Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach [1 ed.] 1786302691, 9781786302694

Citation preview

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

This work is dedicated to our son Romain and to the memory of our parents

Series Editor Jean-Paul Bourrières

Analysis, Modeling and Stability of Fractional Order Differential Systems 1 The Infinite State Approach

Jean-Claude Trigeassou Nezha Maamri

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Jean-Claude Trigeassou and Nezha Maamri to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019940902 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-269-4

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Part 1. Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs) . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1. The Fractional Integrator . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Simulation and modeling of integer order ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Simulation with analog computers . . . . . . . . . . . 1.2.2. Simulation with digital computers . . . . . . . . . . . . 1.2.3. Initial conditions . . . . . . . . . . . . . . . . . . . . . 1.2.4. State space representation and simulation diagram . . 1.2.5. Concluding remarks . . . . . . . . . . . . . . . . . . . 1.3. Origin of fractional integration: repeated integration . . . . 1.4. Riemann–Liouville integration . . . . . . . . . . . . . . . . 1.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2. Laplace transform of the Riemann–Liouville integral . 1.4.3. Fractional integration operator . . . . . . . . . . . . . . 1.4.4. Fractional differentiation . . . . . . . . . . . . . . . . . 1.5. Simulation of FDEs with a fractional integrator . . . . . . 1.5.1. Simulation of a one-derivative FDE . . . . . . . . . . . 1.5.2. FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3. Simulation of the general linear FDE . . . . . . . . . . A.1. Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1. Lord Kelvin’s principle . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

3 3 3 5 6 7 9 10 12 12 13 14 15 17 17 18 18 20 20

vi

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

A.1.2. A brief history of analog computing . . . . . . . . . . . . . . . . . . . . . A.1.3. Interpretation of the RK2 algorithm . . . . . . . . . . . . . . . . . . . . . A.1.4. The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 23

Chapter 2. Frequency Approach to the Synthesis of the Fractional Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.2. Frequency synthesis of the fractional derivator . . . . 2.3. Frequency synthesis of the fractional integrator . . . 2.3.1. Objective. . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Direct method . . . . . . . . . . . . . . . . . . . . 2.3.3. Indirect method . . . . . . . . . . . . . . . . . . . 2.3.4. Frequency synthesis of 1 / s n . . . . . . . . . . . 2.4. State space representation of I dn (s ) . . . . . . . . . . 2.5. Modal representation of I dn (s ) . . . . . . . . . . . . 2.6. Numerical algorithm. . . . . . . . . . . . . . . . . . . 2.7. Frequency validation . . . . . . . . . . . . . . . . . . 2.8. Time validation . . . . . . . . . . . . . . . . . . . . . 2.9. Internal state variables . . . . . . . . . . . . . . . . . . A.2. Appendix: design of fractional integrator parameters A.2.1. Definition of Gn . . . . . . . . . . . . . . . . . . A.2.2. Definition of α and η . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

25 26 28 28 29 30 32 33 36 40 41 44 47 49 49 51

Chapter 3. Comparison of Two Simulation Techniques . . . . . . . . . . . .

55

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Simulation with the Grünwald–Letnikov approach . . . . . . . . . . 3.2.1. Euler’s technique . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. The Grünwald–Letnikov fractional derivative . . . . . . . . . . 3.2.3. Numerical simulation with the Grünwald–Letnikov integrator . 3.2.4. Some specificities of the Grünwald–Letnikov integrator . . . . 3.2.5. Short memory principle . . . . . . . . . . . . . . . . . . . . . . 3.3. Simulation with infinite state approach . . . . . . . . . . . . . . . . 3.4. Caputo’s initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Numerical simulations. . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Comparison of discrete impulse responses (DIRs) . . . . . . . . 3.5.3. Simulation accuracy . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. Static error caused by the short memory principle . . . . . . . . 3.5.5. Caputo’s initialization . . . . . . . . . . . . . . . . . . . . . . . 3.5.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

55 56 56 58 60 61 63 66 68 69 69 70 72 74 75 78

Contents

vii

. . . . .

. . . . .

78 78 79 79 80

Chapter 4. Fractional Modeling of the Diffusive Interface . . . . . . . . . . .

81

A.3. Appendix: Mittag-Leffler function . A.3.1. Definition . . . . . . . . . . . . . A.3.2. Laplace transform . . . . . . . . n A.3.3. Unit step response of 1/ (s + a) A.3.4. Caputo’s initialization . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Heat transfer and diffusive model of the plane wall . . . . . . . . . 4.2.1. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Physical model of the diffusive interface . . . . . . . . . . . . . 4.2.3. Frequency analysis of the diffusive phenomenon . . . . . . . . 4.2.4. Time analysis of the diffusive phenomenon . . . . . . . . . . . 4.2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Fractional commensurate order models . . . . . . . . . . . . . . . . 4.3.1. Physical origin . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Analysis of physical commensurate order models . . . . . . . . 4.4. Optimization of the fractional commensurate order model . . . . . 4.4.1. The proposed frequency approach . . . . . . . . . . . . . . . . . 4.4.2. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Fractional non-commensurate order models. . . . . . . . . . . . . . 4.5.1. Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Parameter estimation of Hn1,n2(s) . . . . . . . . . . . . . . . . . 4.5.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Appendix: estimation of frequency responses – the least-squares approach . . . . . . . . . . . . . . . . . . . A.4.1. Identification of the commensurate order model H N −1, N ( j ω ) A.4.2. Parameter estimation of the non-commensurate model H n 1 ,n2 ( j ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

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

102 103

. . . . . .

