X. Wang, V.T. Pham, G. Grassi, T. Ziar, Co-existence of some synchronization types between non-identical commensurate and incommensurate fractionalorder chaotic systems with different dimensions, Advances in Difference Equations 21 (2018). [77] A. Ouannas, G. Grassi, V.T. Pham, T. Ziar, Co-existence of function-based hybrid synchronization types between non-identical fractional-order chaotic systems, Advances in Difference Equations 1 (2018) 1–16. [78] A. Ouannas, Z. Odibat, A study on Q-S synchronization in coupled chaotic incommensurate fractional order systems, Chinese Journal of Physics 56 (5) (2018) 1940–1948. [79] A. Ouannas, N. Debbouche, X. Wang, V.T. Pham, O. Zehrour, Secure multipleinput multiple-output communications based on F–M synchronization of fractionalorder chaotic systems with non-identical dimensions and orders, Applied Sciences 8 (10) (2018) 1746. [80] A. Ouannas, O. Zehrour, Z. Laadjal, Nonlinear methods to control synchronization between fractional-order and integer-order chaotic systems, Nonlinear Studies 25 (1) (2018) 1–13. [81] A. Ouannas, S. Bendoukha, C. Volos, A. Karouma, Synchronization of commensurate and incommensurate fractional hyperchaotic Rabinovich systems via linear and nonlinear control law, International Journal of Control, Automation, and Systems 17 (9) (2019) 2211–2219. [82] A. Ouannas, X. Wang, V.T. Pham, G. Grassi, V. Van Huynh, Synchronization results for a class of fractional-order spatiotemporal partial differential systems based on fractional Lyapunov approach, Boundary Value Problems 2019 (1) (2019) 74. [83] A. Ouannas, S. Bendoukha, A. Karouma, S. Abdelmalek, A general method to study the co-existence of different hybrid synchronizations in fractional-order chaotic systems, International Journal of Nonlinear Sciences and Numerical Simulation 20 (3–4) (2019) 351–359. [84] A. Ouannas, X. Wang, V.T. Pham, T. Ziar, Dynamic analysis of complex synchronization scheme between integer-order and fractional-order chaotic systems with different dimensions, Complexity (2017). [85] A. Ouannas, A.A. Khennaoui, G. Grassi, S. Bendoukha, Q-S chaos synchronization of different dimensional systems, Discrete Dynamics in Nature and Social (2018). [86] A. Ouannas, S. Bendoukha, Generalized and inverse generalized synchronization of fractional-order discrete-time chaotic systems with non-identical dimensions, Advances in Difference Equations 1 (2018) 1–14. [87] S. Bendoukha, A. Ouannas, X. Wang, A.A. Khennaoui, V.T. Pham, G. Grassi, V.V. Huynh, The co-existence of different synchronization types in fractional-order discrete-time chaotic systems with non-identical dimensions and orders, Entropy 20 (9) (2018) 710.

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CHAPTER FIVE

Stabilization of different dimensional fractional chaotic maps Adel Ouannasa , Amina-Aicha Khennaouib , Iqbal M. Batihac , and Viet-Thanh Phamd a Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria b Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria c Department of Mathematics, Faculty of Science, The University of Jordan, Amman, Jordan d Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

Chapter points •

This chapter provides a contribution to the topic by presenting nonlinear and linear control laws to stabilize the dynamics of 2D and 3D fractional chaotic maps.

•

All results have been obtained by exploiting new theorems based on the linearization and Lyapunov methods.

•

Simulation results are presented to show the eﬀectiveness of the proposed linear control approach.

5.1. Introduction Fractional calculus is an ancient classical mathematical field that has been successfully used in many fields of engineering and science [1–3,83]. Very recently, attention has been focused on discrete fractional calculus (DFC) and fractional difference operators [32–34,97]. Several studies have attempted to develop a complete framework for DFC and generalize the stability theory of conventional discrete calculus to the fractional domain [6,7,98,104]. For example, in reference [6], Abdeljawad discusses the discrete-time counterparts of the well-known Riemann and Caputo derivatives. Successively, several types of difference operators have been proposed including fractional h difference operators, which represent a further generalization of the fractional difference operators [100,102]. Namely, since fractional operators are nonlocal, they are more suitable for modeling natural phenomena characterized by a memory effect. In particular, Wu Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00010-X All rights reserved.

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and Baleanu [8] used the Caputo left difference operator to define a fractional form of the logistic map outlined by fractional difference equations where the discrete memory was confirmed. They have claimed that the fractional logistic map shows more complex and unpredictable behaviors, which are heavily dependent on the fractional order. The exploration of chaotic dynamics has received considerable attention during the past few years. Numerous attempts have been dedicated to analyze the classical systems (outlined by differential or difference equations of integer order) as well as fractional-order systems (outlined by differential or difference equations of fractional order) [10,11,78,79,101,106,107,110]. With the discovery of chaos in nonlinear systems, several efforts have been devoted to the study of control methods for effectively stabilizing the chaotic dynamics at the origin [4,50,51,99]. Some interesting results have been recently published regarding this challenging topic [21–23,25]. Another important aspect of chaotic systems is their synchronization [66–69]. Synchronization in general requires some form of control strategy [12–15,17]. Various kinds of chaos synchronization have been presented, such as, Q-S synchronization [16,18,59], hybrid projection synchronization [9,24,31,49,56,60,63,74], and complete synchronization [19,20,109]. Many types of chaos synchronization can be found in the literature aimed at synchronizing integer-order discrete systems [26–30,52–55,57,58,61,62, 64,65], integer-order continuous chaotic systems [66–69], fractional-order continuous chaotic systems [70–77,80–82,84–88], and fractional-order discrete chaotic systems [90,93,95]. Since nonlinear maps can have potential application in many areas such as secure communication and image encryption [5], many fractional-order maps based on integer-order maps have been extensively investigated in the literature [41,42]. For example, Ouannas et al. proposed a fractional form of the Ikeda map [40]. In [37] a new fractional-order double scroll map is introduced and a method for controlling its chaotic behavior is presented. Control laws for stabilizing the chaotic dynamics of three maps were studied in [38,95]. In [39] a new 3D fractional map is proposed, whereas in [36] a fractional form of the Tinkerbell map is studied. The system has high complexity, so it can have potential application in the field of encryption. Reference [37] shows that the complexity of the fractional-order maps is related to the fractional order value of systems. It is worthy to state that all fractional-order maps show more complex chaotic behavior than their integer-order counterparts, which make them more suitable to applications in the encryption of data. On the other hand, researchers constructed

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new fractional chaotic maps with hidden attractors using a Caputo-like difference operator [35]. Hidden chaotic attractors are attractors in unusual systems, for example, systems with no equilibria [43,46], with close curve fixed points [48], or with only one stable equilibrium [47], for which the initial conditions can only be found via extensive numerical search [44,45]. Consequently, these studies have proven the significant role of hidden attractors in theoretical problems and engineering applications. Referring to synchronization of fractional-order maps, a number of papers have been published [91,94]. In [92] the synchronization of integer- and fractionalorder chaotic maps with different dimension is studied. Similarly, in [89] two control schemes for the Q-S synchronization of two different dimensional fractional maps are proposed and analyzed in detail. In [105] the control properties of a 3D fractional map have been investigated. Similarly, in [103] bifurcations and control of a quadratic fractional map without equilibrium points have been studied. In [108] the chaotic dynamics of the fractional Duffing map have been investigated. This chapter provides a contribution to the topic by presenting nonlinear and linear control laws to stabilize the dynamics of 2D and 3D fractional chaotic maps, namely the 2D fractional Lozi map and the 3D fractional Wang map. All the results have been obtained by exploiting new theorems based on the linearization and Lyapunov methods. Simulation results are presented to show the effectiveness of the proposed stabilization approaches. The chapter is organized as follows. In Section 5.2, some tools of discrete fractional calculus are reported. In Section 5.3, a novel theorem is proved which enables the dynamics of the fractional Lozi map to be stabilized at the origin by a 1D nonlinear control law. In Section 5.4, a 2D linear control law is proposed to stabilize at zero the fractional chaotic Wang map. Finally, simulation results are reported throughout the chapter, with the aim to show the effectiveness of the proposed approaches.

5.2. Basic tools A brief summary of the mathematical background related to the Caputo delta difference and h-difference operators is provided. In addition, we present some theorems in discrete fractional calculus to study the stability of fractional maps.

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5.2.1 Caputo delta difference operator and stability Discrete fractional calculus is a recent research topic, which is known because of the development of integer-order calculus. There are three different definitions for fractional difference operators, namely, Grunwald– Letnikov, Caputo, and Riemann–Liouville operators, which have indeed played a practical role in engineering applications. Definition 5.1. Let X : Na → R with Na = {a, a + 1, a + 2, ...}. The forward difference operator for function X is defined by [6] X (s) = X (s + 1) − X (s).

(5.1)

Definition 5.2. The μ-Caputo type delta difference of a function X (s) : Na → R, which is of the form C

m−μ) m μa X (s) = −( X (s) a

=

s−( m−μ) 1 (s − τ − 1)(m−μ−1) m τ X (τ ) , (m − μ) τ =a

(5.2)

where μ ∈ N is the fractional order, s ∈ Na+m−μ , and m = μ + 1. The term denotes the Gamma function, given by (μ) =

+∞

e−s sμ−1 ,

(μ + 1) = (μ),

(5.3)

0

and s(μ) is the falling function, defined as s(μ) =

(s + 1) . (s + 1 − μ)

(5.4)

Definition 5.3. [33] The μth fractional sum of mτ X (s) is defined as −μ a X (τ ) =

1 (μ)

s−μ

(s − τ − 1)(μ−1) X (τ ) ,

(5.5)

τ =a

with s ∈ Na+μ , μ > 0. The following theorems provide the basis for the numerical method and stability analysis that we will be requiring later in the study when dealing with the proposed fractional-order discrete-time systems.

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Theorem 5.1. [34] For the fractional difference equation

C μ X (s) = f (s + μ − 1, X (s + μ − 1)), a b X (a) = Xb , m = μ + 1, b = 0, 1, ..., m − 1,

(5.6)

the unique solution of the initial value problem (5.6) is given by X (s) = X0 (s) +

s−μ

1

(μ) τ =a+m−μ

(s − σ (τ ))(μ−1) f (τ + μ − 1, X (τ + μ − 1)),

s ∈ Nμ+m ,

(5.7)

where X0 (s) =

m −1 b=0

(s − a) b b X (a). b+1

(5.8)

Theorem 5.2. [96] Consider the following linear fractional discrete-time system: C

μa X (s) = AX (s + μ − 1) ,

X (0) = X0 ,

(5.9)

where 0 < μ ≤ 1, X (s) ∈ Rn , and A ∈ Rm×m . The zero equilibrium point of system (5.9) is asymptotically stable ∀s ∈ Na+1−μ if and only if |arg λ| > μ

and

π

(5.10a)

2

|arg λ| − π |λ| < 2 cos 2−μ

μ

(5.10b)

are satisfied for all eigenvalues λ of matrix A.

5.2.2 Caputo h-difference operator and stability In this section, some basic concepts related to the Caputo h-difference operator are briefly summarized. In this context, we first introduce some definitions and notation. Definition 5.4. Let h be a strictly real positive number and f : (hN)a → R. The forward h-difference operator is introduced as h X (s) =

where σ sh = (s + 1) h.

X (σh (s)) − X (s) , h

(5.11)

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Definition 5.5. The h-falling factorial function of real order μ is defined by (μ)

sh = h

μ

s h

s

+1

, +1−μ h

(5.12)

in which s ∈ R. As one can see when h = 1, the h-falling factorial function is equivalent to the falling factorial function (5.4) defined in the previous section.

Definition 5.6. [97] Let X : hN a → R and 0 < μ be given, and let a be a starting point. The μth-order h-sum is given by −μ

h a

X ( s) =

h (μ)

s −μ

h

s − σ τh

(μ−1)

X τh ,

h

τ = ha

σ τ h = (τ + 1) h, a ∈ R, s ∈ hN a+μh ,

with the h-falling factorial function, where hN

a + (2 − μ) h, . . . .

a+(1−μ)h

(5.13) = a + (1 − μ) h,

Based on the above definition of the h-fractional sum, it is possible to define the h-Caputo-like difference operator.

Definition 5.7. [6] For X (s) defined on hN Caputo-like difference is defined by −(m−μ) m C μ X h a X (s) = a

where X (s) =

X s+h −X (t) h

(s) ,

a

and 0 < ν , μ ∈/ N, the

s ∈ hN

a+(m−μ)h

,

(5.14)

and m = μ + 1. If μ = 1, then we have

C μ h a X

= m X ,

t ∈ (hN)a .

(5.15)

In order to derive in the following the numerical solution of the discrete fraction map defined in the h-Caputo sense, now a theorem is briefly summarized. Theorem 5.3. The delta fractional difference equation C

μh X (s) = f (s + (μ − 1)h, X (s + μ − 1))

(5.16)

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is equivalent to equation (5.17) when s ∈ (hN)a+(1−ν)h and μ ∈ ]0, 1]: X (s) = X (a) +

s −μ

h

hμ (μ)

1) (s − σ (τ h))(μ− f (τ h + (μ − 1)h, X (τ h + (μ − 1)h)), h

τ = ha +1−μ

s ∈ (hN)a+h .

(5.17)

Note that for the purpose of numerical calculation, equation (5.17) can be written as X (n + 1) = X (0) +

n hμ (n − j + μ) f (j + 1, X (j + 1)), (μ) j=0 (n − j + 1)

(5.18)

where X (0) is the initial value. From the numerical formula (5.18) we can see that the state X (n + 1) depends on X (0), X (1) . . . X (n + 1). Now a stability result is briefly illustrated, with the aim to identify the stability conditions of the zero equilibrium point for the fractional nonlinear difference system in the h-Caputo sense, written in the form (5.19) (s) = f s + μh, X s + μh , where X (s) = (X1 (s) , X2 (s) , ..., Xn (s))T , s ∈ hN a+(1−μ)h , and f is a nonC μ h a X

linear function.

Theorem 5.4. [98] Let X = 0 be an equilibrium point of the nonlinear discrete fractional system (5.19). If there exists a positive definite and decreasing scalar C ν function V (s, X (s)) such that h a V (s, X (s)) ≤ 0, s ∈ hN a+(1−μ)h , then the equilibrium point is asymptotically stable. In the following, a useful inequality for Lyapunov functions is introduced.

Theorem 5.5. [98] For any discrete time s ∈ hN following inequality holds: C μ h a

X T (s) X (s) ≤ 2X T s + μh

C h

a+(1−μ)h

, 0 < μ ≤ 1, the

μa X (s) .

(5.20)

5.3. Stabilization of 2D fractional maps In this section a novel theorem is proved, which enables to stabilize the dynamics of fractional maps to zero asymptotically. The aim of stabi-

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lizing chaotic systems is to derive a 1D control law such that both of the map trajectories are controlled to zero asymptotically. Here, we will realize stabilization of the 2D fractional Lozi map by inserting a time-varying parameter to the state x1 (t). Referring to the fractional Lozi map, it was constructed in [23] based in the μ-Caputo delta definition and is given by

C μ x (s) = −α |x (s − 1 + μ)| + x (s − 1 + μ) + 1 − x (s − 1 + μ) , 1 1 2 1 a 1 C μ x (s) = α x (s − 1 + μ) − x (s − 1 + μ) , 2 1 2 a 2

(5.21) in which 0 < μ ≤ 1, s ∈ Na+1−μ , x1 and x2 are state variables, and α1 and α2 are bifurcation parameters. When μ = 1, the system (5.21) degenerates into the classical Lozi map given by

x1 (n + 1) = −α1 |x1 (n)| + x2 (n) + 1, x2 (n + 1) = α2 x1 (n) .

(5.22)

This integer-order map, whose phase portrait is depicted in Fig. 5.1, was introduced firstly by Lozi. To assess the dynamics of the fractional Lozi map (5.21), we need to define a discrete numerical formula to evaluate the states of the system using MATLAB® . For that, we use Theorem 5.1 as a tool to obtain the following discrete solution: ⎧ 1 s−μ (μ−1) x1 (s) = x1 (a) + (μ) (−α1 |x1 (τ − 1 + μ)| ⎪ τ =a+1−μ (s − τ − 1) ⎪ ⎪ ⎨ + x2 (τ − 1 + μ) + 1 − x1 (τ − 1 + μ)) , 1 s−μ (μ−1) ⎪ x2 (s) = x2 (a) + (μ) (α2 x1 (τ − 1 + μ) ⎪ τ =a+1−μ (s − τ − 1) ⎪ ⎩ − x2 (τ − 1 + μ)) .

(5.23) Let the new variable n = s − a, a = 0, and since the discrete kernel function (s − τ − 1)μ−1 / (μ) is equal to (s − τ )/ (s − τ − μ + 1), it follows that

1 x1 (n) = x1 (0) + (μ)

x2 (n) = x2 (0) + (μ) 1

n−1

(n−i+μ) i=0 (n−i+1) (n−i+μ) i=0 (n−i+1)

n−1

(−α1 |x1 (i)| + x2 (i) + 1 − x1 (i)) , (α2 x1 (i) − x2 (i)) ,

(5.24) where x1 (0) and x2 (0) are the initial states. The fractional Lozi map (5.21) shows a long memory effect, which means that the present state depends on all the past states x1 (0), . . . x1 (n) as described in (5.24). This is not the

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Figure 5.1 The phase portrait of the integer-order Lozi map (5.22).

case with the integer-order Lozi map (5.22). As one can see, the fractional Lozi map expressed by equation (5.21) is a generalization of the integerorder Lozi map given with difference system (5.22). It is always helpful to examine the bifurcation diagram corresponding to a specific critical parameter in order to gain a comprehensive understanding of the dynamics of the fractional map. We have produced the bifurcation diagram of (5.21) with α2 ∈ [−0.6, 0.6] and α1 = 1.5, as shown in Fig. 5.2(a) for the fractional order μ = 0.97. It shows that the state of the fractional Lozi map (5.21) changes from periodic to chaotic at ] − 0.5, 0.5[. Additionally, the dynamic behavior of the fractional Lozi map (5.21) can also be illustrated by calculating the largest Lyapunov exponent (LLE). For the same bifurcation parameters and μ = 0.97, the LLE diagram is drawn in Fig. 5.2(b). The chaotic attractor of the fractional Lozi map under system parameters α1 = 1.5, α2 = 0.3 with initial conditions x1 (0) = 0, x2 (0) = 0 is shown in Fig. 5.2(c). To obtain our results, the following theorem is presented.

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Figure 5.2 (a) Bifurcation diagram of x2 and (b) largest Lyapunov exponent of the fractional Lozi map (5.21) calculated for α2 ∈ [−0.6, 0.6] and α1 = 1.5 with fractional order value μ = 0.97 using initial condition [x1 (0), x2 (0)] = [0, 0]. Additionally, (c) represents the chaotic attractor for α2 = 0.3.

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Theorem 5.6. The fractional Lozi map (5.21) can be stabilized under the 1D control law

1 1 C (s) = x1 (s) + α1 |x1 (s)| − 1 − + 1 x2 (s) . 2 α2 16

(5.25)

Proof. Since the nonlinearity term is in state x1 , we add the time-varying control parameter C to the first equation from system (5.21). Then, the controlled fractional-order Lozi map is given by ⎧ ⎪ ⎨ ⎪ ⎩

C μ x (s) = −α |x (s − 1 + μ)| + x (s − 1 + μ) + 1 − x (s − 1 + μ) 1 1 2 1 a 1

+ C (s − 1 + μ) ,

C μ x (s) = α x (s − 1 + μ) − x (s − 1 + μ) . 2 1 2 a 2

(5.26) Substituting the proposed control law (5.25) into (5.26) yields the simplified system

C μ x (s) = − 1 x (s − 1 + μ) − 1 x (s − 1 + μ) , a 1 2 1 α2 16 2 C μ x (s) = α x (s − 1 + μ) − x (s − 1 + μ) . 2 1 2 a 2

(5.27)

As we mentioned earlier, the aim is to show that the zero equilibrium of (5.27) is asymptotically stable, which means that the system states converge towards zero as time progresses. Asymptotic stability can be established using the linearization method as described in Theorem 5.2. The controlled system can be written in the compact form C

μa (x1 (s) , x2 (s))T = A × (x1 (s − 1 + μ) , x2 (s − 1 + μ))T ,

where

A=

− 12 α2

−1 α2 16

(5.28a)

−1

.

(5.28b)

The characteristic equation of matrix A is given by 3 2

λ2 + λ +

9 = 0. 16

(5.29)

It is easy to see that the eigenvalues λi of matrix A are negative real numbers, that is, |arg λi | = π >

νπ

2

(5.30a)

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and

|arg λi | − π |λi | < 2 cos 2−ν

ν ,

for i = 1, 2.

(5.30b)

Therefore, from Theorem 5.2, the states of the controlled fractional Lozi map are stabilized to zero. The proof is completed To verify the theoretical results obtained above, numerical simulations were preformed using MATLAB. We start by employing Theorem 5.1 to obtain the numerical formula of the controlled dynamical system (5.27) as follows:

x2 (n) = x2 (0) +

n−1

(n−i+μ) i=0 (n−i+1) 1 n−1 (n−i+μ) i=0 (n−i+1) (μ)

1 x1 (n) = x1 (0) + (μ)

− 12 x1 (i) −

1

x (i) α2 16 2

(α2 x1 (i) − x2 (i)) ,

,

(5.31)

where x1 (0) and x2 (0) are the initial states. The parameter values are taken as α1 = 1.5 and α2 = 0.3 to ensure the existence of chaos. Figs. 5.3 and 5.4 show the state trajectories and phase space plots of the controlled system (5.8) when the fractional order values are taken as μ = 0.97 and μ = 0.95, respectively. These plots clearly show that the chaotic dynamics of the fractional Lozi map (5.6) are controlled to equilibrium point (0, 0) by control law (5.25).

5.4. Stabilization of 3D fractional maps By exploiting a novel theorem based on a suitable Lyapunov function, in this section a 1D linear control law is illustrated, with the aim to stabilize at zero the chaotic dynamics of the fractional Wang map. The fractional Wang system is a 3D nonlinear discrete-time system that can exhibit chaotic behavior. In [95], the fractional discrete Wang system has been studied and discussed. In this study, the authors have considered the ν -Caputo version of a standard map that returns to Wang which is defined as ⎧ ⎪ ⎪ ⎨x1 (n) = β3 x2 (n − 1) − (β4 + 1)x1 (n − 1), x2 (n) = β1 x1 (n − 1) + x2 (n − 1)β2 x3 (n − 1), ⎪ ⎪ ⎩x (n) = (β + 1)x (n − 1) + β x (n − 1)x (n − 1) + β , 3 7 3 6 2 3 5

(5.32)

where x1 , x2 , x3 are the states of the map and βi for i = 1, 7 are bifurcation parameters. By introducing the Caputo-like difference operator of order μ ∈ ]0, 1], the fractional Wang map is introduced as

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Figure 5.3 (a) Evolution of the ﬁrst state x1 (t) of the controlled fractional Lozi map (5.26) with fractional order μ = 0.97. (b) Evolution of the second state x2 (t). (c) Phase portrait of the controlled Lozi map (5.26).

⎧ C μ ⎪ ⎪ ⎨ a x1 (s) = β3 x2 (s + μ − 1) − β4 x1 (s + μ − 1), C μ x (s) = β x (s + μ − 1) + β x (s + μ − 1), 1 1 2 3 a 2 ⎪ ⎪ ⎩C μ x (s) = β x (s + μ − 1) + β x (s + μ − 1)x (s + μ − 1) + β , 7 3 6 2 3 5 a 3

(5.33)

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Figure 5.4 (a) Evolution of the ﬁrst state x1 (t) of the controlled fractional Lozi map (5.26) with fractional order μ = 0.95. (b) Evolution of the second state x2 (t). (c) Phase portrait of the controlled Lozi map (5.26).

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where C μa is the μ-Caputo operator and a is the starting point. As demonstrated in [95], the fractional map (5.34) exhibits a chaotic behavior. Plots of the evolution of states and chaotic attractor of the fractional Wang map (5.33), obtained with (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9) and initial condition x1 (0) = 0.1, x2 (0) = 0.05, and x3 (0) = 0.02, are reported in Figs. 5.5 and 5.6 for the fractional order μ = 0.999. Herein, the fractional Caputo h-difference operator is considered, in order to derive a different mathematical model of the fractional Wang map. Namely, the following equations are proposed: ⎧ C μ ⎪ x1 (s) = β3 x2 (s + μh) − β4 x1 (s + μh), ⎪ ⎪ ⎨h a

C μ h a x2 (s) = β1 x1 (s + μh) + β2 x3 (s + μh), ⎪ ⎪ ⎪ ⎩C μ x (s) = β x (s + μh) + β x (s + μh)x (s + μh) + β , 7 3 6 2 3 5 h a 3

(5.34)

μ where s ∈ hN a+(1−μ)h , C h a denotes the fractional h-difference operator, and {β1 , β2 , β3 , β4 , β5 , β6 , β7 } are parameter values. The solution of the fractional Wang map (5.34) is obtained by introducing the fractional h-sum operator. According to Theorem 5.4, the equivalent implicit discrete formula can be written in the form

⎧ ⎪ x1 (n + 1) = x1 (0) + ⎪ ⎪ ⎪ ⎪ ⎨x (n + 1) = x (0) + 2

2

n

(n−i+μ) i=0 (n−i+1) (β3 x2 (i + 1) − β4 x1 (i + 1)), hμ n−1 (n−i+μ) i=0 (n−i+1) (β1 x1 (i + 1) + β2 x3 (i + 1)), (μ) μ n−1 (n−i+μ) h i=0 (n−i+1) (β7 x3 (i + 1) (μ)

hμ

(μ)

⎪ ⎪ x3 (n + 1) = x3 (0) + ⎪ ⎪ ⎪ ⎩ + β6 x2 (i + 1)x3 (i + 1) + β5 ),

(5.35) where x1 (0), x2 (0), and x3 (0) are the initial state values. Based on the predictor-corrector method, the implicit equation (5.35) is transformed into its explicit form, which can be used for investigating the dynamic behavior of the proposed map numerically. The predictor-corrector can be summarized as follows: ⎧ ⎪ x1 (n + 1) = X 1 (0) + ⎪ ⎪ ⎨

x2 (n + 1) = X 2 (0) + ⎪ ⎪ ⎪ ⎩x (n + 1) = X (0) + 3

3

hμ (μ)

hμ (μ)

hμ (μ)

Mn,n (β3 X 2 (i + 1) − β4 X 1 (i + 1)), Mn,n (β1 X 1 (i + 1) + β2 X 3 (i + 1)), Mn,n (β7 X 3 (i + 1) + β6 X 2 (i + 1)X 3 (i + 1) + β5 ), (5.36)

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Figure 5.5 (a) Evolution of the ﬁrst state x1 (t) of the controlled fractional Wang map (5.33) with fractional order μ = 0.999. (b) Evolution of the second state x2 (t) of the controlled fractional Wang map (5.33). (c) Evolution of the third state x3 (t).

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Figure 5.6 Different phase space projection of the chaotic attractor of the fractional Wang map (5.33) obtained for the fractional order value μ = 0.999 with initial condition [x − 1(0), x2 (0), x3 (0)] = [0.1, 0.05, 0.02] and system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, −0.5, 2.3, 2, −0.6, −1.9).

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where ⎧ hμ n−1 ⎪ X 1 (n + 1) = x1 (0) + (μ) i=0 Mi,n (β3 x2 (i + 1) − β4 x1 (i + 1)), ⎪ ⎪ ⎪ ⎪ μ n−1 ⎨X (n + 1) = x (0) + h 2 2 i=0 Mi,n (β1 x1 (i + 1) + β2 x3 (i + 1)), (μ) μ ⎪ n −1 h ⎪ X 3 (n + 1) = x3 (0) + (μ) ⎪ i=0 Mi,n (β7 x3 (i + 1) ⎪ ⎪ ⎩ + β6 x2 (i + 1)x3 (i + 1) + β5 ),

(5.37) with Mj,n = (n − j + μ − 1)(μ−1) ,

(5.38)

and X 1 , X 2 , X 3 refer to the predictor values. By taking the initial state values x1 (0) = 0.1, x2 (0) = 0.05, and x3 (0) = 0.02, h = 0.1, the fractional order value μ = 0.99, and the system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9), the fractional map (5.35) displays the strange attractor reported in Fig. 5.7. Fig. 5.8 shows the bifurcation diagram obtained by varying the system parameter β3 from 0.2 to 1. Figs. 5.7 and 5.8 clearly highlight the chaotic behavior of the fractional Wang map (5.35) for μ = 0.99. Note that the evolution of states of the fractional map (5.35), which involves the adoption of the Caputo h-difference operator, is different from that of the map reported in [95] using a Caputo-like difference operator (see Fig. 5.9). Namely, the adoption of two different fractional operators has led to different shapes in the chaotic attractors. Here, a controller is presented in order to stabilize at zero the chaotic trajectories of the state variables of the h-Caputo Wang map (5.35). The objective is achieved by adding two linear terms into both first and third equations of the fractional difference system (5.35). The controlled fractional Wang map is expressed as follows: ⎧ C μ ⎪ x1 (s) = β3 x2 (s + μh) + β4 x1 (s + μh) + L1 (s + μh), ⎪ ⎪ ⎨h a

C μ h a x2 (s) = β1 x1 (s + μh) + β2 x3 (s + μh), ⎪ ⎪ ⎪ ⎩C μ x (s) = β x (s + μh) + β x (s + μh)x (s + μh) + β + L (s + μh), 7 3 6 2 3 5 2 h a 3

(5.39)

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Figure 5.7 Bifurcation diagram of the fractional Wang map (5.33) versus bifurcation parameter β3 and for fractional order value μ = 0.99 with initial condition [x − 1(0), x2 (0), x3 (0)] = [0.1, 0.05, 0.02] and system parameters (β1 , β2 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, −2.3, 2, −0.6, −1.9).

where L1 and L2 are suitable controllers to be determined. To this purpose, a theorem is now given to rigorously ensure that the dynamics of (5.41) can be stabilized at zero. Theorem 5.7. The 3D fractional Wang map (5.34) can be stabilized under the following 2D control law:

L1 (s) = −(β3 + β1 )x2 (s), L2 (s) = −β5 − (|β6 b|)x3 (s) − β2 x2 (s),

where |x2 (s)| ≤ b, s ∈ hN

a+(1−μ)h

.

(5.40)

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Figure 5.8 Different phase space projection of the chaotic attractor of the fractional Wang map (5.33) obtained for the fractional order value μ = 0.99 with initial condition [x − 1(0), x2 (0), x3 (0)] = [0.1, 0.05, 0.02] and system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, −0.5, 2.3, 2, −0.6, −1.9).

Proof. The controlled fractional Wang map can be expressed as in equation (5.39): ⎧ C μ ⎪ ⎪ ⎨h a x1 (s) = β3 x2 (s + μh) + β4 x1 (s + μh) + L1 (s + μh),

C μ h a x2 (s) = β1 x1 (s + μh) + β2 x3 (s + μh), ⎪ ⎪ ⎩C μ x (s) = β x (s + μh) + β x (s + μh)x (s + ν h) + β + L (s + μh). 7 3 6 2 3 5 2 h a 3

(5.41)

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Figure 5.9 Evolution of states of the fractional Wang map with h-Caputo difference operator for fractional order value μ = 0.99 with initial condition [x − 1(0), x2 (0), x3 (0)] = [0.1, 0.05, 0.02] and system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, −0.5, 2.3, 2, −0.6, −1.9).

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Substituting the control law (5.39) into (5.40), the system dynamics become ⎧ C μ ⎪ ⎪ h a x1 (s) = −β1 x2 (s + μh) + β4 x1 (s + μh), ⎪ ⎪ ⎨C μ x (s) = β x (s + μh) + β x (s + μh), 1 1 2 3 h a 2 C μ ⎪ x ( s ) = (β − |β | b ) x ( s + μh) + β6 x2 (s + μh)x3 (s + μh) ⎪h a 3 7 6 3 ⎪ ⎪ ⎩ − β2 x2 (s + μh).

(5.42)

We take the Lyapunov function in the form 1 1 1 V = x21 (s) + x22 (s) + x23 (s). 2 2 2

(5.43)

The adoption of the Caputo h-difference operator implies that C μ h a V

μ 2 C μ 2 C μ 2 =C h a x1 (s) +h a x2 (s) +h a x3 (s),

(5.44)

and by using Theorem 5.5, we obtain C μ h a V

μ C μ ≤ x1 (s + μh)C h a x1 (s + μh) + x2 (s + μh)h a x2 (s) μ + x3 (s + μh)C h a x3 (s)

= − β1 x1 (s + μh)x2 (s + μh) + β4 x21 (s + μh) + β1 x2 (s + μh)x1 (s + μh) + β2 x2 (s + μh)x3 (s + μh) + (β7 − |β6 |b)x23 (s + μh) + β6 x2 (s + μh)x23 (s + μh) − β2 x3 (s + μh)x2 (s + μh)

(5.45)

≤ β4 x21 (s + μh) + (β7 − |β6 |b)x23 (s + μh) + β6 x2 (s + μh)x23 (s + μh) ≤ β4 x21 (s + μh) + (β7 − |β6 |b)x23 (s + μh) + β6 x2 (s + μh) x23 (s + μh) ≤ β4 x21 (s + μh) + (β7 − |β6 |b)x23 (s + μh) + |α6 | bz2 (s + μh) = β4 x21 (s + μh) + β7 x23 (s + μh) < 0.

This is because β4 = β7 = −1.9. From Theorem 5.5 it can be concluded that the zero equilibrium of (5.42) is asymptotically stable. As a consequence, it is proved that the dynamics of the proposed 3D fractional discrete Wang system (5.34) are stabilized by the linear control law (5.35). Remark 5.1. The existence of the constant b is justified by the boundlessness property of chaotic map states.

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Now, numerical simulations using MATLAB are presented to demonstrate the theoretical results in the previous theorem. We start by employing Theorem 5.4 to obtain the numerical formula of the controlled dynamical system (5.41) as follows: ⎧ hμ n (n−i+μ) ⎪ x1 (n + 1) = x1 (0) + (μ) i=0 (n−i+1) (−β1 x2 (j + 1) + β4 x1 (j + 1)), ⎪ ⎪ ⎪ ⎪ μ n (n−i+μ) ⎨x (n + 1) = x (0) + h 2 2 i=0 (n−i+1) β1 x1 (j + 1) + β2 x3 (j + 1), (μ) μ n ⎪ n−i+μ) h ⎪ x3 (n + 1) = x3 (0) + (μ) i=0 ( (β7 − |β6 |b)x3 (j + 1) ⎪ (n−i+1) ⎪ ⎪ ⎩ + β6 x2 (j + 1)x3 (j + 1) − β2 x2 (j + 1),

(5.46) where x1 (0), x2 (0), and x3 (0) are the initial states of the controlled system. The predictor-corrector method with h = 0.1 is used to simulate the states of the fractional-order Wang system. The parameter values are taken as (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9) to ensure chaotic behavior. The initial values are chosen as x1 (0) = 0.1, x2 (0) = 0.05, x3 (0) = 0.02 with μ = 0.99. The evolution of the states and the phase space projections are shown in Fig. 5.10 and Fig. 5.11, respectively. These plots clearly show that the fractional discrete Wang system is driven to the origin by linear control laws in the form of (5.35). Furthermore, we have plotted the evolution of states and phase space of the controlled Wang system with the same bifurcation parameters and fractional order μ = 0.98. Figs. 5.12 and 5.13 illustrate the results. Clearly, the state variable of the controlled system goes to zero. Thus, the numerical results show very well the effectiveness of the proposed controllers.

5.5. Summary and future works In this chapter, we have reviewed the recently published fractional maps referred to as 2D fractional Lozi map and 3D fractional Wang map. These fractional maps are simple and exhibit rich dynamics. We have shown that the states of the two fractional maps can be stabilized by means of very simple nonlinear and linear controllers. The convergence of the proposed controllers is shown by means of the linearization method and the Lyapunov approach. Numerical results showed the convergence of the stabilized states. The proposed control schemes require less control effort with respect to the nonlinear approaches introduced in the literature. Since the literature seems to agree that due to the added degrees of freedom, fractional chaotic maps are more suitable for data encryption and

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Figure 5.10 Evolution states of the controlled fractional Wang map (5.39) for fractional order value μ = 0.99 with system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9). (a) Evolution of the ﬁrst state x1 (t). (b) Evolution of the second state x2 (t). (c) Evolution of the third state x3 (t).

secure communications, it is worth investigating the randomness of bit sequences generated by the standard and fractional Lozi-based maps. A not very accurate way of doing so is by calculating the approximate entropy of the fractional maps. For future work, we aim to calculate these randomness measures and assess the suitability claim.

Acknowledgments The result of this approach is to produce a top-down view of a framework that standardizes the manner in which organizations can refer to complex data content,

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Figure 5.11 Difference phase space projection of the controlled fractional Wang map (5.39) for fractional order value μ = 0.99 with system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9).

thereby reducing the overhead support and providing an incremental approach to improving business applications and operating efficiency. It can be applied as and when needed, depending on the manner in which organizations refer to complex data content, thereby facilitating business applications and the adoption of new systems.

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Figure 5.12 Evolution states of the controlled fractional Wang map (5.39) for fractional order value μ = 0.98 with system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9). (a) Evolution of the ﬁrst state x1 (t). (b) Evolution of the second state x2 (t). (c) Evolution of the third state x3 (t).

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Figure 5.13 Difference phase space projection of the controlled fractional Wang map (5.39) for fractional order value μ = 0.98 with system parameters (β1 , β2 , β3 , β4 , β5 , β6 , β7 ) = (−1.9, 0.2, 0.5, −2.3, 2, −0.6, −1.9).

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CHAPTER SIX

Observability of speed DC motor with self-tuning fuzzy-fractional-order controller Arezki Fekika,b , Mohamed Lamine Hamidab , Hamza Houassinea , Hakim Denounb , Sundarapandian Vaidyanathanc , Nacera Yassaa , Ahmed G. Radwand,e , and Lobna A. Saide a Electrical

Engineering Department, Akli Mohand Oulhadj University-Bouira, Bouira, Algeria of Advanced Technologies of Electrical Engineering (LATAGE), Mouloud Mammeri University Tizi-Ouzou, Tizi-Ouzou, Algeria c Research and Development Centre, Vel Tech University, Chennai, India d Engineering Mathematics and Physics Department, Cairo University, Cairo, Egypt e Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt b Laboratory

6.1. Introduction The increasing use of electronically powered and controllable systems in the industrial sector, driven by improved performance, has led to a proliferation of static converters. Today, the research on electronic power converters for the control and command of AC or DC electric motors is constantly increasing [4–6,8,9,18,21–34,38–42]. The use of DC or AC electric motors is constantly increasing in the industrial sector thanks to the development of control techniques for pulse-width modulation converters. The production of electrical energy and its transformation into mechanical energy are the basis of the industrial structure in the world [16,61]. This transformation is possible thanks to AC and DC electric motors [3,35]. Despite the increasing use and great rivalry with AC motors, which, as three-phase motors, are also part of electric motors, DC motors still play a very important role today [46]. They are often used in industrial applications, because of their properties – in particular the possibility that they offer to adjust their extremely variable torque and speed precisely. This is the case, for example, with the PMA series servo actuators from Harmonic Drive AG, which consist of a high dynamic current motor and a permanent magnet generator reduction mini-cartridge with incremental encoder [19,54,56]. They are ideal for applications in the semiconductor industry, medical technology, and measurement and testing machines. For position Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00011-1 All rights reserved.

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or speed control, one can use closed-loop drives; this requires the speed of return information from pulse encoders or tachometers. As a result, the use of a transducer may be limited to the work and may affect the performance balance of the engine. For hardware complexity and cost reduction [57], intense research has been conducted on the development of speed estimation. For example, using knowledge of motor parameters, motor voltage, and input current, speed estimates can be calculated. However, in these methods, the open-loop estimators have a lower accuracy than the closed-loop estimators. In this chapter, we take the DC motor with separate excitation into consideration, of which the field circuit is isolated from the circuit of the armature [2,37,51]. The downside of this electric system is that its speed changes are very little with the load fluctuations. This function makes it useful for any pace stability requirements. The DC motor as nonlinear mathematical model has encouraged the usage of various control methods, from open-loop control to nonlinear control. The two open-loop strategies [55,58] are focused on both model predictive control (MPC) and nonlinear predictive techniques; techniques such as fluid tuning PI controllers [44], the singular perturbation method [45], feedback linearism [17,49], and equivalence implementation have been established for closed-loop control strategies [13,62]. DC motors are particularly observable at zero current, so careful consideration should be paid to the measurement of the speed near this stage. In [11,12,43], the authors suggest the implementation in the measurable field of expanded Kalman adaptive filters, with high gain relative to the predicted Kalman filter; however, no approach has been created to tackle the nature of the observability and no regulation without sensors can be performed. A nonlinear observer was presented in [36,49] indicating that DC motor sensorless control is feasible. There is, however, no approach to observability, including the singularity. This chapter presents a separate excitation DC motor control without speed sensor with an adaptive observer to estimate the rotational speed of the rotor (closed-loop estimator) and analyzes the speed estimation stability using the Routh–Hurwitz criterion [7,14]. The modified direct command action is introduced into the adaptive observer in order to eliminate the redundancy from the model and simplify setting up artwork. Furthermore, a conception guideline for gaining feedback and a speed controller with a fractional integral proportional regulator whose parameters are automatically adjusted using a fuzzy logic controller are provided to ensure system stability for the whole operating regime.

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Figure 6.1 Armature controlled circuit diagram of the DC motor.

This chapter is organized into six sections. Section 6.1 is introductory. In Section 6.2, a mathematical model of the DC motor is given. In Section 6.3, stability of speed estimation is described. In Section 6.4, the proposed speed controller is presented. Simulation results and a discussion are presented in Section 6.5. In Section 6.6, concluding remarks are given.