104

Chapter 5. Modeling of Physical Systems with Fractional Models: an Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

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

. . . . . . . .

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

. . . . .

81 82 82 83 85 86 87 88 88 89 91 91 96 97 97 97 98 101 102

. . . . . . . .

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

. . . . .

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

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Modeling with mathematical models: some basic principles 5.3. Modeling of the induction motor . . . . . . . . . . . . . . . . 5.3.1. Construction of the induction motor. . . . . . . . . . . . 5.3.2. Principle of operation . . . . . . . . . . . . . . . . . . . . 5.3.3. Induction motor knowledge model . . . . . . . . . . . . 5.3.4. Park’s model. . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5. Fractional Park’s model . . . . . . . . . . . . . . . . . .

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

. . . . .

. . . . . . . .

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

. . . . . . . .

. . . . . . . .

107 108 109 109 109 110 112 114

viii

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

5.4. Identification of fractional Park’s model . . . . . . . . . . . . . 5.4.1. Simplified model . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Identification algorithm . . . . . . . . . . . . . . . . . . . . 5.4.3. Nonlinear optimization . . . . . . . . . . . . . . . . . . . . . 5.4.4. Simulation of yˆ k and σ k . . . . . . . . . . . . . . . . . . . 5.4.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6. Application to the identification of fractional Park’s model

. . . . . . .

. . . . . . .

117 117 118 119 122 122 123

Part 2. The Infinite State Approach . . . . . . . . . . . . . . . . . . . . . . . . . .

127

Chapter 6. The Distributed Model of the Fractional Integrator . . . . . . . .

129

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Origin of the frequency distributed model . . . . . . . . . . . . . . 6.3. Frequency distributed model . . . . . . . . . . . . . . . . . . . . . 6.4. Finite dimension approximation of the fractional integrator . . . . 6.5. Frequency synthesis and distributed model . . . . . . . . . . . . . 6.6. Numerical validation of the distributed model . . . . . . . . . . . 6.6.1. Reconstruction of the weighting function . . . . . . . . . . . . 6.6.2. Reconstruction of the impulse response. . . . . . . . . . . . . 6.7. Riemann–Liouville integration and convolution . . . . . . . . . . 6.7.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8. Physical interpretation of the frequency distributed model . . . . 6.8.1. The infinite RC transmission line . . . . . . . . . . . . . . . . 6.8.2. RC line and spatial Fourier transform . . . . . . . . . . . . . . 6.8.3. Impulse response of the RC line . . . . . . . . . . . . . . . . . 6.8.4. General solution . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.5. Initialization in the time and spatial domains . . . . . . . . . . A.6. Appendix: inverse Laplace transform of the fractional integrator

. . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

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

. . . . . . .

159

. . . . . . . . . .

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

. . . . . . .

Chapter 7. Modeling of FDEs and FDSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . .

129 130 133 134 136 138 138 140 142 147 147 147 149 151 153 155 156

. . . . . . . . . .

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

. . . . . . .

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

7.1. Introduction . . . . . . . . . . . . . . . . . . . 7.2. Closed-loop modeling of an elementary FDS . 7.3. Closed-loop modeling of an FDS. . . . . . . . 7.3.1. Modeling of an N-derivative FDS . . . . . 7.3.2. Distributed state . . . . . . . . . . . . . . . 7.4. Transients of the one-derivative FDS . . . . . 7.4.1. Numerical simulation . . . . . . . . . . . . 7.4.2. Initialization at t = t 1 . . . . . . . . . . . 7.4.3. Initialization at different instants . . . . . 7.5. Transients of a two-derivative FDS . . . . . .

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

. . . . . . .

. . . . . . . . . .

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

. . . . . . . . . .

. . . . . . . . . .

159 160 162 162 165 168 168 169 171 173

Contents

7.6. External or open-loop modeling of commensurate fractional order FDSs . . . . . . . . . . . . . . . . . . . . . 7.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . 7.6.2. External model of an elementary FDE . . . . . . 7.6.3. External representation of a two-derivative FDE. 7.6.4. External representation of an N-derivative FDE . 7.7. External and internal representations of an FDS . . . 7.8. Computation of the Mittag-Leffler function . . . . . . 7.8.1. Introduction . . . . . . . . . . . . . . . . . . . . . 7.8.2. Divergence of direct computation . . . . . . . . . 7.8.3. Step response approach . . . . . . . . . . . . . . . 7.8.4. Improved step response approach . . . . . . . . . A.7. Appendix: inverse Laplace transform of 1 / (s n + a )

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

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

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

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

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

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

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

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

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

175 175 176 179 180 182 183 183 184 185 186 189

Chapter 8. Fractional Differentiation . . . . . . . . . . . . . . . . . . . . . . . . .

193

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Implicit fractional differentiation . . . . . . . . . . . . . . . . . . . . 8.3. Explicit Riemann–Liouville and Caputo fractional derivatives . . . 8.3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Theoretical prerequisites . . . . . . . . . . . . . . . . . . . . . . 8.3.3. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Initial conditions of fractional derivatives . . . . . . . . . . . . . . . 8.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. Implicit derivative . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. Caputo derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4. Riemann–Liouville derivative . . . . . . . . . . . . . . . . . . . 8.4.5. Relations between initial conditions . . . . . . . . . . . . . . . . 8.5. Initial conditions in the general case . . . . . . . . . . . . . . . . . . 8.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Implicit derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3. Caputo derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4. Riemann–Liouville derivatives . . . . . . . . . . . . . . . . . . 8.5.5. Relations between initial conditions . . . . . . . . . . . . . . . . 8.6. Unicity of FDS transients . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1. Transients of the elementary FDE . . . . . . . . . . . . . . . . . 8.6.2. Unicity of transients . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Numerical simulation of Caputo and Riemann–Liouville transients 8.7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2. Simulation of Caputo derivative initialization . . . . . . . . . . 8.7.3. Simulation of Riemann–Liouville initialization . . . . . . . . .

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

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

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

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

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

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

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

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

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

ix

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

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

193 194 195 195 197 198 199 199 200 201 203 204 205 205 205 206 207 207 208 208 209 210 212 212 212 215

x

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Chapter 9. Analytical Expressions of FDS Transients . . . . . . . . . . . . . . 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Mittag-Leffler approach . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. Free response of the elementary FDS . . . . . . . . . . . . . 9.2.2. Free response of the N-derivative FDS . . . . . . . . . . . . 9.2.3. Complete solution of the N-derivative FDS . . . . . . . . . 9.3. Distributed exponential approach . . . . . . . . . . . . . . . . . 9.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Solution of Dn ( x(t ) ) = ax(t ) using frequency discretization 9.3.3. Solution of Dn ( x(t ) ) = ax(t ) using a continuous approach . 9.3.4. Solution of Dn ( x(t ) ) = ax(t ) using Picard’s method . . . . 9.3.5. Solution of D n ( X (t ) ) = AX (t ) . . . . . . . . . . . . . . . . 9.3.6. Solution of D n ( X (t ) ) = AX (t ) + Bu(t ) . . . . . . . . . . . 9.4. Numerical computation of analytical transients . . . . . . . . . . 9.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Computation of the forced response . . . . . . . . . . . . . . 9.4.3. Step response of a three-derivative FDS . . . . . . . . . . .

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

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

219 221 221 223 225 227 227 227 230 232 235 237 237 237 238 240

Chapter 10. Infinite State and Fractional Differentiation of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Calculation of the Caputo derivative . . . . . . . . . . . . . . . . 10.2.1. Fractional derivative of the Heaviside function . . . . . . . . 10.2.2. Fractional derivative of the power function . . . . . . . . . . 10.2.3. Fractional derivative of the exponential function . . . . . . . 10.2.4. Fractional derivative of the sine function . . . . . . . . . . . 10.3. Initial conditions of the Caputo derivative . . . . . . . . . . . . . 10.4. Transients of fractional derivatives . . . . . . . . . . . . . . . . . 10.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2. Heaviside function. . . . . . . . . . . . . . . . . . . . . . . . 10.4.3. Power function . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4. Exponential function . . . . . . . . . . . . . . . . . . . . . . 10.4.5. Sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Calculation of fractional derivatives with the implicit derivative 10.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2. Fractional derivative of the Heaviside function . . . . . . . . 10.5.3. Fractional derivative of the power function . . . . . . . . . . 10.5.4. Fractional derivative of the exponential function . . . . . . . 10.5.5. Fractional derivative of the sine function . . . . . . . . . . . 10.5.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

219

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

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

243 244 245 246 248 249 250 253 253 254 255 256 256 257 257 258 259 260 261 262

Contents

10.6. Numerical validation of Caputo derivative transients . 10.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . 10.6.2. Simulation results . . . . . . . . . . . . . . . . . . A.10. Appendix: convolution lemma . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

xi

. . . .

262 262 264 266

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285

Foreword

The synthesis of non-integer differentiation or integration has given a new impulse to this operator by extracting it from the mathematicianʼs drawer: enabling its application and, thus, the discovery of its remarkable properties in system dynamics. Indeed this operator has overcome the mass-damping dilemma in mechanics and the stability degree-precision dilemma in automatic control. The invalidation of these dilemmas is also inscribed in the more general context of the damping robustness in the CRONE approach. Solving these issues by passing to non-integer theory, and therefore by changing our way of thinking, is an excellent illustration of Albert Einsteinʼs quotation “We cannot solve problems with the thinking that created them”. It is true that noninteger differentiation or integration does not escape the slogan “different operator, different properties and performances”. That amounts to saying that this operator merits specific development. Therefore, one more book is not one book too many, especially if the book in question is a carefully thought-out monograph as proposed by the authors. So, I am delighted and sincerely thank Jean-Claude Trigeassou for benefiting the community with, his researcher and pedagogue qualities, by providing important contributions likely to clarify delicate subjects, which deserve to be discussed and even revisited. Such scientific qualities can certainly be attributed to his education, a French Agrégation in Applied Physics; however, he benefits from an additional quality that prevails over his education. It is a broadly tested common sense that has enabled him to construct an in-depth view of non-integer theory, which is reinforced by a constant reflection on both theoretical and practical foundations. He shares this view with us in these exemplary volumes.