6.2. Mathematical model of DC motor The DC motor with distinct excitation is studied in this application. Fig. 6.1 shows a schematic of the DC motor circuit, where R is armature resistance and L is armature inductance [8,38]. The resistance and inductance of the inductor winding are denoted Rf and Lf , respectively. In addition, E is the electromotive force in the armature circuit, ω is the motor speed, kb is an electromotive force constant, Tem and TL are the motor and the load torque, respectively, B is the motor viscous friction constant, and J is the moment of inertia of the rotor. Electrical and mechanical equations are used in the DC motor model. In the linear domain of the magnetization continuum, DC motors are commonly used where the interval flux is equal to the field current: = kf ∗ If , with kf being constant.

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The motor torque, Tem , is proportional to the current of the armature and magnetic field power: Tem = K1 kf If ,

(6.1)

Tem = KT If .

(6.2)

where K1 , kf , If are constants. Thus,

The rear electric motor force (EMF), Eb , is directly proportional to the speed as follows: dθ = kb ω(t). dt By using the voltage law of Kirchhoff, we get Ea = kb

V (t) = RI (t) + L

(6.3)

di + E(t). dt

(6.4)

The motor torque can be obtained by Newton’s law as Tem − TL = J

d2 θ dθ dω +B =J + Bω. 2 dt dt dt

(6.5)

Using Laplace transformation for Eqs. (6.3), (6.4), and (6.5), we get Ea (s) = kb ω(s),

(6.6a)

V (s) = (R + Ls)I (s) + Ea (s),

(6.6b)

Tem (s) − TL (s) = (B + Js)ω(s).

(6.6c)

Using the previous equations, Fig. 6.2 shows the DC motor function block diagram. If TL = 0, then Eq. (6.6c) becomes Tem (s) = (B + Js)ω(s).

(6.7)

Also, the transfer function can be obtained as C (s) =

KT (R + Ls)(B + Js) + KT kb

.

(6.8)

Using the voltage and armature current given by Eq. (6.9), the DC motor observer establishes that the state variable (in our case, armature current)

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Figure 6.2 Schematic block of the DC motor.

can be constructed: R KT V (t) di(t) = − i(t) − ω(t) + . dt L L L

(6.9)

According to Eq. (6.9), a constructed adaptive observer that estimates the speed is given by R KT V (t) dˆi(t) = − ˆi(t) − ω( ˆ t) + G[ˆi(t) − i(t)] + , dt L L L

(6.10a)

ω( ˆ t) = (kp + ki

(6.10b)

dt)e(t),

where kp > 0, kI > 0, and G is the adaptive observer feedback gain. According to Eqs. (6.9) and (6.10a), the error adaptive observer derivative is given by R KT de(t) d ˆ = [i(t) − i(t)] = − [ˆi(t) − i(t)] − [ω( ˆ t) − ω(t)] + G[ˆi(t) − i(t)]. dt dt L L (6.11) The derivative of the output error equation induced by the speed estimation error is given by E(s) = Iˆ (s) − I (s) =

KT L

[ω( ˆ s) − ω(s)]

s + RL − G

= P (s)[ω( ˆ s) − ω(s)],

(6.12)

with P (s) =

s+

KT L . R L −G

(6.13)

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6.3. Stability of speed estimation The functional diagram given in Fig. 6.3 is obtained from Eqs. (6.11) and (6.12).

Figure 6.3 Speed estimation error.

The basic diagram used in the stability analysis is the result of the diagram in Fig. 6.3, which can be described by the open-loop transfer functions. Using the Laplace transform, we get

P (s) kp + ksi

K

T ) ω( ˆ s) L (kp s + ki = = K k ω(s) 1 + P (s) k + ki s2 + RL + TL p − G s + p s

KT k p L

.

(6.14)

The following equation gives the stable pole condition for the adaptive observer to analyze the stability of the speed estimated with the Routh– Hurwitz criterion: R KT k p . + (6.15) G< L L

6.4. Proposed speed controller 6.4.1 Literature review The development of fractional-order calculus is based on the generalization of the integration and differentiation operators into a single fundamental operator Dαt , where α is the order of the operator, which is a noninteger

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number [10,52]. The differential integration operator is defined by

Dt = α

⎧ ⎪ ⎨

dα dt

1

⎪ ⎩ Iα t

for α > 0 for α = 0 . for α < 0

(6.16)

The operators Dαt and Iαt represent the operators of noninteger-order derivation and integration, respectively. Fractional integration and derivation are the generalization of integration and the integer classical derivation to any irrational or complex noninteger orders. Indeed, in 1947, Riemann proposed to replace the factorial function by the Gamma function, to a real noninteger α ∈ R∗+ . In addition, the generalization of integer derivation to noninteger orders has given rise to several definitions, which are not all equivalent. In the discrete case, as in the continuous case, there are different definitions of the fractional differentiation operator. The most used are those of Riemann–Liouville, Caputo, and Grunwald–Letnikov (GL) [50], which are represented in what follows.

6.4.1.1 Riemann–Liouville fractional difference The function f has a real value defined on Na, with a ∈ R. The difference in the sense of Riemann–Liouville is given [60] for 0 < α < 1 and f : Na → R by the following equation: 1−α) f )(t), (α αh f )(t) = h (−( h

(6.17)

t ∈ Na + (1 − α), with Na = {a, a + 1, . . .}.

6.4.1.2 Caputo fractional difference Caputo [1] defined fractional-order differences as follows. For 0 < α < 1 and f : Na → R, the Caputo type difference is defined as follows: 1−α) (α αC f )(t) = h (−( f )(t) = h

1

t−( 1−α)

(1 − α)

s=α

(t − s − 1)(f )(s), (6.18)

where f (s) = f (s + 1) − f (s) is the classical difference, t ∈ Na + (1 − α), with Na = {a, a + 1, . . .}, and is Euler’s Gamma function.

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6.4.1.3 Grunwald–Letnikov fractional difference The definition proposed by GL consists of the generalization of the definition of the derivative integer of a time function f with real-order derivatives α ∈ R∗+ . Consider the sampling time t = kh for k = 0, 1, 2, . . . . The sampling period which we assume, h, without loss of generality, in the following equals 1 (h = 1). The difference of finite order 1 is given as follows: 1 f (k + 1) = f (k + 1) − f (k).

(6.19)

Eq. (6.19) also represents Euler’s discrete approximation of the integer derivative df (t)/t; the generalization to one order α ∈ R∗+ is formulated as follows [20,50]: k 1 j α f (kh) = α (−1) f ((k − j)h), h j=0 j α

(6.20)

where is the discrete fractional derivative operator with an initial time equal to zero and αis a fractional value with α ∈ R∗+ . The notation αj indicates the binomial of Newton defined by α

j

=

1,

α(α−1)...(α−j+1)

j!

,

for j = 0, for j > 0.

(6.21)

Note that for the integer derivative, the derivative of order 1, apart from j = 0 and j = 1, the weighting coefficients (−1)j αj are zero, thus implying a local characterization of the function. On the other hand, for noninteger orders, these weighting coefficients do not cancel each other out. Moreover, the noninteger derivative at each instant is a linear combination of all the values of the function f ((k − j)h), j = 0, 1, . . . k. Indeed, the fractional derivative of a function at a time t takes into account the values of the function at all past times. This shows that unlike the integer derivative, the noninteger derivative gives a global characterization of the function. Therefore, nonwhole systems are often thought of as long-memory systems.

6.4.2 Fractional PID controller In [53], Podlubny introduced the notion of the fractional PID controller. This simplified controller is referred to as a PIλDμ regulator, which has

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an order λ integrator and an order μ differentiator. In the same paper, Podlubny showed that when used in a control loop with a fractionalorder warehouse, the fractional-order controller has a stronger response than integer-order controllers. It has been verified in more recent research [15,47] that the fractional controller is superior to the full PID controller order. The PIλDμ controller control operation can be expressed as follows: u(t) = kp e(t) + ki D−λ e(t) + kd Dμ e(t).

(6.22)

The error signal is where e(t) is. The following equation is obtained after applying the Laplace transform to (6.22) assuming zero initial conditions: CPID (s) = kp +

ki + kd sμ . s

(6.23)

If we assume that λ = μ = 1, the classical integer-order PID controller is obtained. With more freedom in controller tuning, the four-point PID diagram can now be seen as a PID controller plane, which is shown in Fig. 6.4.

Figure 6.4 PID controller plane.

6.4.3 Fractional-order PI controller I is a fractional order operator in FOPI controllers. Thus, one more degree of independence needs to be applied to variables kp and ki by regulating the fractional integration operator. This variable, λ, is the fractional integrator. The added degree-of-freedom makes the FOPI controller better than the classical PI controller. In the time domain, the signal control of FOPI can

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be described as u(t) = kp e(t) + ki D−λ e(t),

(6.24)

where D is an operator of differentials. It is possible to write the transfer function as ki (6.25) C (s) = kp + λ . s Fig. 6.5 indicates the block diagram of the fractional-order controller.

Figure 6.5 Fractional-order PI controller.

6.4.4 Self-tuning PI fractional-order controller with fuzzy logic Fuzzy logic regulation provides a formal theory of control and heuristic learning on how to control a device. This is also a beneficial approach to designing nonlinear controls by using heuristic data as a matter of fact. The advantages of fuzzy and PI controllers can then be merged into a controller to achieve high control efficiency. The proposed adaptive fuzzyPI controller is used to modify the PI controller parameters by integrating a fuzzy inference system with a PI controller based on a rotational speed error of the DC motor and an error adjustment using fuzzy logic. The speed error and its variance of the DC motor are the input of the fuzzy model; the outputs of this controller are the two parameters Kp and Ki of the fractional PI controller. The proposed fuzzy logic PI controller relies on the fuzzy logic concept of Mamdani [48,59]. Fig. 6.6 indicates the structure for the proposed controller. The fuzzy inference method is used to tune the PI controller based on the monitoring of the error’s nonlinear mapping and its rate of change

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Figure 6.6 Fuzzy FOPI controller structure.

Figure 6.7 Fuzzy error membership functions and change of error.

to the benefits of the PI controller. A few parts are used in the fuzzy-PI controller framework: fuzzification, fuzzy rules, aggregation, and defuzzification. The fuzzy linguistic variables for tracking error e(t) and its time derivative e˙(t) are defined as NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium), and PB (positive big). Five triangular membership functions and two trapezoidal membership functions in the range [−1, 1] represent each input, as shown in Fig. 6.7. According to the Mamdani rule as shown in Fig. 6.8, each of the outputs of this controller is represented by five triangular membership functions and two trapezoids membership functions within the range [0, 5]. Tables 6.1 and 6.2 outline the fuzzy management laws, which are formulated between the seven linguistic values of error and error change.

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Figure 6.8 Fuzzy output membership functions of gains Kp and Ki . Table 6.1 Fuzzy rules for Kp . Kp E NB NM

NS

ZE

PS

PM

PB

PB PB PM PM PS ZE ZE

PM PM PM PS ZE NS NM

PM PS PM ZE NS NM NM

PS PS ZE NS NS NM NM

ZE ZE ZE NS NM NM NB

ZE ZE ZE NM NM NB NB

Table 6.2 Fuzzy rules for Ki . Ki E NB

NM

NS

ZE

PS

PM

PB

NB NM NS ZE PS PM PB

NB NB NM NM NS ZE ZE

NM NM NM NS ZE PS PS

NM NS NS ZE PS PS PM

NS NS ZE

ZE ZE PS PM PM PB PB

ZE ZE PS PM PB PB PB

E

E

NB NM NS ZE PS PM PB

PB PB PM PM PS PS ZE

NB NB NB NM NM ZE ZE

PS PM PM

The fuzzy rule surface of the self-tuning fuzzy fractional-order PI controller is represented in Fig. 6.9. The effect shows the relationship between Kp and Ki and the DC motor’s two inputs, error and fractional error change rate. Fig. 6.9 displays the overall block diagram of the proposed controller, where wref and w are the speed relation and calculation, respectively, and Ge and Ge are error fuzzification gains and change of error, respectively.

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Figure 6.9 Fuzzy control surface of self-tuning fuzzy FOPI controller.

The advanced FLC control structure exploits the inputs of error and derivative error and measures the scaling element for proportional and integral terms, and then these values are used to refresh the FOPI controller gain parameters. The final benefit values are then calculated from the following expression for the FOPI controller:

kp = kp + kp , ki = ki + ki ,

(6.26)

where Kp and Ki are the gain value of the FOPI controller and the scaling variables calculated from FLC. Fig. 6.6 displays the frame of a standard fuzzy FOPI control system. For input and output fuzzy sets, the triangle mem-

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bership functions are used and the Mamdani type fuzzy inference method is implemented.

6.5. Results and discussion The DC motor parameters used in this study are as given in Table 6.3. Table 6.3 The DC motor parameters. Parameter Value

Voltage armature: V Voltage field: Vf Armature resistance: R Field resistance: Rf Armature inductance: L Field inductance: Lf Back Emf constant: kb Viscous friction coefficient: B Total inertia: J

300 V 110 V 1.35 65.15 0.0059 H 8.359 H 1.41 V/rd/S 0.0045 N/m/s 0.036 K/m2

To show the validity and effectiveness of the proposed method, several tests are considered.

6.5.1 Test 1 The reference speed is set at wref = 1500 rpm. Fig. 6.10 displays the results of the motor speed and the estimation speed according to Fig. 6.10. The measured rotation speed and that estimated are identical thanks to the good regulation provided by the proposed controller as well as the good observation of the estimated speed, which shows the efficiency of the studied observer. Fig. 6.11 depicts the error between the measured speed and the estimated speed, which is small (error = 1,7466 rpm), which shows that the observer is efficient in static conditions.

6.5.2 Test 2 The reference speed varies as follows: from 0 s to 3 s, wref = 1000 rpm; from 3 s to 6 s, w ref = 1225 rpm; from 6 s to 9 s, wref = 1500 rpm; from 9 s to 12 s, w ref = 1225 rpm; from 12 s to 15 s, wref = 1000 rpm.

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Figure 6.10 DC motor speed with speed reference (wref = 1500 rpm).

Figure 6.11 Error between measured speed and estimated speed with speed reference (wref = 1500 rpm).

In Fig. 6.12, one can see that despite the changes in the rotation speed of the DC motor the estimated speed follows its imposed reference. Fig. 6.13 shows that the proposed observer is robust and efficient despite the sudden change in reference speed.

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Figure 6.12 Motor velocity with its reference.

Figure 6.13 Error between measured speed and estimated speed with speed variation.

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6.5.3 Test 3 The load torque changes as follows: from t = 0 s to 3 s, TL = 5 Nm; from t = 3 s to 6 s, TL = 10 Nm; from t = 6 s to 9 s, TL = 15 Nm; from t = 9 s to 12 s, TL = 10 Nm; from t = 12 s to 15 s, TL = 5 Nm. Fig. 6.14 shows the corresponding results where the DC motor has experienced variations in the load torque (TL) that occur over a different time period. It can be observed from Figs. 6.14 and 6.15 that, considering the difference in the load torque with a brief transient period, the DC motor’s rotation speed matches its relation. For both the suggested selfadjusting FOPI and the observer, when the disturbance is added as well as the stabilization time, the rise time is decreased.

Figure 6.14 Speed of the motor with load torque variance.

6.5.4 Test 4 In this test, a comparison is carried out of the performance between two classic PI regulators and that proposed in this study, with the reference speed of the motor changing according to the profile in the second test. In order to demonstrate the effectiveness and validity of the proposed system relative to the classic approach, Fig. 6.16 offers a distinction between the two types of regulators. The error between the observed speed and that

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Figure 6.15 Error between measured speed and estimated speed with load variation.

Figure 6.16 Comparison between two regulators with the change of rotation speed.

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Figure 6.17 Error between speed estimation and measured speed using two controllers.

determined is negligible relative to the PI controller from the point of view of speed and accuracy, as seen in Fig. 6.17, as can be shown by using the proposed FOPI regulator.

6.6. Conclusions In this chapter, we proposed an adaptive observer to estimate the rotor speed of a separately excited DC motor without speed sensor. We analyzed the stability of the estimated speed by the mean of the Routh–Hurwitz criterion. The DC motor speed is controlled by a hybrid controller which combines fuzzy logic and fractional PI controllers. In the proposed control method, the fuzzy rules are adjusted online to optimize the gains of the fractional-order controller PI as a function of the error and its variation between the reference rotation speed and its measurement. The obtained results demonstrate that the overshoot, the peak rise, and the stabilization time of the proposed controller to integrate with the speed observer are lower than those of the conventional PI controller. In future research work, we aim to use this intelligent controller in the field of renewable energies and to compare this regulator with other conventional regulators (Fuzzy,

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Neurone, Fuzzy-PSO, . . . ) for electric machines and in the design of an adaptation algorithm for the freedom factor of the fractional regulator with fuzzy rules.

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[45] D.P. Iracleous, A.T. Alexandridis, Fuzzy tuned PI controllers for series connected DC motor drives, 1995 Proceedings of the IEEE International Symposium on Industrial Electronics, vol. 2, IEEE, 1995, pp. 495–499. [46] N.G. Jabson, K.G.B. Leong, S.W. Licarte, G.M.S. Oblepias, E.M.J. Palomado, E.P. Dadios, The autonomous golf playing micro robot: with global vision and fuzzy logic controller, International Journal on Smart Sensing and Intelligent Systems 1 (4) (2017). [47] Y. Luo, Y. Chen, Fractional-order [proportional derivative] controller for robust motion control: tuning procedure and validation, in: 2009 American Control Conference, IEEE, June 2009, pp. 1412–1417. [48] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Human-Computer Studies 51 (2) (1999) 135–147. [49] S. Mehta, J. Chiasson, Nonlinear control of a series DC motor: theory and experiment, IEEE Transactions on Industrial Electronics 45 (1) (1998) 134–141. [50] D. Mozyrska, E. Girejko, M. Wyrwas, Comparison of h-difference fractional operators, in: Advances in the Theory and Applications of Non-Integer Order Systems, Springer, Heidelberg, 2013, pp. 191–197. [51] R. Nagarajan, A. Sathishkumar, S. Deepika, G. Keerthana, J.K. Kiruthika, R. Nandhini, Implementation of chopper fed speed control of separately excited DC motor using PI controller, International Journal of Engineering and Computer Science 6 (3) (2017) 20629–20633. [52] I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science & Business Media, 2011. [53] I. Podlubny, Fractional-order systems and PIλDμ controllers, IEEE Transactions on Automatic Control 44 (1) (1999). [54] D. Portugal, G. Cabrita, B.D. Gouveia, D.C. Santos, J.A. Prado, An autonomous all terrain robotic system for field demining missions, Robotics and Autonomous Systems 70 (C) (2015) 126–144. [55] J. Santana, J.L. Naredo, F. Sandoval, I. Grout, O.J. Argueta, Simulation and construction of a speed control for a DC series motor, Mechatronics 12 (9–10) (2002) 1145–1156. [56] C. Semini, HyQ-design and development of a hydraulically actuated quadruped robot, Doctor of Philosophy (Ph.D.) University of Genoa, Italy, 2010. [57] P.C. Sen, Electric motor drives and control-past, present, and future, IEEE Transactions on Industrial Electronics 37 (6) (1990) 562–575. [58] I.I. Siller-Alcalá, J.U. Liceaga-Castro, R. Alcántara-Ramırez, J. Jaimes-Ponce, Speed nonlinear predictive control of a series dc motor for bidirectional operation, Recent Researches in Mathematical Methods in Electrical Engineering and Computer Science 30 (2011) 182–187. [59] A. Tepljakov, Fractional-Order Modeling and Control of Dynamic Systems, Springer, Berlin, Germany, 2017. [60] G.C. Wu, D. Baleanu, W.H. Luo, Lyapunov functions for Riemann–Liouville-like fractional difference equations, Applied Mathematics and Computation 314 (2017) 228–236. [61] C. Zhang, W. Tang, C. Han, F. Fan, Z.L. Wang, Theoretical comparison, equivalent transformation, and conjunction operations of electromagnetic induction generator and triboelectric nanogenerator for harvesting mechanical energy, Advanced Materials 26 (22) (2014) 3580–3591. [62] D. Zhao, N. Zhang, An improved nonlinear speed controller for series DC motors, IFAC Proceedings Volumes 41 (2) (2008) 11047–11052.

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CHAPTER SEVEN

Chaos control and fractional inverse matrix projective difference synchronization on parallel chaotic systems with application Pushali Trikha, Lone Seth Jahanzaib, and Ayub Khan Jamia Millia Islamia, Department of Mathematics, New Delhi, Delhi, India

Chapter points •

A novel synchronization technique, fractional inverse matrix projective diﬀerence synchronization, is introduced.

•

The basic dynamics of the parallel fractional-order chaotic systems are examined using Lyapunov exponents, the Kaplan–Yorke (KY) dimension, and phase diagrams.

•

The results of achieved synchronization are compared with some other published synchronization results.

•

The eﬀect of the fractional order on the synchronization error convergence rates and parallel phase diagrams is discussed.

•

The application of the achieved synchronization in secure communication is illustrated.

•

Assuming uncertainties and disturbances chaos is contained using an adaptive sliding mode control technique about two of the system’s stagnation points.

7.1. Introduction Chaos, which has been studied extensively in the recent past, is found in most nonlinear systems. In its nascent stage, chaos was just considered as a hindrance to the performance of physical systems. Chaos was first observed by Henri Poincare while searching for a solution to the three-body problem. The weather system was the first chaotic system to be modeled by Lorenz [1] in 1963. Many theories and tools were developed to study this chaotic behavior, such as the study of symbolic dynamics, bifurcation analysis, their parameter values, their rate of divergence from nearby initial Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00012-3 All rights reserved.

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points, the intersection of orbits with subspace, the system properties along solutions that do not change with time, i.e., about their equilibrium points, the entropy character, and their hidden or self-excited attractors [2]. Much later it was realized that the chaotic properties of such systems, i.e., high sensitivity to initial conditions, continuous bandwidth spectrum, unpredictability, and nonperiodic and pseudorandom behavior, can be exploited. The chaotic nature of these systems has found applications in cryptography, signal processing, secure communication [3,4], image encryption [5–8], and various other fields. During the last decade many studies have been carried out on fractional calculus, such as fractional differential and difference equations. Their numerical solutions have been explored and analyzed. Also fractional calculus has been used to describe various processes like mechanical systems, robotics, signal processing, thermal systems [9,10], geophysical systems, and many more. While modeling real-life systems too, fractional calculus has shown tremendous success due to their flexibility and hence better memory properties. These properties have been best utilized to model biological models [11–14] such as brain activity, the heart rhythm, and pumping of the lungs. Though many standard chaotic systems exist such as the Lorenz system, the Chen system, and the Lu system, novel nonstandard chaotic systems of integer and fractional order are also being constructed by researchers all over the world and made applicable in areas of secure communication, image and voice encryption, etc. The novel chaotic systems are constructed either by modifying the standard systems, for example by adding or deleting terms, or by adding exponential, sine, or cosine terms. These nonstandard chaotic systems [15–17] have an edge over the standard chaotic systems as these can not be guessed by the intruders. Therefore hacking of the secret message in fields of secure communication is not possible. The chaotic systems show mesmerizing properties in spite of their simple structures. The different shapes of the chaotic attractors are also a characteristic of such systems. These systems are also explored for their stagnation points. The number and stability of stagnation points is also of significance in such systems; for example, Rucklidge [18] introduced the two-wing-shaped attractor with three equilibrium points, Letellier et al. [19] introduced the cord-shaped attractor with one equilibrium point, Sen et al. [20] introduced the three-winged attractor with five equilibrium points, Wang et al. [21] introduced the sphere-shaped attractor with one equilibrium point, Yu et al. [22] introduced a multifolded torus attractor with three equilibrium points, Matsumoto [23] introduced a double-folded torus with one equilibrium point, etc.

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Chaos control and synchronization [24] have become a hot topic of research ever since Pecora and Carroll (1990) achieved success in synchronizing two chaotic systems by developing suitable controllers. Chaotic systems that may seem only slightly different from each other are capable of showing entirely different dynamics. In such a situation, making one system follow the path of other asymptotically involves designing suitable controllers, choosing carefully the desired synchronization type using some control methods. The positive Lyapunov exponent in the spectrum of such systems stops the chaotic systems from synchronizing with one another. It is beneficial to the efficacy of the designed controller that two chaotic systems are synchronized. Nowadays more than two chaotic systems are also synchronized by designing novel synchronization schemes. Some popular synchronization schemes are difference synchronization [25,26], dislocated synchronization, projective synchronization, lag synchronization, etc. [27–29]. Complete synchronization [31] was introduced in 1990 by Pecora and Carroll, generalized synchronization [32] by Rulkov, Sushchik, and Tsimring (1995), phase synchronization [33] (1996) and lag synchronization [34] (1997) by Rosenblum, Pikovsky, and Kurths, and projective synchronization [35] (1999) by Mainieri and Rehacek. Sun et al. introduced compound synchronization [36] in 2013 and Dongmo et al. introduced difference synchronization [37] in 2018. Various control methods have been designed for synchronization and chaos control of chaotic systems. A few of them are sliding mode control, adaptive control, tracking control, active control [30], and optimal control. However the active control method is the simplest to apply, sliding mode control is the most robust method for synchronization and control of chaos. The chattering effect that occurs in sliding mode control is considered as a disadvantage of the method, so a number of methods have been designed to reduce the chattering effect. Motivated by the above discussion this chapter introduces a novel synchronization technique, “fractional inverse matrix projective difference synchronization.” In this technique the difference synchronization technique cooperates with the fractional inverse matrix projective synchronization. Though difference synchronization can be thought of as a suitable replacement of the combination synchronization technique, it increases the diversity in the list of possible synchronization methods. The synchronization technique is applied to the parallel systems of the fractional-order chaotic system. The dynamics of parallel chaotic systems such as Lyapunov

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exponents [38] and the KY dimension are discussed. The effect of the fractional order on phase diagrams and the synchronization error convergence rate is studied. Results of the achieved synchronization are compared with some previously published literature and are illustrated as an example in the field of secure communication. The chaos present in the chaotic system is also stabilized about its two stagnation points by designing controllers using an adaptive sliding mode control technique. The external disturbances and uncertainties have been considered while controlling chaos. The efficacy of the designed controllers is also verified based on the results of simulations performed using MATLAB® .

7.2. Preliminaries 7.2.1 Deﬁnition Being the generalization of integer calculus, fractional calculus with its better memory properties is being used to model various real-life systems. Though a number of fractional derivatives exist, such as Riemann– Liouville’s derivative, Grunwald–Letnikov’s derivative, and Caputo’s derivative, we have used Caputo’s definition [39]: Q a Dx g(x) =

1 (n − Q)

a

x

g(n) (τ )dτ , (x − τ )Q−n+1

(7.1)

where n is an integer, Q is a real number, (n − 1) ≤ Q < n, and (.) is the Gamma function. Recently the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative have also been defined.

7.2.2 Stability criterion The dynamical system Q a D t Zi

= Hi (Z1 , Z2 , . . . , Zn ), 0 < Q < 1, i = 1, 2, . . . , n,

(7.2)

is asymptotically stable [40] at its stagnation point if the eigenvalues of the Jacobi matrix J = ∂∂ZHi fulfill the following criterion: |arg(eigenvalues)| >

Qπ , 2

where Zi are state variables and Q is the fractional order.

(7.3)

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7.3. The fractional inverse matrix projective difference synchronization 7.3.1 Problem formulation Consider the following two n-dimensional fractional-order chaotic master systems: Master System I, DQ U = B1 U + G1 (U ); dtQ

(7.4)

DQ V = B2 V + G2 (V ). dtQ

(7.5)

Master System II,

Here B1 , B2 ∈ Rn×n are the matrices corresponding to the linear part of the chaotic systems, G1 (U ), G2 (V ) are the matrices corresponding to the nonlinear part of the systems, and U , V ∈ Rn are the state variables. Consider the following slave system: Slave System, DQ W ¯. = B3 W + G3 (W ) + U dtQ

(7.6)

Here B3 ∈ Rn×n is the matrix corresponding to the linear part of the slave system, G3 (W ) is the matrix corresponding to the nonlinear part of the slave system, and W ∈ Rn is the state variable of the slave system with U¯ ∈ Rn being the controller to ensure synchronization of the systems. Define the fractional inverse matrix projective difference synchronization error as E = (U − V ) − MW .

(7.7)

Then master systems (7.4) and (7.5) are in fractional inverse matrix projective difference synchronization with slave system (7.6) if lim ||E|| = lim ||(U − V ) − MW || = 0,

t→∞

t→∞

M being the projective matrix with the ||.|| Euclidean norm.

(7.8)

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Theorem 7.1. Master systems (7.4) and (7.5) achieve fractional inverse matrix projective difference synchronization with slave system (7.6) provided the controller U¯ is chosen as U¯ = M −1 [(B1 − B2 + K )E + B1 U + G1 − B2 V − G2 ] − B3 W − G3 . (7.9) Proof. The fractional inverse matrix projective difference synchronization error is given as E = (U − V ) − MW .

(7.10)

Differentiating the above, the error dynamical system is obtained as DQ E = (DQ U − DQ V ) − MDQ W .

(7.11)

From (7.4)–(7.6), we have DQ E = (B1 U + G1 (U ) − A2 V − G2 (V )) − M (B3 W + G3 (W ) + U¯ ). (7.12) Substituting the controllers, we get DQ E = (B1 − B2 + K )E.

(7.13)

Suitably choosing gain matrix K, by the stability lemma in Section 7.2 the synchronization is attained.

7.3.2 System description We first describe the considered master and slave systems. The phase diagrams of the master and slave systems are given in Fig. 7.1. The considered systems are parallel to each other in the sense that their phase diagrams are parallel to one another. This is shown in Fig. 7.2. Master System I: D Q U1 = AU1 − U2 U3 , dtQ d Q U2 = −BU2 + U1 U3 , dtQ D Q U3 = C − U3 + U1 U2 + DU2 U3 , dtQ

(7.14)

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Figure 7.1 Phase diagram of (a) Master System I, (b) Master System II, and (c) the Slave System.

where A, B, C, D are parameters describing the system. For A = 0.7, B = 0.1, C = 0.001, D = 0.1 and initial values (0.1, 0.2, 0.3) and Q = 0.95 the system is chaotic.

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Figure 7.2 Parallel phase diagrams of the master and slave systems.

Master System II: We have D Q V1 = AV1 − 2V2 V3 , dtQ D Q V2 = −BV2 + 2V1 V3 , dtQ D Q V3 = C − V3 + 2V1 V2 + 2DV2 V3 , dtQ

(7.15)

where A, B, C, D are parameters describing the system. For A = 0.7, B = 0.1, C = 0.001, D = .1 and initial values (0.2, 0.4, 0.6) and Q = 0.95 the system is chaotic. Slave System: We have DQ W1 = AW1 − 3W2 W3 , dtQ DQ W2 = −BW2 + 3W1 W3 , dtQ DQ W3 = C − W3 + 3W1 W2 + 3DW2 W3 , dtQ

(7.16)

where A, B, C, D are parameters describing the system. For A = 0.7, B = 0.1, C = 0.001, D = 0.1 and initial values (0.3, 0.6, 0.9) and Q = 0.95 the system is chaotic. The positive Lyapunov spectrum component which gives the rate of separation of infinitesimally close lying trajectories is given in Table 7.1.

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Table 7.1 Maximal Lyapunov exponent (LE) and chaotic Kaplan–Yorke (KY) dimension for master and slave systems. System Maximal LE KY dimension

Master System I Master System II Slave System

0.1159 0.1134 0.1073

2.154436 2.15 2.125813

The positive maximal positive component value confirms that the considered system is chaotic. The KY dimension, which gives the dimension of the phase attractor using the Lyapunov spectrum component values, is also given in the second column of Table 7.1. The KY dimension for the chaotic phase diagram is a fractional value, i.e., the chaotic phase attractor has a fractional dimension.

7.3.3 Simulations and discussions As in Section 7.3, for the above described systems we have the following: 0.7 0 0 0.7 0 0 0.7 0 0 B1 = 0 −0.1 0 , B2 = 0 −0.1 0 , B3 = 0 −0.1 0 , −1 −1 −1 0 0 0 0 0 0 ⎡ ⎢

G1 (U ) = ⎣ ⎡ ⎢

G2 (V ) = ⎣ ⎡ ⎢

G3 (W ) = ⎣

−U 2 U 3

⎤ ⎥

U1 U3 ⎦, 0.001 + U1 U2 + 0.1U2 U3 −2V2 V3

⎤ ⎥

2V1 V3 ⎦, 0.001 + 2V1 V2 + 0.2V2 V3 −3W2 W3

⎤ ⎥

3W1 W3 ⎦. 0.001 + 3W1 W2 + 0.3W2 W3

The controller to be designed according to Theorem 7.1 is ⎡ ⎤

u1

⎢u ⎥ ⎢ 2⎥ ⎢ ⎥ U¯ = ⎢u3 ⎥ . ⎢ ⎥ ⎣u4 ⎦

u5

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We choose (M), the projective matrix, as ⎡

⎤

2 1 0 ⎢ ⎥ M = ⎣0 1 1⎦ . 0 0 1

(7.17)

Therefore we get the error function as E1 = U1 − V1 − 2W1 − W2 , E2 = U2 − V2 − W2 − W3 ,

(7.18)

E3 = U3 − V3 − W3 . On differentiating the above, the error dynamical system is obtained as DQ E1 = DQ U1 − DQ V1 − 2DQ W1 − DQ W2 , DQ E2 = DQ U2 − DQ V2 − DQ W2 − DQ W3 ,

(7.19)

D E3 = D U3 − D V3 − D W3 . Q

Q

Q

Q

From (7.14)–(7.16) the controllers are designed as in Theorem 7.1, and choosing gain matrix K as ⎡ ⎢

−1

K =⎣ 0 0

⎤

0 0 ⎥ −2 0 ⎦ , 0 −3

the simplified error dynamical system is obtained as DQ E1 = −E1 , DQ E2 = −2E2 ,

(7.20)

D E3 = −3E3 . Q

The stability criterion given in Section 7.2 is satisfied by the eigenvalues of the Jacobian matrix of the above error system, implying stability of the system about its stagnation point. Also, the inverse matrix projective difference synchronization is obtained between the master and slave systems. The syn-

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Figure 7.3 The fractional inverse matrix projective difference synchronized trajectories.

chronized trajectories are given in Fig. 7.3. The inverse matrix projective difference synchronization error converging to zero is shown in Fig. 7.4 for initial conditions for the error dynamical system as (−1.3, −1.7, −1.2).

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Figure 7.4 The fractional inverse matrix projective difference synchronization error plot.

7.3.4 Comparison with published literature In [41] the authors have investigated fractional matrix and inverse matrix projective synchronization between two chaotic systems. In the case of fractional inverse matrix synchronization they have obtained a synchronization error approaching zero at t = 0.1, 1, 1.5, and 5 units. In [42] the authors have performed inverse matrix projective combination synchronization, where synchronization is achieved at 1, 2, 2, and 5 units. In this chapter we have considered three chaotic systems for attaining fractional inverse matrix projective difference synchronization. Amid such complex systems we have obtained synchronization errors approaching zero at t = 2, 2, and 4.5 units. This shows that the designed controllers are efficient in the synchronization process. Fig. 7.5 shows the error convergence rates for different fractional orders. Clearly from the figure the effect of the fractional order on the error convergence rate is not very significant. The parallel phase diagrams for different fractional orders are also given in Fig. 7.6.

7.3.5 Chaos control about the stagnation points in the presence of uncertainties and disturbances The stagnation points of the system are determined by equating the following equations to 0: AU1 − U2 U3 = 0, −BU2 + U1 U3 = 0,

C − U3 + U1 U2 + DU2 U3 = 0.

(7.21)

Chaos control and fractional inverse matrix projective diﬀerence synchronization

Figure 7.5 Error convergence rates for Q (a) 0.9, (b) 0.93, (c) 0.95, or (d) 0.97.

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Figure 7.6 Parallel phase diagrams for Q (a) 0.93, (b) 0.95, (c) 0.97, and (d) 0.99.

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For A = 0.7, B = 0.1, C = 0.001, D = 0.1 and initial values (0.1, 0.2, 0.3) and Q = 0.95 we obtain the following stagnation points: E1 = (0.33033, −0.87397, −0.264575), E2 = (−0.303872, 0.80397, −0.264575), E3 = (0.302678, 0.800811, 0.264575),

(7.22)

E4 = (−0.329135, −0.870811, 0.264575), E5 = (0, 0, 0.001). In this section we will control chaos about two stagnation points E2 and E5 . Chaos in fractional chaotic systems exposed to uncertainties and disturbances is contained using an adaptive sliding mode control technique. We design controllers to approach stagnation points (p1 , p2 , p3 ). The disturbed fractional chaotic system assuming uncertainties and external disturbances is D Q U1 = 0.7U1 − U2 U3 + H1 + D1 + v1 , dtQ d Q U2 = −0.1U2 + U1 U3 + H2 + D2 + v2 , dtQ D Q U3 = 0.001 − U3 + U1 U2 + 0.1U2 U3 + H3 + D3 + v3 , dtQ

(7.23)

where Hi and Di are system uncertainties and external disturbances and vi are control laws to be designed about equilibrium point. Let | Hi | < Ci and Di < Fi , where Ci and Fi are unknown positive values, and Cˆ i and Fˆi are estimates of Ci and Fi , respectively. The state trajectories and phase plots of the disturbed system are displayed in Figs. 7.7 and 7.8, respectively. Define the control error about the fixed point (p1 , p2 , p3 ) as e1 = U1 − p1 , e2 = U2 − p2 , e3 = U3 − p3 .

(7.24)

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Figure 7.7 State trajectories of disturbed system (7.23).

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Figure 7.8 Phase portraits of disturbed system (7.23).

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Taking Caputo’s derivative of (7.24) we get the error dynamical system as DQ e1 = 0.7(e1 + p1 ) − (e2 + p2 )(e3 + p3 ) + H1 + D1 + v1 , DQ e2 = −0.1(e2 + p2 ) + (e1 + a)(e3 + c ) + H2 + D2 + v2 , DQ e3 = 0.001 − (e3 + p3 ) + (e1 + a)(e2 + b) + 0.1(e2 + b)(e3 + c )

(7.25)

+ H3 + D3 + v3 .

The sliding surface is considered as si (t) = D

Q−1

ei (t) + λi

t

ei (ξ )dξ.

(7.26)

0

To have (7.25) slide, the following is necessary: si (t) = 0, s˙i (t) = 0.

(7.27)

s˙i (t) = DQ ei (t) + λi ei (t), i = 1, 2, 3.

(7.28)

Differentiating (7.26),

Then from (7.27), we have DQ ei (t) = −λi ei (t).

(7.29)

From Matignon’s theorem [43], stability of (7.29) is achieved. Designing controllers based on (7.29), (7.25), and sliding mode control theory, v1 = −0.7(e1 + p1 ) + (e2 + p2 )(e3 + p3 ) − λ1 e1 − (Cˆ1 + Fˆ1 + r1 )sign(s1 ), v2 = 0.1(e2 + p2 ) − (e1 + p1 )(e3 + p3 ) − λ2 e2 − (Cˆ2 + Fˆ2 + r2 )sign(s2 ), v3 = −0.001 + (e3 + p3 ) − (e1 + p1 )(e2 + p2 ) − 0.1(e2 + p2 )(e3 + p3 ) − λ3 e3 − (Cˆ3 + Fˆ3 + r3 )sign(s3 ),

(7.30) where sign(.) (signum function) and ri are positive constants. Parameter laws are C˙ˆ i = ci | si |, E˙ˆ i = fi | si |, where ci , fi > 0 are constants.

(7.31)

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Theorem 7.2. Chaos in the trajectories of fractional chaotic systems assuming uncertainties and disturbances can be contained about the stagnation point (p1 , p2 , p3 ) using control and update laws as in (7.30)–(7.31). Proof. Using Lyapunov’s direct method [44], stability is obtained considering the Lyapunov function as V = V1 + V2 + V3 ,

(7.32)

1 1 ˆ 1 ˆ (C1 − C1 )2 + (F1 − F1 )2 , V1 = s21 + 2 2c1 2f1 1 1 ˆ 1 ˆ V2 = s22 + (C2 − C2 )2 + (F2 − F2 )2 , 2 2c2 2f2 1 1 ˆ 1 ˆ V3 = s23 + (C3 − C3 )2 + (F3 − F3 )2 . 2 2c3 2f3

(7.33)

where

Differentiating (7.33), 1 ˆ 1 ˆ ˙ˆ ˙ˆ , (C1 − C1 )C 1 + (F 1 − F1 )F 1 c1 f1 1 1 V˙ 2 = s2 s˙2 + (Cˆ 2 − C2 )C˙ˆ 2 + (Fˆ 2 − F2 )F˙ˆ 2 , c2 f2 1 1 V˙ 3 = s3 s˙3 + (Cˆ 3 − C3 )C˙ˆ 3 + (Fˆ 3 − F3 )F˙ˆ 3 . c3 f3 V˙ 1 = s1 s˙1 +

(7.34)

From (7.28), we have 1 ˆ 1 ˆ ˙ˆ ˙ˆ , (C1 − C1 )C 1 + (F 1 − F1 )F 1 c1 f1 1 1 V˙ 2 = s2 (DQ e2 + λ2 e2 ) + (Cˆ 2 − C2 )C˙ˆ 2 + (Fˆ 2 − F2 )F˙ˆ 1 , c2 f2 1 1 V˙ 3 = s3 (DQ e3 + λ3 e3 ) + (Cˆ 3 − C3 )C˙ˆ 3 + (Fˆ 3 − F3 )F˙ˆ 3 . c3 f3 V˙ 1 = s1 (DQ e1 + λ1 e1 ) +

(7.35)

˙ˆ ˙ˆ in (7.35), Substituting DQ ei , C i , and F i

V˙ i = s1 [(Hi + Di ) − (Cˆ i + Fˆ i + ri )signsi ] + (Cˆ i − Fi )|si | + (Cˆ i − Fi )|si | ˆ i − Fi )|si | + (C ˆ i − Fi )|si | ≤ (| Hi | + |Di |)|si | + (C ˆ i + Fˆ i + ri )|signsi | + (C ˆ i − Fi )|si | + (C ˆ i − Fi )|si | < (Ci + Fi )|si | − (C = −Ti |si |.