xiv

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Jean-Claude Trigeassou has taken an interest in non-integer theory from the synthesis founded on recursivity, an idea that I then drew from fractality. Even if he was initially inspired by this idea, his scientific production was not limited to the synthesis. He has indeed extended his contributions to non-integer domains out of my field of expression, so that for a long time, our regular and frequent scientific exchanges have constituted, for us, a genuine mutual enrichment. By inscribing their contributions in analysis, modeling, initialization, observation and stability of fractional representations, through an infinite state approach, Jean-Claude Trigeassou and Nezha Maamri bring, among others, clarifying answers to the initial value problem and to stability analysis. This book (in two volumes) is all the more important as the author contributions take place beyond a mere overview on fractional systems. Their contributions indeed constitute a synthesis of the works they have led, for twenty years, in the framework of their infinite state approach. They themselves interpret this approach as a frequency distributed approach or as a fractional integrator approach: such an interpretation enables them to show the close relation between frequency and diffusive approaches, while highlighting their difference in a closed-loop context. Knowing that the thematics tackled in this book fall, in essence, into the category of experimental sciences (applied physics oblige), the authors inscribe the establishment of their results in the context of these sciences, by successively borrowing from: experimentation for phenomenology, mathematics for modeling, and numerical simulation for validation. By combining physical systems, numbered examples, comparisons and reachable calculus, the authors use all the ingredients liable to answer the readerʼs expectations and to convince him of the specificity and the interest of fractional systems, and this, in relation to integer systems that firmly constitute the substrate of all our educations. Finally, by associating a truly asserted pedagogical will with this conviction, the authors have achieved a reference work that I recognize with satisfaction, which honors not only the authors themselves but also the community as a whole. Alain OUSTALOUP Emeritus Professor at Bordeaux INP

Preface

This book in two volumes is dedicated to the analysis, modeling and stability of fractional order differential equations and systems using an original methodology entitled the infinite state approach. During a long period, since the early works of Liouville, Grünwald, Letnikov and Riemann (see the historical surveys in [OLD 74] and [MIL 93]), fractional calculus has remained a mathematical topic interesting a limited circle of researchers. More recently, after pioneering books [OLD 74, SAM 93, MIL 93], we observe an exponential increase in research works, as well as in the theoretical domain or applications, as reported in several monographs [POD 99, DIE 10, DAS 11, PET 11, ORT 11, OUS 15] and in many journal papers (Fractional Calculus and Applied Analysis, etc.) and Conferences (FDA, FSS, etc.). Fractional calculus is no longer a specialized topic of mathematics; it concerns henceforward many domains such as viscoelasticity [BAG 85, CHA 05], thermics [BAT 02], electricity in capacitors [WES 94] and in electrical machines [RET 99], electrochemistry [OLD 72], fractals [LEM 83] and biology [COL 33, MAG 06]. On the contrary, in the engineering domain, the works of Oustaloup on robust control [OUS 83], after the early works of Bode [BOD 45] and Manabe [MAN 60], have motivated a great interest in the automatic control community, as related by a great number of papers. Several monographs have intended to present an overview of these research works, such as the exemplary work of Podlubny [POD 99]. Many researchers have tried to generalize linear and nonlinear system theory to the fractional domain. However, some issues have been recognized as difficult problems, such as the initialization of fractional differential equations and systems.

xvi

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Moreover, some researchers have highlighted theoretical incoherences in these research works. For example, state variables in fractional differential equations are not equivalent to their integer order counterparts since they are no longer able to predict future behaviors, based on the initial conditions of Riemann–Liouville or Caputo derivatives. The Mittag-Leffler function is considered as the reference mathematical tool for the analysis of fractional systems. It is well adapted to formulate input–output dynamical transients expressed in terms of pseudo-state variables. Nevertheless, it has not been able to express dynamics due to internal state variables. In this book, we do not intend to propose a supplementary overview on fractional systems theory and applications. On the contrary, our essential objective is to provide a synthesis of research works related to an original methodology entitled the infinite state approach, which could have also been called the frequency distributed approach or the fractional integrator approach. This methodology provides solutions to the previous theoretical issues, and particularly to the initial value problem. It also provides original solutions to the stability analysis of fractional systems based on the Lyapunov technique or to their state control. Initially, this technique has been introduced to allow fractional system identification based on the output error technique. As this method requires the simulation of a differential model, the concept of closed-loop representation with fractional integrators has been proposed. This integrator has been approximately realized thanks to a frequency approach already used by Oustaloup for the fractional differentiator [OUS 00]. Although the concept of internal state variables was not initially a major concern, comparisons with the diffusive representation introduced by Montseny [MON 98], Matignon and Heleschewitz [HEL 00] have revealed a close relationship between frequency and diffusive approaches. This equivalence gave birth to the concept of frequency distributed variables. However, it is necessary to note that the fractional closed-loop representation, generalization of the integer order approach, is different from the diffusive representation, which is in fact an open-loop representation, as specified in Chapter 7. Thus, this book provides a synthesis of research works realized during 20 years, the first one published in 1999 [TRI 99]. It is important to warn the reader that this book is not based on mathematical proofs and theorems. The approach used by the authors will be certainly criticized by theoreticians: it is based essentially on numerical experimentations used to validate intuitive concepts in a first step, which are modeled and theorized in a second step. All important results are systematically verified by numerical simulations in order to validate their applicability. This approach is commonly used in electronics and applied physics; it was recommended

Preface

xvii

by the Nobel Prize winner G. Charpak as “la main à la pâte”. Moreover, it has been deeply influenced by works on analog computing and numerical techniques in the 1970s. The research works related to this new methodology have been carried out in collaboration with T. Poinot, N. Maamri and PhD students at Poitiers University (France), with K. Jelassi and PhD students at ENI of Tunis (Tunisia) and with A. Oustaloup and his colleagues at Bordeaux University (France). Moreover, fructuous exchanges with T.T. Hartley (Akron University, USA) and C.F. Lorenzo (NASA, USA) have allowed theoretical advances in system initialization and fractional energy. A reader of this book does not require knowledge of sophisticated high-level mathematics. Necessary prerequisites concern Laplace transform, complex variables, ordinary differential equations and classical numerical analysis. Every time a specific topic is required for understanding, it is revised in an appendix of the concerned chapter. This book in two volumes is composed of four parts, with each one divided into five chapters. Volume 1 Part 1 is dedicated to the simulation of fractional differential equations with fractional integrators and to the modeling and identification of physical systems with fractional order models. In Chapter 1, we review the fundamental principle of differential system simulation with integrators and define the fractional integrator concept. In Chapter 2, we propose a realization of this simulation operator thanks to a frequency approach. In Chapter 3, the simulation technique based on the Grünwald–Letnikov derivative is compared to the fractional integrator method. It is demonstrated in Chapter 4 that fractional order differential systems are mathematical tools adapted to the modeling of diffusive processes. Finally, in Chapter 5, we propose an identification methodology based on the association of integer and fractional order models for the modeling of the induction machine. After this introduction to simulation and modeling, in Volume 1 Part 2, we treat the theoretical problems related to the infinite state approach, i.e. to the concept of frequency distributed state variables. The frequency distributed model of the fractional integrator is defined in Chapter 6 and its equivalence with the intuitive frequency model of Chapter 2 is demonstrated, with a particular interest in the fundamental convolution concept. In Chapter 7, the closed-loop representation of fractional differential systems is revisited within a theoretical framework and compared to the diffusive representation. The fractional Riemann–Liouville and Caputo derivatives are defined and analyzed in Chapter 8 with particular attention

xviii

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

to the unicity of fractional systems transients. Chapter 9 is dedicated to the analytical formulation of fractional transients using the Mittag-Leffler technique and the frequency distributed exponential approach, specificity of the distributed model of fractional differential systems. Finally, we demonstrate in Chapter 10 that the frequency distributed concept provides an original solution to the fractional differentiation of functions; moreover, we introduce the notion of transients caused by the truncation of the differentiation process. In Volume 2 Part 1, we treat the fundamental issues related to initialization, observation and control of the distributed state. Chapter 1 Volume 2 is dedicated to the initialization of a fractional system with two approaches: the first one in an input–output framework and the second one using a closed-loop frequency distributed representation. Fractional system observability and controllability concepts are treated in Chapter 2 Volume 2 using the frequency distributed representation and revisiting the approaches of the integer order case. Chapter 3 Volume 2 is dedicated to the observation of the distributed state, which is then applied to derive an improved initialization technique. In Chapter 4 Volume 2, we are interested by state control of the fractional system distributed state. Finally, in Chapter 5 Volume 2, this methodology is applied to the control of the internal distributed state of a diffusive system based on the identification and state control of a non-commensurate order fractional model. In Volume 2 Part 2, we treat stability issues related to fractional differential systems. In Chapter 6 Volume 2, the closed-loop representation concept is used to perform stability analysis of non-commensurate order fractional differential equations using the Nyquist criterion. The fundamental concept of fractional energy is defined and analyzed in Chapter 7 Volume 2; its comparison with integer order energy highlights its physical significance. Chapter 8 Volume 2 is dedicated to the Lyapunov stability analysis of commensurate order fractional systems based on fractional energy used as the Lyapunov function. The Lyapunov stability of noncommensurate order fractional systems is treated in Chapter 9 Volume 2 within a physical framework related to the passivity approach. Finally, in Chapter 10 Volume 2, we propose an introduction to the Lyapunov stability analysis of nonlinear fractional systems using the Van der Pol oscillator example. Jean-Claude TRIGEASSOU Nezha MAAMRI May 2019

PART 1

Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs)