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Finally, we get V˙ =

3

V˙ i

i=1 3 T .

i=1

Then we have

V˙ < −T s21 + s22 + s23

(7.37)

< 0.

From Lyapunov stability theory si → 0 as t → ∞. Hence, asymptomatic stability is achieved. Simulations

For simulations, choose A = 0.7, B = 0.1, C = 0.001, D = 0.1 and initial values (0.1, 0.2, 0.3) and Q = 0.95, H1 = 0, D1 = sin(8t), H2 = 8 sin(U1 ), D2 = 0, H3 = 0, D3 = cos(t). Here v1 , v2 , v3 are controllers to be designed about the stagnation point (p1 , p2 , p3 ), λ1 = 1, λ2 = 2, λ3 = 3, r1 = 1, r2 = 2, r3 = 3 for computations. The controlled trajectories, control errors, and converging sliding surface about the stagnation points E2 = (−3.01662, −2.51385, 15.1667) and E5 = (0, 0, 0.001) with disturbance estimates are shown in Figs. 7.9 and 7.10, respectively.

7.4. Illustration in secure communication The peculiar sensitivity of chaotic systems to initial conditions and parameter values is utilized in the field of secure communication. Until and unless the exact parameters and initial conditions of the chaotic system are known to the intruders, the secret message which is hidden among the chaotic signals cannot be retrieved. Therefore, hackers who try to hack the secret message by approximating systems can never reach out to the secret

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Figure 7.9 (a) Controlled states about E2 = (−0.303872, 0.80397, −0.264575). (b) Error. (c) Surfaces. (d) Disturbance estimates. (e) Disturbance estimates.

message, as even the closest approximated parameter values and initial condition values have entirely different results. The following is an illustration of an application of the novel synchronization technique to chaotic sys-

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Figure 7.10 (a) Controlled states about E5 = (0, 0, 0.001). (b) Error. (c) Surfaces. (d) Disturbance estimates. (e) Disturbance estimates.

tems by designing suitable controllers as above. Here the above performed fractional inverse matrix projective difference synchronization has been utilized.

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Figure 7.11 Illustration in secure communication.

Illustration

Consider the secret message to be transmitted to be p(t) = sin2 (t) cos(t). We mix it with the chaotic signals obtained from the difference of two chaotic master systems. The mixed message is now transmitted as p1 (t). At the receiving end the slave system synchronizes with the difference of the signals from the master systems after applying a suitable controller at the receiving end. The message is recovered as p2 (t). Fig. 7.11 shows the numerical simulation results of the illustration.

7.5. Conclusions In this chapter a novel synchronization technique, fractional inverse matrix projective difference synchronization, has been achieved between parallel fractional-order chaotic systems – two master systems and one slave

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system. The basic dynamics of the parallel systems have been discussed using the Lyapunov spectrum, the KY dimension, phase diagrams, etc. Results of the achieved synchronization are compared with some published results to establish the efficacy of the designed controllers. The effect of the fractional order on the error convergence rates and parallel phase diagrams has been discussed. The application of fractional chaotic systems and the novel synchronization method is illustrated with the help of an example in secure communication. The chaos present in the system is controlled about two stagnation points assuming uncertainties and disturbances using adaptive sliding mode control. The estimation of disturbances and uncertainties is also achieved using this method. The future direction of this work lies in the application of the introduced synchronization technique in image and voice encryption. Also its realization in electronic circuits would be interesting.

References [1] G. Grassi, F.L. Severance, D.A. Miller, Multi-wing hyperchaotic attractors from coupled Lorenz systems, Chaos, Solitons and Fractals 41 (1) (2009) 284–291. [2] V.T. Pham, S. Vaidyanathan, C. Volos, T. Kapitaniak, Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors, vol. 133, Springer, 2018. [3] K. Murali, M. Lakshmanan, Secure communication using a compound signal from generalized synchronizable chaotic systems, Physics Letters A 241 (6) (1998). [4] M. Feki, An adaptive chaos synchronization scheme applied to secure communication, Chaos, Solitons and Fractals 18 (1) (2003) 141–148. [5] M.S. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems, Physics Letters A 367 (1–2) (2007) 102–113. [6] N.K. Pareek, V. Patidar, K.K. Sud, Image encryption using chaotic logistic map, Image and Vision Computing 24 (9) (2006) 926–934. [7] C.K. Volos, I.M. Kyprianidis, I.N. Stouboulos, Image encryption process based on chaotic synchronization phenomena, Signal Processing 93 (5) (2013) 1328–1340. [8] S. Sun, A novel hyperchaotic image encryption scheme based on DNA encoding, pixel-level scrambling and bit-level scrambling, IEEE Photonics Journal 10 (2) (2018) 1–14. [9] D. Baleanu, A. Jajarmi, S.S. Sajjadi, J.H. Asad, The fractional features of a harmonic oscillator with position-dependent mass, Communications in Theoretical Physics 72 (5) (2020), IOP publishing. [10] D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Boundary Value Problems 2020 (1) (2020) 1–16, Springer. [11] M. Javidi, N. Nyamoradi, Dynamic analysis of a fractional order prey-predator interaction with harvesting, Applied Mathematical Modelling 37 (20–21) (2013) 8946–28956. [12] B. Inan, M.S. Osman, T. Ak, D. Baleanu, Analytical and numerical solutions of mathematical biology models: the Newell-Whitehead-Segel and Allen-Cahn equations, Mathematical Methods in the Applied Sciences (2019), Wiley Online Library.

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[13] S.S. Sajjadi, D. Baleanu, A. Jajarmi, H.M. Pirouz, A new adaptive synchronization and hyperchaos control of a biological snap oscillator, Chaos, Solitons and Fractals 138 (2020) 109919, Elsevier. [14] D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative, Chaos, Solitons and Fractals 134 (2020) 109705. [15] A. Khan, S.J. Lone, P. Trikha, Analysis of a novel 3-D fractional order chaotic system, in: ICPECA, IEEE, 2019, pp. 1–6. [16] A. Khan, L.S. Jahanzaib, P. Trikha, Secure communication: using parallel synchronization technique on novel fractional order chaotic system, IFAC-PapersOnLine 53 (1) (2020) 307–312, Elsevier. [17] P. Trikha, L.S. Jahanzaib, Dynamical analysis of a novel 5-d hyper-chaotic system with no equilibrium point and its application in secure communication, Differential Geometry—Dynamical Systems 22 (2020) 269–288. [18] A.M. Rucklidge, Chaos in models of double convection, Journal of Fluid Mechanics 237 (1992) 209–229. [19] C. Letellier, L.A. Aguirre, Required criteria for recognizing new types of chaos: application to the “cord” attractor, Physical Review E 85 (3) (2012) 036204. [20] S. Zhang, Y. Zeng, Z. Li, One to four-wing chaotic attractors coined from a novel 3d fractional-order chaotic system with complex dynamics, Chinese Journal of Physics 56 (3) (2018) 793–806. [21] Z. Wang, Y. Sun, S. Cang, A 3-D spherical attractor, 2011. [22] S. Yu, J. Lu, G. Chen, Multifolded torus chaotic attractors: design and implementation, Chaos: An Interdisciplinary Journal of Nonlinear Science 17 (1) (2007) 013118. [23] L. Chua, M. Komuro, T. Matsumoto, The double scroll family, IEEE Transactions on Circuits and Systems 33 (11) (1986) 1072–1118. [24] A.K. Singh, V.K. Yadav, S. Das, Synchronization between fractional order complex chaotic systems with uncertainty, Optik-International Journal for Light & Electron Optics 133 (2017) 98–107. [25] A. Khan, P. Trikha, Compound difference anti-synchronization between chaotic systems of integer and fractional order, SN Applied Sciences 1 (2019) 757, https:// doi.org/10.1007/s42452-019-0776-x. [26] A. Khan, P. Trikha, Study of earths changing polarity using compound difference synchronization, GEM-International Journal on Geomathematics 11 (1) (2020) 7. [27] A. Khan, L.S. Jahanzaib, T. Khan, P. Trikha, Secure communication: using fractional matrix projective combination synchronization, AIP Conference Proceedings 2253 (2020) 020009. [28] P. Trikha, L.S. Jahanzaib, Secure communication: using double compoundcombination hybrid synchronization, in: Proceedings of International Conference on Artificial Intelligence and Applications, Springer, 2020, pp. 81–91. [29] E.E. Mahmoud, L.S. Jahanzaib, P. Trikha, M.H. Alkinani, Anti-synchronized quadcompound combination among parallel systems of fractional chaotic system with application, Alexandria Engineering Journal (2020), Elsevier. [30] A. Khan, P. Trikha, S.J. Lone, Secure communication: using synchronization on a novel fractional order chaotic system, in: ICPECA, IEEE, 2019, pp. 1–5. [31] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Physical Review Letters 64 (8) (1990) 821. [32] N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, H.D.I. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems, Physical Review E 51 (2) (1995) 980. [33] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators, Physical Review Letters 76 (11) (1996) 1804.

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CHAPTER EIGHT

Aggregation of chaotic signal with proportional fractional derivative execution in communication and circuit simulation Najeeb Alam Khana , Saeed Akbara , Muhammad Ali Qureshib , and Tooba Hameeda a Department b Department

of Mathematics, University of Karachi, Karachi, Pakistan of Physics, University of Karachi, Karachi, Pakistan

Chapter points •

To obtain new chaotic ﬂow using aggregation of two well-known systems.

•

The eﬀects of various physical parameters on the aggregated system are incorporated through graphs and tables.

• •

Digitally, a circuit of the aggregated system is designed using Multisim. Diﬀerent data formats were scrambled using the designed aggregated system.

8.1. Introduction Chaotic systems can be described as complex dynamic systems that depend on system parameters and initial states. Even insignificant changes in these values can lead to a change in the system outcomes. These systems are usually the focus of various studies in the fields of engineering science. The most recent developments have shown that chaotic systems are widely used in electronic engineering and cryptographic applications. Correspondingly, novel chaotic systems are being presented every day, and their broad fields of usage are attractive. Some most familiar dynamical systems having chaotic behavior are the Lorenz system, which was first constructed by American meteorologist Lorenz [1], the attractor, an innovative Rössler system designed by Rössler [2] in 1976, and a chaotic oscillation circuit named Chua Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00013-5 All rights reserved.

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system, proposed in 1986 by Chua and Komuro [3]. During the last two decades many researchers discovered and modified various chaotic systems along with the qualitative properties named fixed point, eigenvalues [4], Lyapunov exponent [5], and Poincaré section [6]. The Lorenz attractor also arises naturally in models of dynamos and lasers, ocean movement, rotation of the Earth, atmospheric rotation, secure communications with security analysis, and electronic realization. The Liu chaotic system and its modified forms are also found in circuit realization and cryptographic implementation. In various fields of real sciences, the fractional-order derivative (FOD) has received more and more attention due to its extraordinary properties and simplicity [7]. From the last four decades, the fractional derivative has been used in mathematical modeling of different types of dynamical phenomena having an asymptotic- or chaotic-based nature. Due to its generality, the fractional-order derivative can be used to describe the complete memorized history, achieving more exact and stronger performance. Fractional derivatives having a distinct nature of kernels are the generalized form of fractional-order differential systems (FODSs) that satisfy ordinary differential equation systems (ODESs) at standard values. Most properties of ODESs cannot be extended to FODSs; therefore, FODS have received more attention [8,9]. Fractional-order derivatives have also been applied on chaotic systems, and the family of attractors are observed with the corresponding properties of occurring fluctuation. Few of them are cited here; multiple scrolls in the fractional-order Chua system [10,11], data enhancement and image encryption using fractional-order chaotic systems and maps [12], chaos and hyperchaos in fractional-order Rössler systems [13], the fractional-order jerk system having double scroll [14], the fractional-order multiwing system having multistability [15], the Lorenz system of fractional order [16], and the realization of a fractional-order Liu system [17,18]. The fluctuated nature of these systems can be tested using the statistical test suite NIST 800-22 [19], and the desired randomness allows us to use the system in required applied fields. Digitalized world users always need some strong secure communication system for their confidential communications in every field [20]. Cryptographic techniques successfully provide security on various types of data format from sender to receiver in public and private sectors [21]. However, invaders always put much effort into apprehensively decrypting the encrypted data between communication networks. In the presence of abstract third parties, data transformation may lead to the destruction of any

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organized firm or educational institute and compromise the state’s documentations, etc. During the last two decades, researchers have shown more interest in chaos-based cryptography due to their high randomness nature [22–24]. The immense progress in mobile communication networks has attracted extra attention to the area of voice encryption to stop attacks by spies [25]. Due to this, a complex-level security will be required to achieve secure data transformation for users having no sense of security. To give a powerful solution for secure communications, chaos-based cryptosystems are used to design the algorithms for different types of data [26–31]. Chaos and chaotic systems are not only used in cryptographic applications, but also in various fields of science and engineering, such as biology, chemistry, the design of pseudorandom number generators (PRNGs), and business [32]. In this chapter, we present an aggregation of two chaotic systems having commensurate-order proportional fractional derivative (PFD). The chaotic behavior of the aggregated system is observed through phase portrait, Lyapunov exponents, and eigenvalues at different fractional values. Circuit simulation is also realized along with passive components like capacitors, resistors, operational amplifiers, and multipliers. Through security software the system is analyzed via multiple factors of data type (format).

8.2. Fractional-order chaotic systems and their properties Consider the following 3D generalized Lorenz system, presented by Grigorenko and Grigorenko [33], having a chaotic nature at sensitive values of system parameters: ⎛

⎞

⎛

⎞⎛

⎞

⎛

ϕ1 −δ δ Dγ ϕ1 0 0 ⎟⎜ ⎜ γ ⎟ ⎜ ⎟ ⎜ ⎝ D ϕ2 ⎠ = ⎝ r −1 0 ⎠ ⎝ ϕ2 ⎠ + ϕ1 ⎝ 0 Dγ ϕ3 ϕ3 0 0 −μ 0

0 0 1

⎞⎛

⎞

ϕ1 0 ⎟⎜ ⎟ −1 ⎠ ⎝ ϕ2 ⎠ , ϕ3 0 (8.1) where ϕ1 , ϕ2 , and ϕ3 are state variables and δ , r, and μ are positive parameters with the values δ = 10, r = 25, μ = 83 and initial state values ϕ1 (0) = 0.1, ϕ2 (0) = 0.1, and ϕ3 (0) = 0.1. Next, consider the following generalized form of the simplest structure but with complex behavior of the Liu system, proposed by Jun-Jie and

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Chong-Xin [18]: ⎛

⎞

⎛

⎞⎛

⎞

⎛

⎞⎛

⎞

−a a 0 ϕ1 ϕ1 Dγ ϕ1 0 0 0 ⎜ γ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ D ϕ2 ⎠ = ⎝ b 0 0 ⎠ ⎝ ϕ2 ⎠ + ϕ1 ⎝ 0 0 −k1 ⎠ ⎝ ϕ2 ⎠ , Dγ ϕ3 ϕ3 h 0 0 ϕ3 0 0 −c (8.2) where ϕ1 , ϕ2 , and ϕ3 are state variables and a, b, c, k, and h are positive parameters with the values a = 10, b = 40, c = 2.5, k = 1, h = 4 with initial state values ϕ1 (0) = 2.2, ϕ2 (0) = 2.4, and ϕ3 (0) = 38. The aggregation of Eq. (8.1) and Eq. (8.2) leads to the following generalized 3D system with two nonlinear terms: ⎛

⎞

⎛

⎞⎛

⎞

−aδ aδ ϕ1 Dγ ϕ1 0 ⎜ γ ⎟ 1⎜ ⎟⎜ ⎟ = −1 0 ⎠ ⎝ ϕ2 ⎠ ⎝ D ϕ2 ⎠ ⎝ br 2 0 0 − cμ Dγ ϕ3 ϕ3 ⎛

⎞⎛

⎞

ϕ1 0 0 0 ⎜ ⎟⎜ ⎟ + ϕ1 ⎝ 0 0 −(k + 1) ⎠ ⎝ ϕ2 ⎠ , h 1 0 ϕ3

(8.3)

where aδ = 20, br = 65, cμ = 31 6 , k = 1, and h = 4 are parameters with initial conditions ϕ1 (0) = 1.15, ϕ2 (0) = 1.25, and ϕ3 (0) = 19.05. Here we implement the following property proportional fractional derivative of the differential operator Dγ ϕ (t) of order γ , proposed by Jarad et al. [34]: Dγ ϕ (t) = η1 (γ , t) ϕ˙ (t) + η0 (γ , t) .

(8.4)

The expression of Eq. (8.4) transforms the fractional-order derivative into the form where the presence of the integer-order derivative minimizes the accuracy of different complex computations. Upon substitution of property (8.4), we transform the aggregated system (8.3) into the following equivalent integer form: ⎛

⎞ ⎛ γ − a2δ − 1 ϕ˙ 1 ⎜ ⎟ 1⎜ br ⎝ ϕ˙ 2 ⎠ = ⎝ 2 γ ϕ˙ 3 0 ⎛

0

ϕ1 ⎜ + ⎝ 0 γ h 2

0 0 1 2

⎞ ϕ1 ⎟⎜ ⎟ γ − 32 ⎠ ⎝ ϕ2 ⎠ cμ ϕ3 0 γ − 2 −1 ⎞⎛ ⎞ 0 ϕ1 ⎟ k+1 ⎟ ⎜ − 2 ⎠ ⎝ ϕ2 ⎠ . ϕ3 0 aδ 2

0 0

⎞⎛

(8.5)

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Table 8.1 Fixed (equilibrium) points and eigenvalues of system (8.5). γ Fixed points Eigenvalues 1.00 O(0, 0, 0) (−23.893, 13.393, −2.58333) (−16.51, 1.71158 ± 9.86069i) P1 (5.75, 5.75, 32) P2 (−5.75, −5.75, 32) (−16.51, 1.71158 ± 9.86069i) (−24.1445, 13.5182, −2.61953) 0.99 O(0, 0, 0) P1 (5.76, 5.76, 31.99) (−16.68, 1.71877 ± 9.96029i) P2 (−5.76, −5.76, 31.99) (−16.68, 1.71877 ± 9.96029i) (−26.6589, 14.77, −2.98148) 0.90 O(0, 0, 0) P1 (5.85, 5.90, 31.89) (−18.4517, 1.79064 ± 10.9563i) P2 (−5.85, −5.90, 31.89) (−18.4517, 1.79064 ± 10.9563i)

To obtain the fixed points and eigenvalues, we substitute (ϕ˙1 , ϕ˙2 , ϕ˙3 ) = (0, 0, 0) in Eq. (8.5) and calculate O, P1 , and P2 at γ = 1.00, 0.99, and 0.90, tabulated in Table 8.1. Eigenvalues of system (8.5) are two negative one and a real one at γ = 1.00 and a negative real one and a complex value with a positive real part at γ = 0.99 and 0.90. To compare graphical results of Lorenz, we use Liu and a new aggregated system for integer- and fractional-order values of γ . Fig. 8.1(a)–(c) shows phase plots of (ϕ1 , ϕ2 ), Fig. 8.2(a)–(c) of (ϕ1 , ϕ3 ), and Fig. 8.3(a)–(c) of (ϕ2 , ϕ3 ), at γ = 1.00. Fig. 8.4(a)–(c) shows phase plots of (ϕ1 , ϕ2 ), Fig. 8.5(a)–(c) of (ϕ1 , ϕ3 ), and Fig. 8.6(a)–(c) of (ϕ2 , ϕ3 ), at γ = 0.99. Fig. 8.7(a)–(c) shows phase plots of (ϕ1 , ϕ2 ), Fig. 8.8(a)–(c) of (ϕ1 , ϕ3 ), and Fig. 8.9(a)–(c) of (ϕ2 , ϕ3 ), at γ = 0.90. For integer-order γ = 1 and fractional-order γ = 0.99, systems (8.1), (8.2), and (8.5) exhibit chaotic behavior in which we can observe that system (8.5) is an aggregation of systems (8.1) and (8.2) graphically. At γ = 0.90 the Lorenz system exhibits a fixed point solution; the Liu system and new aggregated system exhibit chaotic behavior. In Fig. 8.10(a)–(c), we plot a Poincaré map of system (8.5) for (ϕ1 , ϕ2 ), (ϕ1 , ϕ3 ), and (ϕ2 , ϕ3 ), respectively. The returned point of system (8.5) exclusively lies in the Poincaré section of the bounded trajectory for codimension in one phase space.

8.2.1 Lyapunov spectrum and Kaplan–Yorke dimension To calculate exponential degrees of separation (Lyapunov exponents) at adjacent trajectories of system (8.5), the Wolf algorithm [5] is used. From Table 8.2 we can see that two Lyapunov exponents are negative and one is positive, which shows that system (8.5) is chaotic at γ = 1.00 and 0.90. On the other hand, at γ = 0.99, one Lyapunov exponent is negative and two

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Figure 8.1 Phase portraits in the (ϕ1 , ϕ2 ) plane of chaotic systems at γ = 1. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter system values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

are positive, which indicates that system (8.5) is chaotic. The generalization of Kaplan–Yorke dimensions is also calculated and tabulated in Table 8.2, which shows that dimensions are fractional. Numerical results of Lyapunov spectra and their corresponding Kaplan–Yorke dimension are tabulated in Table 8.2. Liu system (8.2) shows high complexity, and the aggregated system (8.5) is more complex than Lorenz system (8.2).

8.2.2 Dissipativity In vector representation, system (8.5) can be written as ⎡

⎤

f1 (ϕ1 , ϕ2 , ϕ3 ) ⎢ ⎥ ϕ˙ = f (ϕ) = ⎣ f2 (ϕ1 , ϕ2 , ϕ3 ) ⎦ . f3 (ϕ1 , ϕ2 , ϕ3 )

(8.6)

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Figure 8.2 Phase portraits in the (ϕ1 , ϕ3 ) plane of chaotic systems at γ = 1. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter system values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

Here, 1

1 aδ (ϕ2 − ϕ1 ) − (1 − γ ) ϕ1 , f1 (ϕ1 , ϕ2 , ϕ3 ) = γ 2 1 1 f2 (ϕ1 , ϕ2 , ϕ3 ) = br ϕ1 − k + 1 ϕ1 ϕ3 − ϕ2 − (1 − γ ) ϕ2 , γ 2 1 1 2 −cμ ϕ3 + hϕ1 + ϕ1 ϕ2 − (1 − γ ) ϕ3 , f3 (ϕ1 , ϕ2 , ϕ3 ) = γ 2

(8.7)

with parameter values aδ = 20, br = 65, cμ = 31 6 , k = 1, and h = 4. Divergence in the vector fields’ measurements is very beneficial in a variation of applications. The divergence of the vector at a given point in a

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Figure 8.3 Phase portraits in the (ϕ2 , ϕ3 ) plane of chaotic systems at γ = 1. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter system values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

vector field is a scalar and is defined as the amount of flux diverging from a unit volume element per second around that point. The divergence of vector field f on R3 is defined as ∂ fi ∂ f1 ∂ f2 ∂ f3 + + = . div f = ∂ϕ1 ∂ϕ2 ∂ϕ3 ∂ϕi i=1 3

(8.8)

Let be any region in R3 with smooth boundary, dV = dt

(t)

div f dϕ1 dϕ2 dϕ3 ,

(8.9a)

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Figure 8.4 Phase portraits in the (ϕ1 , ϕ2 ) plane of chaotic systems at γ = 0.99. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

−6γ + aδ + cμ + 7 div f = − < 0, 2γ

dV = dt

−6γ + aδ + cμ + 7 − dϕ1 dϕ2 dϕ3 , 2γ

(t)

V (t) = V (0)eηt , where η = −

−6γ +aδ +cμ +7 2γ

.

(8.9c)

−6γ + aδ + cμ + 7 dV = − dt 2γ

(8.9b)

V (t) ,

(8.10a) (8.10b)

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Figure 8.5 Phase portraits in the (ϕ1 , ϕ3 ) plane of chaotic systems at γ = 0.99. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

From (8.10b), we find that V (t) shrinks to zero exponentially as t → ∞. Therefore, the asymptotic flow of the aggregated system (8.5) settles exponentially onto a set of measure zero, i.e., a strange attractor.

8.3. Analog circuit imitation This section presents the Multisim design of an analog circuit, which is a physical approach to realize the feasibility of the mathematical model. For the construction of an analog circuit, system (8.5) is converted to a

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Figure 8.6 Phase portraits in the (ϕ2 , ϕ3 ) plane of chaotic systems at γ = 0.99. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

circuital equation by using Kirchhoff circuital laws:

ϕ1 1 ϕ2 (1 − γ ) ϕ1 aδ − , ϕ˙ 1 = − γ 2 R1 C 1 R2 C 1 R3 C 1 ϕ1 ϕ3 (1 − γ )ϕ2 ϕ2 1 1 ϕ1 ϕ˙ 2 = − , br − k+1 − γ 2 R4 C 2 R5 C 2 R6 C 2 R7 C 1 ϕ1 ϕ2 1 1 ϕ3 ϕ2 (1 − γ ) ϕ3 ϕ˙ 3 = − cμ − , +h 1 + γ 2 R8 C 3 R9 C3 R10 C3 R11 C1

1

(8.11)

where ϕ1 , ϕ2 , ϕ3 are voltages across the capacitors C1 , C2 , and C3 , respectively. Calculated values of resisters and capacitors depend on the timing scale value of the circuit. The timing scale value 2505 was studied by Oppenheim and Cuomo [35]. The designed circuit and calculated component

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Figure 8.7 Phase portraits in the (ϕ1 , ϕ2 ) plane of chaotic systems at γ = 0.90. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

values for γ = 0.99 are 10.101ϕ2 10.111ϕ1 − , R1 C 1 R2 C 1 32.828ϕ1 1.010ϕ1 ϕ3 0.515ϕ2 ϕ˙2 = − − , R3 C 2 R4 C 2 R5 C 2 2.619ϕ3 2.020ϕ12 0.505ϕ1 ϕ2 + + , ϕ˙3 = − R6 C 3 R7 C 3 R8 C 3 ϕ˙1 =

(8.12)

where the calculated resistor and capacitor values are R1 = R2 = R4 = 40K , R3 = 12.16K , R5 = 775K , R6 = 154.53K , R7 = 20K , R8 = 80K , R9 = R10 = R11 = R12 = R13 = R14 = 100K , C1 = C2 = C3 = 1nF. The schematic electronic circuit of aggregated system (8.5) is displayed

Aggregation of chaotic signal with proportional fractional derivative

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Figure 8.8 Phase portraits in the (ϕ1 , ϕ3 ) plane of chaotic systems at γ = 0.90. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

in Fig. 8.11, and the oscilloscope output results are shown in Fig. 8.12(a) and (b). The phase portraits obtained by numerical solution of system (8.5) have the same oscilloscopic graph, which shows the feasibility.

8.4. Security analysis In this section, random numbers generated by system (8.5) are used for security purposes at γ = 0.99 to encrypt and decrypt the selected sound and images. To achieve a big impact on the quality of encryption, a high randomness is used that is measured by calculating P-values via NIST 800-22 [19]. Table 8.3 is arranged for γ = 0.99 for the positive sequence re-

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Table 8.2 Lyapunov spectra and Kaplan–Yorke dimension of aggregated system (8.5). γ System Lyapunov spectrum Kaplan–Yorke dimension 1.00 Lorenz 2.05454 (0.786905, 0.001223, −14.451679)

0.99

0.90

system Liu system Aggregated system Lorenz system Liu system Aggregated system Lorenz system Liu system Aggregated system

(1.622444, 0.000146, −14.113370) (1.205954, −0.001377, −14.229435)

2.11497 2.08465

(0.799018, 0.002499, −14.633132)

2.05477

(1.686880, −0.000239, −14.333856) (1.008689, 0.077865, −14.279456)

2.11767 2.07609

(−0.044428, −0.0053024, −15.418428)

1.99677

(1.865605, 0.001260, −16.071016) (0.860227, −0.176039, −15.496465)

2.11616 2.04415

Table 8.3 The P-values test result (NIST-800-22) of the new aggregated system (8.5), at γ = 0.99. Statistic test P-value Result ϕ1 ϕ2 ϕ3

Frequency monobit Frequency block Runs Longest runs ones 10000 Matrix rank Spectral Nonoverlapping Overlapping Maurer’s universal statistic Complexity Serial 1 Serial 2 Entropy Cumulative sums Random excursions at x = −4 Excursions variant x = −9 Cumulative sums

0.027569 0.030197 0.720644 0.254608 0.494505 0.397104 0.087227 0.871591 0.286112 0.626063 0.846633 0.343019 0.341106 0.032043 0.979024 0.987282 0.023936

0.977413 0.636493 0.185162 0.915282 0.467412 0.128533 0.105124 0.744774 0.896294 0.148224 0.802878 0.857275 0.948834 0.674268 0.166494 0.180192 0.647965

0.425088 0.123023 0.819487 0.213789 0.918883 0.052752 0.234288 0.265395 0.817705 0.523923 0.840838 0.943039 0.657891 0.729546 0.615451 0.388711 0.513244

Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified Qualified

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Figure 8.9 Phase portraits in the (ϕ2 , ϕ3 ) plane of chaotic systems at γ = 0.90. (a) Lorenz system (8.1) with initial condition (0.1, 0.1, 0.1) and parameter values (δ, r, μ) = (10, 25, 8/3). (b) Liu system (8.2) with initial condition (2.2, 2.4, 38) and parameter values (a, b, c, k, h) = (10, 40, 2.5, 1, 4). (c) Aggregated system (8.5) with initial condition (1.15, 1.25, 19.05) and parameter values aδ , br , cμ , k, h = (20, 65, 31/6, 1, 4).

sults. The successful calculation and testing suggest that our system is ready to go for encryption, whereby we did the test on voice data in Fig. 8.13(a) and test images 1 and 2 in Fig. 8.15(a) and 8.17(a). These sound and images are taken and converted into a 32-bit sequence array, and the test result data are plugged at γ = 0.99 into the RNG algorithm, written in python code [36]. Voice encryption 1. Import essential modules like “numpy,” “scipy.integrate,” “scipy.io,” and “matplotlib”. 2. Run the numerical solution using “odeint” from scipy.integrate on our chaotic model generating the values of ϕ1 , ϕ2 , and ϕ3 .

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Figure 8.10 Poincaré plots of aggregated system (8.5) with initial condition (1.15, 1.25, 19.05), parameter values aδ , br , cμ , k, h = (20, 65, 5.1667, 1, 4), and fractional order γ = 0.99 in plane (a) (ϕ1 , ϕ2 ), (b) (ϕ1 , ϕ3 ), and (c) (ϕ2 , ϕ3 ).

3. Now use module “scipy.io” to open the voice file and import the data at a standard sample rate, i.e., 44,100 bits per second. 4. After this, we take XOR between the generated values of ϕ1 , ϕ2 , and ϕ3 . 5. The generated ϕ1 , ϕ2 , and ϕ3 XOR data are then again XOR with the imported voice data of the wave file to do encryption. 6. ϕ1 , ϕ2 , and ϕ3 are then again XOR with the encrypted data to do the decryption. 7. Lastly the “matplotlib” module is utilized to draw graphs of original, encrypted, and decrypted voice data. Image encryption 1. Import essential modules like “numpy,” “scipy.integrate,” “CV2,” and “matplotlib.”

Aggregation of chaotic signal with proportional fractional derivative

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Figure 8.11 Electronic circuit of aggregated system (8.12) designed by Multisim.

2. Run the numerical solution using “odeint” from scipy.integrate on our chaotic model generating the values of ϕ1 , ϕ2 , and ϕ3 . 3. Now using module “CV2” to open the image file and import the data in terms of frames R, G, and B and separate them in different variables. 4. After this, we take XOR between the generated values of ϕ1 with the R frame, ϕ2 with the G frame, and ϕ3 with the B frame. 5. The generated XOR data are then merged using “CV2” in order to combine the scramble data for encryption. 6. ϕ1 , ϕ2 , and ϕ3 are then again XOR with R, G, and B frames of the given image do the decryption. 7. Lastly the “matplotlib” module is utilized to show the results of original, encrypted, and decrypted images in graphical form. The output results have negligibly high noisiness, as can be seen in Fig. 8.13(b). On the other hand, test images also lost their original message

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Figure 8.12 Oscilloscope results ofsystem (8.12) inplane (a) (ϕ1 , ϕ3 ), (b) (ϕ2 , ϕ3 ), and (c) (ϕ1 , ϕ2 ) with parameter values aδ , br , cμ , k, h = (20, 65, 5.1667, 1, 4).

and present encrypted, as shown in Fig. 8.15(b) and 8.17(b). Following the backward procedure of the RNG algorithm, the returned decrypted sound data are plotted in Fig. 8.13(c) and images are shown in Fig. 8.15(c) and 8.17(c). The spectra of original, encrypted, and decrypted tested sound states are displayed in Fig. 8.14(a)–(c). Numerical distributions of test images are displayed in Fig. 8.16(a)–(c) and Fig. 8.18(a)–(c) with the aid of histogram examination. The histogram of the encrypted image shows an

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Figure 8.13 Sound encryption and decryption. (a) Original signals. (b) Encrypted signals. (c) Decrypted signals.

identical number of gray pixel values, where the gray level is homogeneously distributed and randomness is confirmed. The homogeneity of the encrypted image distributions shows the encryption is good. The closer the data distributions are, the more difficult it is to decrypt the encrypted data. From figures we can see that the histogram of encrypted images is uniformly distributed, which shows that the quality of encryption is good. Number of pixels change rate The number of pixels change rate (NPCR) is a very important quantity that increases the power of encryption. NPCR evaluates the percentage of different pixels between original and encrypted images. Mathematically it is calculated as

D k, l =

0 1

C k, l = C ∗ k, l , C k, l = C ∗ k, l .

M N D k, l NPCR = × 100%, M .N k=1 l=1

(8.13a) (8.13b)

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Figure 8.14 Spectrograms of 32-bit test sound ﬁle. (a) Original. (b) Encrypted. (c) Decrypted.

where C k, l and C ∗ k, l characterize the pixel values of the two encrypted images whose plain-images have only a one-pixel difference. Unified average changing intensity The unified average changing intensity (UACI) indicates the average value of the changed pixel. It is defined as N M C1 i , j − C2 i , j × 100%. UACI = 255 × M × N i=1 j=1

(8.14)

For good quality encryption, NPCR values will be at least 99.6% and UACI more than 30%. Correlation Correlation analysis is the assessment for two random generated data variables using the correlation given as j − ηj k − ηk p j , k . Correlation = σj σk j ,k

(8.15)

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Figure 8.15 Encryption and decryption of test image 1. (a) Original test image 1. (b) Encrypted test image 1. (c) Decrypted test image 1.

Mean squared error Mean squared error (MSE) is one of the ways to compute similarity between original and decrypted images by using the following well-defined metric:

MSE q1 , q2 =

M −1 N −1 2 1 q1 i , j − q2 i , j , MN i=0 j=0

(8.16)

where q1 (i, j) and q2 (i, j) indicate the original and decrypted images, respectively. From Table 8.5 we can see MSE is very low; therefore, decrypted images are of good quality.

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Figure 8.16 Histograms of encrypted and decrypted test image 1. (a) Original test image 1. (b) Encrypted test image 1. (c) Decrypted test image 1.

Peak signal-to-noise ratio The peak signal-to-noise ratio (PSNR) is inversely proportional to MSE. For good image quality, its value will be high. Its mathematical form is defined as PSNR = 10 log10

(2n − 1)2

MSE

.

(8.17)

PSNR shows image quality level and noise; for good image quality its value will be high.

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Figure 8.17 Encryption and decryption of test image 2. (a) Original test image 2. (b) Encrypted test image 2. (c) Decrypted test image 2.

Entropy Entropy is a feature that defines the level of uncertainty and randomness in the data and is used to measure the constant distribution of the gray pixel level in an image. Mathematically it is defined as H ( s) = −

N −1 2

j=0

1 p sj log . p sj

(8.18)

In Eq. (8.18) the probability of image pixel and histogram distribution is denoted by p, while the number of bits is denoted by N.

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Figure 8.18 Histograms of encrypted and decrypted test image 2. (a) Original test image 2. (b) Encrypted test image 2. (c) Decrypted test image 2.

Analyses through these parameters are performed on two test images and results of experiments are tabulated in Tables 8.4 and 8.5. NPCR and UACI analyses of 8-bit encrypted images were conducted; the results showed that the encrypted 8-bit house and parrot images passed the NPCR and UACI analyses successfully. The correlations of original and encrypted test image 1 are 0.87612 and 0.16682, and those of test image 2 are 0.80845 and 0.3, indicating that the images are neither identical nor opposite. The mean number pixels of images ensure that entire images have one pixel. The MSE is mostly used for superior image quality; if the MSE is low, image quality is good. Here the obtained MSE values of images 1 and 2 are 0.00001 and 0.00003, which shows that the quality of images is high. The PSNR values of test images 1 and 2 are 27.89837 and 27.89830, respectively. Meanwhile, values of UACI show a small significant change between the test images. Numerical results of entropy of images are listed in Table 8.5, in which the entropy of original test image 1, having the pixel value 256 × 256, is 7.60076 on the red scale, 7.60076 on the green scale, and 6.46055 on the blue scale. On the other hand, the entropy of en-

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Table 8.4 Results of NPCR, UAIC, and correlation. Image NPCR (%) UAIC (%) Correlation Original image Encrypted image

House Parrot

99.63735 99.56767

32.60435 40.97717

0.87612 0.80845

Table 8.5 Results of PSNR, MSE, and entropy. Image Dimension MSE PSNR of pixel

House

256×256

0.00001

27.89837

Parrot

256×256

0.00001

27.89830

0.16682 0.16526

Entropy Original Encrypted

R G B R G B

7.60076 7.60076 6.46055 6.05504 6.91410 6.71754

7.76215 7.79365 7.75283 7.76170 7.78579 7.75250

crypted test image 1 is 7.76215, 7.79365, and 7.75283 on red, green, and blue scales, respectively. Test image 2 has the same pixel value, 256 × 256, and its calculated entropies of original and encrypted images are 6.05504, on the red, 6.91410 on the green, 6.71754 on the blue, and 7.76170 on the red, 7.78579 on the green, and 7.75250 on the blue scale, respectively.

8.5. Conclusion In this chapter, we have presented the aggregation of two chaotic oscillators by using the proportional fractional derivative. The new aggregated system has dissimilar stability points from Lorenz and Liu systems. The dynamical properties of the new aggregated system are described through Lyapunov exponents, random number generation, and phase and Poincaré plots. The calculated Kaplan–Yorke dimension at different values of γ shows that the aggregated system is complex. Digitally, the analog circuit of the new aggregated system is also designed with the advanced components of electronics to validate our results in another way. Next, this chaotic solution was utilized by the RNG algorithm on different informative file extensions, such has (.wav) and (.png), to encrypt and decrypt as well. Security analyses and image quality tools show good quality of encrypted and decrypted images. In the future, the constructed protocol, which is formed for scrambling of voice and images, will be used on other files.

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CHAPTER NINE

CNT-based fractors in all four quadrants: design, simulation, and practical applications Avishek Adhikarya Indian Institute of Technology, Bhilai, Department of Electrical Engineering and Computer Science, Raipur, India

Chapter points •

The objective of this chapter is to understand how a wide-CPZ, long-life fractor can be fabricated using a CNT device.

•

The chapter also discusses how fractance impedance can be realized in any of the four impedance quadrants.

•

It also explores the advantage of four-quadrant fractors in designing practical circuits.

9.1. Introduction The feasibility of practical application of fractional calculus in designing electronic circuits has always spun around a single question – when and how shall we have the first commercial fractional-order (FO) circuit element? With various recent progresses in this field like the fabrication of the first “green” fractor [24] or the IC-based realization of FO immittance [56], possibly the advent is now only a matter of time. The last six decades have witnessed hundreds of research works on practical realizations of FO elements [1,14,15,20,27,38,59] and with contributions from each such work the FO circuit theory has gradually achieved a matured stage [18,35,42,52,53,57]. Even the terminologies associated with FO system analysis have also evolved. What was initially called pseudocapacitance or fractional capacitance has finally claimed its own name as “fractance dea Dr Avishek Adhikary is currently an Assistant Professor in the Dept. of EECS in IIT Bhilai,

India, since 2018. He completed his MTech-PhD from IIT Kharagpur, India, in 2018 and BTech from Jadavpur University, India, in 2007. His research interests include application of fractional calculus in electronic circuits, fractional-order modeling of real-time systems, and fabrication of smart sensors and instruments. Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00014-7 All rights reserved.

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vice” or “fractor” [2]. The initial trend of discussing FO immittances as a deviation of conventional capacitance has subsided [60] and rather a new thought has surfaced out which looks into the FO immittance as a generalized circuit element [3]. As the FO circuit theory develops, new design challenges and research scopes are also emerging. To incorporate new theories into practice it is now more important, if not already done, to realize FO elements with precise specifications. The order of the fractor, its coefficients (fractance), the operating zone (constant phase zone [CPZ]), and the fluctuation in its constant phase (CP) characteristics (phase band [PB]) are a few of such significant parameters of practical fractors. There are different techniques with which an RC ladder network can be designed meeting these specifications [4,59]. However, inclusion of RC ladder networks in practical circuits makes systems bulky as well as noisy. To circumvent this, researchers are now exploring various possibilities for fabricating a fractance device as a single element. The last two decades have witnessed many such researches; a brief account of them is given in Table 9.1. Now, various different devices have been developed for different orders utilizing different physicochemical effects. Nevertheless, they also suffer from several limitations like small CPZ, short lifespan, or large PB. In this context, this chapter presents a typical carbon nanotube (CNT)-based packaged, electrochemical fractor which shows an about 6-decades wide CPZ and a stable life of more than two years. It will be shown that such fractor is readily usable in analog circuits and gives less noisy responses than its RC ladder equivalent. The chapter also explores the concept of generalization of FO immittances in all four quadrants and uses the said CNT-based fractor in a typical Bruton generalized impedance converter (GIC) circuit to realize FO immittances in all four quadrants. Such four-quadrant fractors are then used to design tunable, high-Q FO resonators. Next, Section 9.2 presents necessary terminologies and a brief account on the state-of-the-art of fractor design. Section 9.3 discusses fabrication and a characteristics study of a CNT-based fractance device. A Foster-I-based ladder design technique is presented in Section 9.4 which is used to simulate the proposed CNT fractor in PSpice. A generalized realization topology of FO immittances in all four quadrants is presented in Section 9.5. Advantages of such four-quadrant fractors are demonstrated in Section 9.6 by designing tunable, high-Q FO resonators. Concluding remarks are given in Section 9.7.

a b

d.: decades NA: not available, m.: months

Lifeb

Application

NA 2–6 m. NA NA NA NA NA NA NA 20 m.