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

1 The Fractional Integrator

1.1. Introduction A fundamental topic of mathematics concerns the integration of ordinary differential equations (ODEs). Except particular cases where analytical solutions exist, the search for general solutions for any type of ODE and excitation has been an important issue either theoretically or numerically for engineering applications. The quest for practical solutions has motivated the formulation of approximate numerical solutions (Euler, Runge–Kutta, etc.) and the realization of mechanical and electrical computers based on a physical analogy. The interest of an historical reminder concerning analog computers is to highlight the fundamental role played by the integration operator, which is also present in an implicit form in numerical ODE solvers. As in the integer order case, the integration of fractional differential equations (FDEs) requires a fractional integration operator, which is the topic of this chapter. After a reminder of the integer order ODE simulation, the Riemann–Liouville integration or fractional order integration is presented as a generalization of repeated integer order integration. Then, the fractional integration operator is defined and applied to the FDE simulation. The realization of the fractional integrator will be addressed in Chapter 2. 1.2. Simulation and modeling of integer order ordinary differential equations 1.2.1. Simulation with analog computers Analog computers are nowadays obsolete, but they present a fundamental interest: they exhibit the key role played by integrators in simulation. According to

4

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

the principle formulated by Lord Kelvin [THO 76] (see Appendix A.1.1), only integrators are required to simulate an ODE. This principle has been used in analog computers, i.e. mechanical computers from 1931 to 1945 and electronic computers from 1945 to 1970 (see Appendix A.1.2). Some analog computers are still used in electronics and automatic control education. It is important to note that the only active operator proposed to the user is the integrator, in association with adders, multipliers and so on, but with no differentiation operator! Of course, there is a fundamental reason: it is well known that the amplification of a differentiator increases with frequency. Therefore, the noise, characterized by a wide frequency spectrum, is amplified and saturates the operator output. Thus, the first paradox is that differentiators cannot be used to simulate ODEs! In fact, derivatives should not be used explicitly. In order to avoid explicit differentiation, analog computers use the integrator property: t

x(t ) =  v(τ ) dτ + x(0)

[1.1]

0

and reciprocally: v(t ) =

dx(t ) dt

[1.2]

where v(t ) is defined as the implicit derivative of x(t ) . Note that this property does

dx(t ) in an open-loop operation. On the contrary, with dt the closed-loop diagram of Figure 1.1, used to simulate:

not allow the computation of

x(0)

u(t)

x(t)

v(t) s

a0

Figure 1.1. Analog simulation of one-derivative ODE

The Fractional Integrator

dx(t ) + a0 x(t ) = u (t ) dt

5

[1.3]

the signal v(t ) is defined as v(t ) =

dx(t ) = u (t ) − a0 x(t ) dt

[1.4]

Therefore, v(t ) , input of the integrator, represents the derivative of the solution x(t ) , without explicit calculation of this derivative.

1.2.2. Simulation with digital computers

Obviously, analog computers are no longer used to simulate ODEs. Therefore, we may ask the question: is Lord Kelvin’s principle really used with digital computers and numerical algorithms? Consider the previous ODE and Euler’s technique, based on the explicit definition of the derivative: lim xk +1 − xk dx(t ) = Δt → 0 Δt dt

with t = k Δt

[1.5]

The corresponding algorithm is: xk +1 = xk + Δt (uk − a0 xk )

[1.6]

This algorithm is based on the explicit definition of the derivative; so where is the integrator? It is straightforward to write: vk −1 = uk −1 − a0 xk −1 xk = xk −1 + Δt vk −1

[1.7]

which corresponds to (using the delay operator q −1 ): xk = Δt

q −1 vk 1 − q −1

[1.8]

6

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Hence, xk is the numerical integral of vk: it means that integrator

1 of Figure 1.1 has s

been replaced by a numerical integrator: I ( q −1 ) = Δ t

q −1 1 − q −1

[1.9]

Therefore, whatever can be the numerical algorithm used to simulate the ODE, it is possible to demonstrate that it corresponds to a sophisticated numerical integrator (see, for example, Appendix A.1.3 for the interpretation of the RK2 algorithm). We can conclude that the simulation of ODEs is based essentially on integrators. Moreover, explicit differentiation is carefully avoided. If necessary, the input of the integrator provides the implicit derivative. 1.2.3. Initial conditions

The initial condition problem is elementary in the integer order case; on the contrary, it is complex in the fractional order case. It is the reason why we insist on certain points that will facilitate the generalization of the methodology to the simulation of FDSs/FDEs. Therefore, consider the electronic integrator of Figure 1.2. C R

v(t)

x(t)

Figure 1.2. Analog realization of the integer order integrator

It is well known that: x(t ) = −

1 t v(τ )dτ + x(0) RC 0

[1.10]

Moreover, x(t ) is equal to the voltage at C terminals. Hence, if the capacitor is initially charged at t = 0 : q(0) = C x(0)

[1.11]

The Fractional Integrator

7

Therefore, x(0) , the initial condition of the integrator, is not only an abstract mathematical value but also a physical phenomenon. Moreover, it is linked to the initial energy of the integrator since: 1 Ec (t ) = C x 2 (t ) 2

[1.12]

EC (t ) represents the electrostatic energy stored in the capacitor. It can also be interpreted as the Lyapunov function of the integrator, and thus of system [1.3]. Practically, this energy is expressed as: E (t ) = x 2 (t )

[1.13]

REMARK 1.– Energy issues, particularly in relation to fractional order systems, are analyzed in Chapter 7 Volume 2. Note that x(0) only concerns the output of the integrator and is not related to its derivative v(t ) =

dx(t ) . dt

Consider Laplace transform of [1.1]: 1 x(0) X (s) = V (s) + s s

As v(t ) =

[1.14]

dx(t ) , we can write: dt

 dx (t )  L  = s X ( s ) − x(0)  dt 

[1.15]

which is known as the Laplace transform of the derivative: usually, x(0) is considered as the initial condition of the derivative. Of course, an integer order derivative has no initial condition; x(0) is in fact the initial condition of the associated integrator. This remark is essential with the fractional integrator. 1.2.4. State space representation and simulation diagram