NA Integrator NA NA NA Oscillator Filter Oscillator Filter Resonator

CNT-based fractors in all four quadrants: design, simulation, and practical applications

Table 9.1 A comparison between different packaged single-element type fractors reported in literature. Reported by Origin CP PB CPZa CPZ position ◦ ◦ ◦ Haba (1995) [38] Solid state fractal −36 , −45 ±5.0 ≤ 6 d. 104 –1010 Hz Mondal (2011) [46] PMMA+gel −8◦ to −20◦ ±1-2◦ ≈ 1 d. 103 –104 Hz Elshurafa (2011) [33] Graphene-based −31◦ to −67◦ ±3-5◦ ≈ 1 d. 105 –106 Hz ◦ ◦ ◦ Caponetto (2013) [25] IPMC gel −5 , −27 ±2-3 ≤ 2 d. 100 –102 Hz ◦ ◦ ◦ John (2017) [41] CNT-based −55 to −85 ±1–3 ≈ 1 d. 102 –107 Hz ◦ ◦ ◦ Agambayev (2018) [15] MoS2 -ferroelectric −58 , −80 ±4 ≈ 5 d. 102 –107 Hz ◦ ◦ ◦ Buscarino (2018) [23] Carbon black −70 to −85 ±5 ≈ 3 d. 105 –108 Hz ◦ ◦ ◦ Agambayev (2018) [14] CNT-polymer −7 to −65 ±6 ≈ 1 d. 105 –106 Hz Caponetto (2019) [24] Bacterial cellulose ≈ −64◦ NA ≈ 3 d. 103 –106 Hz Adhikary (2020) CNT-polyimide −7◦ to −42◦ ±3–4◦ 4−7 d. 10−2 –105 Hz

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9.2. Fractor: deﬁnitions and state-of-the-art The fundamental theory of fractional calculus presents several definitions for FO derivatives as proposed by Riemann–Liouville, Caputo, and Grunwald–Letnikov [39,48]. For example, as per Riemann–Liouville, a fractional derivative of order α (0 < α < 1) can be defined as

1 dm t h(τ ) , m−1 1. The values of ωc are greater than 1 for α > 1, which means that the filter experiences very strong ripples. This takes place because filter poles lie near the unstable region [54]. The right phase Eq. (12.7) may give one solution or no solution. Solution means that the

Analysis and realization of fractional step ﬁlters of order (1 + α)

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Figure 12.2 Magnitude and phase response of (a) the fractional step low-pass ﬁlter of Eq. (12.3), (b) the fractional step high-pass ﬁlter of Eq. (12.8), (c) the fractional step band-pass ﬁlter of Eq. (12.13), (d) the fractional step all-pass ﬁlter of Eq. (12.18), and (e) the fractional step band-stop ﬁlter of Eq. (12.21).

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Figure 12.3 Critical frequencies vs. α .

phase of filter reaches ±90. From Fig. 12.3 it is observed that the right phase frequency gives solutions for α > 0.4.

12.2.2 Second method In the second method, the FTF is approximated using higher integer-order TFs. FSFs can be designed using many techniques, like FLF [7,8,21,22,29, 34,59,63], IFLF [28,34,35], and the SFG [25]. This method requires no fractional element. In this chapter, the design of FSFs based on the SFG approach is considered.

12.2.2.1 Fractional step low-pass ﬁlter The general FTF given in Eq. (12.3) is considered. The evaluation of the term sα might be procured by a second-order approximation using the CFE method and is specified as follows: sα =

b0 s2 + b1 s + b2 . b2 s2 + b1 s + b0

(12.24)

Here b0 = α 2 + 3α + 2, b1 = 8 − 2α 2 , and b2 = α 2 − 3α + 2. Some past work has been centered around finding the approximations of the coefficients utilizing various techniques acquired in [4,13,18,26,32, 39,59]. In this chapter, the coefficients obtained in [13], are considered: k1 = 1,

k2 = 1.0683α 2 + 0.161α + 0.3324,

k3 = 0.29372α + 0.71216. (12.25)

Analysis and realization of fractional step ﬁlters of order (1 + α)

351

The value of sα is substituted from Eq. (12.24) into Eq. (12.3). Then the TF turns out to be T1LPF +α (s) = k1

s3

c 2 s 2 + c 1 s + c0 . + d 2 s2 + d 1 s + d 0

(12.26a)

Here, α 2 − 3α + 2 8 − 2α 2 , c = , c0 = 1, 1 α 2 + 3α + 2 α 2 + 3α + 2 8 − 2α 2 + k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 d2 = , α 2 + 3α + 2 k2 + k3 (8 − 2α 2 ) + α 2 − 3α + 2 d1 = , α 2 + 3α + 2 2 2 k2 α − 3α + 2 + k3 α + 3α + 2 d0 = . α 2 + 3α + 2

c2 =

(12.26b)

Magnitude and phase plots of Eq. (12.26a) for various values of α are given in Fig. 12.4(a).

12.2.2.2 Fractional step high-pass ﬁlter The general FTF given in Eq. (12.8) is considered. Now, substituting Eq. (12.24) into (12.8), the TF becomes T1HPF +α (s) = k1

s 3 + c2 s 2 + c1 s . s3 + d 2 s2 + d 1 s + d 0

(12.27a)

Here, α 2 − 3α + 2 8 − 2α 2 , c1 = 2 , + 3α + 2 α + 3α + 2 8 − 2α 2 + k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 d2 = , α 2 + 3α + 2 k2 + k3 (8 − 2α 2 ) + α 2 − 3α + 2 d1 = , α 2 + 3α + 2 2 2 k2 α − 3α + 2 + k3 α + 3α + 2 d0 = . α 2 + 3α + 2

c2 =

α2

(12.27b)

Magnitude and phase plots of Eq. (12.27a) for various values of α are given in Fig. 12.4(b).

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12.2.2.3 Fractional step band-pass ﬁlter The general FTF given in Eq. (12.13) is considered. Now, substituting Eq. (12.24) into (12.13), the TF becomes T1BPF +α (s) = k1

c 2 s 2 + c 1 s + c0 . s3 + d 2 s2 + d 1 s + d 0

(12.28a)

Here, α 2 − 3α + 2 8 − 2α 2 , c = , 0 α 2 + 3α + 2 α 2 + 3α + 2 8 − 2α 2 + k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 d2 = , α 2 + 3α + 2 k2 + k3 (8 − 2α 2 ) + α 2 − 3α + 2 d1 = , α 2 + 3α + 2 2 2 k2 α − 3α + 2 + k3 α + 3α + 2 d0 = . α 2 + 3α + 2

c2 = 1,

c1 =

(12.28b)

Magnitude and phase plots of Eq. (12.28a) for various values of α are given in Fig. 12.4(c).

12.2.2.4 Fractional step all-pass ﬁlter The general FTF given in Eq. (12.18) is considered. Now, substituting Eq. (12.24) into (12.18), the TF becomes T1APF +α (s) = k1

s 3 + c 2 s 2 + c 1 s + c0 . s3 + d 2 s2 + d 1 s + d 0

(12.29a)

Here,

8 − 2α 2 − k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 , α 2 + 3α + 2 2 α − 3α + 2 + 8 − 2α 2 −k2 + k3 c1 = , α 2 + 3α + 2 −k2 (α 2 − 3α + 2) + k3 (α 2 + 3α + 2) c0 = , α 2 + 3α + 2 (8 − 2α 2 ) + k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 d2 = , α 2 + 3α + 2 k2 + k3 (8 − 2α 2 ) + α 2 − 3α + 2 d1 = , α 2 + 3α + 2

c2 =

Analysis and realization of fractional step ﬁlters of order (1 + α)

353

k2 α 2 − 3α + 2 + k3 α 2 + 3α + 2 d0 = . α 2 + 3α + 2

(12.29b)

Magnitude and phase plots of Eq. (12.29a) for various values of α are given in Fig. 12.4(d).

12.2.2.5 Fractional step band-stop ﬁlter The general FTF given in Eq. (12.21) is considered. Now, substituting Eq. (12.24) into (12.21), the TF becomes T1NF +α (s) = k1

s 3 + c 2 s 2 + c 1 s + c0 . s3 + d 2 s2 + d 1 s + d 0

(12.30a)

Here,

α 2 − 3α + 2 + k3 8 − 2α 2 8 − 2α 2 + k3 α 2 − 3α + 2 , c = , c2 = 1 α 2 + 3α + 2 α 2 + 3α + 2 (8 − 2α 2 ) + k2 α 2 + 3α + 2 + k3 α 2 − 3α + 2 c0 = k 3 , d 2 = , α 2 + 3α + 2 k2 + k3 (8 − 2α 2 ) + α 2 − 3α + 2 d1 = , α 2 + 3α + 2 2 2 k2 α − 3α + 2 + k3 α + 3α + 2 (12.30b) d0 = . α 2 + 3α + 2

Magnitude and phase plots of Eq. (12.30a) for various values of α are given in Fig. 12.4(e). Table 12.2 shows the SFGs of the TFs given in Eqs. (12.26a), (12.27a), (12.28a), (12.29a), and (12.30a).

12.3. Numerical analysis and simulations of FSFs of order (1 + α) Both design procedures can be applied to active, as well as passive filters. An assortment of active building blocks (ABBs), for example, operational amplifiers (Op-Amps) [4,39], CFOAs [59,63]; operational transconductance amplifiers (OTAs) with adjustable current amplifier (ACA) and current follower (CF) [35], universal voltage conveyors (UVCs) [29], DDCCs [28], ACAs with CF [7,34], and CDBAs [25], are utilized to create FSFs. In this chapter, FSFs based on Method I utilize Op-Amps, whereas FSFs based on Method II utilize CDBAs as ABBs.

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Figure 12.4 Magnitude and phase plots of (a) the fractional step low-pass ﬁlter obtained from Eq. (12.26a), (b) the fractional step high-pass ﬁlter obtained from Eq. (12.27a), (c) the fractional step band-pass ﬁlter obtained from Eq. (12.28a), (d) the fractional step all-pass ﬁlter obtained from Eq. (12.29a), and (e) the fractional step bandstop ﬁlter obtained from Eq. (12.30a).

Analysis and realization of fractional step ﬁlters of order (1 + α)

Table 12.2 Signal ﬂow graphs of different FSFs. Type of FSF

FSLPF

FSHPF

FSBPF

FSAPF

FSBSF

Signal ﬂow graph

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(a)

(b) Figure 12.5 (a) Tow–Thomas ﬁlter [14]. (b) Fourth-order approximation using a ladder network.

12.3.1 Circuit simulations based on Method I The Tow–Thomas FSF depicted in Fig. 12.5(a) based on Method I utilizes six resistors, one traditional capacitor, one FOC, and an Op-Amp as an ABB. The impedance function of C2 of order α gives the magnitude 1/ (ωα C2 ). The FOC utilized in the SPICE simulations is realized as shown in Fig. 12.5(b) by a fourth-order rational approximation for sα [30] using CFEs. By using this rational approximation, the fractional Laplacian operator can be realized physically with the RC ladder network. The impedance (Z) of this RC network is Z = Ra +

1/Cb 1/Cc 1/Cd 1/Ce + + + . (12.31a) s + 1/Rb Cb s + 1/Rc Cc s + 1/Rd Cd s + 1/Re Ce

The values of resistances and capacitors are obtained by equating terms of the fourth-order rational approximation with the terms of Eq. (12.31a) and are given in Table 12.3 designed at a center frequency of 1 kHz. By applying Kirchhoff ’s current law (KCL) and nodal analysis to Fig. 12.5(a),

Analysis and realization of fractional step ﬁlters of order (1 + α)

357

then the simplified FTF is specified as follows:

R6 R4 R1 R5 C1 C2

, T1LPF +α (s) = s1+α + R31C1 sα + R4 R2 RR56C1 C2

T1BPF +α (s) =

1 R1 C1

· sα

s1+α + R31C1 sα + R4 R2 RR56C1 C2

.

(12.31b)

(12.31c)

Comparison of coefficients of Eq. (12.31b) and Eq. (12.31c) with Eq. (12.3) and Eq. (12.13), respectively, gives the following relationship: For LPF, k2 =

1 , C3 R1

k3 =

R6 , R4 R2 R5 C 1 C 2

k1 =

R6 . R4 R1 R5 C 1 C 2

(12.31d)

1 . R1 C 1

(12.31e)

For BPF, k2 =

1 , C3 R1

k3 =

R6 , R4 R2 R5 C 1 C 2

k1 =

The R-C component values to obtain the frequency responses of the lowpass filter and band-pass filter are R1 = R2 = R3 = R4 = R5 = R6 = 1 k, C1 = 1 µF, and C2 = 2 × 10−6 [F/s1−α ]. The simulation (solid) and numerical (dotted) results are depicted in Fig. 12.6(a) and (b) for the Tow–Thomas FLPF and FBPF, respectively. It is observed that the simulated and numerical values of slopes of FSLPF are −35.57 dB/decade and −36 dB/decade for order 1.8 and −31.55 dB/decade and −32 dB/decade for order 1.6, respectively. The simulated and numerical values of FBPF are 17.53 dB/decade and 18 dB/decade for order 1.8 and 15.39 dB/decade and 16 dB/decade for 1.6 at low frequencies and −20 dB/decade at high frequencies for both cases. Fig. 12.6(c) and (d) depicts the absolute relative error between the simulated and numerical results, which shows that the absolute relative error is less than 10% up to 1 MHz for the selected values of α for Tow–Thomas FSFs.

12.3.2 Circuit simulations based on Method II FSFs depicted in Fig. 12.8 are based on Method II, utilizing CDBA as an ABB and are designed by the SFG approach. CDBA is an active device with four terminals, as shown schematically in Fig. 12.7(a). A CDBA based on CFOA given in Fig. 12.7(b) is used in the circuit simulations. The voltages

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Table 12.3 Passive component values to realize FOCs at a center frequency of 1 kHz. Passive Fractional-order capacitor components C = 2 × 10−6 [F/s1 − α ] 2

Ra () Rb () Rc () Rd () Re () Cb (µF) Cc (µF) Cd (µF) Ce (µF)

α = 0 .8

α = 0 .6

8.40 42.997 97.755 475.0993 24266 0.64182 1.43 1.308 5.9698

173.603 2728 527.64 35490.75 935.912 0.1901 0.04379 0.1885 0.1293

Figure 12.6 (a) PSpice simulation of FSLPF. (b) PSpice simulation of FSBPF. (c) Relative error in the magnitude response of FSLPF. (d) Relative error in the magnitude response of FSBPF.

Analysis and realization of fractional step ﬁlters of order (1 + α)

359

(a)

(b) Figure 12.7 (a) Schematic of CDBA. (b) CDBA based on CFOA.

and currents of this active component are characterized by the following equation: ⎡ ⎢ ⎢ ⎢ ⎣

Vp Vn Iz Vw

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

0 0 0 0 1 −1 0 0

0 0 0 0

0 0 0 1

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

Ip In Iw Vz

⎤ ⎥ ⎥ ⎥. ⎦

(12.32)

As described by [25], a CDBA-based FSLPF and FSBPF of the SFG given in Table 12.2 are depicted in Fig. 12.8(a). Although the circuit diagrams for FSLPF and FSBPF are equal, the coefficients of both FSFs are different. The TF from Fig. 12.8(a) can be derived utilizing KCL and nodal methods where R6 = R7 = R8 = R9 . Then the simplified TF is specified as follows: Iout = Iin

1 C3 R3

·

R10 2 R10 1 1 R5 s + C2 R2 C3 R3 · R4 s + C1 R1 C2 R2 C3 R3 s3 + C31R3 s2 + C2 R21C3 R3 s + C1 R1 C21R2 C3 R3

·

R10 R0

.

(12.33a)

FSHPF realization of the SFG given in Table 12.2 is depicted in Fig. 12.8(b). The TF for the FSHPF filter given in Fig. 12.8(b) can be derived utilizing KCL and nodal methods where R7 = R8 = R9 = R10 . Then

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the simplified TF is specified as follows: R11 3 R11 2 R11 1 1 Iout R s + C R · R5 s + C2 R2 C3 R3 · R4 s = 3 6 1 23 3 . Iin s + C3 R3 s + C2 R21C3 R3 s + C1 R1 C21R2 C3 R3

(12.33b)

The SFGs of FSAPFs and FSBSFs are similar, but the coefficients for both filters are different and the circuit for both filters is given in Fig. 12.8(c). The TF for FSAPF/FSBSF given in Fig. 12.8(c) can be derived using KCL and nodal methods where R7 = R8 = R9 = R10 . Then the simplified TF is given as follows: Iout = Iin

R11 3 R6 s

+

R11 2 R11 1 1 1 C3 R3 · R5 s + C2 R2 C3 R3 · R4 s + C1 R1 C2 R2 C3 R3 s3 + C31R3 s2 + C2 R21C3 R3 s + C1 R1 C21R2 C3 R3

·

R11 R0

. (12.33c)

The depicted circuits are dissected at a cut-off frequency of 10 kHz. Although the circuit diagrams of some FSFs are the same, different filters have different coefficients. All resistor values can be obtained in terms of the coefficients of the TF; therefore, the design of the circuit is very simple. Coefficients for the filters have been calculated and are listed in Table 12.4. The simulated (solid) and numerical (dotted) magnitude response of various FSFs using coefficients given in Table 12.4 are given in Fig. 12.9(a)–(e). The presented circuits are analyzed at a cut-off frequency of 10 kHz. The simulated and numerical values of slopes of FSLPF are −24.37 dB/decade and −25 dB/decade for order 1.25, −29.55 dB/decade and −30 dB/decade for order 1.5, and −37.58 dB/decade and −38 dB/decade for order 1.9, respectively. The simulated and numerical values of slopes of FSHPF are 24.72 dB/decade and 25 dB/decade for order 1.25, 29.54 dB/decade and 30 dB/decade for order 1.5, and 37.45 dB/decade and 38 dB/decade for order 1.9, respectively. The simulated and numerical values of slopes of FSBPF are 4.45 dB/decade and 5 dB/decade for order 1.25, 9.96 dB/decade and 10 dB/decade for order 1.5, and 17.3 dB/decade and 18 dB/decade for order 1.9 for low frequencies and −19.88 dB/decade and −20 dB/decade for high frequencies for all cases. Fig. 12.9(f)–(h) depicts the corner investigations of the FSLPF at α = 0.25, α = 0.5, and α = 0.9. Three corners (typical-typical [tt], slow-slow [ss], and fast-fast [ff]) have been reviewed. DC gain demonstrates approximately 3.4% (ff) and 5.1% (ss) change from the tt value for all the cases; therefore, the ss corner can be viewed as the worst case for this circuit.

Passive component

R0 (k) R1 (k) R2 (k) R3 (k) R4 (k) R5 (k) R6 (k) R7 (k) R8 (k) R9 (k) R10 (k) R11 (k) C1 = C2 = C3 (nF)

Low-pass ﬁlter

Fractional step ﬁlters of order (1 + α ) High-pass ﬁlter All-pass ﬁlter

Band-pass ﬁlter

Notch ﬁlter

α = 0.25

α = 0. 5

α = 0. 9

α = 0.25

α = 0. 5

α = 0. 9

α = 0.25

α = 0. 5

α = 0. 9

α = 0.25

α = 0. 5

α = 0. 9

α = 0.25

α = 0. 5

α = 0. 9

9.906 39.33 9.25 2.77 1.39 7.73 2 2 2 2 1 None 1

9.95 32.94 8.70 3.51 1.64 14.26 2 2 2 2 1 None 1

10.03 26.96 9.315 3.97 2.34 126.2 2 2 2 2 1 None 1

21.23 39.34 9.25 2.77 1.39 3.61 2 2 2 2 1 None 1

2 32.94 8.70 3.51 1.64 2.85 2 2 2 2 1 None 1

502.5 26.96 9.32 3.97 2.34 2.52 2 2 2 2 1 None 1

None 39.33 9.25 2.77 8.35 1.29 1 2 2 2 2 1 1

None 32.95 8.70 3.51 16.39 1.43 1 2 2 2 2 1 1

None 26.96 9.32 3.97 135.51 2.17 1 2 2 2 2 1 1

17.11 39.34 9.25 2.77 2.73 1.32 1 2 2 2 2 1 1

13.74 32.94 8.70 3.51 5.92 1.047 1 2 2 2 2 1 1

10.57 26.96 9.32 3.97 6.67 15.19 1 2 2 2 2 1 1

12.61 39.33 9.25 2.77 1.46 1.14 1 2 2 2 2 1 1

11.58 32.94 8.70 3.51 1.71 1.31 1 2 2 2 2 1 1

10.27 26.96 9.31 3.96 2.35 2.14 1 2 2 2 2 1 1

Analysis and realization of fractional step ﬁlters of order (1 + α)

Table 12.4 Passive component values to realize FSFs based on Method II.

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(a)

(b)

(c) Figure 12.8 (a) Circuit for FSLPF/FSBPF. (b) Circuit for FSHPF. (c) Circuit for FSAPF/FSBSF.

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Figure 12.9 PSpice simulation of (a) FSLPF, (b) FSHPF, (c) FSBPF, (d) FSAPF, and (e) FSBSF and the corner frequency of FSLPF for (f) α = 0.25, (g) α = 0.50, and (h) α = 0.9.

12.4. Stability Stability is a fundamental parameter of FSFs which can be examined by interchanging the fractional model into various models [53,54]. In linear time-invariant systems, if all the roots of its characteristic equation lie on the left half of the complex plane, then the system is said to be stable, whereas the stability of FSFs is inspected by interchanging the s-plane into the W-plane [53,54], which transforms the FTF into integer plane. Fig. 12.10(a) and (b) shows the stability regions in the s-plane and the W-plane, respectively. As specified in Fig. 12.10(b), the negative axis of the s-plane is mapped onto the line |θw | = π/n and the ±j-axes in the s-plane are mapped onto the lines |θw | = π/2n. The region where |θw | > π/n is not physical and the system is stable if and only if all the roots in the W-plane lie in the region |θw | > π/2n and will oscillate if at least one root lies on the lines |θw | = π/2n. The system will be unstable if at least one root lies in the

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Figure 12.9 (continued)

region |θw | < π/2n [53]. The stability analysis of this filter is carried out by succeeding steps. Interchange the fractional plane into the W-plane by taking the positive integer values of m and n for various values of α and considering s = W n and α = m/n. Solve the transformed TF to estimate the minimum root angle |θw | min. If |θw | min > π/2n, then the filter is stable; otherwise, it is unstable. This method is then used in the denominator of Eqs. (12.3), (12.8), (12.13), (12.18), and (12.21), which offers the following typical equation in the W-plane: W m+n + k2 W n + k3 = 0.

(12.34)

Table 12.5(a) and (b) confirms the stability of the reviewed filters by calculating the minimum root angle using Eq. (12.34) for different estimations of α and coefficients k2 and k3 .

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Table 12.5 Minimum root angle for distinct estimations of α using (a) Method I, (b) Method II. (a) α k2 k3 |θw |min Stability

0.9 0.7 0.6 1 0.5 1.2 1.6 1.8

1.414 1.414 1.414 1.414 1.414 1.414 1.414 1.414

14.086◦ 14.8497◦ 15.1637◦ 12.763◦ 15.49◦ 11.98◦ 8.941◦ 8.50◦

1 1 1 1 1 1 1 1

Stable Stable Stable Stable Stable Stable Unstable Unstable

(b) α

0.25 0.5 0.9

k2 0.4394 0.6800 1.3426

k3 0.7856 0.8590 0.9765

n

m

100 10 10

25 5 9

|θW |min

1.5656◦ 14.1656◦ 13.87◦

Stability

Stable Stable Stable

Fig. 12.10(c) depicts the location of poles in the integer (W) plane from Eq. (12.34) for different values of α with coefficients k2 = 1.414 and k3 = 1, which are intended for both PSpice and MATLAB simulations, confirming the stability of FSFs.

12.5. Sensitivity analysis For computing the performance of active filters, sensitivity investigation is one of the most significant parameters. Sensitivity is computed as

Sxy

x = y

δy , δx

(12.35)

i.e., fractional change within the parameter (y), i.e., TF normalized by the fractional change in the value of the x-component. Change in TF sensitivity relative to component variation is complicated and hard to compute [58]. The sensitivity analysis of FSFs signifies the relative change in filters’ responses with respect to the used circuit components.

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(a)

(b)

(c) Figure 12.10 Stability regions in (a) the s-plane, (b) the W-plane. (c) Pole locations in the W-plane for various values of α .

12.5.1 Sensitivity analysis of Method I The sensitivity of FTFs based on Method I with respect to different components such as FO (α ) and FOC can be established by applying Eq. (12.35) to the TF of the FSFs specified in Eqs. (12.3), (12.8), (12.13), (12.18), and (12.21): LPF

Sα

SαHPF SαBPF

−α s1+α + k2 sα ln (s) , = 1+α s + k2 sα + k3 α ln (s) , = 1+α s + k2 sα + k3 α k3 ln (s) , = 1+α s + k2 sα + k3

(12.36a) (12.36b) (12.36c)

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−2α k2 k3 (sα ) ln (s) , s1+α − k2 sα + k3 s1+α + k2 sα + k3 −α k2 k3 (sα ) ln (s) . = 1+α s + k3 s1+α + k2 sα + k3

SαAPF =

(12.36d)

SαBSF

(12.36e)

The transfer sensitivity of Tow–Thomas FSFs based on Method I has been calculated to specify the effect of FOCs. It is written as follows:

SCTLPF 2

= −

s1+α R3 C1 + sα

s1+α + R31C1 sα + R4 R2 RR56C1 C2

,

(12.37a)

R6 R4 R2 R5 C1 C2

. = SCTBPF 2 1 +α s + R31C1 sα + R4 R2 RR56C1 C2

(12.37b)

The effect of FOC and FO (α ) on the TF sensitivity of Tow–Thomas FSFs is given in Fig. 12.11(a)–(d). It is observed that TF sensitivities of Tow–Thomas FSFs with respect to FOC are less than 1, while the Tow– Thomas FSFs are sensitive to FO (α ).

12.5.2 Sensitivity analysis of Method II Sensitivity investigations for TFs of FSLPF, FSHPF, FSBPF, FSAPF, and FSBSF specified in Eqs. (12.26a), (12.27a), (12.28a), (12.29a), and (12.30a) with respect to coefficients d2 , d1 , and d0 have been calculated as SRd21 ,R2 ,C1 ,C2 = 0,

SRd23 ,C3 = −1,

SRd01 R2 ,R3 ,C1 ,C2 ,C3 = −1.

SRd12 ,R3 ,C2 ,C3 = −1,

SRd11 ,C1 = 0, (12.38)

From Eq. (12.38), it is found that the sensitivity of coefficients d2 , d1 , and d0 is less than or equal to 1 and may be viewed as low.

12.5.3 Monte Carlo simulations The impact of passive component deviation in the performance of FSLPFs based on Method I is examined through Monte Carlo analysis by taking 150 samples. Magnitude response simulations due to 5% of resistance and capacitance tolerances for FSLPF of orders 1.6 and 1.8 are depicted in Fig. 12.12(a) and (b), respectively. The maximum spread in passband/stopband magnitude of order 1.6 or 1.8 is observed to be 1.9 dB/3 dB and 1.7 dB/2.7 dB, respectively.

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TBPF TLPF Figure 12.11 TF sensitivity. (a) |STLPF C2 | versus C2 . (b) |SC2 | versus C2 . (c) |Sα | ver-

sus α . (d) |STBPF α | versus α .

Figure 12.12 Magnitude response as determined by Monte Carlo analysis for FSLPFs of order (a) 1.6 and (b) 1.8.

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12.6. Conclusion The analysis and realization of FSFs with order (1 + α ) has been explored. Two different design techniques to realize the FSFs with mathematical formulations and circuit results were investigated. The critical frequencies of various FSFs were computed and the effect of fractional step α on the critical frequencies was explored. MATLAB results and SPice simulations for both design methods were introduced to approve the hypothetical results. The SFG approach was employed to realize the FSFs, which made the design easy. Stable and unstable regions were explored from the stability investigations for the distinct values of α . Sensitivity investigations for both designs had been explored. It has been observed from hypothetical and simulated results that the FSFs provide an additional level of freedom in controlling the frequency response.

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CHAPTER THIRTEEN

Fractional-order identiﬁcation and synthesis of equivalent circuit for electrochemical system based on pulse voltammetry Sanjeev Kumara and Arunangshu Ghoshb a Department of Electrical Engineering, School of Engineering & Technology, Sandip University, Madhubani, Bihar, India b Department of Electrical Engineering, National Institute of Technology Patna, Patna, Bihar, India

13.1. Introduction The concept of a fractional-order system is based upon fractional calculus. The idea of calculus dealing with fractional orders sprouted in the mind of Gottfried Wilhelm Leibniz, a prominent German mathematician. In 1695, in a letter to his contemporary mathematician L’Hôpital from France, Leibniz discussed the idea of derivatives with nonintegral order as a generalization to the derivatives with integer orders [5]. The 300-year-old conversation between the two great mathematicians on the noninteger-order derivative has been an ongoing topic that is now known as fractional-order calculus or simply fractional calculus. Later, in 1823, the first-ever application of fractional-order calculus was proposed by Abel in which he presented the solution of a problem via an integral in the form of a derivative with order equal to one half [32]. In fractional-order calculus, the process of differentiation and integration is considered as a generalization of ordinary differentiation and integration with an arbitrary (noninteger) order. The application of fractional calculus has remained somewhat dormant over the years in the domain of system analysis. However, with the advancement in today’s computers, which are capable of handling complex calculations, fractional-order systems analysis has now become very popular. A comprehensive survey on fractional-order calculus and its application in the engineering domain may be found in [8]. The preference of fractional calculus for solving modern-day control and modeling problems may be observed in the literature due to the extra degree of freedom present in it when compared to its integral-order counterpart. This extra degree Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00018-4 All rights reserved.

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of freedom is always available with fractional calculus because it considers nonintegral values as the order of derivatives and integrals. In system engineering domain, the systems are mathematically given by a transfer function which represents the linear time-invariant (LTI) dynamics [34]. The number and magnitude of poles and zeros in the transfer function carry significant information about the system dynamics as they are responsible for the transient and the steady-state behavior. In a transfer function, the poles and zeros are simply the roots of the denominator and numerator polynomials, respectively. These polynomials have integral exponents on the complex variable “s,” where the highest integral power in the denominator expression is known as the order of the system. However, with the introduction of fractional calculus, the system order is not restricted to integral values and thus, fractional-order systems have come into the picture. Simply ignoring the fractional order is as ridiculous as ignoring the existence of fractional-valued numbers in mathematics. The systems showing signs of fractional-order dynamics are categorized as fractionalorder systems (FOSs). The transfer function that represents such systems is known as fractional-order transfer function (FOTF). The concept of fractional calculus has become an indispensable part of system control and identification in the field of science and engineering. These days, the conventional integral order of the process controllers is being shifted towards fractional orders on account of better system control and overall performance [32]. Typical proportional integral derivative (PID) controllers have now become fractional PID controllers [46]. This is mainly due to the additional degree of freedom that fractional-order dynamics possess inherently. As far as modeling and identification of systems with fractional order, several significant contributions have been made so far [9,15,19]. Especially, the identification of bio-impedance models of systems is based upon the idea of fractional-order models [43]. Literature also suggests that there has been a trend to represent the electrochemical systems with fractional-order models and circuits [29,35]. One such bioimpedance model is a Cole–Cole model [1,31], which is widely used to model biological as well as electrochemical systems. Significant progress has also been made towards the development of optimization algorithms to estimate parameters of fractional-order models. The parameter estimation of the fractional-order impedance models has been made possible by both conventional and bio-inspired algorithms [44,45]. Considering the scope of representing an electrochemical system with a fractional-order model, this work is an attempt to find an FOTF for a

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system helpful in extraction of taste-related information from liquid analyte. This chapter presents the development of an FOTF and subsequent synthesis of the equivalent circuit to fit the time-domain response of a voltammetric electrochemical system. The estimation of an FOTF and an equivalent fractional-order circuit for such a system will help investigate those characteristics of analyte which were left unexplored using integralorder models and equivalent electrical circuits [21,24,26]. In this work, system identification has been applied to an experimental dataset to find a fractional-order model for an electrochemical system working on pulse voltammetry. In pulse voltammetry, a train of pulse voltage is applied as an input to the electrochemical system while the current response is measured for analysis. This response current is referred to as a voltammetric response. When an array of electrodes is used as electrochemical sensors for extraction of taste-related information from a liquid (food) sample, the setup is called an electronic tongue. In this work, a voltammetric electronic tongue has been used with five working electrodes, a detail of which is presented in Section 13.2 of this chapter. This electronic tongue was developed mainly for the analysis of black tea samples in previous works [10,11]. Therefore, the analysis presented in this chapter is restricted to the voltammetric response obtained from various black tea samples. As far as the existing work on the modeling of the voltammetric system is concerned, the integral-order transfer function (IOTF) model was identified and utilized in the synthesis of equivalent circuits [25,26]. The circuits obtained earlier were linear, which advocated the presence of linear dynamics in the systems with all the circuit elements behaving ideally. However, a certain amount of model misfit was also observed that highlighted the limitations of the proposed equivalent circuit with linear passive elements. The linear models of the electrochemical system are mainly based upon the Randles equivalent circuit [17], which consists of linear passive electrical elements. In the circuit, the solution resistance and charger transfer phenomena are represented by resistors. On the other hand, the formation of a double layer on the electrode–analyte interface is explained by a capacitor in the Randles circuit. However, the literature survey on the redox phenomenon of the electrochemical system suggests that the double layer capacitance developed on the electrode–analyte interface is considered as an imperfect capacitor [28,33,41]. This is because a pure capacitor often fails to fit the experimental data while modeling the impedance of electrochemical systems. The effects of nonidealities are even more during the diffusion process. However, this phenomenon is represented by a re-

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sistance in the Randles equivalent circuit accounting for only the Faradaic process involved in the electrochemical reaction [17]. Therefore, to model the true voltammetric response contributed by these imperfect elements, the fractional-order equivalent circuit should be introduced and hence, the study and identification of FOSs seem an obligation. The circuit elements with fractional orders have been termed as fractance or fractor elements [2,30]. Such elements sound imaginary but the fact is that they are even being developed in laboratories [6,7,19]. Often, these fractional elements produce a better circuit analogy to represent fractional-order systems when compared with the ideal elements like resistors, inductors, and capacitors [4,6,18]. A generalization of electrical passive elements in terms of fractional elements has been shown by [37] in which the resonance and the quality factor of the fractional circuits were analyzed. As far as the equivalent network of a fractional-order system is concerned, recently, the network synthesis using FOTF has been illustrated in detail [27]. The proposed fractional-order equivalent circuit in this chapter utilizes such a prevailing idea of representing the voltammetric system with the fractional circuit elements by applying the method of network synthesis on the FOTF identified in this work. This work is intended to use a similar methodology adopted in [25] for estimation of an equivalent circuit but with fractional-order elements. The identification technique has been implemented on the dataset obtained from the experimental setup of the multielectrode voltammetric system presented in Section 13.2. The preliminary discussions on the FOTF and fractional circuit elements are presented in Section 13.3. In Section 13.4, the fractional-order system identification techniques are discussed, followed by the derivation of the proposed fractional-order electrical circuit in Section 13.5. The application of this work towards the domain of electronic tongues is illustrated using principal component analysis (PCA) in Section 13.6.

13.2. Experimental setup The electrode assembly of the voltammetric system includes an array of working electrode of noble metals, namely, gold, iridium, palladium, platinum, and rhodium. As a counterelectrode (CE), a stainless steel rod was used, while a standard Ag/AgCl was used as the reference electrode. The electrodes were coupled with a laboratory-made low-cost potentiostat and a commercial data acquisition (DAQ) card USB 6008 from National

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Figure 13.1 Diagram showing a voltammetric electronic tongue system with its potentiostat and the electrode array coupled to a computer (PC).

Instruments, which were also used in earlier works [11,12], to perform taste-related analysis on black tea samples. The experimental setup is shown in Fig. 13.1. The function of the potentiostat is to apply desired voltage pulses to the working electrode with respect to the references electrode. The internal circuitry of the potentiostat instrument allows the response current due to the electrochemical reaction to pass through the CE. The DAQ samples the incoming response current at the rate of 1000 samples per second and passes them to the computer for storage and data processing. A customized software interface on the PC has been developed in LabVIEW software which enables the user to define the parameters of the input voltage to be applied across electrodes. The electrodes are arranged in a circular fashion so that the distances between the reference electrode and each of the working electrodes are equal. This ensures the negligible effect of the geometrical arrangement of the cell on the current response. Therefore, the response is characterized only by the chemical properties of the analyte and the properties of the working electrode.

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Figure 13.2 The applied LAPV voltage signal to each working electrode sequentially.

In the experiment with the voltammetric system, 20 mL of tea liquor samples was used as the analyte and the whole electrode assembly was immersed into the analyte. The voltage signal applied to the working electrode with respect to the reference electrode is the large amplitude pulse voltammetry (LAPV) signal, which has been used earlier for tea quality estimation in [3,11,36]. In this experiment, the LAPV signal encompasses voltages in the range of −0.9 V to +0.9 V, as shown in Fig. 13.2 [25], and it has been demonstrated that the LAPV signal carries significant information as compared to other voltammetric excitation profiles [16] such as differential pulse voltammetry and square wave voltammetry. The LAPV voltage signal is applied to working electrodes sequentially, in the order of gold, iridium, palladium, platinum, and rhodium. The process of switching across successive working electrodes is controlled by an electrode switching circuit present in the potentiostat itself. A total of 7400 current data points from five working electrodes (1480 data points × 5 electrodes) are recorded in one observation and the same number of input data points are obtained from the applied LAPV signal. These data points fulfill the requirement of input-output dataset of the system to be used for the FOS identification purpose.

13.3. Fractional-order models In systems theory, the dynamical systems are generally modeled by differential equations which are ultimately governed by calculus that deals with integral orders. These equations are basically constituted by integral

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multiples of integral-order terms. However, the discussions in Section 13.1 reveal the fact that many systems are better modeled by differential equations with fractional orders. Such systems are called FOSs and the calculus used to represent the dynamics of these systems is fractional-order calculus. One such mathematical representation of an FOS is the FOTF explained in Section 13.3.1, which is actually the fractional-order counterpart of the transfer function for a typical LTI system.

13.3.1 Fractional-order transfer function In LTI systems, the mathematical function relating the output to the input of the system is called its transfer function. In the continuous-time domain, the transfer function of an LTI system with output y(t) and input u(t) can be represented as Y (s) bm sm + bm−1 sm−1 + · · · + b1 s + b0 , = G(s) = U (s) an sn + an−1 sn−1 + · · · + a1 s + a0

(13.1)

where integer n ≥ m (condition for a causal and physically realizable system); a0 , a1 , . . . , an−1 , an and b0 , b1 , . . . , bm−1 , bm are unknown coefficients. Here, G(s) is the transfer function of the system provided all initial conditions are zero. The highest power on the complex variable (s = σ + jω) in the denominator expression is called the order of the system (nth-order system). The order n (also m) takes up integer values and therefore, specifically, this transfer function is an IOTF. However, as far as the FOTF is concerned, the values of m and n are fractional. Such a transfer function that represents a system with fractional exponents on “s” is considered as FOTF and may be represented as GF (s) =

bm sβm + bm−1 sβm−1 + · · · + b0 sβ0 . an sαn + an−1 sαn−1 + · · · + a0 sα0

(13.2)

If a and b are considered as the coefficient vectors and na are nb the order vectors of the FOTF, then these vectors may be given by a = [an , an−1 , . . . , a0 ],

(13.3a)

b = [bm , bm−1 , . . . , b0 ],

(13.3b)

na = [αn , αn−1 , . . . , α0 ] ,

(13.3c)

nb = [βm , βm−1 , . . . , β0 ] .

(13.3d)

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13.3.2 Fractional-order circuit elements A system represented by an IOTF when decomposed into an equivalent circuit results in a circuit that contains pure electrical elements like resistor, inductor, and capacitor. But when it comes to an equivalent circuit of a system represented by an FOTF, the scenario does not remain the same. In this case, the circuit elements will not be ideal; instead, they become fractional elements or fractances. The constant phase element (CPE) is one such representation of circuit elements used to model fractional-order systems. During equivalent circuit modeling (Randles model) of electrochemical systems, an ideal capacitor element does not always fit the experimental data, which obligates the use of CPE as a replacement to the double layer capacitor. From this perspective, the CPE is considered as a generalized representation of a fractional-order circuit element with the ability to model the imperfect circuit components in practical scenarios. Any circuit element is represented by its magnitude of impedance and the phase angle. For instance, apart from their magnitude, a resistor has a phase angle of 0 degrees while the inductors and capacitors are characterized by a phase angle of ±90 degrees as applicable. Similarly, in the Laplace domain and with zero initial conditions, the impedance function of a CPE may be represented as ZCPE (s) = Qs−α ,

(13.4)

where Q is the coefficient related to the magnitude of the impedance and α is a fractional exponent. Here, the value of α makes a decision on the true characteristic of a CPE in an electrical circuit as shown in Table 13.1. The phase angle of the CPE is simply represented by φCPE = −90α degrees [6]. From Table 13.1, it is observed that the CPE may be changed to act as an ideal circuit element just by changing the phase angle. Therefore, the CPE may be seen as a global depiction of an electrical circuit element. Using Eq. (13.4), ZCPE =

Q (jω)α

.

(13.5)

,

(13.6a)

Thus, the magnitude of ZCPE is given by |ZCPE | =

Q ωα

and the phase angle is given by ∠ZCPE = (−90α)◦ .

(13.6b)

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Table 13.1 Generalization of CPE as ideal circuit elements on the basis of phase angle. α ZCPE (s) Element Symbol Phase angle (φ CPE )

Fractional value

Qs−α

CPE

(−90α )°

−1

Qs

Inductor, Q

+90°

0

Q

Resistor, Q

0°

+1

Q/s

Capacitor, 1/Q

−90°

When the value of α is exactly 0.5, the impedance takes the form of the Warburg element [17], which is widely used to fit the experimental data. Thus, in the case of a Warburg element, the phase angle becomes −45 degrees. Based upon this, it may be concluded that the Warburg element is a special type of CPE used in modeling predominantly when the fractional order is exactly 0.5. Any other fractional order value is therefore modeled by a CPE in the system’s equivalent circuit. As mentioned earlier, the models of electrochemical systems are mainly based upon the Randles circuit and its transfer function equivalents. As shown in Fig. 13.3, the Randles circuit is a combination of resistors that represents the electrolytic resistance (R ) cascaded in series with a parallel combination of charge transfer resistance (Rct ) and a double layer capacitor (Cd ). However, the electrochemical experiments are not so simple and involve complications on account of some imperfect phenomena such as diffusion and charge transfer leading to imperfections in the circuit elements like Rct and Cd . The applicability of the fractional entities lies in modeling such imperfections in the equivalent circuit models of the electrochemical systems. Next, in Sections 13.4 and 13.5, the identification of FOTF and the estimation of its equivalent circuit are presented for a voltammetric system with various tea samples.

13.4. Identiﬁcation of fractional-order transfer function This section presents the complete procedure adopted in this work for the identification of a fractional-order model using the applied input and the data obtained from the voltammetric system experimentally. Firstly, the model structure is defined and secondly, the procedure of parameter optimization is carried out to model the experimental response of the system.

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Figure 13.3 Randles equivalent circuit in ideal cases.

Apart from characterizing the system, these parameters carry information about the tea liquor samples as well. It is also envisaged that with a given working electrode material and tea sample, the structure of the fractionalorder model proposed in this work does not vary while the parameters change with tea samples. These variations in the parameters convey information about the chemical matrix of tea samples. The input-output observations obtained during the experiment with the voltammetric system described in Section 13.2 are used in this work to identify the fractionalorder model of the system with various tea samples.

13.4.1 Structure of the proposed fractional-order transfer function The Randles equivalent circuit model for ideal electrochemical scenarios has been explored to find the initial model structure [22]. Further, in the Randles circuit shown in Fig. 13.3, let us assume that V (s) represents the applied voltage while the response current is given by I (s). Thus, the corresponding transfer function may be given by sCd Rct + 1 I (s) = , V (s) sCd R Rct + (R + Rct )

(13.7)

where R is solution resistance, Cd is double layer capacitance, and Rct is charge transfer resistance. Eq. (13.7) may be obtained by decomposing the impedance shown in Fig. 13.3 into the corresponding admittance function (V (s)/I (s)). The IOTF is written in a simplified form as 1 + Ps I (s) , = V (s) B + As where A = Cd R Rct , B = R + Rct , and P = Cd Rct .

(13.8)

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Here, Eq. (13.8) shows a one-pole-one-zero IOTF with the unit integral exponent on the complex variable (s) and represents a first-order model of the voltammetric system. In an earlier work [25], the model order was increased to 2 after adding a pole to the system transfer function shown by Eq. (13.8) so that it can better model the voltammetric response from tea samples. The significance of the fractional-order modeling lies in the fact that it will not restrict the exponent on “s” to attain an integral value and lead to higher-order dynamics. Instead, with an additional degree of freedom, the exponent may settle to some fractional value (α ) rather than acquiring a higher-order integral value. The structure of IOTF as given by Eq. (13.8) has been used as a basis to determine the proposed structure of the FOTF to model the voltammetric system under observation. The generalized FOTF as shown earlier in Eq. (13.2) has fractional values of exponents raised on “s.” Taking these two equations into consideration, the structure of the identified FOTF may be given by I (s) b0+1 sβ0+1 + b0 sβ0 , = V (s) a0+1 sα0+1 + a0 sα0 b0+1 sβ0+1 + b0 G(s) = a0+1 sα0+1 + a0 1 + (b0+1 /b0 )sβ0+1 . G(s) = (a0 /b0 ) + (a0+1 /b0 )sα0+1

G(s) =

(13.9a) (for β0 = 0; α0 = 0),

(13.9b) (13.9c)

Now, Eq. (13.9c) may be represented with notations similar to that of Eq. (13.8): G(s) =

I (s) 1 + Psx . = V (s) B + Asy

(13.10)

In Eq. (13.10), x and y are the unknown exponential terms of the FOTF with three additional unknowns coefficients; A, B, and P. An important observation is found when the identified structure of FOTF (Eq. (13.10)) is compared with that of the IOTF (one-pole-one-zero) given by Eq. (13.8). It seems as if the integral unit exponents on “s” in the case of an integral transfer function have been replaced by the fractional exponents in the identified FOTF. Among various possibilities, there may be a condition in which either x or y is kept fixed in the FOTF, but here, both x and y are treated as free parameters, just like A, B, and P, whose optimal values are obtained for best possible model performance.

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13.4.2 Parameter estimation The parameter estimation in this work has been carried out for the proposed FOTF shown by Eq. (13.10). Due to the unconventional fractional exponents, the ease of parameter optimization in FOTF is not similar to that for the integral-order model presented in Eq. (13.8). There is a constraint in writing the FOTF as the fractional exponent on “s” is not supported in the MATLAB® environment. To overcome this, a command called “fotf ” has been adopted from a fractional-order modeling and control (FOMCON) toolbox developed by Aleksei Tepljakov and his team [40]. This facilitated writing any FOTF structure in MATLAB so that further identification and optimization may be carried out. For a given set of the working electrode and the tea samples as an analyte, let us suppose that the input voltage (LAPV) and response current are given by v(t) and i(t), respectively. The parameters of the identified FOTF may be summarized as a coefficient vector denoted by σ = [A B P ] and an exponent vector denoted by θ = [x y]. The parameters of these vectors are obtained iteratively and the initial value assigned to each of them is 1. Let Gn (s) be the FOTF obtained after the nth iteration. After every iteration, it is assumed that the intermediate response of the model is in (t) and the residual error is evaluated as follows: en (t) = i(t) − in (t).

(13.11)

Here, the root mean square error between experimental and simulated responses is the cost function, J (σ, θ ), which is given as N 1 J (σ, θ ) = [en (t)]2 ,

N

(13.12)

1

where N is the number of data points in the waveforms of the inputoutput signal. In this work, the nonlinear least-squares solver with the Levenberg–Marquardt algorithm has been used for the optimization of model parameters. The solver evaluates the minimum of the cost function and returns optimal values of the parameter vectors σ and θ . In the end, the FOTF with the optimized value of parameters is simulated with the voltage input v(t) and the final response is recorded for performance analysis of the identified model.

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13.4.3 Results: performance evaluation of the identiﬁed FOTF The structure of IOTF is based upon the Randles equivalent circuit shown in Eq. (13.8) with three unknown parameters (A, B, and P). It should be noted that this transfer function represents a first-order system with one pole and one zero. The method of system identification is used to find the parameters of the IOTF. Model identification is done for various combinations of tea samples and working electrodes. The same degree of analysis and identification of parameters has been used for the identified FOTF given by Eq. (13.10), which is supposed to exhibit better model performance. In this regard, both IOTF and FOTF models are simulated by applying the LAPV signal as the input. The parameters of the identified FOTF are shown in Table 13.2 for various combinations of working electrode and tea sample (S1 to S5). The simulated waveforms with these parameters are compared with the experimental measurements obtained from the system in the case of various electrodes and tea samples. The comparisons of waveforms in the case of tea sample S1 for various working electrodes are shown in Fig. 13.4. In the case of tea sample S1 and gold electrode, it may be observed that the curve fitting obtained using the FOTF (82.22%) is much better than that of the IOTF (75.38%). Similarly, the superiority of FOTF over IOTF in terms of model fit may be observed with the rest of the working electrodes. These model fits are based on the normalized root mean square error (NRMSE) criterion [25]. The model fit is the measure of goodness of fit, which is calculated in this work as model fit = (1 − NRMSE) × 100,

n

NRMSE =

(ymeas_k − ymodel_k )2 , 2 k=1 (ymeas_k − ymeas )

k=1 n

(13.13a) (13.13b)

where ymeas_k is the measured system output (kth data point), ymodel_k is the estimated output (kth data point) of the model, ymeas is the mean of measured data points, and n is the number of data points in the measurement. The model fit in the case of all tea samples for various electrodes is presented in Table 13.3. The analysis based on the model fit clearly shows that the identified FOTF outperforms the IOTF for all electrode–analyte combinations. The box-plot shown in Fig. 13.5 also supports the efficacy of the identified FOTF over IOTF. In Fig. 13.5, individual boxes are the indication of the variation in NRMSE-based model fit (%) with various electrodes for a given tea sample. It may be observed that all the boxes concerning the FOTF lie above those of IOTF.

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Table 13.2 Parameters of the identiﬁed FOTF for various combinations of electrode and tea sample. Parameters of the identiﬁed FOTF shown in Eq. (13.10) Sample/working electrode A B P x y (S1, Au) 1594.59 167.57 1.04 × 103 0.558 0.975 0.543 0.983 (S1, Ir) 816.61 62.30 7.63 × 104 0.599 1.000 (S1, Pd) 573.78 66.27 3.56 × 104 (S1, Pt) 976.76 101.10 9.03 × 104 0.556 1.000 0.541 0.977 (S1, Rh) 596.65 46.64 5.81 × 104

(S2, Au) (S2, Ir) (S2, Pd) (S2, Pt)

1906.88 6651.78 427.71 571.34

138.69 378.51 41.88 44.17

1.26 × 103 8.95 × 103 2.41 × 104 4.66 × 104

0.548 0.454 0.627 0.563

0.960 0.948 1.015 0.996

(S2, Rh) (S3, Au) (S3, Ir) (S3, Pd) (S3, Pt) (S3, Rh)

893.05 2994.43 6981.42 507.24 708.10 1001.55

89.18 222.70 374.22 48.95 48.73 71.43

1.11 × 103 1.67 × 103 8.14 × 103 3.08 × 104 4.60 × 104 9.18 × 104

0.512 0.593 0.469 0.604 0.611 0.535

1.000 0.982 0.937 1.000 0.997 0.954

(S4, Au) (S4, Ir) (S4, Pd) (S4, Pt) (S4, Rh)

4221.32 5287.63 603.71 837.47 1042.58

305.27 309.74 65.47 61.78 78.79

2.20 × 103 5.90 × 103 4.25 × 104 5.83 × 104 9.30 × 104

0.582 0.454 0.550 0.574 0.519

0.982 0.945 1.000 1.000 0.963

(S5, Au) (S5, Ir) (S5, Pd) (S5, Pt) (S5, Rh)

3566.10 6164.01 634.84 828.30 835.39

291.45 408.75 50.70 58.48 75.24

2.18 × 103 8.73 × 103 2.44 × 104 6.43 × 104 1.08 × 103

0.585 0.444 0.696 0.573 0.494

0.986 0.950 1.011 0.989 0.971

The comparative study in this work has also been performed based on the mean squared error (MSE), which is given by 1 (ymeas_k − ymodel_k )2 . MSE = n k=1 n

(13.14)

The plot of residuals of the identified FOTF in comparison with that of IOTF identified for tea sample S1 with gold electrode is shown in Fig. 13.6. As expected, the FOTF shows clear performance improvement compared

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Figure 13.4 Comparison between the response of identiﬁed IOTF and FOTF (with fractional orders x and y, as shown in Table 13.2) for LAPV input with the experimental response obtained in the case of various working electrodes for tea sample S1.

to IOTF, especially at the response peaks. For the case shown in Fig. 13.6, the MSE obtained with FOTF is almost half of the MSE corresponding to IOTF. This is true for all the cases of tea samples, as evident from Fig. 13.7. The boxes for MSE related to FOTF lie at a much lower level compared to those of IOTF for all the tea samples. The model performance based on MSE also leads to the conclusion that FOTF has a better ability to model the experimental waveforms measured from various electrodes of the voltammetric system. The number of data points present in the response of these integraland fractional-order models for each electrode is equal to 1480. Thus, the

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Table 13.3 Model ﬁt based on NRMSE obtained for the identiﬁed models. Identiﬁed Name of Model ﬁt (%) for working electrodes models tea sample Gold Iridium Palladium Platinum Rhodium

Integralorder transfer function (IOTF)

S1 S2 S3 S4 S5

75.38 75.75 71.81 71.20 72.33

74.29 71.98 73.43 72.5 73.18

73.60 73.76 74.18 73.48 74.14

75.06 73.32 52.36 72.49 72.72

52.64 74.34 76.39 75.17 74.71

Fractionalorder transfer function (FOTF)

S1 S2 S3 S4 S5

82.22 84.49 78.52 78.48 79.27

86.89 92.66 91.97 91.61 90.71

79.36 82.42 81.39 81.63 80.70

83.90 86.80 85.24 85.50 84.34

82.02 86.23 85.78 85.62 84.76

Figure 13.5 Box-plots showing model ﬁt (%) for the identiﬁed models. For a given tea sample, each box represents the range of in model ﬁt values with working electrodes.

correlation study has been done for point-to-point comparison of such a large experimental response and the simulated responses of FOTF and IOTF. The degree of correlation has been represented by the correlation coefficient given by n

correlation coefficient = nk=1

(ymeas_k − ymeas )(ymodel_k − ymodel )

k=1 (ymeas_k

− ymeas )2 (ymodel_k − ymodel )2

,

(13.15)

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Figure 13.6 The residual plots of the model (FOTF and IOTF) responses obtained for tea sample S1 with gold electrode.

Table 13.4 Comparison of FOTF and IOTF on the basis of various performance parameters. Identiﬁed model Average correlation Average NRMSE Average MSE model ﬁt IOTF 0.9611 72.01% 5.66×10−11 A FOTF 0.9905 84.50% 1.49×10−11 A

where ymodel is the mean of data points (response) of the estimated model. In Fig. 13.8, for each tea sample, the ranges of the correlation values obtained from the identified models with various working electrodes are shown as box-plots. The box-plots clearly indicate that the correlations obtained with the FOTF are better than that of the IOTF, irrespective of the tea sample analyzed. As shown in Table 13.4, the average value of correlation calculated over various combinations of tea and electrode in the case of FOTF is 0.9905, which is less when compared to that of IOTF. The averages of the remaining performance parameters (MSE and model fit) are also presented in Table 13.4, which confirm the efficacy of the identified FOTF over the IOTF based on all the criteria of performance evaluation.

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Figure 13.7 Box-plots showing mean squared error (MSE) for the identiﬁed models. For a given tea sample, each box represents the range of MSE values with working electrodes.

Figure 13.8 Box-plots showing correlation between experimental response and model response of the identiﬁed models. For a given tea sample, each box represents the range of correlation values with working electrodes.

The results of the FOTF have been presented so far in comparison with the IOTF to validate the fact that the fractional models have better capability to model the voltammetric electrochemical system. Based on the various

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performance measures, the results establish that the identified FOTF has an upper hand compared to its integral-order counterpart (IOTF). Therefore, the fractional-order electrical circuit derived from the proposed FOTF may be considered valid. The synthesis of such fractional-order equivalent circuit and related analysis are presented in Section 13.5.

13.5. Proposed circuit with fractional-order elements An electrical circuit equivalent of an electrochemical system that is comprised of only resistors, capacitors, and inductors does not often follow the system’s real-world behavior. The inability of such circuits to represent an electrochemical reaction particularly in the case of the voltammetric system under analysis can be mitigated by the presence of fractal elements in the circuit. The CPE is the representation of such a fractal element which can act as a nonideal circuit element in practical scenarios. In this section we derive an equivalent electrical circuit with fractional-order elements using the identified FOTF of the voltammetric system [23].

13.5.1 Network synthesis for fractional-order circuit The electrical circuit with fractional-order elements may be obtained by applying a network synthesis procedure on the identified FOTF in Eq. (13.10) given by I (s) 1 + Psx . (13.16) = G(s) = V (s) B + Asy The driving point impedance function Z (s) is written as V (s) B + Asy , = I (s) 1 + Psx B Asy Z (s) = + ≡ Z1 (s) + Z2 (s). 1 + Psx 1 + Psx

Z (s) =

(13.17a) (13.17b)

In Eq. (13.17b), Z1 (s) and Z2 (s) represent the series impedances present in the equivalent circuit with total impedance equal to Z (s) as shown in Fig. 13.9(a). The corresponding admittances Y1 (s) and Y2 (s) are given as follows: 1 + Psx , B 1 + Psx . Y2 (s) = Asy Y1 (s) =

(13.18a) (13.18b)

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Figure 13.9 (a) Equivalent circuit showing lumped impedances Z1 (s) and Z2 (s) in series. (b) Final equivalent circuit with all the circuit elements.

From Eq. (13.18a), Y1 (s) =

1 Psx 1 1 + ≡ + , B B R ZCPE1

(13.19)

where R is a resistance and ZCPE1 is a CPE in the circuit. Thus, R = B, B ZCPE1 = s−x = QCPE1 s−x . P

(13.20a) (13.20b)

Here, R and ZCPE1 are parallel to each other. Similarly, the other admittance Y2 (s) from Eq. (13.18b) is written as Y2 (s) =

1 Psx 1 1 + ≡ + . y y As As ZCPE2 ZCPE3

(13.21)

The two parallel CPEs from Eq. (13.21) are given by ZCPE2 = Asy = QCPE2 sy , A ZCPE3 = sy−x = QCPE3 sy−x . P

(13.22a) (13.22b)

The final equivalent circuit with all the circuit elements is shown in Fig. 13.9(b). The synthesized equivalent electrical circuit has R and ZCPE1

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393

in parallel combination. The other series impedance consists of two CPEs, ZCPE2 and ZCPE3 , connected in parallel.

13.5.2 Analysis with fractional circuit parameters The estimation of circuit parameters has been done using the identified parameters of the FOTF in Section 13.5.1. The parameters (A, B, P, x, and y shown in Table 13.2) identified for the proposed FOTF are utilized to estimate the elements of the fractional-order circuit. For the circuit shown in Fig. 13.9(b), the resistance (R) is calculated using Eq. (13.20a) and the three CPEs are calculated with the help of Eqs. (13.20b), (13.22a), and (13.22b). The circuit parameters shown in Table 13.5 have been calculated for all tea samples with various working electrodes. As mentioned in Section 13.3.2, the phase angle of a CPE may be calculated as φCPE = −90α degrees. Likewise, for CPEs listed in Table 13.5, the phase angles have been estimated from the values of x and y using Eqs. (13.23a), (13.23b), and (13.23c) as φCPE1 = (−90x)◦ , ◦

φCPE2 = (−90y) , φCPE3 = (−90(y − x))◦ .

(13.23a) (13.23b) (13.23c)

The phase angles are the indication whether a given CPE is dominated by a resistive, capacitive, or inductive effect in the proposed fractional equivalent network. Therefore, to recognize the actual behavior of these CPEs, their phase angles (φCPE1 , φCPE2 , φCPE3 ) are calculated for five tea samples (S1 to S5) with each working electrode. As a result, 25 such phase angles are obtained in total for one CPE, which are shown as a stem plot in Fig. 13.10, which shows very interesting results. In Fig. 13.10, all phase angle values (blue; dark gray in print version) for ZCPE1 are located in the region between 0 degrees and −90 degrees and none of them even touches the line of the pure capacitor (φCPE = −90 degrees). This concludes that ZCPE1 is not an ideal circuit element, but it actually is an imperfect capacitor. This is found true for all the cases with various combinations of tea sample and working electrode. Conversely, the phase angles (orange; light gray in print version) associated with ZCPE3 lie between 0 degrees and +90 degrees. All of them approach the line of pure inductor, which confirms that the behavior of ZCPE3 is similar to that of an imperfect inductor. A similar analysis for ZCPE2 is also worth mentioning as the phase angles (green; mid gray in print version) corresponding to it lie almost on the pure inductor line

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Table 13.5 Estimated parameters of the proposed fractional-order electrical circuit. CPE1 CPE2 CPE3 Sample/working R (k) electrode Q φ Q φ Q φ (S1, Au) (S1, Ir) (S1, Pd) (S1, Pt) (S1, Rh)

167.57 62.30 66.27 101.10 46.64

1.61 × 105 8.17 × 104 1.86 × 105 1.12 × 105 8.03 × 104

CPE1

CPE1 −50.18 −48.91 −53.93 −50.07 −48.72

1594.59 816.61 573.78 976.76 596.65

CPE2

87.77 88.48 89.98 89.99 87.97

CPE2

1533.28 1070.36 1613.47 1080.82 1027.49

CPE3

37.59 39.57 36.05 39.92 39.26

CPE3

(S2, Au) (S2, Ir) (S2,Pd) (S2, Pt) (S2, Rh)

138.69 378.51 41.88 44.17 89.18

1.10 × 105 4.23 × 104 1.74 × 105 9.47 × 104 8.00 × 104

−49.34 −40.83 −56.47 −50.68 −46.11

1906.88 6651.78 427.71 571.34 893.05

86.36 85.33 91.36 89.63 89.99

1506.93 743.83 1776.82 1225.11 800.84

37.02 44.51 34.89 38.96 43.88

(S3, Au) (S3, Ir) (S3, Pd) (S3, Pt) (S3, Rh)

222.70 374.22 48.95 48.73 71.43

1.33 × 105 4.60 × 104 1.59 × 105 1.06 × 105 7.78 × 104

−53.41 −42.24 −54.33 −55.00 −48.19

2994.43 6981.42 507.24 708.10 1001.55

88.39 84.35 90.00 89.76 85.83

1786.12 858.01 1648.20 1545.45 1090.23

34.97 42.11 35.66 34.76 37.64

(S4, Au) (S4, Ir) (S4, Pd) (S4, Pt) (S4, Rh)

305.27 309.74 65.47 61.78 78.79

1.39 × 105 5.25 × 104 1.54 × 105 1.06 × 105 8.47 × 104

−52.41 −40.82 −49.53 −51.70 −46.70

4221.32 5287.63 603.71 837.47 1042.58

88.38 85.07 90.00 89.99 86.65

1917.08 897.02 1417.37 1431.10 1121.22

35.98 44.25 40.47 38.30 39.95

(S5, Au) (S5, Ir) (S5, Pd) (S5, Pt) (S5, Rh)

291.45 408.75 50.70 58.48 75.24

1.34 × 105 4.68 × 104 2.08 × 105 9.10 × 104 6.94 × 104

−52.64 −39.96 −62.61 −51.56 −44.48

3566.10 6164.01 634.84 828.30 835.39

88.72 85.47 91.02 89.00 87.41

1644.13 706.07 2609.07 1288.65 770.46

36.08 45.50 28.41 37.44 42.93

(φCPE = +90 degrees). This indicates that ZCPE2 has characteristics almost similar to those of an inductor in the final circuit. Therefore, the closing statement for the identified fractional-order electrical network is that the resistance R is connected in parallel with an imperfect capacitor (CPE1). CPE2 is nearly an inductor, while CPE3 is an imperfect inductor, and both of them are connected in parallel (Fig. 13.9(b)). The appearance of an inductor in the proposed equivalent circuit for the electrochemical system is not often observed. Additionally, using an inductor to model an electrochemical system is sometimes considered doubtful due to the association of an inductor with the magnetic field. However, there are research reports in which inductors have been used to model the electrochemical systems. Almost all of the electrochemical reaction systems

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Figure 13.10 Estimated values of the phase angles of CPEs of the proposed fractionalorder equivalent circuit.

require the presence of one or more adsorbed intermediate species (either charged or uncharged) to explain the results. A work presented in [39] has considered the inductances as an essential component in the equivalent circuit models of adsorption-reaction kinetics which are basically inertial systems. In such electrochemical phenomena, the Faradaic impedance presented in [39] also comprises an inductive component that represents the impedance associated with the adsorbed intermediates. Thus, it has been shown that the inductors are an appropriate choice of representation in equivalent circuit modeling of electrochemical reactions. In other words, the inertia of the intermediate adsorbing species corresponds to the inductance. It may also be stated that inductors are readily used to model inertial systems like mass–spring–damper systems. Similarly, to model an adsorption-reaction process which is also an inertial system, the inductor may be used as an analogy even without the presence of any associated magnetic field [39]. The work proposed in [3,14] also reports a fractional inductor in the circuit model of an electrochemical system. These discussions justify that inductors may be used to model the electrochemical systems and should be treated as an analogy rather than associating the inductor with the magnetic field. Also, the proposed work here introduces a fractional inductor which itself does not have the typical property of a pure inductor.

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13.6. Principal component analysis: towards electronic tongue application An intrinsic characteristic of a voltammetric electrochemical system is to generate a uniquely distinguishable current response for the different liquid samples. This property of voltammetric systems is extensively exploited in the area of the electronic tongue. The preliminary analysis is done mainly by using data clustering and visualization techniques. One such technique is PCA, which is used extensively with the electronic tongue to assess the clustering ability of food samples (in liquid medium) under observation [36,42]. PCA [20] is a statistical technique generally implemented on multivariate data obtained from different sources for visualization in reduced dimension and clustering analysis. It reduces the data dimension and transforms them into rich information which is projected along the direction of maximum data variance. These directions are called principal components (PCs). Therefore, as per the philosophy, the response of the proposed model should produce distinct clusters in a PCA plot for various tea samples. This is an indication of dissimilarity in the nature of the analyte as they represent their unique characteristics. The PCA score plot is said to produce better clusters if there is less intracluster distance and more intercluster distance, and the measure of separation of the clusters is denoted by the separation index. The higher the separation index magnitude, the better is the clustering in PCA. The separation measure has been calculated on the basis of the Fisher criterion [13] for discrimination of two classes of clusters. A similar separation measure was used earlier to quantify the level of separation among the clusters observed in a PCA plot when an electronic tongue was used to analyze various tea samples [36,38]. To assess the ability of the proposed fractional-order model to produce separable clusters for different tea samples, PCA has been applied to the model response. In this work, the entire simulated response waveforms obtained from the identified system have been used to extract quality information from the tea samples. This particular study will affirm whether the LAPV responses of the identified fractional-order model do contain characteristic information of the tea samples leading to separable clusters. Therefore, the feature set is now the entire model response, which consists of 7400 data points per observation. The PCA plot showing the relative position of the clusters corresponding to various tea samples is presented in Fig. 13.11 in the case of the fractional-order model. It is visible from the

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Figure 13.11 PCA score plot obtained with simulated waveforms of the fractionalorder model showing distinct clusters for ﬁve tea samples.

PCA plot that these clusters are clearly distinguishable for five tea samples (S1 to S5), which is also supported by the very high value of the separation index (55.52). When the similar analysis was done with the integral-order model, the PCA plot displayed overlapping clusters (Fig. 13.12). These are said to be overlapping clusters as some of the observations from a few tea samples fall in the clusters of other tea samples. In Fig. 13.12, tea samples S2 and S5 show condensed clusters but the rest of the tea samples show overlapping behavior. As a result, the separation index of the clusters produced with the integral-order model falls to only 1.19. The higher value of the separation index in the case of the proposed fractional model confirms its ability to produce more separable clusters than its integral-order counterpart. Additionally, in the 2D PCA plot, the percentage variability explained (92.81%) in the response data obtained from the proposed fractional model is also on the higher side. The cluster analysis using PCA, therefore, confirms that the proposed model is able to better mimic a voltammetric electronic tongue system. These display the supremacy of the proposed model and confirm that the model can capture the intrinsic property of a voltammetric electronic tongue to produce distinct clusters for various tea samples.

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Figure 13.12 PCA score plot obtained with simulated waveforms of integral-order model showing overlap of clusters for ﬁve tea samples.

A good clustering tendency of the simulated waveforms corresponding to various tea samples is an indication of better classification and regression potential of the voltammetric system. With the help of this, classification and pattern recognition algorithms may be implemented to classify the tea samples, which has been an utmost priority for electronic tongues. The regression algorithm may also be implemented on the simulated data for the prediction of biochemical markers in black tea samples. As a future scope, regression analysis may be carried out by training the statistical models using a large set of experimental data along with the corresponding target biochemical markers. After successful training, testing data may be obtained from a set of independent samples and used as input to the trained model to get an estimation of the actual biochemical markers in tea. Such analysis may be extended to other liquid food samples as well. As future work, the methodology may be used to classify and quantify the food samples with the help of an electronic tongue. Further efforts may also be employed towards advancement in system design. In this direction, the geometrical arrangement of the electrodes of the array may be altered for optimal performance of the electronic tongue system. Additionally, research towards the selection of appropriate electrode material may also be employed in the future to use voltammetric systems electronic tongues.

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13.7. Conclusions The inclusion of fractional-order elements in the equivalent electrical circuit of the voltammetric system has been the key contribution of this chapter. The work estimated an FOTF for the system, followed by a typical procedure of network synthesis to obtain the equivalent fractionalorder circuit for various tea samples and electrode materials. This chapter illustrated an endeavor to model the voltammetric system with the help of FOS identification using the experimental system response in the time domain. Based on various model performance measures, the proposed FOTF model has been found preferable over its integral-order counterpart. The average value of the model fit calculated over various tea samples for FOTF is 84.50%, in contrast to a fit of 72.01% in the case of IOTF. Therefore, the FOTF results in a true model of a voltammetric system as it has a better ability to explain the imperfect nature of the redox reactions that occur at the electrode–analyte surface. As far as the contribution of this chapter towards estimation of fractional-order equivalent circuits is concerned, the proposed FOTF has been used for network decomposition. The resultant fractional circuit consists of a resistance R in parallel with a CPE denoted by CPE1. The other two CPEs connected in parallel are denoted by CPE2 and CPE3 and they are connected in series with the parallel connection of R and CPE1. The phase angles of the CPEs identified in this work demonstrated the true behavior of the fractional-order circuit elements. CPE1 and CPE3 behaved like an imperfect capacitor and imperfect inductor, respectively. On the other hand, the CPE2 exhibited characteristics similar to those of a pure inductor. PCA was applied to the model response of various tea samples, and it was observed that the fractional-order model produces the more distinguishable clusters in the PCA plot. This behavior of the proposed fractional-order model has a much closer resemblance to that of an electronic tongue system. This work on fractional-order identification therefore finds a potential application in the domain of the electronic tongue.

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CHAPTER FOURTEEN

Higher-order fractional elements: realizations and applications Neeta Pandey, Rajeshwari Pandey, and Rakesh Verma Department of Electronics and Communication Engineering, Delhi Technological University, Delhi, India

14.1. Introduction Fractional-order circuit and system design has gained significant research attention in the recent past due to the extra degree of freedom provided by fractional-order elements (FOEs). This has created a huge opportunity to investigate design flexibility, which is not possible in the narrow and finite subset of conventional integer-order circuits. Thus, the fractional-order approximations available in the literature are being extensively used in areas like signal processing, control systems, biomedical instrumentation, and many others [1–4]. A fractional order Laplacian operator is used to represent fractional-order circuits and systems. The Laplace transformation method of fractional-order operators can easily be simplified into an integer approximation form where it is made physically realizable with FOEs. The impedance of an FOE can be represented as Z (s) = Fsα [5], where the coefficient F is called fractance and α denotes the exponent of FOE and is known as the order of the FOE. The value of α is a fractional number and may assume any fractional value in the range −n < α < n, where n is an integer number. The impedance function with positive value of α represents impedance of a fractional inductor (FI), whereas a negative α -value corresponds to impedance of fractional-order capacitor (FOC). Thus, the magnitude and phase angle of an FOE are given by |Z | = F ω α , π , θ =α

2

(14.1a) (14.1b)

where the SI unit of F is /sα . Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00019-6 All rights reserved.

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The phase of an FOE is independent of frequency; hence, the FOE is also referred as the constant phase element (CPE). In literature the FOEs having their fractional order in the range −1 < α < 1 are also termed lower-order FOEs, whereas those having α as a compound value are called higher-order FOEs. In the field of circuits and systems, the FOEs are the most basic fractional-order systems (FOSs) which facilitate the development of further complex FOSs. FOEs are not commercially available and research on hardware realization of FOEs [6–20] is still in the nascent stage. However, a variety of rational approximations [15,21–23] are available in literature for FOC (−1 < α < 0) behavior emulation. These approximations provide a finite-order transfer function (TF) that can be realized with RC networks using methods such as Cauer and Foster realizations or by using a functional block diagram (FBD)-based approach. Once we realize FOC, the FI (0 < α < 1) can be emulated using various active RC methods similar to their integer-order counterparts. Alternately, the FOEs can also be realized using active blocks such as operational amplifiers (Op-Amps), current mirrors, current feedback operational amplifiers (CFOAs), and operational transconductance amplifiers (OTAs) [6,24–31], along with few passive components. It has been established in the literature that whenever the order of the FOS increases, the system accuracy increases [16]. This leads to design exploration of higher-order FOSs. The higher-order fractional-order circuits can be designed using different approaches, and system design using higherorder FOEs is one among those. Therefore, this chapter presents a detailed description of various realization methods [30–32] of higher-order FOEs and their application in designing higher-order filters. The passive realization methods for higher-order FOEs essentially require an FOC of order −1 < α < 0. Therefore, a brief description of realization methods of FOEs with fractional order < 1 is presented first.

14.2. Realization of FOEs with fractional order < 1 The fractional-order circuits are realized either through physically implemented FOCs [33–41] or through emulated FOCs [10,27–29,42–51] based on various structures such as passive resistor-capacitor (RC) elements arranged in the form of an RC tree [5,12], a cross RC ladder [52],

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a domino ladder [52], etc. The passive RC networks are obtained on the basis of different approximations such as Oustaloup recursive approximation [21], Carlson approximation [21], Matsuda approximation [21], Chareff approximation [21], continued fraction expansion (CFE) [21], modified Oustaloup [22], and El-Khazali reduced-order approximations [23]. Alternatively, the FOEs can also be emulated using an FBD-based approach and are implemented using active blocks and resistive components. The following subsections describe both passive and active emulation methods of FOEs.

14.2.1 CFE approximation-based FOC emulation The CFE approximation method is used widely in the literature for FOC emulation and therefore is briefly described here. The CFE method uses series expansion of (1 + x)α [21] as given by (1 + x)α =

1 α x (1 + α) x (1 − α) x (2 + α) x (2 − α) x . 1− 1+ 2+ 3+ 2+ 5 + ...

(14.2)

On the substitution of x = (s − 1), an integer-order approximation form with infinite terms for the fractional-order Laplacian operator sα is obtained. Depending upon the accuracy requirement we may retain a finite number of terms for representing sα . The order of CFE approximation of sα depends upon the power of s in the integer-order approximation forms. If terms up to sn are retained, it is termed nth-order approximation. Using this method, first-, second-, third-, and fourth-order approximation functions [21] are presented by (14.3a)–(14.3d), respectively, for the ease of comprehension: (1 + α) s + (1 − α) , (1 − α) s + (1 + α) 2 α + 3α + 2 s2 + −2α 2 + 8 s + α 2 − 3α + 2 α , s = 2 α − 3α + 2 s2 + −2α 2 + 8 s + α 2 + 3α + 2 3 α + 6α 2 + 11α + 6 s3 + −3α 3 − 6α 2 + 27α + 54 s2 3 + 3α − 6α 2 − 27α + 54 s + −α 3 + 6α 2 − 11α + 6 α , s = 3 −α + 6α 2 − 11α + 6 s3 + 3α 3 − 6α 2 − 27α + 54 s2 + −3α 3 − 6α 2 + 27α + 54 s + α 3 + 6α 2 + 11α + 6

sα =

(14.3a) (14.3b)

(14.3c)

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4 α + 10α 3 + 35α 2 + 50α + 24 s4 + −4α 4 − 20α 3 + 40α 2 + 320α + 384 s3 + 6α 4 − 150α 3 + 864 s2 + −4α 4 + 20α 3 + 40α 2 − 320α + 384 s 4 + α − 10α 3 + 35α 2 − 50α + 24 . sα = 4 α − 10α 3 + 35α 2 − 50α + 24 s4 + −4α 4 + 20α 3 + 40α 2 − 320α + 384 s3 4 + 6α − 150α 3 + 864 s2 + −4α 4 − 20α 3 + 40α 2 + 320α + 384 s 4 + α + 10α 3 + 35α 2 + 50α + 24

(14.3d) In general, any nth-order CFE approximation function represents an impedance function and may be physically implemented with an RC ladder network using a partial fraction method. Thus, a CFE-approximated FOC can be obtained by using the reciprocal of respective nth-order function. The realization of a fourth-order approximation form for FOC is explained below. The impedance function of FOC is obtained by the reciprocal of (14.3d) as given by 4 α − 10α 3 + 35α 2 − 50α + 24 s4 + −4α 4 + 20α 3 + 40α 2 − 320α + 384 s3 4 + 6α − 150α 3 + 864 s2 + −4α 4 − 20α 3 + 40α 2 + 320α + 384 s + α 4 + 10α 3 + 35α 2 + 50α + 24 1 . = 4 sα α + 10α 3 + 35α 2 + 50α + 24 s4 + −4α 4 − 20α 3 + 40α 2 + 320α + 384 s3 4 + 6α − 150α 3 + 864 s2 + −4α 4 + 20α 3 + 40α 2 − 320α + 384 s 4 + α − 10α 3 + 35α 2 − 50α + 24

(14.4) This impedance function may be realized by the partial fraction method outlined in [53,54]. The domino RC ladder network of Fig. 14.1 is one of the realizations used widely by researchers to verify the proposals. The impedance function of Fig. 14.1 is given by Z C α = Ra +

1/Cb 1/Cc 1/Cd 1/Ce + + + . s + Rb1Cb s + Rc1Cc s + Rd1Cd s + Re1Ce

(14.5)

The poles of the TF of (14.4) determine the normalized component values of the network. The desired value of FOC having scaled frequency ωc can be determined with the help of magnitude (km ) and frequency (kf ) scaling factors, giving the following relationships between unscaled and scaled

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Figure 14.1 A fourth-order domino RC ladder circuit [55].

component values: Rs = km . R and Cs =

C , kf km

(14.6)

where (R, C ) and (Rs , Cs ) are unscaled and scaled component values, respectively. The scaling factors km and kf are given by km =

1 Cα ωcα

(14.7a)

,

kf = ωc .

(14.7b)

Thus, the desired scaling impedance function can be written as Z C α |s = R1 +

1/C2 s+

1 R2 C2

+

1/C3 s+

1 R3 C3

+

1/C4 s+

1 R4 C4

+

1/C5 s + R51C5

.

(14.8)

Comparing (14.6) and (14.8), the scaled components (Ri , Ci ; i = 1..5) may be obtained from unscaled components (Rj Cj ; j = a, b, c , d, e) as R 1 = Ra · k m , R2 = Rb · k m , R3 = Rc · k m , R4 = Rd · k m , R5 = Re · k m ,

C2 = Cb / kf km , C3 = Cc / kf km , C4 = Cd / kf km , C5 = Ce / kf km .

(14.9)

The magnitude and phase of the FOC impedance function are 1/(ωα Cα ) and −απ/2, respectively. Therefore, the magnitude response of the FOC of order α would show a negative slope of −20α log10 ω , while its phase remains constant. To illustrate this further, three FOCs of value 1 µF/sα each, varying in order (0.1, 0.5, and 0.9) with a scaled frequency of 1 kHz, are designed and

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Table 14.1 Component setting of FOC. Order R1 () R2 () R3 () R4 () 0.1 274.7k 81.9k 56.1k 66.3k 0.5 1.402k 3.17k 4.78k 11.2k 0.9 2.6 16.5 49.9 255.9

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R5 () 154.1k 92.9k 55789.8

C2 (nF) 0.165 6.64 1846

C3 (μF) 0.0015 0.023 2.97

C4 (μF) 0.0052 0.043 2.69

C5 (μF) 0.015 0.055 0.544

Figure 14.2 Simulated and theoretical (a) magnitude and (b) phase responses for FOC.

Figure 14.3 Error in simulation results for (a) magnitude and (b) phase responses for FOC.

simulated using SPICE. The computed component values of the domino RC ladder network of Fig. 14.1 are listed in Table 14.1 [16]. The corresponding theoretical and simulated magnitude and phase responses are shown in Fig. 14.2. Further error plots are drawn to visualize the deviation between theoretical (dashed line) and simulated (solid line) responses as shown in Fig. 14.3. A deviation of ±1.5 dB is observed between simulated and theoretical magnitude responses of FOCs of orders 0.1, 0.5, and 0.9 in the frequency ranges of 2.7 Hz to 365 kHz, 9 Hz to 105 kHz, and 4 Hz to 230 kHz, respectively. The simulated phase deviates from the theoretical phase by ±0.30 degrees for FOCs of orders 0.1, 0.5, and 0.9 in the frequency ranges of 170 Hz to 6.8 kHz, 180 Hz to 5.5 kHz, and 150 Hz to 7.5 kHz, respectively.

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Figure 14.4 The impedance inverter circuit (IIC). (a) Integer domain. (b) Fractional domain.

Figure 14.5 OTA-based (a) impedance inverter circuit [56] and (b) generalization in the fractional domain [30].

14.2.2 FI emulation An impedance inverter circuit (IIC) gives input impedance, which is inversely proportional to the impedance connected at its other end as shown in Fig. 14.4(a). It is primarily used for simulating inductors for IC applications via inverting capacitive reactance and impedance matching circuits. This method may be extended to obtain fractional-order inductors by using the fractional-order capacitance as depicted in Fig. 14.4(b). The fractional-order capacitance can be realized using the method outlined in Section 14.2.1. The schematic of OTA-based IIC [56] is shown in Fig. 14.5(a). The input impedance of the circuit is given as Zin =

1 , gm1 gm2 Z1

(14.10)

where gm1 and gm2 correspond to transconductances of OTA1 and OTA2, respectively. The circuit of Fig. 14.5(a) can be generalized to the fractional domain by replacing Z1 by Z1α , where Z1α represents the fractional-order impedance

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of order α . The resulting circuit is shown in Fig. 14.5(b), and its input impedance is expressed as Zinα =

1 gm1 gm2 Z1α

(14.11)

.