A state space representation is not specific to automatic control and system theory. It has long been used in mathematics and mechanics (see, for example,

8

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

the phase plane technique and limit cycles of Poincaré [POI 82]). However, the simulation diagrams used on analog computers refer explicitly to state space equations. It is certainly one of the reasons of the emergence of state space theory in the 1950s, in parallel with the development of analog computing for the simulation of automatic control systems [LEV 64]. Moreover, the first monographs [DOR 67, WIB 71, FRI 86] on state space theory referred explicitly to flow diagrams, which are the simulation diagrams, exhibiting integrators and their initial conditions. Consider, for example, the two-derivative ODE: d 2 x(t ) dx(t ) + a1 + a0 x(t ) = u (t ) dt dt 2

[1.16]

which requires two integrators for its simulation (see Figure 1.3). x1(0)

x2 (0)

u(t)

x] (t) = x(t)

x2 (t) = v] (t)

v2 (t) s

s

a] a0

Figure 1.3. Analog simulation of a two-derivative ODE

This diagram can be replaced by a state space representation, with x1 (t ) = x(t ) and x2 (t ) =

dx(t ) . dt

Then:  dx1 (t )  dt = x2 (t )   dx2 (t ) = −a x (t ) − a x (t ) + u (t ) 0 1 1 2  dt

[1.17]

corresponds to the well-known model: d X (t ) = A X (t ) + B u (t ) dt

[1.18]

The Fractional Integrator

9

with X (t )T = [ x1 (t ) x2 (t ) ]

[1.19]

This connection between simulation/flow diagrams and state space models is well recalled in Kailath [KAI 80], who insists on the physical meaning of initial conditions and our tribute to Lord Kelvin’s principle. Using Laplace transform, the solution of the differential system is: X ( s ) = [ sI − A] x(0) + [ sI − A] B U ( s ) −1

−1

[1.20]

where x(0) is the initial state of [1.17], i.e. of the different integrators. The first term represents the free response, i.e. the initialization of the differential system. REMARK 2.– State is related to energy, as mentioned previously. It is also defined as the essential information required to predict the future system behavior. Therefore, we have to recall that state definition and system initialization are coupled problems. REMARK 3.– Equation [1.18] is interesting because it is a compact writing of flow diagrams. On the contrary, it represents some risk because it does not refer to integrators which are implicit. Of course, there is no risk of confusion with integer order differential systems. However, direct generalization of equation [1.18] to fractional order systems, without reference to the characteristics of their specific integrators, has certainly led to questionable definitions of fractional state variables. REMARK 4.– The simulation diagram presented in Figure 1.3 is a closed-loop system: it is a closed-loop representation of system [1.16]. As with any closed-loop system, its stability is conditional: of course, its stability condition is the same as that of the ODE [1.18]. An elementary modification of this diagram shows that its stability can be analyzed using the Nyquist criterion [NYQ 32]: this approach gives the same results as the Routh criterion [ROU 77], with the advantage of stability margins, giving indications on the stability degree of the ODE. This methodology is presented in Chapter 6 Volume 2. It has been published in [TRI 09c] and generalized to the stability of FDEs and time delay systems [MAA 09]. 1.2.5. Concluding remarks

The interest of this reminder has been to insist on some fundamental principles which will be generalized to fractional differential systems (FDSs):

10

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

– only integrators are required for the simulation of ODEs; t

– integrators perform the integrals: x i (t ) =  vi (τ ) dτ + xi (0) ; 0

– vi (t ) is the implicit derivative of xi (t ) : vi (t ) =

dxi (t ) ; dt

– xi (0) is the initial condition of the ith integrator; it corresponds to the initial charge of capacitor Ci for an analog integrator, i.e. its initial energy; – the state variables xi (t ) are the outputs of the different integrators; explicit differentiation is not required for the simulation of ODEs. 1.3. Origin of fractional integration: repeated integration

We can define the fractional integration of the order n (n real) or the Riemann–Liouville integration using the repeated integration of the order N (N integer) [MAT 94, POD 99]. Let f (τ ) be a function defined for τ ≥ 0 . Moreover, let us define t

I 1 (t ) =  f (τ )dτ 0

[1.21]

I 1 (τ ) is a new function, defined for τ ≥ 0 , with I 1 (0) = 0 .

Then again, we can define t

I 2 (t ) =  I 1 (τ )dτ 0

[1.22]

I 2 (τ ) is a repeated integral, defined for τ ≥ 0 , with I 2 (0) = 0 .

If we continue this repeated integration process until step N , we can define the function I N (τ ) with the recursive relation t

I N (t ) =  I N −1 (τ )dτ 0

defined for τ ≥ 0 , with I N (0) = 0 .