An FI of order α may be obtained from (14.11) if Z1α corresponds to an FOC of order α . The impedance of the resulting FI can be expressed as Zinα =

sα C α . gm1 gm2

(14.12)

14.2.3 Functional block diagram-based emulation The FOEs can also be realized using an FBD-based approach. This approach [29] relies on the substitution of the Laplacian operator sα (0 < α < 1) by an equivalent integer (m)-order approximation form. The TFs of the fractional-order differentiator and integrator, with scaling frequency ω0 = 1/τ , can be expressed as H (s)FOD = (sτ )α , H (s)FOI =

1 (sτ )α

(14.13a) (14.13b)

.

Now (sτ )α can be expressed using CFE approximation as given by (14.2). A higher order of approximation offers a wider frequency range of operation but the number of elements used for the realization of (sτ )α will be higher. A second-order approximation of (sτ )α as given by (14.14) is considered for further elaboration: 2 α + 3α + 2 (sτ )2 + 8 − 2α 2 (sτ ) + α 2 − 3α + 2 . (sτ ) = 2 α − 3α + 2 (sτ )2 + 8 − 2α 2 (sτ ) + α 2 + 3α + 2 α

(14.14)

Substitution of (sτ )α from (14.14) in (14.13a) and (14.13b) yields a generic TF for both FOD and FOI expressed as (14.15) [29]. The mapping of coefficients of si (i = 1, 2) for FOD and FOI is given in Table 14.2. We have H ( s) =

G2 s2 + Gτ11 s + τG1 τ02 s2 + τ11 s + τ11τ2

.

(14.15)

The TF represented by (14.15) is a second-order function and can be implemented using FBD as given in Fig. 14.6(a) [29]. The OTA-based

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Table 14.2 Mapping of coefﬁcients of si (i = 1, 2). Circuit G2 G1 G0 τ2

FOD FOI

2 α +3α+2 2 α 2 −3α+2 α −3α+2 α 2 +3α+2

1 1

2 α −3α+2 2 α 2 +3α+2 α +3α+2 α 2 −3α+2

8−2α 2 τ α2 +3α+2 2 8−2α τ α 2 −3α+2

τ1

2 α −3α+2 τ 2 28−2α α +3α+2 τ 8−2α 2

Figure 14.6 (a) Functional block diagram of FOD/FOI. (b) OTA-based realization.

realization of FBD is given in Fig. 14.6(b), where gm2 = gm4 , gm5 = G0 gm1 , gm6 = G1 gm2 , and gm7 = G2 gm2 . The FBD of FOD/FOI along with an OTA-based V-to-I converter may be used for FOE realization as depicted in Fig. 14.7 [29]. The impedance of the resulting FOE is given by Zin =

1 . gm0 (sτ )q

(14.16)

It may be noted that the FBD of Fig. 14.7 implements FOC for q = α , whereas, for q = −α , it represents an FI.

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Figure 14.7 FBD for FOE realization. (a) FOE for order 0 < |q| < 1. (b) FOC when q = α . (c) FI when q = −α .

14.3. Realization of fractional-order element with 1 < fractional order < n Researchers are exploring new design methodologies for implementing higher-order FOSs as such systems result in increased system accuracy. In the context of higher-order fractional-order circuits, two design approaches are available in the literature – the first employs higher-order FOEs, while the second approach relies on the substitution of the Laplacian operator sα (0 < α < 1) by an equivalent integer (m)-order approximation form. This modifies the fractional-order circuit of order (1 + α ) to an integer-order function of order (m + 1) [57–73], which can be realized using an FBD approach. The key focus of this chapter is to present higher-order FOE design methodologies available in the literature. They are described in subsequent subsections.

14.3.1 IIMC-based realization The IIC scheme presented in Section 14.2.2 can be extended to obtain an impedance inverter multiplier circuit (IIMC) as depicted in Fig. 14.8. The concept of IIMC was first proposed in [30], where it was realized using OTA. If this circuit uses (n + 1) V-to-I converters and n impedances, then Zin for this circuit can be computed as Zin−IIMC =

1

gm0 gm1 gm2 . . . gmn (Z1 Z2 . . . Zn )

.

(14.17)

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Figure 14.8 Generic schematic of IIMC.

Furthermore, this scheme can be generalized to the fractional domain if Z1 is replaced by an FOE of order α while other impedances remain the same. The input impedance is then modified to Zin−IIMC =

1

gm0 gm1 gm2 . . . gmn (Zα Z2 . . . Zn )

.

(14.18)

The realization of IIMC using OTA [30] is depicted in Fig. 14.9(a), and the input impedance of IIMC is given by Zin−IIMC =

1

gm0 gm1 gm2 . . . gmn (Z1 Z2 . . . Zn )

.

(14.19)

The generalization of OTA-based IIMC in the fractional domain is shown in Fig. 14.9(b), and the input impedance is expressed as Zin−IIMC =

1 . gm0 gm1 gm2 . . . gmn (Zα Z2 . . . Zn )

(14.20)

An FI of order (n − 1 + α ) can be developed using the IIMC scheme if in Fig. 14.9 Z1 is replaced by an FOC of order α (represented as C1α ) and remaining Zi ’s are replaced by capacitive elements as depicted in Fig. 14.10. The impedance of an FI of order (n − 1 + α ) is given by input impedance Zin−IIMC(n−1+α) , which can be expressed as Zin−IIMC(n−1+α) =

sn−1+α C1α C2 . . . Cn . gm0 gm1 gm2 . . . gmn

(14.21)

To realize an FOC of order (n − 1 + α ) the circuit of the fractional domain IIC (Fig. 14.5(b)) is used, where Z1α is to be replaced by an inductor having impedance as represented by (14.21).

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Figure 14.9 (a) OTA-based IIMC [30]. (b) Fractional domain generalization of OTAbased IIMC [30].

Figure 14.10 The FI realization of order (n − 1 + α ) [30].

14.3.2 GIC-based realization Inductors are an integral part of electronic circuit design; however, designing inductors on a chip has given problems in terms of the usage of space, weight, cost, and tunability. Thus, inductors are emulated using active RC circuits. The generalized impedance converter (GIC) [53] is widely used for realizing inductors in integer-order circuits. Its use may be extended to the fractional domain to realize fractional-order impedances also.

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Higher-order fractional elements: realizations and applications

Figure 14.11 The Op-Amp-based generalized impedance converter [53].

The Op-Amp-based GIC is shown in Fig. 14.11, where Z1 to Z4 form the part of GIC and it is terminated by load impedance Z5 . The input impedance Zin is computed as Zin (s) =

Z1 (s) Z3 (s) Z5 (s) . Z2 (s) Z4 (s)

(14.22)

By appropriate selection of Zi ’s, FOEs of different fractional order may be realized [53]. These are listed in Table 14.3. It may therefore be concluded that using GIC for impedance implementation, the FOEs of order in the range −5 < α < 5 can be emulated.

14.3.3 FBD-based realization The FBD-based approach presented in Section 14.2.3 can be extended to obtain FOEs of higher order ( ϒ ). At low frequencies (100 Hz to 1 kHz), dielectric loss in α -PVDF is larger than that in β -PVDF and γ -PVDF. At intermediate frequencies (∼1 kHz), dielectric loss in β -PVDF dominates over other polymorphs. Above 1 MHz, the dielectric loss of α PVDF and γ -PVDF again becomes large mainly due to dielectric dipole relaxation at high frequencies [20,104]. To accurately evaluate the dielectric performance of a material, dipole alignment needs to be analyzed under varying field strengths. Displacement vs. field plots or the D-E hysteresis loops (frequency = 10 Hz) for the three polymorphs (α , β , and γ ) of PVDF indicate that all three phases possess remnant polarization. For a field strength of 150 MV/m, β -PVDF exhibits high remnant polariza-

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tion, characteristic to a normal ferroelectric, which is evident from the rectangular D-E loop (hysteresis curve). Comparatively, α - and γ -PVDF have lower remnant polarization, even under a field strength as high as 200 MV/m. The D-E loops of α and γ -PVDF show similarity in structure and displacement. Under similar field strength, β -PVDF has the largest maximum displacement due to the all-trans (TTTT [or GGGG]) conformation [59,60]. In addition to the advantage of having different polar and nonpolar polymorphs, PVDF has the advantage of existing as binary and ternary copolymers. PVDF can be easily copolymerized with monomers of chlorotrifluoroethylene (CTFE), hexafluoropropylene (HFP), TrFE, tetrafluoroethylene (TFE), CFE, etc. PVDF and its copolymers have remarkable dielectric, piezoelectric, pyroelectric, ferroelectric, and electrooptic properties compared to pure PVDF. PVDF copolymers undergo changes in the configuration of atoms and as a result change their structure with the addition of comonomers (functionalization). A binary copolymer is formed with the addition of a single type of monomer, while a ternary polymer consists of two comonomers functionalizing the main PVDF chain. The interchain distance in a binary copolymer (e.g., PVDF-TrFE) is larger than that in PVDF, and similarly the interchain distance of ternary copolymer (e.g., PVDF-TrFE-X, where X = CFE or CTFE) is larger than that of binary copolymer (PVDF-TrFE) and PVDF. It is believed that the expansion of interchain distance and pinning from the larger third comonomers supports the existence of a polar β - or γ -phase. The structures of single blocks of various copolymers are shown in Fig. 15.2. A comparison of physical, chemical, and electrical properties of PVDF and PMMA is given in Table 15.3 [104].

15.3.1.2 Inducing β-phase PVDF The β -phase of PVDF polymer has the highest piezoelectric, pyroelectric, and ferroelectric coefficients of polarization among all its polymorphs. A remnant polarization of 110 mC/m2 is reported in [46]. It is usually obtained from mechanical stretching/poling of α -phase PVDF or directly from polymer melting under extreme conditions like high electric field and pressure, rapid cooling, solution crystallization under 70◦ C, and addition of nucleating fillers like BaTiO3 , clay, PMMA, TiO2 , and nanoparticles of ferrite, palladium, gold, CNTs, graphene, etc. [44,46]. However, some binary and ternary copolymers of PVDF have an inherent β -phase. For example, PVDF-TrFE always exists as a ferroelectric “β ” crystalline phase which

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Figure 15.2 Some common copolymers of PVDF. Table 15.3 Comparison of physical, chemical, and electrical properties of PVDF and PMMA [11,15,26,43,49,53,62,63,83]. Properties PVDF PMMA

Young’s modulus (GPa) Density (g/cm3 ) Thermal conductivity (W/m-K) Melting temperature (◦ C) Dielectric constant “ ” (at 1 MHz) Dielectric loss “tan δ ” (at 1 MHz) Surface resistivity () Volume resistivity (-cm) Dielectric strength (kV/mm) Water absorptivity

2 1.78 0.13–0.2 154–184 7.25 0.18 1014 1013 22 120◦ C > 55◦ C, respectively. Pure PMMA could exist in any of the three conformations. Commercial PMMA is available as a combination of isotactic/syndiotactic PMMA, atactic/syndiotactic PMMA, and atactic/isotactic PMMA [24].

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15.3.2.1 Dielectric properties of PMMA Dielectric properties of PMMA were evaluated in [8,73]. Nyquist data were plotted for different pore sizes. It was observed that as the pore size approaches the nanometer-size regime, PMMA develops a resistive behavior and can be modeled as a capacitor in parallel with a resistor. A PMMA matrix will then represent a network of RC blocks. It is evident that porosity affects dielectric properties. Reduction of pore size immobilizes the polymer chains, which decreases the number of free dipoles for rotation with field polarity reversal. This reduces capacitive behavior and conveys a fractional order to PMMA. The loss tangent “tan δ ” increases at low and medium frequencies for nanoporous PMMA.

15.4. Conductive ﬁllers To incorporate and increase fractional-order behavior in capacitors, an increase in dissipation losses or leakage is recommended. This is usually accomplished by altering the conductivity of the capacitor dielectric material by addition of impurities. Polymer dielectrics being discussed here are made partially conductive by compositing with nanometer-sized inorganic highly conductive materials called fillers. Conductive fillers have been reported to increase the disorder in polymeric dielectrics. Materials such as different forms of carbon (CNTs, graphene, etc.), TMDCs, and MXenes have been used by researchers to fabricate capacitors with fractional-order behavior known as FOCs. Conductive fillers of these materials embedded in a ferroelectric or porous polymer have shown tremendous success in designing FOCs [2–5,18,19,21,23,32,51,57,68,87,88,94]. Here we describe various types of materials which have the potential of being employed as fillers in polymeric dielectric matrices for FOC fabrication. The classification of filler materials is based on individual particle dimensions, which are in the nanometer range (≤ 100 nm) [7,44]. • 0D: All the dimensions are in nanometer scale, i.e., no dimension is larger than 100 nm. The most common example of 0D nanomaterial is fullerene. Quantum dots of various metallic and semiconducting materials can be used. • 1D: Two of the dimensions are in the nanometer range. Rod/needlelike structures form this class of nanomaterials. 1D nanomaterials include nanoplatelets, nanorods, nanoclays, and nanotubes, e.g., graphene nanoribbon (GNR), CNTs, carbon nanofiber (CNF), etc.

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2D: The material is nanometer-sized in at least one dimension. 2D nanomaterials include nanofibers, nanotubes, nanorods, and whiskers, e.g., GNSs, TMDCs, boron nitride (BN), MXenes, etc. • 3D: Materials having none of the three dimensions in the nanometer range. 3D materials include graphite, carbon fibers, carbon cloth, carbon aerogels, and hierarchical porous structures. Nanofillers are fundamental components in the fabrication of polymer NCs. Metallic nanoparticles have different chemical and physical properties than their bulk counterparts (e.g., lower melting temperatures, specific properties, high surface area, mechanical strength, etc.). Some important nanofillers for FOC fabrication are described below. •

Fullerenes

Fullerenes were first produced by the arc discharge method from graphite. They can also occur naturally, possessing various shapes like a hollow sphere (buckyballs), an ellipsoid, or a tube. CNTs and graphene belong to the fullerene family. Carbon nanotubes

A single-walled CNT can have a diameter in the range of 0.6–5 nm, whereas a multiwalled CNT has an inner diameter of 1.5–15 nm and an outer diameter of 2.5–50 nm. CNTs of various aspect ratios and lengths are synthesized by varying the processing technique [103]. CNTs exhibit a wide range of electronic, thermal, and structural properties, which are decided by the nanotube diameter, length, and chirality (or twist) [14,27]. A single-walled CNT can be metallic or semiconducting according to the chiral structure [103]. An electron mobility of ∼10,000 cm2 /(V-s) and an electric current density of 4 × 109 A/cm2 have been reported for singlewalled CNTs, which surpass the maximum values drastically [47,110]. Multiwalled CNTs with electrical conductivities up to 105 S/cm and current densities at the level of 106 A/cm2 have been reported [91]. CNTs possess Young’s moduli in the order of 270–950 GPa and a tensile strength in the range of 11–63 GPa. Multiwalled and single-walled CNTs show higher thermal conductivities than copper, approximately 3000 and 3500 W/mK along the axial direction. CNTs can withstand temperatures as high as 2800°C in vacuum. The high aspect ratio and high surface area are also unique to CNTs [91].

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Graphene

Graphene-based composite research papers were first published around 2006 and have experienced exponential growth since 2010. Graphenebased composite is attractive as it leads to improved mechanical properties, electrical conductivity, thermal conductivity, and thermal stability compared to neat polymers (similar to CNTs). [14] Graphene is composed of a planar hexagonal monolayer of conjugated sp2 carbon atoms. The singlelayer thickness of graphene can be as low as 1 nm, with lengths ranging from tens of nanometers to centimeters. Monolayer graphene exhibits a Young’s modulus of ∼1100 GPa and a tensile strength of 130 GPa, making it the strongest material existing [58]. Graphene possesses a high carrier mobility of 15,000 cm2 /(V-s) at room temperature and also a high current density [25]. The quantum Hall effect at room temperature and ferromagnetism due to the lone electron pair at the edges and defects are some unique properties of graphene layers [99]. Thermal conductivity of graphene is ∼5000 W/mK, higher than those of CNTs and metals such as gold, silver, and copper. A comparison of useful properties and inherent limitations of 1D, 2D, and 3D conductive nanofillers is given in Table 15.4 [91]. Carbon black

Carbon black (also called acetylene) has a bulk density of 170–230 g/L. It consists of amorphous carbon atoms in the form of colloidal particles. It is usually produced by incomplete combustion or thermal decomposition of fossil fuels. It is in the form of soot, pellet, or powder, generally black in color, and having a particle size in the range of 10–300 nm [88,91,96]. Transition metal dichalcogenides

To overcome the limitations of zero gap graphene, a new class of materials known generally as TMDCs have widely been explored by researchers. They usually possess a layered structure and can be represented by the generic formula MX2 , where M is a transition metal (Mo, W, etc.) and “X” is a chalcogen atom (e.g., S, Se, or Te). Also TMDC layers can be exfoliated and exhibit varying properties (electrical, mechanical, optical, etc.) with the changing number of layers [85]. The bandgap of TMDCs also varies with the number of layers, e.g., multilayered MoS2 possesses an indirect bandgap (∼1.39 eV) and a monolayer presents a different-valued (1.56 eV) direct bandgap [36,54,98,108,109]. TMDCs thus have a possibility to tune the bandgap and the electronic and optical properties, which also change with

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Table 15.4 Comparison of useful properties and inherent limitations of 1D, 2D, and 3D conductive nanoﬁllers. Dimensions Filler type Features Limitations

1D

CNTs, CNF, GNR

• •

2D

GO, rGO, GNS, GNF, TMDCs, MXenes

• • • •

3D

Graphite, carbon fibers, carbon cloth, carbon aerogels, hierarchical porous structures

•

• • •

Dispersion and distribution is difficult Agglomeration-prone Higher cost

High electrical conductivity at low concentration High thermal stability Large surface area Large interlayer spacing

•

Very expensive

High surface area Good conductivity Good electrolyte Interconnected architecture Better conductive network formation Cost effective relatively

• •

Poor filler–matrix adhesion Higher loading required for evident change Zaid Mohammad Shah et al.

• • • • •

Improved electrical conductivity at low concentration Excellent mechanical properties

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applied heat and pressure [13,61]. MoS2 is the most widely researched metal chalcogenide, followed by others like MoSe2 , WSe2 , WS2 , PtS2 , and SnS2 . The electronic properties and characteristic XRD and Raman peaks for MoS2 , MoSe2 , WSe2 , and WS2 are presented in Table 15.5. Solid state MXenes

MXenes are another new class of 2D layered materials. MXenes form a family of early transition metal carbides, carbonitrides, and nitrides [66,69,70,79,97,105] extracted from the “MAX” phase by etching the A layers, where A is a group IIIA or IVA element. M refers to early transition metal and X is C or N [31–34]. The MAX phase receives its name from the general formula of MXenes, i.e., Mn+1 AXn , where n = 1, 2, or 3, e.g., Ti3 AlC2 , Ti2 C, Ta4 C3 , and Ti3 CN [66]. MXenes possess good metallic conductivity (6500 S/cm) [70,86,97], low surface water adsorption, and high packing densities [79]. MXenes are 2D layered nanosheets resembling graphene. According to [69], over 60 MAX phases are known to exist with the possibility of using combinations of M, A, or X atoms, e.g., (Ti0.5 Nb0.5 )2 C, (V0.5 Cr0.5 )3 C [66].

15.5. Methods of synthesis As is evident from the availability of a wide variety of combinations of polymers and fillers for synthesizing polymer NCs, the methods of synthesis adopted also vary. A particular method of synthesis may be followed depending on some particular properties of polymer and filler. Structural orientation, chemical composition, and dimensions of the fillers are the major factors influencing the selection of method of synthesis for polymer NCs. Polymer NCs can be synthesized using chemical or mechanical processes. A major factor deciding the quality of NCs is the degree of uniformity and homogeneity attained in dispersing the fillers. The aggregation and reagglomeration tendency of fillers hampers the dispersion process and deteriorates the NC properties. Various techniques have been followed by scientists to attain homogeneous dispersion of fillers in polymer matrices. These include chemical reaction of the constituent materials, polymerization during mixing, or modification of filler bond structure. The microstructure of the composite depends on the nature of polymer and filler. The polymer matrix interacts with the filler material in such a way that it may form a composite where the participating phases are separated by grain boundaries called unintercalated structure, the phases may

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Table 15.5 Electronic and physical properties of graphene and common TMDCs. MoSe2 Parameter Graphene MoS2

Mobility (cm2 /(V-s)) Young’s modulus (GPa) Thermal conductivity (W/m-K) Density (g/cm3 ) Melting temperature (◦ C) Eg (bulk) (eV) Eg (bilayer) (eV) Eg (monolayer) (eV) Raman peaks

WSe2

60–200 (bulk) >200 (monolayer) 197–330

160–250 (bulk) 50 (monolayer) 177–224

20–100 (bulk) 0.2 (monolayer) 236–272

120–150 (bulk) 30–180 (monolayer) 163–167

5000

103

54

142

53

2.267 3652–3697

5.06 1185

6.98 ∼1300

7.5 1250

9.32 1200

– –

0.788–1.679 1.198–1.710 1.8

0.852–1.393 1.194–1.424 1.56

0.917–1.636 1.338–1.658 1.659

0.910–1.407 1.299–1.442 1.444

409 (A1g) 383 (E 2g) 14.5◦ (002) 68.8◦ (200) 69.4◦ (108) 69.7◦ (201)

244 (A1g) 286 (E 2g) 31.89◦ (100) 38.21◦ (103) 56.50◦ (110)

–

–

–

–

∼0

– 26.35◦ (110)

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XRD peaks (2θ )

WS2

10,000 (bulk) >140,000 (monolayer) 430–1000

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be intertwined, forming an intercalated and/or flocculated structure, or the filler material completely disperses in the matrix, forming an exfoliated or delaminated structure [35,55,56]. Major routes for polymer NC synthesis are listed below.

15.5.1 Intercalation Layered nanofillers form good composites with polymers when the layers are separated physically and polymer chains are inserted in increased spacing between layers. It is a top-down approach where the bulk filler material may be disintegrated as one of the new dimensions of the filler is in the nanometer range. Intercalation may be achieved through separation of the layers of nanofillers and mixing with polymer chains and postmixing treatment if necessary. The separation of layers is induced by mechanical- or heat-assisted surface modification with radical groups. The approaches followed are based on the separation technique adopted. The entropy loss due to confinement of polymer chains within the layered nanofiller structure is balanced by entropy gain associated with layer separation [35].

15.5.1.1 Chemical intercalation In this technique layered nanofiller material is dispersed in a monomer solution, followed by in situ polymerization by radiation or a catalyst, linking the two phases. This type of synthesis is usually done in solution form [89].

15.5.1.2 Mechanical intercalation Intercalation of polymer chains in filler material layers is achieved via solution mixing. It is a well-known fact that interlayer spacing increases in solution phase. Polymer solution is then mixed under shear stress with nanofiller solution. This method requires a common solvent or miscible solvents for dispersing both constituents [56].

15.5.1.3 Melt intercalation Another method of mixing polymer chains with layered nanofiller is where the mixing of the melts of constituent takes place. The constituents are individually heated to the sublimation temperature and then mixed through kneading or under an extreme shear rate and at a uniformly maintained temperature. The molten composite so formed can be molded into the desired shape before annealing, which is usually done under high pressure

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conditions. This process is widely adopted for industrial-scale synthesis. It is considered an environmentally friendly technique due to the lack of usage of solvents. However, some limitations exist due to the high processing temperature, which may modify the bonds on surface of fillers. The compatibility of filler and polymer may also hamper the quality of the final NC [35,76]. In some cases, melt intercalation can be further improved by using microwave irradiation [89]. Stresses generated during melt blending can break up the nanofiller aggregates. The extent of affinity of the filler and polymer matrix also determines the quality of dispersion of fillers. Affinity is usually increased by functionalization. A highly viscous melt will facilitate better dispersion due to higher stresses present [14,76].

15.5.2 Sol-gel method As per the definition, “sol” is a colloidal suspension of solid nanometersized particles in monomer solution and a “gel” is a 3D interconnected network formed between phases. Solid nanoparticles are dispersed in monomer solution forming colloidal suspension. Gel is then formed by a polymerization reaction followed by hydrolysis. Polymer molecules act as nucleation sites for growth of the composite crystal network. The network so formed has polymer intercalated within the layered structure of fillers [14,56].

15.5.3 Direct mixing Direct mixing for polymer NC synthesis is a technique where the constituents are mixed without any preprocessing. It is a top-down approach since the nanofillers break down during the mixing process. There are two common ways to mix the constituents of a polymer NC directly, as described in the following subsections [56].

15.5.4 Melt compounding Being considered as the most economical and environmentally friendly synthesis methods for polymer NCs, melt compounding is fit for industrialscale production [14]. In this method the nanofillers are mixed with polymers above the glass transition temperature of polymers. Nanofiller aggregates are broken down and dispersed in the molten matrix with the aid of hydrodynamic force produced due to applied shear stress. In the compounding procedure, the polymer and the nanofiller mixture are subjected to shear and elongational stress along with heat to form a melt in

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a single or twin-screw extruder. Dispersion of filler is achieved by changing the screw configuration of the extruder, which disintegrates the filler agglomerates and facilitates the adoption of a homogeneous and uniform nanofiller distribution in the polymer matrix [55]. Dispersion improves at high shear rates, high production rates, good material throughput (less wastage), and solvent-free synthesis. However, high shear rates can degrade the polymer and nanofiller. The compounded nanoparticle–polymer composite can be further processed using other polymer-processing techniques such as injection molding, profile extrusion, and blow molding. Optimal process parameters (e.g., temperature, screw speed, time of processing before extrusion, and shear stress) for a particular blend of filler and polymer differ for each combination and need tuning via an experimental approach [14].

15.5.5 Solution blending Mixing of solutions has always proven to yield homogeneously dispersed mixtures. The mechanical intercalation method discussed previously is based on the solution mixing principle. Here the constituents of an NC are dissolved in solvent using either ultrasonication or mechanical stirring. The constituents are then mixed under shear stress (lower than that required for melt compounding). The polymer chains in solution intercalate and displace the solvent within the layers of the filler. The polarity of the solvent is critical in facilitating the intercalation of the polymer chains into the interlayer spacing of layered nanofillers [14]. Postmixing, the mixture is controllably heated to undergo solvent evaporation or solvent coagulation [14,55]. This technique is also called exfoliation adsorption, which mostly involves using water as a solvent, although aqueous and organic solvents can also be used. Exfoliation takes place due to entropy gain during desorption of solvent. In contrast to melt compounding, a low viscosity of the filler–polymer solution aids better dispersion of filler after agitation or ultrasonication. Both thermoplastic and thermoset materials have been produced. However, it is not an environmentally friendly process due to the evaporation step involved for solvents, which may be toxic, during synthesis.

15.5.6 In situ polymerization This method involves polymerization after mixing the filler material with the desired monomer. A monomer as described at the start of this chapter

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is a low-molecular weight single part of a polymer chain. Monomers are easily transported to the interlayer spacing of the filler in solution form. Polymerization of the mixture is attained through heating, through radiation (microwave), or using initiators (organic or monomer initiator – mixing two monomers) [35]. Monomers undergo polymerization in between layers of the filler to form intercalated or exfoliated NCs. Emulsion polymerization is a subtechnique where the monomer and nanofiller are mixed in water and emulsifier. In situ polymerization is an alternative to melt intercalation when the constituents are thermally unstable and also to exfoliation adsorption where the polymer matrix is insoluble in most solvents [35,55]. This technique can be used to produce both thermoset and thermoplastic materials and also produces better exfoliated structures.

15.6. Percolation threshold In polymer NCs for FOC fabrication, knowledge of the critical loading percentage of nanofiller by weight or volume, beyond which the NC dielectric exhibits considerable conduction, is necessary for optimal synthesis. An increase in conduction has been reported to happen due to the formation of a continuous network of conductive nanofillers within the polymer matrix. This critical loading is called the EPT. Beyond the EPT, conductivity increases strongly for even a small increment in filler loading. The increase in conductivity is not regular with the increasing filler content as it becomes constant beyond a certain level of filler loading. The EPT is increased or decreased by varying the aspect ratio, size and shape, electrical conductivity and functionalization of the nanofillers, polymer dielectric constant and conductivity, and also the synthesis parameters like processing technique, dispersion method, etc. Theoretically, the EPT for rod-like structures is half of that for platelets [14,16,56,75,78,87,107]. According to classical percolation theory, σ = σ0 (Vf − Vc )s ,

(15.3)

where σ is the electrical conductivity of the composite, σ0 is the electrical conductivity of the filler, Vf is the filler volume fraction, Vc is the EPT, and s is the critical percolation conductivity exponent. For a single percolation system, i.e., materials percolated by single type of filler, s depends on the dimensionality of composites. For example, s lies between 1.6 and 2 for a 3D filler network and s takes a value of around 1 to 1.3 for 2D filler

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networks [75,106]. EPT is dependent on a number of parameters of the fillers, e.g., size, shape, orientation, number of layers, etc. The percolation threshold of graphene is comparable to that of CNTs [107]. The conductivity depends on the method of synthesis and on the surface modification, particularly on the number of defects (reactive sites) generated during the oxidation-reduction process on the surface [14,56].

15.7. Factors affecting properties of polymer NCs To eradicate the anomalies introduced in the obtained results, a defect- and impurity-free NC is sought for FOC fabrication. This section discusses the major drawbacks to polymer NC FOC dielectrics. Apart from the properties of the filler and polymer matrix, the quality of the composite and eventually the reliability of the results depend on [14,45,55,95]: • the type of nanofiller, • alignment and orientation of nanofiller, • the size, shape, and defects of nanostructures, • the NC fabrication process, • the volume fraction of the nanofiller, • the dispersion process and processing time, • the interface characteristics of nanofiller and polymer matrix. Nanofillers mentioned in this chapter having varying physical and chemical properties shall produce polymer NCs with different properties. The modulus of NCs usually increases with increasing weight loading fraction. However, at higher nanofiller content, composite properties decrease, indicating poor dispersion of fillers. The interfacial interaction which determines the electron transport properties depends on the nanofiller type, the type of functional group on the nanofiller, the aspect ratio, the weight percentage of filler, the type of polymer matrix, and the method of processing. Alignment of nanofiller in the matrix increases dissipation losses and theoretically shall increase the CP zone. The alignment is much easier in the case of 2D layer and 1D particle nanostructures. The layer-by-layer (LBL) assembly technique produces defect-free NCs. Solution-based polymer NCs provide better electrical characteristics than melt-based ones [14]. A description of the important factors is given below.

15.7.1 Alignment of the ﬁller Nanofillers having anisotropic properties (dependent on orientation) when employed in composite preparation need to be aligned properly to produce

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desired results. FOC fabrication involves the use of highly conductive and semiconductive fillers, most of which are 2D layered nanostructures, e.g., TiO2 , MoS2 , GNS, hexagonal BN, or 1D, e.g., CNTs, CNFs, GNRs, etc. Alignment of the filler is usually accomplished by various techniques like melt compound extrusion [92], fiber spinning, shear force, surface acoustic waves, application of a strong electric or magnetic field [22,39,64,90], and mechanical stretching and poling [14,50]. However, magnetic- and spin-assisted alignment is most preferred, since other techniques may align superficial layers only. This process should precede the polymerization step due to immobilization of filler in a polymer matrix. A good alignment of fillers is usually effectively obtained until a maximum concentration (6000 rpm) can alter the surface morphology of both filler layers and polymer chains. However, lower shear rates will demand longer time. Ball milling involves the crushing of bulk material with ceramic balls over a considerable amount of time. The material to be milled needs to be in powder form. Optimum treatment duration is important in order to avoid damage due to processing conditions.

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15.7.3 Interfacial bonding between ﬁller and the polymer matrix For efficient energy transfer within the polymer NC dielectric, interaction between nanofiller and the polymer matrix is crucial. As mentioned in the previous subsection, a moderate interaction is optimum for good dispersion. The interaction between the interconnecting phases depends on the ratio of surface energy of filler and matrix. A high surface area due to a high aspect ratio of the nanometer-sized filler dominates the interface properties. The width of the interphase region is decided by the flexibility, the energy of adsorption, and the extent of polymer chain entanglement. The extent of interface determines the energy transfer from matrix to filler. Interaction is also affected by the shape of the nanofiller. Asymmetric nanoparticles such as layered materials (graphene, TMDCs, BN, etc.) or CNTs produce higher-quality polymer NCs than symmetrical nanoparticles with spherical shape, etc. To improve this interaction, the nanofiller is functionalized. Functional groups can undergo covalent bonding (defect and side wall functionalization) and noncovalent bonding (π –π interactions, hydrogen and ionic bonding). However, covalent bonding modify the hybridization from sp2 to sp3 , reducing conjugated bonds and lowering its electrical properties. Noncovalent functionalization can be used as an alternative, which may improve the composite strength while maintaining the electrical properties of the nanofiller. The strength of noncovalent functionalization can be determined by the combined effect of attractive forces (electrostatic, dispersive, and inductive interactions) and repulsive forces (exchange repulsion). In carbonaceous materials, the noncovalent functionalization depends on the ππ interactions (e.g., H-ππ , ππ -ππ , cation-ππ , and anion-ππ ) [45].

15.8. A GNS/PVDF FOC To validate the guidelines provided in this chapter to fabricate FOCs, GNS/PVDF NCs with various different weight percentages of GNS are employed as dielectrics between copper electrodes. This section discusses the work done in [87] and the steps followed to fabricate and characterize the device. The FOC utilizes a PVDF polymer dielectric percolated by GNS fillers above 3% by weight of the NC. The polymer NC is synthesized using a solution blending technique. Post synthesis, the NC is encapsulated within the cavity of concentric circular copper electrodes (diameter, 1.2 cm and 0.6 cm). The circular electrode design to fabricate FOCs has been

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used in [21] as well. An actual photograph of the same device is given in Fig. 15.5. The analysis techniques include measuring the impedance “|Z|” as given in Eq. (15.1) and phase angle “” of impedance vs. a sweeping frequency “ω” (impedance spectroscopy [IS]) and plotting the data in various forms, e.g., “log |Z| vs. ω,” “ vs. ω,” or Nyquist (“Zimaginary vs. Zreal ”) and Bode plots (“Zreal vs. ω” and “Zimaginary vs. ω”). From the plots, the following FOC characteristics can be derived: 1. the slope α of log |Z|, decreasing −20α dB/decade; 2. the frequency over which remains constant (the CP zone); 3. equivalent circuit parameters in terms of resistance and pseudocapacitance. The dielectric material characterization is done using XRD and Raman spectroscopy data. The graphs are obtained for individual thin films of the same composite material as used in FOC fabrication. The materials used, the process parameters, and the results obtained along with methods of interpretation are given here.

15.8.1 Materials and methods The dielectric NC employed is prepared by blending various different weight percentages of GNS with a weighed quantity of PVDF polymer (density, 1.78 g/cm3 ). For each sample, 3 g of PVDF in pellet form is mixed with 10 mL DMF and stirred at 4000–4500 rpm for 2 hours using a magnetic stirrer (at 25–30°C), followed by increasing the temperature and stirring at 80°C for 1 hour. As a result, a homogeneous solution of PVDF polymer chains loosely dispersed in the solution is obtained. The interchain separation occurs in solution form for the PVDF polymer for intercalation of the filler particles. The concentration of the final solution is 300 mg/mL, which decides the ease with which intercalation of filler particles takes place. GNSs (σ ∼ 106 S/cm) in powder form are weighted and added to suitable amounts of DMF solvent to yield 5 mg/mL GNS/DMF (filler) solution. The filler solution is subject to ultrasonication and stirring to attain uniform dispersion of graphene sheets in DMF. The dispersion solution is then added to the polymer solution and stirred for 2 hours at room temperature, and then most of the solvent is evaporated via stirring at 80°C. Five samples of GNS/PVDF NC were synthesized with 3%, 5%, 7%, 9%, and 13% loading by weight. The onset of partial conductivity in the composite is tracked by initially using minimal loading of filler (∼3%) and gradually increasing the concentration of filler. Beyond 13% the resistive nature will

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dominate the dielectric properties. The highly viscous GNS/PVDF solution is then poured into concentric cylindrical copper tubes. The polymer solution forms the dielectric between the inner surface of the larger diameter (1.2 cm) tube and the outer surface of the tube with smaller diameter (0.6 cm). The design parameters have been adopted to fabricate carbon black/epoxy resin-based FOC in [21]. The setup so formed has the geometry of a capacitor as shown in Fig. 15.4(a) (front view) and Fig. 15.4(b) (top view).

Figure 15.4 Cylindrical FOC with GNS/PVDF NC dielectric sandwiched between electrodes. Connecting wires (tin-coated copper) are soldered to the clean copper surface of the electrodes. (a) Front view. (b) Top view.

Terminal wires of tin-coated copper were soldered to inner and outer electrodes of the structure to carry out IS in the frequency range of 100 Hz to 20 MHz using a Wayne Kerr precision impedance analyzer (model 6500B).

15.8.2 Results and discussion IS data are plotted for all the samples, and the most common electrical phenomena taking place within the NC dielectric are explained. Modulus of impedance |Z| versus log of frequency plots (Fig. 15.5) provide the fractional order α , which is the slope of |Z|, decreasing at −20α dB per decade of frequency. The variation of phase angle from −84 degrees for 3 wt% of GNS to −46 degrees for 13 wt% of GNS with a phase ripple of ±5 degrees is plotted in Fig. 15.6. The CP zone for most samples can be found in the frequency range of 100 kHz of 2 MHz. The shift of the phase angle is attributed to the formation of a microscale resistive (of

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interconnecting conductive GNS clusters) and capacitive (dielectric polymer chains interfaced with GNS) network. This network will assume the characteristics of infinite RC network with a large number of individual time constants characteristic of various RC sections. As a result, a distributive time constant is introduced in the current–voltage relationship. Charge transport takes place in the conductive network formed by GNS between the circular electrodes, due to random transport (walk) and short-range hopping between NC clusters.

Figure 15.5 |Z| vs. ω for FOCs with GNS/PVDF NC dielectric. GNS loading is varied from 3% to 13%. Impedance at lower frequencies is seen to decrease drastically beyond 9% loading.

Figure 15.6 Phase angle vs. ω plots for FOC having GNS/PVDF NC dielectric. GNS loading used was 3%, 5%, 7%, 9%, and 13%. The CP zone extends in the range from ∼100 kHz to 2 MHz.

Nyquist (Z vs. Z ) plots for all five samples (3 wt%, 5 wt%, 7 wt%, 9 wt%, and 13 wt% GNS/PVDF composites) are plotted in Fig. 15.7(a)–(e). In order to evaluate the variation of electrical parameters, Nyquist plots are subject to fitting curves using an equivalent circuit. The equivalent circuit is chosen as per the shape of the plot. Various types of modeling circuits

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for nonideal and ideal dielectric response are mentioned in [87], the most common being the widely discussed Cole–Cole (CC) model. Nyquist plots (pink line) are fitted using the simulation software Multiple Electrochemical Impedance Spectra Parameterization (MEISP). The best-fit curve is plotted as a solid black dotted line. The origin in the plots represents highfrequency behavior, with frequency decreasing in the positive x-direction. From the plots, non-Debye behavior is evident in the dielectric of FOC. Deviation from ideal dielectric characteristics is attributed to relaxation phenomena taking place within the partially conducting GNS/PVDF NC. For an ideal dielectric, the Z vs. Z plot is a semicircle with center on the real (Z ) axis. However, deviation from ideality is often evident as depression of the semicircle, the center of which makes an angle of απ/2 with the real axis. Some additional features in the graph, e.g., low-frequency tails or Levy type distributions, indicate formation of conductive networks within the dielectric. The plots in Fig. 15.7(b)–(d) indicate more than one relaxation phenomenon possessing a semicircle with an almost linear tail at low frequencies. Peak frequency (ωp ) represents the characteristic frequency of relaxation where dissipation reaches its maximum value. The semicircle is attributed to short-range hopping of carriers at the interfaces between conductive GNS and PVDF matrix. This process is known as dielectric interfacial relaxation dominating at intermediate frequencies. At higher frequencies, charge carriers do not respond instantly to the changing field polarity, thus reducing the mean displacement. The current in such cases follows ideal (Debye) exponential models. At low frequency, molecular dipole moments follow the field polarity. Dielectric relaxation follows to balance the diffusive currents due to separation of charge. The conductive (linear) behavior is dominant until a certain frequency called the critical frequency (fc ). This critical frequency increases with filler content. Levy type distributions at the lower end of the frequency range indicate Warburg type behavior. Warburg impedance can be seen in graphs for 5%, 7%, and 9% GNS/PVDF FOCs. The formation of a conductive network within the NC structure increases the leakage currents and thereby the dissipative nature of the dielectric. The low-frequency tail extends to higher frequencies at higher GNS loading. The high- and low-frequency behavior of the GNS/PVDF dielectric as given in the Nyquist plots is modeled using an electrical equivalent circuit for the CC model. In [87], the plots follow a power-law distribution at low frequencies and a stretched exponential function at intermediate to high frequencies. At low frequencies, complex permittivity ( ∗ ) and tan δ are

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high and decrease with increasing frequency. The low-frequency conductivity is now considered complex and is expressed in complex permittivity terms: σ ∗ (ω) = jωo ∗ (ω) = ωo + jωo ,

(15.4)

where o is the permittivity of free space, i.e., 8.85e−12 Farad-meter.

Figure 15.7 (a)–(e) Nyquist (Zimag vs. Zreal ) plots for GNS/PVDF FOCs. GNS content varied from 3% to 13% by weight of the PVDF matrix. ωp is peak frequency; loss tangent is maximum at ωp .

Deviation from ideal Debye characteristics in heterogeneous media is a consequence of MWS relaxation phenomena occurring at interfaces of different phases due to accumulation of charge [87]. The Cole–Cole equivalent circuit model is shown in Fig. 15.8. The CC equivalent circuit parameters for the Nyquist plots extracted in MEISP are plotted in Table 15.6. The variable parameters in the circuit are the high- and lowfrequency permittivity/capacitance values (“εs or Cs ” and “ε∝ or C∝ ”), the

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Figure 15.8 The Cole–Cole equivalent circuit model. Table 15.6 Fit values for equivalent circuit parameters of GNS/PVDF FOC. The GNS weight percentage is 3%, 5%, 7%, 9%, or 13% by weight. α is the fractional exponent, C∞ is the high-frequency response, and Cs is the low-frequency response. GNS wt%

α (experimental)

α (simulated)

Rseries

3 wt% 5 wt% 7 wt% 9 wt% 13 wt%

0.93 0.85 0.75 0.65 0.51

0.91 0.875 0.73 0.61 0.56

500 1500 4600 4900 7500

Cα

()

F s1−α

1 × 10−9 9 × 10−8 2.1 × 10−6 9.5 × 10−6 590 × 10−6

C∞

Co (Cs )

10 pF 33 pF 68 pF 330 pF 500 pF

5 nF 20 nF 230 nF 400 nF 590 µF

pseudocapacitance (Cα ), the fractional order (α ), and the series resistance (Rser ). The extracted value of α shows an error of ±0.05 with the experimental values from phase angle vs. frequency plots. A consequence of the increasingly conductive nature of the dielectric composite as the filler content increases from 3% to 13% is the increase in dissipation losses due to Rser and a shift of the fractional order α from 0.93 to 0.51. EIS data can be used to calculate real and imaginary parts of permittivity and capacitance as follows [16,87]: =

−Z |Z(ω)|2 ωCref

and

=

Z |Z(ω)|2 ωCref

,

(15.5)

where Cref is the capacitance of an empty cell of similar dimensions and identical electrodes. Similarly, C =

−Z |Z(ω)|2 ω

and

C =

Z |Z(ω)|2 ω

.

(15.6)

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15.9. Conclusion In this chapter, single-component designs for FOC fabrication based on polymer NC dielectrics have been discussed in detail. The chapter provides insight into the properties of the various polymers and conductive fillers used by researchers as FOC dielectrics. Properties of materials having potential in FOC dielectric synthesis are also discussed. PVDF (β -phase) and its binary and ternary copolymers, especially PVDF-TrFE-CFE, have been of special interest due to their high remnant polarization, large dielectric constant, large dissipation loss, and high percentage of β -phase polymorph. A CP zone of 5 decades is possible using a PVDF-TrFE-CFE and MoS2 composite dielectric. Porous polymers are equally important due to their hierarchical porous structure which could imitate the characteristics of an infinite RC network and provide a fractional order to the material and device. The maximum CP zone achievable for PMMA-based FCs has been 4 decades, using MWCNT filler. PVDF and porous polymers are considered suitable candidates for FOC fabrication. However, conductive polymers which already find application in supercapacitors have not yet been explored for FOC dielectrics. The important properties of various fillers used for FOC fabrication have also been explained. Layered nanofillers induce ferroelectric β -phase in PVDF, facilitating fractional-order dynamics. Most of the designs employ a layered structure; however, the number of works is far lower than the number of possible combinations of conductive filler and polymer. Graphene, CNTs, and metallic particles provide flexibility of having a low percolation threshold. TMDCS and TMCs (MXenes) can be explored to fabricate externally tunable devices. Various electronic and physical properties of conductive fillers are tabulated for reference purposes. This chapter also discusses the techniques to synthesize NCs. It is well known that the properties of NCs are process-dependent. FOC behavior hence also varies with fabrication process applied. A particular process may produce high-quality composites for one combination of polymer/filler but may not yield similar results for other combinations. An account of the techniques is provided for reference. The properties of a polymer change beyond a certain loading of filler particles. This critical loading, also called EPT, varies with size, shape, orientation, etc., of filler and also with the type of polymer and filler used. The properties are also affected by alignment of filler, dispersion efficiency, polymer matrix– nanofiller interaction, and structural defects. To support the literature provided in this chapter, samples of GNS/ PVDF NCs with loading of 3–13% have been synthesized using a solu-

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tion blending technique. Cylindrical FOCs employing the GNS/PVDF NC have been analyzed using an impedance analyzer and the datafitting software MEISP. The samples show fractional-order characteristics in impedance and phase plots beyond 3% loading of filler. The value of α varied from 0.93 for 3% GNS loading to 0.51 for 13% GNS loading. A CP zone extending over around 3 decades of frequency (100 kHz to 2 MHz) was obtained for all samples. These guidelines may be followed for future attempts at implementation of FOCs.

Acknowledgments The Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India supported the research presented in this chapter, under the Extra Mural Research (EMR) scheme (EMR/2016/007125).

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CHAPTER SIXTEEN

Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor Dina Anna Johna and Karabi Biswasb Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal, India

Chapter points • •

Fabrication procedures of solid-state fractional capacitors are given in detail.

•

Performance of the solid-state fractional capacitors is correlated with the material characterization of the nanocomposite.

The parameter combination for obtaining the predeﬁned speciﬁcations is elaborated.

16.1. Introduction Fractional calculus is more than 300 years old. In layman terms, it is defined as the branch of mathematical analysis which deals with integration and differentiation of an order that can take real or non-integer values. In order to express fractional integration and differentiation, some mathematical representations are required. For that, the three widely used definitions are Grünwald–Letnikov, Riemann–Liouville, and Caputo [85]; all these definitions in the s-domain will have a fractional operator sα . In the initial years, the studies on fractional calculus were restricted to mathematicians. Later, fractional calculus started to gain popularity in the fields of engineering and science. With regard to this, some of the applications of fractional calculus in the engineering domain include fractional circuits, which are primarily comprised of filters, controllers, and oscillators. In addition, fractional calculus is also utilized in the area of signal processing, like fast encryption for big data privacy protection [1,19,113], fractional difference models to diagnose tumor growth and other diseases [17,66], a Research scholar at the Department of Electrical Engineering. b Associate Professor at the Department of Electrical Engineering.

Fractional-Order Design: Devices, Circuits, and Systems Copyright © 2022 Elsevier Inc. https://doi.org/10.1016/B978-0-32-390090-4.00021-4 All rights reserved.

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noise detection and image denoising [106,107], etc. It is also seen that fractional calculus is used as an important mathematical tool to describe many complex problems such as seismic signals [110], the heat equation [14,103], the viscoelastic wave equation [16,82], fluid dynamics, energy harvesting in dynamic systems [20,44,95], economics [34,96], physics [45,111], neuron dynamics [99], chaotic behavior [94], and electrochemistry [48,67]. Generally, fractional circuits are defined by the fractional differential equation in the Laplace domain so that it can be solved algebraically with ease when compared to the time domain equation, where the solution steps are a little bit more complex. To approximate the fractional operator sα in the fractional equation, there are many approximation methods available in the literature which help to convert the fractional equation to an integerorder equation. The main fractional approximation methods are: 1. the Oustaloup method [81,114] – where the magnitude response of the fractional transfer function is approximated by a number of poles and zeros; 2. the continued fraction expansion (CFE) method [74,91] – where the transfer function is converted to a rational transfer function by CFE; 3. Matsuda’s method [64,105] – which uses CFE and the set points are logarithmically placed; 4. the stability boundary locus (SBL) fitting method [36] – which employs fitting of the rational transfer functions with the SBL curves of fractional transfer functions. Besides the above, there are improved versions of these methods, like Carlson’s method and its variants [65,97], Charef ’s method [33], state-space approximation methods [68,86], vector fitting approximation [39], smallerorder approximate models [112], and the curve-fitting approximation algorithm [22]. Apart from the approximation method, the fractional operator can also be implemented in the analog domain; then the realized device is termed fractional-order element. The fractional-order element can be labeled as fractional capacitor or fractional inductor according to the value of α . For the fractional capacitor, the current–voltage relationship can be written mathematically as i(t) = F1

dα v(t) , dtα

(16.1)

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where F1 is the fractance of the device and α , the fractional order of differentiation. The Caputo fractional derivative [85] in the time domain is C α a Dt

=

dα f (t) 1 = dtα (n − α)

t

f (n) (τ )dτ (t − τ )α+1−n

a

,

(16.2)

where t is an independent variable, α is the fractional order, n − 1 < α < n, n ∈ N, and the Laplace transform of the above derivative with the lower bound a = 0 is α α L {C a Dt f (t )} = s F (s) −

n−1

sα−k−1 f (k) (0).

(16.3)

k=0

When the initial conditions are taken as zero, Eq. (16.3) is simplified as α α L {C a Dt f (t )} = s F (s).

(16.4)

Using Eq. (16.2) and n = 1, Eq. (16.1) becomes i(t) = F1

F1 dα v(t) = dtα (1 − α)

t 0

v (τ )dτ . (t − τ )α

(16.5)

Taking the Laplace transform of Eq. (16.5) by employing Eq. (16.4), I (s) = F1sα V (s).

(16.6)

Therefore, the impedance of the fractional capacitor in the s-domain can be given as V (s) 1 , (16.7) = I (s) F1sα where α varies between 0 and 2. These fractional capacitors are required for implementing the fractional circuits in the analog domain. The significance of the fractional capacitor is that it is useful for system modeling (since it is governed by the fractional-order differential equation as seen in Eq. (16.5)), which provides a better description of the system like speech signals, fruits, vegetables, bio-impedance measurements, batteries, and supercapacitors [10,15,54,72,109]. It enables designing the filter [38,90,102,108] with non-integer order and achieve the required pass band/stop band characteristics, avoiding circuit complexity. Whereas, in the case of an oscillator [42,43,73,83,87], the same circuit can provide signals with higher frequencies without changing the component. The fractional controller can Z (s) =

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achieve better output regulation and rejection of the disturbances, and it also provides better control for many natural systems, which are inherently fractional in nature [9,12,25,84,93,98]. In the last 60 years, numerous attempts have been made to realize fractional capacitors using different combinations of available circuit components [3,4,18,37,38,46,47,63,80,91,104]. Generally, the realization of a fractional capacitor can be categorized into one of the following two methods: 1. Multicomponent approach: The truncated CFE/rational approximated transfer function can be used to find the values of passive electrical components (like a resistor, capacitor, or inductor) or a combination of active elements like operational amplifiers (Op-Amps), current feedback amplifiers (CFOAs), MOS transistors, and passive components for the realization of a fractional capacitor [21,38,79]. 2. Single component approach: Here, the material properties are modulated to achieve the constant phase angle (CPA) for a particular frequency range [2,23,27,40,49]. Both methods have their own advantages and disadvantages. The main limitation of the multicomponent approach is the high number of required components [80,91], and chances of noise introduction make the prototype difficult to be practically implemented. On the other hand, there are some reports on single component realization of fractional capacitors; the details of those will be discussed in this paragraph. The fractal-based fractional capacitor was first introduced by Haba et al. in 1997 and it has a fractal structure made on a metallic surface [49]. However, the fabrication requires photolithography, wet chemical etching, and microsoldering, and the CPA obtained is in the high-frequency region. The fractional capacitor made by Bohannan is made of the material LiN2 H5 SO4 , and due to the anomalous transport of ions, a CPA was attained [27,28]. The fractional capacitor has a larger size with a three-plate structure and requires hermetic sealing, and only two values of α were reported. It has found utility in motor and temperature control applications [28]. Another work was on the liquid fractional capacitor [23] and based on poly(methyl methacrylate) (PMMA)-coated copper or platinum electrodes in an ionic solution. The fractional capacitor was bulky, and afterwards, it was made in a packaged form [76] so that it could be used to design electronic circuits, avoiding the spillage of liquid medium. The developed fractional capacitor has found applicability in milk adulteration detection [35] and soil moisture measurement [32] systems.

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In 2009, Jesus et al. [53] proposed a fractal electrode-based electrolytic fractional capacitor, which has disadvantages like liquid spillage and bulkiness. Again, a fractal structure on silicon providing fractional behavior of order 1/2 was reported by Haba et al. [50,51] but lacked definite guidelines for changing the value of α . Elshurafa et al. [40] introduced a fractional capacitor based on graphene polymer nanocomposite, where the CPA can be varied (−29.7 to −65.7 degrees) by changing the wt% of loaded graphene from 12% to 2.5%. In the same year, an ionic polymer metal composite reported by Caponneto et al. [31] was found to impart fractional behavior, and this work provided the guideline for changing the constant phase zone (CPZ). But to get fractional behavior, the fractional capacitor needs to be connected with an input series resistor. In 2015, Adhikary et al. made an electrochemical fractional capacitor with BPADA-mPD polymer and 1% CNT, which has the advantages of long CPZ and high yield rate, but again faces the issue of no proper guideline for tuning the value of α [2]. Other fractional capacitors can be seen in [6–8,11,29,30], where different polymer nanocomposites are used to get fractional behavior. The disadvantages posed in the above single component realizations are mainly the size, spillage of liquid, and the lack of guidelines for tuning the value of α ; this has motivated us to develop a new fractional capacitor named “solidstate fractional capacitor.” The solid-state fractional capacitor was developed at the Department of Electrical Engineering, Indian Institute of Technology Kharagpur. We have realized the solid-state fractional capacitor [57,58] by modifying the dielectrics between the electrodes. For the fabrication, multiwalled carbon nanotube (MWCNT)-polymer nanocomposite is employed as it contains both conducting material (MWCNT) and insulating material (epoxy resin). It has been noticed that most of the single component fractional capacitors lack guidelines to obtain different CPA and F1 values; hence the other objective is to provide directions for fabricating the solid-state fractional capacitor with predefined specifications. To define the specifications, the following equation of impedance [3,69], which is obtained by taking s = jω in Eq. (16.7), is used: Z (jω) =

1 , F1(jω)α

(16.8)

where F1 is fractance, having the unit .sα or F.sα−1 , α is the fractional exponent (0 < α < +1), and ω is the angular frequency in rad/s. It is to be noted that the single component fractional capacitors have α -values in

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between 0 and 1. From Eq. (16.8), the fractance F1 and α can be expressed as 1 F1 = , (16.9) |Z (jω)| × (ω)α ∠Z (jω) = −α × 90◦ ,

i.e.,

α = −∠Z (jω)/90◦ .

(16.10)

From Eq. (16.10), one can see that α depends on the phase angle and remains constant (ideally), whereas the fractance (F1) depends on the magnitude of impedance, the fractional exponent α , and the frequency. By using Eqs. (16.9) and (16.10) on the experimental results of impedance, the guidelines for the solid-state fractional capacitors are framed. Here, α and F1 have been considered as the primary specifications to formulate the guidelines. It will be worth to mention here that to get the value of capacitance, C, of the fractional capacitor in Farad [37], the following equation can be used: F1 C= . (16.11) 1−α (ω)

Besides the fabrication guidelines, material characterization is also carried out to illustrate the charge transport mechanism from one electrode to the other. Hence, in this chapter, the structure of the fractional capacitor is first elaborated, followed by the steps of making the device, which includes the preparation of MWCNT-epoxy nanocomposite and assembling of the device in Section 16.2. After that, batch analysis has been carried out to design the fabrication guidelines of the solid-state fractional capacitor, which is given in Sections 16.3 and 16.4, respectively. In Sections 16.5 and 16.6, material characterization of the nanocomposite is done to find out its relation with the performance of the solid-state fractional capacitor. The chapter is concluded in Section 16.7.

16.2. Solid-state fractional capacitors This section discusses the structure of fractional capacitor and the procedure to make the device.

16.2.1 Structure of the fractional capacitor The structure of the fractional capacitor is shown in Fig. 16.1(i) and the photograph of the fabricated device can be seen in Fig. 16.1(ii). The

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Figure 16.1 Illustration of a solid-state fractional capacitor.

fractional capacitor is realized by sandwiching MWCNT-epoxy nanocomposite and a porous middle plate in between the two copper electrodes [57,58,61]. Thus, it is mainly constituted of: 1. Nanocomposite: The nanocomposite is MWCNT-epoxy (where MWCNT acts as the conducting material and epoxy resin as the insulating material). This forms a combination of resistors and nanocapacitors in between the electrodes. 2. Porous middle plate: The porous middle plate enhances the distributive nature of the charge flow from one electrode to the other. A middle plate (e.g., 1 cm × 1 cm or 1.5 cm × 1.5 cm) of porous in nature, like a copper plate coated with poly(methyl methacrylate) (PMMA) film or Whatman filter paper of different pore sizes, is inserted between the upper and lower electrodes. 3. Electrodes: These serve to make this device as an electronic component like a capacitor/resistor. For acquiring the fractional behavior, one needs to modulate the conductivity of the nanocomposite. This in turn alters the resistance and capacitance values of the electrical network formed between the electrodes to get fractional capacitors of different specifications. The conductivity is increased by having a nanocomposite whose wt% of MWCNT is greater than the percolation threshold [70] and a middle plate of size less than the upper and lower electrodes. To get the percolation threshold, we have measured the DC conductivity of the nanocomposites with different wt% of MWCNT. From the conductivity studies, the percolation threshold was found to lie between 0.5% and 0.75%, where the sharp increase in conductivity is noticed [58]. Therefore, a wt% of MWCNT loading from 0.75% onwards is used for manufacturing the device. The distributive nature of the charge flow from one plate to the other is augmented by the nanocom-

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posite clusters (as seen from Fig. 16.1(i)) formed by the introduction of the porous middle plate. The charge movement in the nanocomposite clusters provides a noninteger value of α and in turn yields the desired fractional behavior. This is verified by drawing the normalized imaginary part of impedance −Z /Zmax and the normalized imaginary part of electric modulus M /Mmax plots. From the normalized −Z and M plots of a solid-state fractional capacitor, a difference in their peaks is observed, which means both long-range conduction and short-range hopping happen inside the disordered material [52]. The explanation of the charge transport mechanism and the normalized −Z and M plots are given in Section 16.6. The solid-state fractional capacitors which are discussed in this chapter contain a PMMA-coated middle plate. For that middle plate, the SEM image of the porous surface is shown in Fig. 16.2 (taken with the help of a SUPRA 40 field emission scanning electron microscope (FESEM), Carl Zeiss SMT AG). The average pore size was found to be 16 µm.

Figure 16.2 SEM image of the surface of 1% PMMA-coated middle plate.

16.2.2 Fabrication procedure We have carried out the fabrication of the devices at different curing temperatures, i.e., from 25◦ C to 140◦ C. From the experiments, it was found that 60◦ C cured solid-state fractional capacitors provided wider bandwidth. Therefore, the fabrication procedure at 60◦ C is taken into account for the batch analysis. The fabrication process including the nanocomposite preparation is illustrated in Fig. 16.3. For the preparation of MWCNT-epoxy nanocomposite, two main materials – epoxy resin and MWCNT are required. The epoxy resin employed for the fabrication is Araldite standard epoxy adhesive from Huntsman International (India) Private Limited [13].

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Figure 16.3 Depiction of the fabrication process of a solid-state fractional capacitor at 60◦ C.

The pack has both resin and hardener, which need to be mixed in a ratio of 2:1. The MWCNT used in this study is Nanocyl NC 7000TM series obtained from Nanocyl SA, Belgium [77]. The steps for fabricating the fractional capacitor by the temperature curing method are summarized as follows. 1. A copper plate of dimension 1.5 cm × 1.5 cm × 0.06 cm is cut from the copper sheet to make the electrodes. 2. The electrodes are cleaned with 25% H2 SO4 solution to remove any oxide coating present on it. 3. The middle plate of size 1 cm × 1 cm is also cut from the same copper sheet. 4. The middle plate is cleaned and then coated with 1% PMMA solution. The coated copper plates are kept in open air for 24 h to dry the solvent in the PMMA [24]. 5. Six different combinations of nanocomposites by mixing MWCNT in epoxy (0.75%, 1%, 1.5%, 2%, 3%, and 4% MWCNT in the nanocomposites) have been prepared [58] following the process shown in Fig. 16.3.

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6. The measured amount of nanocomposite is coated on the lower plate; and the porous middle plate is placed over it. The arrangement is then kept in hot air at 60◦ C for 30 minutes, followed by keeping it at room temperature for 24 h. 7. After 24 h, the upper plate is coated with the required amount of nanocomposite and placed over the dried structure. 8. The entire structure is kept in hot air at 60◦ C for 30 minutes and then kept at room temperature for 24 h before making the electrical contacts. Now the fractional capacitor is ready for characterization.

16.3. Batch analysis of the solid-state fractional capacitors for deﬁning the guidelines In this section, the process that has been performed to formulate the specification is detailed. For this purpose, 10 numbers of fractional capacitors for each % of MWCNT nanocomposite (total 60) are made following the above procedure.

16.3.1 Characterization To frame the specifications of the fractional capacitor, first, the magnitude and phase angle (θ ) of impedance of the fabricated fractional capacitors are recorded with the help of an Alpha A analyzer (Novocontrol Technologies), which excites the fractional capacitor with a 1 Vrms sinusoidal signal in the frequency range 0.01 Hz to 20 MHz. The entire measurement has been carried out at a controlled room temperature of 25◦ C to minimize the temperature effect if any. The reading of each fractional capacitor is taken at different times to analyze the repeatability. Out of these 60 fractional capacitors, three from each % of MWCNT are taken for plotting the graphs ([FC1_0.75, FC2_0.75, FC3_0.75 for 0.75%], [FC1_1, FC2_1, FC3_1 for 1%], [FC1_1.5, FC2_1.5, FC3_1.5 for 1.5%], [FC1_2, FC2_2, FC3_2 for 2%], [FC1_3, FC2_3, FC3_3 for 3%], and [FC1_4, FC2_4, FC3_4 for 4%]). Figs. 16.4 to 16.7 show the magnitude and phase angle plots of the 18 fractional capacitors. In the ideal case, the fractional capacitor should have a constant phase over the entire frequency range, but practically it has CPA for a particular bandwidth. The average values of F1, C, and α are calculated from the corresponding magnitude and phase angle values in their CPZ. Those values are tabulated in Table 16.1.

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Figure 16.4 Variation of magnitude with % MWCNT (for 0.75% to 2%) in the frequency range 10 Hz to 200 kHz.

Figure 16.5 Variation of magnitude with % MWCNT (for 3% and 4%) in the frequency range 100 Hz to 300 kHz.

It is worth mentioning that all the 60 capacitors do not exhibit CPA at the same frequency zone. Some show CPA in two different frequency zones, and some do not show CPA at all. Possible reasons are the following: the MWCNT did not disperse uniformly to allow the charge to hop from one electrode to the other through nanocomposite clusters and the number of nanocomposite clusters formed is different due to the manual manufacturing process. A statistical study has been carried out to find the yield rate of the fabricated fractional capacitor and is presented in the next section.

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Table 16.1 Average values of α , F1, and C obtained from magnitude and phase angle data of the impedance for different % of MWCNT. F1 (nsα ) C (nF) Details of thickness % Sample CP zone CPA (in deg) α a MWCNT No. Thickness of Middle plate MWCNT the sample epoxy thickness thickness (mm) (mm) (mm) 0.75

1

1.5

2

4

a

1 kHz–118 kHz

1.7 kHz–118 kHz

117 Hz–10 kHz 19.6 Hz–3.32 kHz 12.6 Hz–2.13 kHz 38.3 Hz–2.13 kHz 697 Hz–118 kHz 4.15 kHz–184 kHz 228 Hz–75.6 kHz 10 kHz–118 kHz 10 kHz–231 kHz 12.7 kHz–231 kHz

Fractional exponent from phase plot.

−85.2 ± 0.475 −85.6 ± 0.381 −85.8 ± 0.283 −82.0 ± 1.735 −80.2 ± 2.061 −82.3 ± 1.888 −77.7 ± 0.558 −77.7 ± 0.643 −76.3 ± 0.669 −53.2 ± 2.14 −52.2 ± 2.346 −51.2 ± 1.315 −45.0 ± 2.976 −44.0 ± 2.795 −45.2 ± 2.841 −38.8 ± 1.16 −39.2 ± 2.90 −39.3 ± 1.917

0.947 ± 0.005 0.951 ± 0.004 0.954 ± 0.003 0.911 ± 0.019 0.892 ± 0.022 0.914 ± 0.020 0.863 ± 0.006 0.863 ± 0.007 0.848 ± 0.007 0.591 ± 0.023 0.580 ± 0.026 0.569 ± 0.014 0.501 ± 0.033 0.489 ± 0.031 0.503 ± 0.031 0.431 ± 0.012 0.436 ± 0.032 0.436 ± 0.021

0.0258± 0.0015 0.0335±0.0015 0.0338±0.0011 0.0577±0.0118 0.0696±0.0174 0.0671±0.0149 0.1210±0.0055 0.1040±0.0081 0.1450±0.0109 4.59±0.9120 4.55±0.8550 5.64±0.6350 125±44.3 157±53.8 164±52.9 417±51.2 436±159 503±126

0.0144±0.0011 0.0196±0.0013 0.0203±0.0013 0.0203±0.0026 0.0196±0.0030 0.0244±0.0029 0.0370±0.0067 0.0313±0.0055 0.0382±0.0076 0.2710±0.1710 0.2920±0.1780 0.2560±0.1340 0.6400±0.458 0.3800±0.2400 1.3400±1.0100 0.3850±0.1840 0.3890±0.2340 0.4490±0.2320

0.243 ± 0.002 0.231 ± 0.011 0.248 ± 0.002 0.251 ± 0.004 0.227 ± 0.010 0.264 ± 0.005 0.231 ± 0.008 0.243 ± 0.000 0.234 ± 0.003 0.228 ± 0.008 0.234 ± 0.001 0.222 ± 0.002 0.220 ± 0.002 0.253 ± 0.002 0.245 ± 0.001 0.251 ± 0.008 0.237 ± 0.012 0.234 ± 0.003

1.035 0.940 1.038 1.15 0.885 0.785 1.150 1.091 0.99 0.998 1.003 0.947 2.305 2.103 1.689 1.733 1.701 1.599

1.684 ± 0.192 1.578 ± 0.195 1.694 ± 0.143 1.817 ± 0.094 1.538 ± 0.065 1.463 ± 0.086 1.735 ± 0.063 1.737 ± 0.092 1.645 ± 0.018 1.662 ± 0.137 1.668 ± 0.185 1.601 ± 0.013 2.938 ± 0.437 2.772 ± 0.072 2.355 ± 0.200 2.391 ± 0.0365 2.343 ± 0.212 2.253 ± 0.152

Dina Anna John and Karabi Biswas

3

FC1_0.75 FC2_0.75 FC3_0.75 FC1_1 FC2_1 FC3_1 FC1_1.5 FC2_1.5 FC3_1.5 FC1_2 FC2_2 FC3_2 FC1_3 FC2_3 FC3_3 FC1_4 FC2_4 FC3_4

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Figure 16.6 Variation of phase angle with % MWCNT (for 0.75% to 2%) in the frequency range 10 Hz to 200 kHz.

Figure 16.7 Variation of phase angle with % MWCNT (for 3% and 4%) in the frequency range 100 Hz to 300 kHz.

16.3.2 Yield rate Table 16.2 presents the number of fractional capacitors having the same value of phase angle (average) and the corresponding F1 over a particular frequency range. The above details are obtained from a batch of 10 devices

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Table 16.2 CPA and F1 obtained for different % of MWCNT. % CPA (in deg) CP zone F1 (sα ) MWCNT 0.75

−85 to −86 −83 to −84

1

−82 to −83 −80 to −81

1.5

−77 to −78 −76 to −77

2

−53 to −54 −52 to −53 −51 to −52

3

−45 to −46 −44 to −45

4

−39 to −40 −38 to −39 −37 to −38

1 kHz–118 kHz 1 kHz–118 kHz 1.7 kHz–118 kHz 1.7 kHz–118 kHz 117 Hz–10 kHz 117 Hz–10 kHz 19.6 Hz–3.32 kHz 12.6 Hz–2.13 kHz 38.3 Hz–2.13 kHz 228 Hz–118 kHz 4.15 kHz–184 kHz 10 kHz–231 kHz 10 kHz–118 kHz 8.11 kHz–231 kHz

0.0258 n to 0.033 n 0.0386 n 0.0577 n to 0.0704 n 0.0696 n 0.104 n to 0.120 n 0.121 n to 0.145 n 4.59 n 4.55 n 5.64 n 117 n to 164 n 157 n 436 n to 503 n 417 n 769 n to 771 n

No. of fractional capacitors

Yield (%)

3 1 4 1 2 2 1 1 1 3 1 2 1 2

40 50 40

30 40

50

Figure 16.8 Bar chart of deviation from the mean value of phase angle in the CP zone.

for each % of MWCNT. From the table, it can be seen that the values of the CPA and F1 increase with % of MWCNT. To observe the phase variability of the 18 fractional capacitors mentioned in Section 16.3.1, the average value of the absolute deviation of phase angle (in the given CP zone) has been calculated and plotted as bar graph in Fig. 16.8. From the figure, it is found that six fractional capacitors have a deviation less than 1 degree, five have a deviation in between 1 and 2 degrees, and seven have a deviation in between 2 and 3 degrees.

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Considering ±3 degrees deviation as the acceptable limit in the CP zone, the yield rates of the usable fractional capacitors are: for 0.75% MWCNT – 40%; for 1% MWCNT – 50%; for 1.5% MWCNT – 40%; for 2% MWCNT – 30%; for 3% MWCNT – 40%; and for 4% MWCNT – 50%. It is important to mention that ideally, the fractional capacitor should exhibit CPA for the entire frequency range. However, the reported bandwidth is about 3 decades only (can be seen in Table 16.1). The following guidelines can be designed from the above observations. 1. To get an F1 value between 2 × 10−11 and 4 × 10−11 sα and α from 0.927 to 0.954, use 0.75% MWCNT. 2. To obtain an F1 value between 5 × 10−11 and 8 × 10−11 sα and α from 0.880 to 0.920, use 1% MWCNT. 3. To acquire an F1 value between 1 × 10−10 and 2 × 10−10 sα and α from 0.844 to 0.866, one can choose 1.5% MWCNT. 4. To have an F1 value between 4 × 10−9 and 6 × 10−9 sα and α from 0.566 to 0.600, use 2% MWCNT. 5. To get an F1 value between 1 × 10−7 and 2.5 × 10−7 sα and α from 0.488 to 0.511, one can utilize 3% MWCNT. 6. To obtain an F1 value between 4 × 10−7 and 8 × 10−7 sα and α from 0.420 to 0.442, use 4% MWCNT. It was also found that the variation of phase angle and fractance (F1) was around 3% over a period of two years. Next, the effect of dielectric thickness was studied to see whether it has any effect on α and F1.

16.3.3 Effect of thickness of the nanocomposite and the middle plate A measured amount of nanocomposite has been placed between the two electrodes. Similarly, all the middle plates are coated with 1% PMMA solution and used to make the device. However, as the fabrication has been done manually, the thickness of the nanocomposites may differ slightly in the same batch. Hence, we have investigated whether the thicknesses of the nanocomposite and the middle plate have any effect on the specifications. The thickness of the different parts of the fabricated fractional capacitor are measured using a Mitutoyo 293-821-30 micrometer (range: 0 to 25 mm; resolution: 0.001 mm) before and after the fabrication. And from that measurement, the thickness of the nanocomposite has been derived. However, the PMMA-coated middle plate thickness is measured

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directly. The thickness values are shown in the last three columns of Table 16.1. From Table 16.1, it is apparent that the middle plate thickness is almost the same. The thickness of the nanocomposite varies for different devices of the same batch but the fractional exponent α follows no pattern. Hence, we can conclude that there is no dependency of thickness (if it is in the range as mentioned in Table 16.1) on the fractional exponent of the solid-state fractional capacitor. The sixth and seventh columns of Table 16.1 show the values of F1 and C. Here also, we do not find an effect of thickness on these parameters. It must be pointed out that the capacitance value is shown to get an idea about the nature of the device. We can see that the capacitance value is in the nF range but it does not follow any particular relationship with the % of MWCNT or its thickness.

16.4. Validation of the deﬁned guidelines To validate the guideline for fabrication given in Section 16.3.2, 10 fractional capacitors are fabricated with 3% of MWCNT. The 10 newly fabricated fractional capacitors are characterized with the same impedance analyzer. Values of F1 and α have been calculated from the magnitude and phase angle data using Eqs. (16.9) and (16.10), respectively. The bandwidth is noted from the phase data collected with the impedance analyzer, and it is the frequency range where the fractional capacitors exhibit a CPA. The values of F1 and α are tabulated in Table 16.3. The CPAs obtained for 3% MWCNT fractional capacitors are: −40 to −42 degrees (one), −43 to −45 degrees (four), −45 to −47 degrees (three), and −50 to −52 degrees (two). That is, the value of α varies from 0.446 to 0.577 when rounded to three decimal places. For all the fractional capacitors, the deviation in α is ≤ 0.052. Again, the value of F1 is between 1 × 10−7 and 2.5 × 10−7 sα in the case of 3% except FCB7_3. But taking into account the specifications as mentioned in Section 16.3.2, 5 out of 10 fractional capacitors fall in the category of α = 0.488 to 0.511 and F1 = 1×10−7 to 2.5 ×10−7 sα . The graphs of parameters F1 and α for all fractional capacitors (belonging to 3% MWCNT) are shown in Fig. 16.9(i) and 16.9(ii). From the figures, it is seen that both the parameters are almost constant in the CPZ. A related point to consider from Table 16.3 is that the CPA does not have a particular trend with MWCNT epoxy thickness when the % MWCNT is the same.

a

Fractional exponent from phase plot.

Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor

Table 16.3 α and F1 values obtained from magnitude and phase angle data of fractional capacitors having a middle plate of pore size 16 μm. F1 (nsα ) MWCNT % Sample CP zone CPA (in deg) αa Sample epoxy MWCNT No. thickness thickness (mm) (mm) 3 FCB1_3 1 kHz–118 kHz −45.2 ± 2.970 0.502 ± 0.033 1.20 × 10−7 ± 4.02 × 10−8 2.290 2.910 FCB2_3 10 kHz–118 kHz −40.2 ± 1.694 0.446 ± 0.018 2.49 × 10−7 ± 5.61 × 10−8 2.276 2.918 FCB3_3 285 Hz–118 kHz −43.8 ± 4.707 0.486 ± 0.052 1.76 × 10−7 ± 9.51 × 10−8 1.753 2.389 FCB4_3 228 Hz–75.6 kHz −44.3 ± 3.300 0.492 ± 0.036 1.40 × 10−7 ± 5.20 × 10−8 1.990 2.641 FCB5_3 720 Hz–94.4 kHz −44.0 ± 1.620 0.488 ± 0.018 2.4 × 10−7 ± 4.6 × 10−8 2.100 2.760 FCB6_3 3.3 kHz–118 kHz −43.1 ± 3.154 0.479 ± 0.035 1.94 × 10−7 ± 6.60 × 10−8 2.135 2.764 FCB7_3 446 Hz–48.4 kHz −50.9 ± 4.603 0.565 ± 0.051 6.30 × 10−8 ± 3.42 × 10−8 1.628 2.242 FCB8_3 446 Hz–75.6 kHz −45.7 ± 3.516 0.508 ± 0.039 1.17 × 10−7 ± 4.65 × 10−8 1.901 2.552 FCB9_3 285 Hz–60.4 kHz −45.0 ± 2.831 0.500 ± 0.031 1.55 × 10−7 ± 4.29 × 10−8 1.700 2.350 FCB10_3 6.49 kHz–48.4 kHz −51.9 ± 2.888 0.577 ± 0.032 2.30 × 10−7 ± 1.09 × 10−7 2.139 2.772

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Figure 16.9 Fractional capacitors fabricated with nanocomposite having 3% MWCNT.

From the above discussion, it can be concluded that the fabrication of solid-state fractional capacitors with predefined fractance, F1, and α -values by varying the wt% of MWCNT is possible with a yield rate of at least 50%. However, an automated fabrication procedure may help to achieve a higher yield rate.

16.5. Material characterization 16.5.1 Details of the analysis As stated in Section 16.2, the solid-state fractional capacitors fabricated at 60◦ C curing temperature provided wide bandwidth. Now, to understand the reason behind the wide CPA for 60◦ C cured solid-state fractional capacitors, material characterization of nanocomposites cured at two different temperatures is conducted. They are: one cured at 25◦ C (for 24 h) and the other at 60◦ C (the nanocomposite is placed in a hot air oven at 60◦ C for 30 min, followed by 24 h at 25◦ C). The electrical model for the solid-state fractional capacitors is reported in [59]. The model consists of a parallel combination of resistor and constant phase element (CPE) connected in series (as shown in Fig. 16.10), and its impedance is represented as Z (jω) =

R1 R2 + , α 1 + R1 F1(jω) 1 1 + R2 F2(jω)α2

(16.12)

where R1 , R2 are the resistors, F1, F2 are the fractances, and α1 , α2 are the fractional exponents of CPEs in the model. Following the model,

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Figure 16.10 Electrical equivalent model of the solid-state fractional capacitor.

two charge transport mechanisms are happening inside the solid-state fractional capacitor. The parallel combination of R1 , F1, and α1 represents the charge transport in the grain/bulk (MWCNT/epoxy resin) [5]. Usually, it is predominant in the frequency region > 100 kHz, whereas the parallel combination of R2 , F2, and α2 is because of the movement of charges across the grain–bulk interface (i.e., MWCNT–epoxy resin/MWCNT–epoxy resin– PMMA interface), which is prominent in the low-frequency region (up to 100 kHz). The above electrical model and the wide CPA has given the motivation to apprehend the fractional behavior in terms of random walk of charges at the grain–bulk interface and the grain/bulk, and later correlate it with the material analysis carried out on the nanocomposites. For this study, first, the nanocomposites are prepared, and then the correlation between the parameters (mainly the CPA and CPZ) of a solid-state fractional capacitor with the chemical structure of the nanocomposite is explained by performing the below techniques: 1. Fourier transform infrared spectroscopy (FTIR) – to identify the functional groups in the nanocomposites; 2. Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) images – to comprehend the state of dispersion of MWCNT in epoxy resin; 3. Electrical property analysis – to get a better understanding of the charge transport in the solid-state fractional capacitor and the correlation of CPA with material characterization. The particulars of the material characterization conducted are provided in this paragraph. The FTIR spectrum (from 4000 to 500 cm−1 ) is scanned using an FTIR spectrometer. As mentioned earlier, this analysis is carried out to recognize the functional groups on the surface of the MWCNTs and the prepared nanocomposites. The same spectrum can also be used to extract the details about the dispersion of MWCNT (taking different wt% values of MWCNT) in epoxy resin and hardener. Many peaks are visible in the wavenumbers between 4000 and 500 cm−1 , and the particulars of the peaks which are taken for this analysis are provided in Table 16.4.

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Table 16.4 Normalized peak intensity values at various wavenumbers in FTIR spectra of the nanocomposites cured at 25◦ C and 60◦ C. Nanocom- WaveBand description Normalposite number ized peak type (cm−1 ) intensity IIR

0%

1.5%

3%

0%

1.5%

3%

917

25◦ C cured nanocomposites Epoxy group in epoxy resin

1.11

1036

OH stretching vibration from the re- 1.03 action of the amine with epoxy ring

3299

−OH group on the surface of epoxy 1.09 resin [78] Epoxy group in epoxy resin 1.13

916 1035.6

OH stretching vibration from the re- 1.03 action of the amine with epoxy ring

3300

−OH group on the surface of epoxy 1.12 resin and MWCNT Epoxy group in epoxy resin 1.21

915 1034

OH stretching vibration from the re- 1.07 action of the amine with epoxy ring

3287

−OH group on the surface of epoxy 1.22 resin and MWCNT 60◦ C cured nanocomposites Epoxy group in epoxy resin 1.1

917 1036

OH stretching vibration from the re- 1.02 action of the amine with epoxy ring

3299

−OH group on the surface of epoxy 1.08 resin Epoxy group in epoxy resin 1.03

916 1033

OH stretching vibration from the re- 1.0 action of the amine with epoxy ring

3300

−OH group on the surface of epoxy 1.02 resin and MWCNT Epoxy group in epoxy resin 1.04

916 1033

OH stretching vibration from the re- 1.01 action of the amine with epoxy ring

3300

−OH group on the surface of epoxy 1.03 resin and MWCNT

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Figure 16.11 FTIR spectrum of Nanocyl NC 7000TM MWCNTs.

The SEM analysis done on the fractured surface of the nanocomposites is measured using a MERLIN field emission scanning electron microscope (FE-SEM); a JEM-2100 HRTEM takes the TEM images of the nanocomposites made by the ultramicrotomy method.

16.5.2 Results from material characterization 16.5.2.1 FTIR spectra In this section, the FTIR spectra of MWCNT nanocomposites prepared at 25◦ C and 60◦ C are discussed. The FTIR spectrum of the Nanocyl NC 7000TM MWCNTs, which we have used for the nanocomposite preparation, is displayed in Fig. 16.11. The vibrational bands noted for MWCNT are 3422 cm−1 and 1046 cm−1 . The peaks at the above specified wavenumbers are because of the hydroxyl groups (−OH) present on the surface of MWCNT [62,116]. Now, moving on to the nanocomposites spectrum, this gives a picture of the curing reaction occurring inside it. The data of the FTIR spectra seen for the nanocomposites prepared at 25◦ C (RT) and 60◦ C (TC) are provided below. To make a comparison for both temperatures (RT and TC), 0, 1.5, and 3 wt% of MWCNT nanocomposites are prepared. (a) FTIR spectra of nanocomposites cured at 25◦ C (RT) The spectra of these nanocomposites are presented in Fig. 16.12(i). The information of the peaks with its normalized peak intensity taken for this analysis are presented in Table 16.4. Taking into account the normalized peak intensity at 917 cm−1 [71], its value has increased with the addition of MWCNT. This indicates the presence of more unreacted epoxy rings in the nanocomposite in comparison with the 0% MWCNT nanocomposite (i.e., the epoxide rings in the epoxy resin have not opened and reacted with the hardener).

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Figure 16.12 FTIR spectra of the nanocomposites with different wt% of MWCNT cured at (i) 25◦ C (RT) and (ii) 60◦ C (TC); R is the reference peak (with intensity IR ) used for ﬁnding the normalized peak intensity.

Thus, the unreacted epoxy ring can form hydrogen bonds with the −OH groups on the surface of MWCNT. The peak at 1036 cm−1 represents the OH stretching vibration [115], which has resulted from the reaction of amine groups (in hardener) with the epoxy ring. For this wavenumber, the normalized intensity has increased with the MWCNT addition, which implies more hardener amine groups are observed as unreacted with the epoxy resin, confirming the hydrogen bonding of epoxy ring with −OH groups of MWCNT. Again, the same trend of increasing peak intensity ratio with wt% of MWCNT is identified at 3299 cm−1 . This illustrates that all the −OH groups in MWCNT have not bonded with epoxy resin. In addition to this, a broadening of the spectrum around this wavenumber is spotted due to the inclusion of MWCNT which denotes the presence of agglomerates. (b) FTIR spectra of nanocomposites cured at 60◦ C The FTIR spectra of the nanocomposites prepared at 60◦ C are presented in Fig. 16.12(ii), and its normalized peak intensity data are shown in Table 16.4. The normalized intensities at wavenumber 917 for different wt% of MWCNT nanocomposites were assessed; the value has a decreasing nature with the addition of MWCNT, which signifies that the epoxide ring in the epoxy resin opens and reacts with the hardener along with the hydrogen bonding with MWCNT. Therefore, it can be inferred that for 1.5% and 3% nanocomposites,

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the reaction of epoxy resin with hardener is accelerated by a higher temperature. Consequently, the intensity ratios of the above wt% values of nanocomposites are smaller than those in the 0% case. Furthermore, the intensity ratio at 1036 cm−1 drops with the incremental rise in MWCNT loading, which implies that there is almost complete reaction of hardener with the epoxy resin. Likewise, a similar trend is seen at 3299 cm−1 , which signifies a lower number of unreacted hydroxyl groups on the surface of MWCNT; as a result, better dispersion of MWCNT in the nanocomposite is attained. In the case of nanocomposites cured at 60◦ C, the peak area in the FTIR spectra is smaller than that for the 25◦ C temperature cured nanocomposites (where the peak area indicates a higher presence of agglomerates in the nanocomposite). It is evident from the above examination that the reaction of epoxy ring with hardener has improved at 60◦ C.

16.5.2.2 SEM and TEM images The images obtained from SEM are from the nanocomposites’ fractured surfaces and are shown in Fig. 16.13. From the SEM images, less agglom-

Figure 16.13 SEM images of MWCNT nanocomposites with (i) 1.5 wt% at 25◦ C, (ii) 3 wt% at 25◦ C, (iii) 1.5 wt% at 60◦ C, and (iv) 3 wt% at 60◦ C.

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eration is visible in the nanocomposites cured at 60◦ C. The same information is confirmed from the TEM figures presented in Fig. 16.14, that is, the MWCNTs are uniformly dispersed (less agglomeration) in the specimens cured at 60◦ C, whereas, agglomeration is more in the 25◦ C cured nanocomposites.

Figure 16.14 TEM images of MWCNT nanocomposites with (i) 1.5 wt% at 25◦ C, (ii) 3 wt% at 25◦ C, (iii) 1.5 wt% at 60◦ C, and (iv) 3 wt% at 60◦ C.

It can be concluded from the above material characterization results that a more exfoliated structure with less agglomeration is found in the 60◦ C cured nanocomposites compared to the 25◦ C cured nanocomposites.

16.6. Correlating the material characterization with the CPA of a solid-state fractional capacitor In this part, an attempt to correlate the above attributes of nanocomposite with the electrical transport of charges from one electrode to the other is demonstrated, thus providing the fractional behavior information. The solid-state fractional capacitors taken for this analysis have a copper middle plate of 1 cm2 with 1% poly(methyl methacrylate) (PMMA)

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coating. The porous middle plate causes the formation of nanocomposite clusters (as shown in Fig. 16.1 and discussed in Section 16.2.1), and these provide the distributive time constants by the hopping of charges in these clusters. For this study, first, the normalized imaginary part of impedance −Z /Zmax and the normalized imaginary part of electric modulus M /Mmax plots with respect to frequency are drawn from the impedance, phase angle, and permittivity data recorded using the impedance analyzer. M is found from the complex electric modulus (M ∗ ), which is given by M ∗ = M + jM =

1 ∗

,

(16.13)

and ∗ denotes the complex permittivity. In [52], it has been reported that, whenever a difference in the peaks of normalized −Z and M is noticed, both long-range conduction and short-range hopping occur inside the disordered material. The same has been observed in the normalized −Z and M plots of solid-state fractional capacitors with 1.5% and 3% of MWCNT fabricated at 25◦ C (Fig. 16.15) and 60◦ C (Fig. 16.16), i.e., a difference in their peaks is seen. By correlating the difference in peaks with the concept of long and short-range conduction, it can be specified that there are two possible charge transport mechanisms happening in 1.5% and 3% fractional capacitors (i.e., when wt% of MWCNT > percolation threshold), namely: 1. charge transport in the nanocomposite clusters (which follows a random walk) generated by the agglomerates in between the electrodes, and

Figure 16.15 (i) M /Mmax and (ii) −Z /Zmax plots for solid-state fractional capacitors cured at 25◦ C.

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Figure 16.16 (i) M /Mmax and (ii) −Z /Zmax plots for solid-state fractional capacitors cured at 60◦ C.

Table 16.5 CP values and , (at 100 Hz) of solid-state fractional capacitors fabricated at 25◦ C and 60◦ C. Nanocom CP zone (Hz) Constant Maximum posite phase (in deg) deviation type from average CP for 2 years

0% 1.5% 3%

0% 1.5% 3%

25◦ C cured 117–1 M 0.226–10 1 k–60 k 30.6–228 8.1 k–184 k 60◦ C cured 117–1 M 1 k–148 k 697–118 k

solid-state fractional capacitors −89.0 ± 0.271 1 0.735 −11.3 ± 0.533 2 65.5 −50.8 ± 1.296 0.5 −9.09 ± 2.180 2 154 −46.8 ± 1.406 1.5 solid-state fractional capacitors −89.0 ± 0.271 1 0.735 −78.2 ± 1.821 1 7.31 −42.3 ± 1.909 1 1110

0.011 104 646

0.011 32.1 3980

2. short-range hopping of charges (or the random walk of charges) in the nanocomposite clusters introduced by the porous middle plate. Hence, the short-range hopping [92] of charges imparts the fractional behavior to the solid-state fractional capacitor. The specification of the solid-state fractional capacitors fabricated at 25◦ C and 60◦ C, i.e., the CP range as well as the CPA, are shown in Table 16.5. It is derived from the data obtained by the impedance analyzer. Now, we will illustrate the random walk of charges in the solid-state fractional capacitors. Firstly, taking into account the 25◦ C cured solid-state

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fractional capacitors, there is a possibility of obtaining two CP zones all the time. One of the CP zone tends to lie between 600 Hz and 500 kHz, while the other one will appear in the frequency values lower than 600 Hz. By relating the CP zone attained in the frequencies less than 600 Hz with the material characterization results, it can be stated that this CP zone is primarily caused by the charge transport in the agglomerates (which are seen between the electrodes) [56]. The charge transport is also improved by the conductive interface layer between the MWCNTs due to the hydrogen bonding of the epoxy ring in epoxy resin with MWCNT. Whereas, the CP zone obtained between 600 Hz and 500 kHz is because of the random walk of charges in the nanocomposite clusters contributed by the porous middle plate [58]. Also, for 60◦ C cured solid-state fractional capacitors, there are fewer agglomerates and better dispersion of the MWCNT in the nanocomposite, which eventually results in a single wide CP zone between 600 Hz and 500 kHz. This wide CP zone is mainly due to the hopping of charges in the nanocomposite clusters introduced by the middle plate. Besides, the CP zone seen between 600 Hz and 500 kHz is larger in solid-state fractional capacitors cured at 60◦ C than in 25◦ C cured solid-state fractional capacitors. This CP zone is located between the peaks of −Z and M . It can be noted from Table 16.5 that the CPA has increased with % MWCNT as there is a higher number of MWCNTs in the nanocomposite clusters, which promotes the charge transport due to hopping. The electrical model parameters (according to Eq. (16.12)) for the fractional capacitors used in this analysis are given in Table 16.6, and these parameters are extracted using MEISP software. From this table, it is noticed that the resistor values R1 , R2 have increased and the fractance values F1, F2 have decreased for the 60◦ C cured fractional capacitors in comparison with 25◦ C cured fractional capacitors. When correlating this with nanocomposite characterization, the increase in the model parameters is caused by the uniform dispersion and more reaction of the epoxy ring with diamine (making the interface layer more insulating), thus contributing more resistance to the flow of charges between two MWCNTs. Interrelating the electrical model parameters with the charge transport mechanisms, R2 , F2, and α2 are linked with the charge transport in the nanocomposite clusters due to agglomerates in between the electrodes, and the charge transport in the nanocomposite clusters introduced by the porous middle plate is associated with R1 , F1, and α1 . The values of and at 100 Hz are also specified in Table 16.5 to obtain insight into the behavior of the fractional capacitors at differ-

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Table 16.6 Electrical model parameters of solid-state fractional capacitors fabricated at 25◦ C and 60◦ C. wt% of MWCNT R1 () F1 (sα ) α1 R2 () F2 (sα ) α2

1.5% 3% 1.5% 3%

25◦ C cured 995 k 33.6 k 60◦ C cured 8.56 M 15.5 k

solid-state fractional capacitors 0.821 μ 0.592 1.38 M 14.7 n 1μ 0.858 93.1 k 95.5 n solid-state fractional capacitors 0.743 n 0.895 1.36 × 1010 0.120 n 3.24 n 0.746 220 k 9.54 n

0.626 0.553 0.888 0.777

ent MWCNT loading. For this, we assume that there are many parallel combinations of resistors and nanocapacitors (developed in between two MWCNTs) connected in series between the electrodes. From Table 16.5, it can be seen that for 0% MWCNT, > , which evidences low leakage current through the resistors, because of which the solid-state fractional capacitor has a capacitive behavior, whereas > for solid-state fractional capacitors (including both RT and TC cases) with 1.5% and 3% loading, which indicates that the leakage current through the resistors and the capacitive effect result in the fractional behavior. In addition, the maximum deviation of the average CPA for the fabricated fractional capacitors is ±2◦ over a period of two years, and this variation is less for the fractional capacitors cured at 60◦ C (seen in Table 16.5). A comparison chart of some of the single component fractional capacitors reported in the literature, including the fabricated solid-state fractional capacitor, is given in Table 16.7.

16.7. Conclusion In this chapter, the fabrication procedure of the solid-state fractional capacitors is described in detail. The fractional capacitor using MWCNTpolymer nanocomposite is easy to manufacture, compact, and size comparable to a discrete capacitor and does not require hermetic sealing to keep the plates intact. Later, fabrication guidelines of the fractional capacitor with the desired exponent (α ) and fractance (F1) have been discussed with the help of batch analysis. Based on the proposed guidelines, identical fractional capacitors with a yield rate of 50% can be manufactured. Batch production and automated fabrication processes will help to improve the yield rate and the performance; hence, they are considered future works. Even though particular values of F1 and α were attained, the CPZ is not the

Fractional capacitor [26,27]

2000

Fractal based [50]

2005

Liquid fractional capacitor [23]

2005

(Experimental) Ripple = ±2.7◦ CPA = −36◦ CPZ = 20 Hz to 400 kHz CPA = −27◦ and −45◦ Ripple = ±6◦ CPZ = 10 kHz to 1 MHz (Simulation) CPZ = 2 kHz to 1 MHz (Experimental) Ripple = ±2.7◦ CPA = −36◦ or −45◦ CPZ = 100 Hz–1 MHz Ripple = ±3◦ CPA = −9◦ to −76.5◦

thickness and metal thickness. Large dimension. Analog fractional controller for temperature and motor control applications [28] (Experimental studies) Analog integrator, analog differentiator and study of fractional PID controller [51] (Simulation and experimental studies)

Vary CPA and CPZ by dipping length, coating thickness, pH, conductivity. Spillage issue. Low yield rate, low shelf life (max = 6 months). No correlation of α with parameters.

Analog integrator [24], analog differentiator [75], milk adulteration detection [35], soil moisture measurement [32], Wien bridge oscillator [89], Sallen Key LP, HP and BP filter [88], fractional PLL [100], fractional inductor, fractional band-pass filter [101] (Simulation and experimental studies) continued on next page

513

No spillage, long life. Hermetic sealing required. No parameter to change α . Size = 3.5 cm×3.5 cm×0.6 cm. Fabrication procedure for α = 0.4 and 0.5 given. Vary the CPZ with dielectric type and its thickness, type and doping of substrate.

Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor

Table 16.7 Comparison of present work with the other works on single-component fractional capacitors. Fractional Year Parameters Advantages & Disadvantages Applications capacitor Not given Fractal based 1997 CPZ = 100 kHz to 100 GHz Fabrication procedure for only [49] (Simulation) α = 0.4 given. CPZ = 100 kHz to 1 MHz Vary the CPZ with dielectric

Fractal electrode based electrolytic type [53] Electrostatic capacitors [40]

Parameters

Advantages & Disadvantages

Applications

2009

CPZ = Not given CPA = −18◦ to −54◦ Ripple = Not given

Vary α by changing area of fractal structure and molarity of the solution. Liquid spillage, bulky.

Not given

2013

CPZ = 50 kHz to 2 MHz CPA = −31◦ to −67◦ Ripple = Not given

Hartley oscillator [41] (Experimental studies)

2013

CPZ = 1 Hz to 100 Hz CPA = −4.5◦ and −27◦ Ripple = Not given

2015

CPZ = 20 Hz to 2 MHz CPA = −31◦ Ripple = ±2◦

Vary α by varying loading % of graphene. Fabrication procedure – difficult and tedious. PCB compatible. Vary CPZ by absorption time. No guideline to tune α . Fabrication process requires special care. Long CPZ, high yield rate. No guideline to tune α . Size large, spillage issue.

2017

CPZ = 10 kHz to 10 MHz CPA = −65◦ to −83◦ Ripple = Not given

PCB compatible. Tune α by varying polymers. Simple procedure.

Not given

Not given

Not given

continued on next page

Dina Anna John and Karabi Biswas

Ionic polymer metal composite [31] CNT based electrochemical fractional capacitor [2] Ferroelectric polymer based fractional capacitor [7]

514

Table 16.7 (continued) Fractional Year capacitor

MWCNTpolymer composite based fractional capacitor [8] MoS2 ferroelectric polymer based fractional capacitor [6] Solid-state fractional capacitor [58] (Present work)

Parameters

Advantages & Disadvantages

Applications

2018

CPZ = 150 kHz to 2 MHz CPA = −7◦ to −65◦ Ripple = Not given

PCB compatible. Fabrication procedure – difficult and time consuming.

Hartley oscillator [8] (Experimental studies)

2018

CPZ = 100 Hz to 10 MHz CPA = −58◦ to −80◦ Ripple = ±4◦

PCB compatible. Fabrication procedure – difficult and time consuming.

Not given

2016

CPZ = 0.226 Hz to 500 kHz CPA = −9.09◦ to −89◦ Ripple = ±3◦

Small size = 1.5 cm×1.5 cm×0.06 cm. Fabrication procedure – simple. No hermetic sealing required. Tune α by wt% of MWCNT.

Analog fractional PI controller [60], analog fractional PID controller [55] (Simulation and experimental studies)

Design guidelines for fabrication of MWCNT-polymer based solid-state fractional capacitor

Table 16.7 (continued) Fractional Year capacitor

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same for all the devices with a particular % of MWCNT. This is because the electrical network formed by MWCNTs on different devices is not the same. It is mainly caused by the dispersion of MWCNT in the nanocomposite and the difference in the number of nanocomposite clusters formed for fractional capacitors fabricated with the same wt% of MWCNT. Therefore, one of the limitations of the fabrication process discussed in this work is that CPZ is not the same for the devices with different wt% of MWCNT. This opens up a future direction for modifying the fabrication process of the fractional capacitor or using different combinations of polymers to get the same CPZ for all the devices with different wt% of MWCNT. From the material characterization results, it can be inferred that more agglomerates and less reaction of epoxy resin with diamine are found in the case of nanocomposites cured at 25◦ C temperature when compared to the ones cured at 60◦ C. As a result, the 25◦ C temperature cured fractional capacitors have CPA at two frequency zones. Conclusively, the following points can be taken into account from the analysis carried out in this work. The nanocomposite chosen for the fabrication of a solid-state fractional capacitor should have a minimum wt% of MWCNT greater than the percolation threshold, and a porous middle plate needs to be inserted, which causes the formation of nanocomposite clusters (charge transport in these clusters imparts the fractional behavior). The ideal curing temperature to get a wide CPA for the solid-state fractional capacitor is 60◦ C. The probable wide CP zone for the solid-state fractional capacitors cured at 60◦ C is between 600 Hz and 500 kHz. It is also observed from the results that, with a change in wt% of MWCNT, CPA can be altered. Yet another future work will be focused on inserting different types of middle plates and investigating the performance of the devices by carrying out similar material characterization and correlation studies.

Acknowledgments This chapter was based on work of the European “COST Action CA15225,” which is focused on “Fractional-order systems; analysis, synthesis and their importance for future design.”

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Index

A Active building block (ABB), 65, 353 Adaptive control, 111, 183, 281 Adjustable current amplifier (ACA), 339, 353 Analog circuit, 37, 216, 231, 236 Armature circuit, 159 resistance, 159 Arteriovenous access stenosis (AAS), 279 Automated fabrication procedure, 502 processes, 512 Autoregressive (AR) model, 278 Autoregressive integrated moving average (ARIMA), 278 Autoregressive moving average (ARMA), 278

B Bias currents, 9, 13, 15–18, 26, 28, 29 Bifurcation diagram, 40, 45, 94, 97, 104, 106, 131, 140, 280 Biological chaotic models, 38 networks, 281, 293 systems, 276–278, 284, 430 Black tea samples, 375, 377, 398 Boron nitride (BN), 458, 468, 470 Bulk filler material, 463

C Capacitance fractional, 235, 238 in electronic devices, 455 tolerances, 79 values, 79, 475, 491, 500 Capacitive fractor, 240, 258, 260, 262, 265 Capacitor dielectric, 438, 442, 447 dielectric material, 457 fabrication, 440

fractional, 65, 66, 68, 69, 71, 73–76, 79, 486–491, 493–495, 497, 499, 500 nonideal, 446 response, 438 transfer function, 446 Caputo fractional derivative, 278, 487 difference, 163 Carbon nanotube (CNT), 439 device, 250, 258, 262, 269 fractor, 236, 241, 242, 246–249, 252, 256, 257, 259, 262, 269 fractor device, 250 fractor impedance plot, 246 Cauer networks, 4 RC networks, 6 Causality concept, 293 Cell capacitance, 250 Central pattern generator (CPG), 293 Chaos control, 37, 38, 89, 183, 192, 195 in fractional chaotic systems, 195 fractional rational maps, 90 nonlinear systems, 124 synchronization, 90, 111, 124 theory, 103 Chaotic attractor, 35, 40, 45, 48, 49, 52, 94, 104, 106, 125, 131, 137, 140, 182 behavior, 36, 38–40, 50, 97, 104, 124, 134, 137, 140, 145, 181, 207, 209, 211, 486 circuits, 36–39 combined synchronization, 91 dynamics, 40, 90, 124, 125, 134 flows, 35, 39 fractional discrete-time system, 111 fractional maps, 91, 92, 111 hidden attractor, 40 hyperjerk system implementation, 37 maps, 39, 111, 144 523

524

master systems, 203 model, 36, 221, 223 motion, 105 nature, 182, 209 oscillation circuit, 208 oscillators, 36, 231 oscillators synchronization control, 38 phase attractor, 189 regions, 45 signals, 200, 203 systems, 35–39, 50, 52, 89, 124, 130, 181–183, 185, 207–209, 276 systems chaos control, 183 systems synchronization, 35 trajectories, 140 Characteristic equation, 40, 75, 78, 100, 133, 363, 421 frequency, 474 Charging circuits, 432 Circuit blocks, 251 complexity, 487 components, 76, 79, 365, 380 design, 333 design problems, 325 diagrams, 359, 360 electrical, 36, 71, 375, 376, 380, 391, 399, 432 elements, 64, 235, 236, 240, 250, 256, 260, 375, 376, 380, 381, 392, 393 FO, 235, 236, 239–241, 256, 265, 269 GIC, 240, 260, 269, 339 impedance, 267 implementation, 16, 50, 71 model, 395 oscillatory, 65, 66 parameters, 393 passive, 72 realization, 52, 208 simulations, 209, 356, 357 theory, 238 Circuital equation, 217 CMOS circuits, 439 OTA, 428 Cole–Cole (CC) model, 474

Index

Combination synchronization, 100 synchronization projective, 192 synchronization scheme, 97 synchronization technique, 183 Complete synchronization, 90, 124 Condition of oscillation (CO), 65 Conductive fillers, 440, 446, 457, 469, 477 nanofillers network, 466 network, 473, 474 Constant phase angle (CPA), 240, 241, 488 Constant phase (CP), 236, 238, 240, 242, 438 Constant phase element (CPE), 1, 2, 25, 29, 238, 380, 393, 404, 438, 502 emulator, 7 fractional exponents, 502 parallel, 392 Constant phase zone (CPZ), 236, 239–241, 438, 489 width, 244, 246, 248, 249, 257 Continued fraction expansion (CFE), 2, 68, 239, 338, 405, 486 Control applications, 488 chaos, 37, 38, 89, 183, 192, 195 efficiency, 166 effort, 145 electronic, 7, 8, 10, 14, 16, 28, 29 engineering practice, 306 errors, 195, 200 law, 91, 99, 106, 124, 125, 130, 133, 134, 141, 144, 145, 195 loop, 165 mechanism, 330 methods, 124, 183 nonlinear, 158 parameters, 97, 105 phase, 316, 320 schemes, 90, 125 strategy, 124 synchronization, 91 systems, 403 technique, 157, 184, 195 theory, 1, 198

525

Index

Controllable CPE emulator, 29 systems, 157 Controller fractional, 106, 165, 487 tuning, 165 Conventional capacitance, 1 capacitor, 65 capacitor cell, 439 passive circuit elements phase angles, 72 passive components, 71 Convolutional neural network (CNN), 290, 291 Current differencing buffered amplifier (CDBA), 339, 353, 357 Current feedback operational amplifier (CFOA), 2, 65, 240, 339, 404, 488 Current follower (CF), 353 Cutoff frequency, 437

D DC motor, 157–160, 166 circuit, 159 control, 158 function block diagram, 160 model, 159 parameters, 170 sensorless control, 158 Derivative orders, 307 Derivative orders fractional, 317 Device fabricated, 242, 247, 490 fractance, 236, 238 testing, 445 Differential fractional order, 320 phase control, 316, 320 Differential difference current conveyor (DDCC), 66, 339, 353 Differential voltage current conveyor (DVCC), 65, 66 Dimensional fractional maps, 90, 125 Dipolar relaxation frequency, 442 Discrete fractional calculus (DFC), 123

Dislocated synchronization, 183 Domino RC ladder network, 406, 408 Double layer capacitor, 380, 381

E Electric motor force (EMF), 160 Electrical circuit, 36, 71, 375, 376, 380, 391, 399, 432 circuit equivalent, 391 devices, 446 equivalent circuit, 474 networks, 337 systems, 84 Electrical impedance spectroscopy (EIS), 249 Electrical percolation threshold (EPT), 439 Electrochemical devices, 245, 248 FOCs, 442 fractor, 236 fractor suffer, 245 impedance spectra, 474 system, 242, 249, 374, 375, 380, 381, 391, 394, 395 type fractor, 245 Electroencephalogram (EEG), 278 signal, 280 signal processing, 280 Electronic circuit, 218, 235, 246, 269, 488 circuit design, 414 control, 7, 8, 10, 14, 16, 28, 29 devices, 447 tongue, 375, 376, 396, 398 Emulator characteristics, 8 FI, 6, 10, 16, 18, 20, 28, 29 order, 28 parameters, 28 Encryption, 35, 124, 219, 221–223, 225, 226 Entangled multiqubit systems, 329 qubit states, 320, 325, 327, 330 Equilibrium points, 35, 40, 43, 45, 52, 125, 182

526

Equivalent circuit, 375, 376, 380, 381, 391, 392, 394, 471, 473, 475 circuit electrical, 474 network, 393 Evoked potential (EP), 280

F Fabricated device, 242, 247, 490 fractional capacitors, 494, 512 fractor, 246 Fabrication capacitor, 440 FOC, 439, 444, 446, 447, 457, 467–469, 471, 477 fractor, 259 guidelines, 269, 490, 512 parameters, 257 procedure, 492, 512 process, 492, 516 technique, 258 Faradaic impedance, 395 process, 376 Ferroelectric polymer NCs, 447 PVDF, 447 Field emission scanning electron microscope (FESEM), 492 Filler agglomerates, 465 aggregate, 468 bond, 461 content, 474, 476 layers, 468 loading, 466 material, 439, 457, 461, 463, 465 material layers, 463 MWCNT, 477 network, 466 particles, 468, 469, 471, 477 properties, 446 solution, 468 FitzHugh–Nagumo (FHN) neuron model, 280

Index

Follow-the-leader-feedback (FLF) structure, 339 FOPI controller, 165, 169 Foster networks, 3 Fractance device (FD), 64, 236, 238, 337 Fractance value, 244, 248, 258, 264, 511 Fractional approximation methods, 486 behavior, 429, 489, 491, 492, 503, 508, 510 capacitance, 235, 238 Caputo, 94, 137 chaotic maps, 91, 125, 145 chaotic systems, 90, 195, 199, 204 chaotic Wang map, 125 circuit, 376, 486, 487 circuit elements, 376 circuit parameters, 393 controller, 106, 165, 487 cumulants, 275–277, 281, 284, 287, 293, 295 derivative, 36, 164, 184, 208–210, 231, 305, 306, 315, 324, 325 derivative orders, 317 difference, 108, 140, 485 difference equation, 92, 124, 127 difference equation delta, 128 difference operators, 90, 91, 123, 126 differential equation, 182, 486 differentiation operator, 163 dimension, 189 discrete Wang system, 134, 144, 145 domain, 123, 409, 413, 414, 421, 422 Duffing map, 90, 125 elements, 350, 376 equation, 486 error, 168 exponent, 379, 380, 383, 384, 489, 490, 500 filters, 339 flow map, 93, 97 generalized Hénon maps chaotic behavior, 90 impedance, 68, 72, 77 integral, 158 integration, 163, 165, 485 integrator, 165

Index

inverse matrix projective difference synchronization, 202 Laplacian operator, 356 Lorenz map, 93, 94, 97, 100 Lozi map, 93, 94, 97, 100, 101, 125, 130, 131, 133, 134 maps, 90, 93, 103, 124, 125, 129, 131, 134, 137, 140 maps stability, 125 models, 363, 390 number, 403 operators, 123, 140, 485, 486 order operator in FOPI controllers, 165 orders in signal processing, 276 oscillators, 66, 76 peers, 282 PI controllers, 166, 175 PID controller, 164, 374 powers, 288, 339 process, 280 Rössler map, 103, 104, 111 Rössler system, 104, 111 sine maps, 90 step, 369 step filter, 340 sum, 92 synchronization, 91 system, 38 system controllers, 104 theory, 337 theory concept, 337 value, 164, 189, 209, 383 versions, 278, 293 Wang map, 105, 125, 134, 137, 140–142, 145 Wang system, 134 Fractional ARIMA (FARIMA), 278 Fractional calculus (FC), 63, 65, 90, 91, 123, 125, 126, 182, 184, 235, 238, 275, 301, 302, 306, 373, 374, 485, 486 Fractional Fourier transform (FrFT), 275 Fractional inductor (FI), 74, 395, 403, 432, 486 circuits, 426 emulation, 6, 7, 10, 16, 26, 409 emulator, 6, 10, 16, 18, 20, 28, 29

527

Fractional moment (FM), 275, 281, 282 Fractional step all-pass filter (FSAPF), 346–348, 352, 367 Fractional step band-pass filter (FSBPF), 345, 346, 348, 352, 359, 360, 367 Fractional step band-stop filter (FSBSF), 347, 348, 353 Fractional step filter (FSF), 339, 340, 350, 353, 359, 360, 363, 365, 369 Fractional step filter (FSF) stability, 363, 365 Fractional step high-pass filter (FSHPF), 341, 345, 348, 351, 360, 367 Fractional step low-pass filter (FSLPF), 340, 341, 348, 350, 357, 359, 360, 367 Fractional-order capacitor (FOC), 1, 65, 66, 68, 69, 71, 73–76, 79, 240, 337–340, 367, 403–407, 437–439, 441, 447, 457, 470, 486–491, 493–495, 497, 499, 500 cylindrical, 478 dielectric, 442, 444–447, 468, 477 electrolytic, 441 fabrication, 439, 444, 446, 447, 457, 467–469, 471, 477 Fractional-order chaos system (FOCS), 275, 279 Fractional-order element (FOE), 238–240, 275, 339, 403–405, 410, 415, 417, 432, 437 Fractional-order filter (FOF), 339, 417, 421, 422, 429, 432 Fractional-order (FO) circuit, 235, 236, 239–241, 256, 265, 269 circuit element, 235 circuit synthesis, 269 elements, 1, 2, 29, 30, 235, 256 immittance, 235, 236, 239–241, 250, 253, 254, 256, 260, 261 impedance, 250 integrator, 10, 11, 16, 19, 29 parameters, 42, 45, 52, 246, 247, 262, 265, 314, 315, 330, 333 parameters in CNT fractor, 257 resonators, 265, 267

528

stage, 11, 12, 15–17, 22, 25, 26 system analysis, 235 Fractional-order inductor (FOI), 1, 240, 437 Fractional-order modeling and control (FOMCON) toolbox, 384 Fractional-order modeling (FOM), 275, 278 Fractional-order signal processing (FOSP), 275 Fractional-order system (FOS), 374, 378, 379, 399 Fractional-order transfer function (FOTF), 374, 376, 379–381, 383–385, 387–390, 393 Fractor design, 236 design techniques, 258 electrochemical, 236 fabricated, 246 fabrication, 259 immittance, 238 lifespan, 249 magnitude plot, 239 phase, 248 RC ladder, 262 realization, 252, 261 theory, 259 Frequency band, 238 controlling capability, 66 domain, 238 phase, 341, 350, 421 plots, 472, 476 range, 23, 30, 246, 247, 251, 408, 424, 427, 448, 472, 474, 488, 494, 497, 500 regions, 79, 503 spectra, 279 zones, 495, 516 Frequency of oscillation (FO), 65 Functional block diagram (FBD), 11, 12, 404, 405, 410 Fuzzy FOPI control system, 169 logic controller, 158

Index

G Generalized impedance converter (GIC), 236, 414 circuit, 240, 260, 269, 339 impedance, 261 network, 260 resistors, 262, 264, 265, 267 Granger causality (GC), 277, 280, 287, 288 Graphene nanosheet (GNS), 444, 458, 468, 470, 471 Guidelines fabrication, 490, 512 FOC fabrication, 440

H Hadamard mode (HM), 317–319, 329, 330 Hindmarsh–Rose (HR) neuron model, 280 Hybrid projection synchronization, 90, 124 Hyperchaos, 38, 208 Hyperjerk chaotic systems, 37 systems hyperchaotic behaviors, 37

I Impedance analyzer, 246, 472, 478, 500, 509, 510 characteristics, 247, 256, 261 circuit, 267 converter, 6, 236, 414 expression, 3 FDs, 64 FO, 250 fractional, 68, 72, 77 function, 3, 5, 64, 260, 406 GIC, 261 implementation, 415 magnitude, 9, 23, 27, 28, 424, 444 phase angle, 438 phase response, 427 plane, 240, 259 plots, 254, 261, 262 response, 263 spectroscopy, 471 values, 422 Impedance inverter circuit (IIC), 409 Impedance inverter multiplier circuit (IIMC), 412

529

Index

Imperfect capacitor, 375, 393, 394, 399 inductor, 393, 394, 399 Inductive fractor, 240, 260, 265 Inductor fractional, 74, 395, 403, 432, 486 Inertial systems, 395 Inhibitory connection, 284, 287, 288, 290, 291, 295 Integer-order (IO) capacitors, 1 stages, 17, 22 system, 1 transfer function, 240 Integral-order transfer function (IOTF), 375, 379, 380, 382, 383, 385, 387–390 Interconnected network, 440 Interference circuits, 314, 316, 318, 326, 327 multiqubit, 320, 327 quantum, 303, 319, 327, 329–331 Ionic polymer metal composite (IPMC), 441

J Jerk chaotic system, 37 systems, 36

K Kirchhoff circuital laws, 217

L Ladder capacitor values, 270 networks, 253, 254, 256, 260, 441 RC, 236, 240, 241, 254, 256, 258, 262, 404 resistor, 270 Lag synchronization, 183 Large amplitude pulse voltammetry (LAPV) signal, 378, 385 Largest Lyapunov exponent (LLE), 131 Layered nanofiller, 463, 465, 477

Linear controllers, 145 fractional discrete-time system, 93, 127 Linear time-invariant (LTI) dynamics, 374 systems, 379 Liu chaotic system, 208 Lorenz slave system, 94 Lozi fractional map, 93 Lyapunov exponent (LE), 36, 40, 94, 184, 209, 211, 231 Lyapunov stability theory, 200

M Magnetic resonance imaging (MRI), 278 Magnetoencephalogram (MEG), 278 Master chaotic system, 90 system, 90, 93, 97, 99, 100, 103, 106, 110, 111, 185, 186, 188, 203 Maxwell–Wagner–Sillars (MWS) relaxation, 438 Mean squared error (MSE), 227, 228, 230, 386, 387 Measurement probabilities, 309, 313, 315, 316, 318, 321, 329 Mechanical systems, 182, 276 Mechanical systems quantum, 303 Megastability, 38 Memristive circuits, 38 Model predictive control (MPC), 158 Moving average (MA), 278 Multielectrode voltammetric system, 376 Multioutput current follower (MO-CF), 339 Multiple Electrochemical Impedance Spectra Parameterization (MEISP), 474, 475, 478 Multiqubit interference circuits, 320, 327 quantum interference, 325 quantum interference circuits, 327, 329, 330 quantum interference control circuits, 312 quantum systems, 317, 330 systems, 303, 321, 324

530

Multistability, 35, 38 Multiwalled carbon nanotube (MWCNT), 442, 444, 489, 491, 493–495, 498–500, 503, 505–508, 511 addition, 506 epoxy, 500 filler, 477 fractional capacitors, 500 loading, 444, 491, 507, 512 nanocomposites, 494, 505, 506 MXenes, 439, 457, 458, 461, 477

N Nanocapacitors, 491, 512 Nanocomposite (NC) clusters, 492, 495, 509–511 dielectrics, 439 Nanofiller aggregates, 464 content, 467 for FOC fabrication, 458 mixture, 464 type, 467 Negative resistance RC oscillator, 65 Negative resistor, 260 Network conductive, 473, 474 connectivity, 295 filler, 466 GIC, 260 polymers, 445 RC ladder, 236, 242, 252, 259, 356, 406, 422, 427 resistors, 6 structure, 281, 287 synthesis, 376, 391, 399 system, 281 system connectome, 281 system functionality, 281 Neural networks, 293, 294 Neuronal network, 293 Nonlinear control, 158 control law, 125 discrete fractional system, 129 dynamic systems, 280 systems, 181

Index

Normalized root mean square error (NRMSE), 385 Number of pixels change rate (NPCR), 225, 226, 230

O Offset frequency, 82 Operational amplifier (Op-Amp), 2, 36, 65, 209, 353, 404, 488 Operational transconductance amplifier (OTA), 2, 16, 28, 66, 240, 353, 404, 427 Operational transconductance amplifier (OTA) circuits, 26 Operational transresistance amplifier (ORTA), 65 Ordinary differential equation system (ODES), 208 Oscillation frequency, 73, 75, 76, 79, 82 Oscillatory circuit, 65, 66, 437 systems, 84

P Parallel combination, 381, 393, 503 CPEs, 392 systems, 183, 204 Parallel resonator block (PRB), 417–419 Partial fraction expansion (PFE), 2, 13, 14 Passive capacitors, 9, 26, 29 circuit, 72 components, 66, 71, 73, 75, 79, 84, 209, 404, 488 RC networks, 405 resistors, 3, 7, 8 Peak frequency, 474 Peak signal-to-noise ratio (PSNR), 228 Phase angle, 71, 72, 380, 381, 393, 403, 438, 444, 448, 471, 472, 476, 488, 490, 494, 497–500, 509 control, 316, 320 deviation, 251 difference, 25, 337 effects, 303

531

Index

estimation, 304 fractor, 248 frequency, 341, 346, 350, 421 gates, 303, 304, 306, 313 noise, 66, 82 oscillations, 256 plots, 195, 247, 248, 267, 351–353 portraits, 209, 219 properties, 302 quantum, 305 response, 23, 69, 70, 258, 262, 348, 420, 427 shift, 65, 66, 76, 79, 82, 301–305, 307, 310, 311, 313, 316, 317, 321, 329, 348 shift operators, 302 space, 211, 302, 303 synchronization, 183 term, 302 value, 246, 248 variation, 239 Phase band (PB), 236, 239, 240, 246 Phase shift keying (PSK) applications, 66 PI controller, 158, 165–168, 175 Poles frequency, 422, 429 Polymer NCs, 439, 440, 445, 446, 458, 461, 464, 467, 468, 470 composite, 442 dielectric, 470 FOC dielectrics, 467 for FOC fabrication, 466 matrix, 447 synthesis, 463, 464 Poly(methyl methacrylate) (PMMA), 241, 242, 440, 441, 453, 455–457, 488, 491, 493, 499, 508 chain, 456 coatings, 440 films, 242, 440 matrix, 457 nanoporous, 457 Polyvinylidene fluoride (PVDF), 439, 442–444, 447–449, 451–453, 455, 471, 472, 474 amorphous, 447 chain, 453 composites, 473

copolymers, 453 density, 447 films, 455 FOCs, 470, 474 matrix, 444 monomers, 455 NC, 470, 471, 474, 477, 478 polymer, 453, 470, 471 polymer chains, 471 polymorphs, 452 samples, 449, 451 Porous middle plate, 491, 492, 494, 509–511 Principal component analysis (PCA), 376, 396, 397 Principal component (PC), 377, 396 Projective combination synchronization, 192 difference synchronization, 183, 185, 186, 190, 192, 203 difference synchronization error, 185, 186, 191 synchronization, 183, 192 Proportional fractional derivative (PFD), 209 Proportional integral derivative (PID), 374, 437 controller, 165 controller fractional, 164, 374 Pseudocapacitance, 1, 6, 7, 11, 15, 71, 75, 82, 438, 444, 471, 476

Q Quantum circuit, 303, 309, 319, 330 circuit implementation, 310, 311 gradient circuit, 305 interference circuit, 313 interference circuits, 303, 319, 327, 329–331 mechanical systems, 303 phase, 305 phase gates, 316, 321 phase shift, 311 phase shift circuits, 303 systems, 305, 312, 325, 330, 331

532

Qubit states, 301–304, 309, 312–314, 316–318, 320, 322, 325, 327, 329 Qubit states phase, 303, 313, 329

R Randles circuit, 375, 381, 382 equivalent circuit, 375, 376, 382, 385 Rationalized fractional impedances, 69 impedance function, 68 RC circuits, 414 ladder, 236, 240, 241, 254, 256, 258, 262, 404 circuits, 239, 252 fractor, 262 network, 236, 242, 252, 259, 356, 406, 422, 427 networks, 2, 7, 29, 241, 356, 404, 439, 440, 473, 477 parallel network, 338 Reduced graphene oxide (rGO), 439, 442 Resistors GIC, 262, 264, 267 microscale network, 439 network, 6 parallel combination, 502 passive, 3, 7, 8 Resonating frequency, 267 Response chaotic system, 90 Root mean square error (RMSE), 279 Routine circuit analysis, 418

S Scaling frequency, 410 Scanning electron microscopy (SEM), 503, 505, 507 Second-generation current conveyor (CCII), 65 Secure communication, 35, 103, 124, 146, 182, 184, 200, 204, 208, 209 Series impedances, 391, 393 resistor, 267, 489

Index

Signal flow graph (SFG), 339, 340, 350, 353 Signal processing, 182, 275, 277, 281, 282, 337, 338, 403, 417, 485 Signal processing systems, 63 Single-element fractor (SEF), 241, 256, 257 Sinusoidal oscillators, 63, 71–73, 75, 83, 84 Slave systems, 90, 93, 97, 99–101, 103, 104, 111, 185, 186, 188, 190, 203, 204 Solid-state fractional capacitor, 489, 490, 492, 494, 500, 502, 503, 508–510 Stability analysis, 77, 126, 162, 339, 364, 418, 421, 422, 429 conditions, 92, 129, 422 criterion, 184, 190 FSFs, 363, 365 investigations, 369 regions, 363 result, 92, 129 theorems, 432 theory, 123 Stability boundary locus (SBL), 486 Supercapacitors, 448, 477 Superposed qubit states, 302, 313 Synchronization approaches, 111 chaos, 90, 111, 124 control, 91 controllers, 94, 97 criterion, 91 error, 90, 99, 100, 106, 108, 109, 111, 184, 192 fractional, 91 methods, 110, 183 phase, 183 projective, 183, 192 schemes, 90, 91, 111, 183 technique, 183 Syndiotactic PMMA, 456 Systems biological, 276–278, 284, 430 chaotic, 35–39, 50, 52, 89, 124, 130, 181–183, 185, 207–209, 276 control, 403

533

Index

electrical, 84 fractional, 38 multiqubit, 303, 321, 324 nonlinear, 181 oscillatory, 84 parallel, 183, 204 quantum, 305, 312, 325, 330, 331 theory, 378

T Tank circuits, 438 Tea samples, 381–386, 389, 393, 396–398 Temporal activities, 287, 290 Total harmonic distortion (THD), 81 Transfer function (TF), 240, 340, 404 Transition metal dichalcogenide (TMDC), 439, 446, 457–459, 470, 477 Transmission electron microscopy (TEM), 503, 505, 507, 508 Tunable fractance, 258 fractors, 264

U Ultrasonic frequency, 469 Unified average changing intensity (UACI), 226 Universal voltage conveyor (UVC), 353

V Voice encryption, 209, 221 Voltage differencing inverting buffered amplifier (VDIBA), 66 Voltammetric electronic tongue, 375, 397 system, 375, 376, 378, 381–383, 387, 391, 396, 398, 399 system electronic tongues, 398

W Warburg impedance, 474 Wire mode (WM), 317

X X mode (XM), 317

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