[1.23]

The Fractional Integrator

11

Then, let us express I N (t ) as a function of f (τ ) . Consider the variable change μ = t − τ . Then t

I N (t ) =  I N −1 (t − μ )d μ 0

[1.24]

Let us use the integration by parts technique



b

a

b

udv = [uv ]a −  vdu b

a

[1.25]

For N = 2 , we obtain t

I 2 (t ) =  μ f (t − μ )d μ 0

[1.26]

For N = 3 , a similar procedure provides t

μ2

0

2

I 3 (t ) = 

f (t − μ )d μ

[1.27]

Now consider the general case for N repeated integrals: t

I N (t ) =  I N −1 (t − μ )d μ 0

t

=  μ I N − 2 (t − μ )d μ 0

t

μ2

0

2

=

[1.28]

I N −3 (t − μ )d μ

Using i times the integration by parts technique, we obtain t

μi

0

i!

I N (t ) = 

I N −1−i (t − μ )d μ

[1.29]

Therefore, for N − 1 − i = 0 or i = N − 1 , we obtain t

μ N −1

0

( N − 1)!

I N (t ) = 

I 0 (t − μ )d μ

[1.30]

12

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As I 0 (t − μ ) = f (t − μ ) , we finally obtain the general repeated integration formula t

μ N −1

0

( N − 1)!

I N (t ) = 

f (t − μ )d μ

This formula can be interpreted as a convolution between

[1.31] t N −1 and f (t ) , ( N − 1)!

which can be written as I N (t ) =

t N −1 * f (t ) ( N − 1)!

Moreover, the function

[1.32]

t N −1 can be interpreted as the impulse response of a ( N − 1)!

linear filter hN (t ) =

t N −1 H (t ) ( N − 1)!

N ≥ 0 integer

[1.33]

where H (t ) is the Heaviside function. Therefore, we can conclude I N (t ) = hN (t ) * f (t )

[1.34]

1.4. Riemann–Liouville integration 1.4.1. Definition

The Riemann–Liouville integral generalizes the repeated integral for n real thanks to the gamma function Γ(n) which is the interpolation of the factorial function between the integer values of n . This function Γ(n) ( n real) is defined by the integral (see Appendix A.1.4) ∞

Γ(n) =  e − x x n −1dx 0

[1.35]

The Fractional Integrator

13

Note that (n − 1)! = Γ(n) for n integer. Then, for n real, hn (t ) becomes hn (t ) =

t n −1 H (t ) Γ ( n)

[1.36]

and the repeated integral I N (t ) becomes the fractional integral with order n real (Riemann–Liouville integral) of the function f (t ) , which we express as [POD 99]: t

μ n −1

0

Γ ( n)

I n ( f (t )) = 

f (t − μ )d μ

[1.37]

1.4.2. Laplace transform of the Riemann–Liouville integral

Our objective is to express L { I n ( f (t ))} [SPI 65]. First, we consider L {t n −1 } =  e − st t n −1dt ∞

[1.38]

0

We use the variable change t =

u . s

Then L {t

n −1

}=

∞

0

u e   s −u

n −1

du 1 = n s s



∞

0

e −u u n −1du

[1.39]

∞

As Γ(n) =  e− u u n −1du , we obtain 0

Γ ( n) sn

[1.40]

 t n −1  1 L  = n = L {hn (t )}  Γ( n)  s

[1.41]

L {t n −1} = and

14

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As I n ( f (t )) = hn (t ) * f (t ) , we obtain L {I n ( f (t ))} = L {hn (t )} L { f (t )}

[1.42]

Therefore L { I n ( f (t ))} =

1 F ( s) sn

[1.43]

1.4.3. Fractional integration operator

The case 0 < n < 1 is particularly interesting because it corresponds to the fractional integration operator. Let us recall that for n = 1 , I 1 (t ) corresponds to the integral of f (t ) : t

I 1 (t ) =  f (τ )dτ

[1.44]

0

This integral is provided by the integer order integrator, whose Laplace 1 transform is . It can be expressed by the diagram of Figure 1.4. s

f (t) s

I1 (t)

Figure 1.4. Analog simulation of integer order integration

where 1 = L {h1 (t )} = L { H (t )} s

[1.45]

If we generalize this relation to 0 < n < 1 , the fractional integral I n ( f (t )) can be interpreted as the diagram of Figure 1.5.

The Fractional Integrator

f (t)

sn

15

tn(f (t))

Figure 1.5. Analog simulation of fractional order integration

1 is the fractional integration operator for 0 < n < 1 , whose impulse sn response is hn (t ) :

where

 t n −1  1 = = L h ( t ) L { }   n sn  Γ ( n) 

[1.46]

1.4.4. Fractional differentiation

We can define fractional differentiation as a generalization of integer order differentiation. Let f (t ) be the output of a chain of N integer order integrators is

1 , whose input s

d N f (t ) (see Figure 1.6). dt N dN 1 f(t)

df (t) dt

dtN 1

dN f (t)

f (t)

N

dt

s

s

s

Figure 1.6. Implicit integer order differentiation

We can write f (t ) = h1 (t ) *

df (t ) dt

 d

N −1

[1.47] N

f (t ) d f (t ) = h1 (t ) * dt N −1 dt N

16

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore f (t ) = h1 (t ) * * h1 (t ) *

d N f (t ) dt N

[1.48]

Thus: F (s) =

1  d N f (t )  L  s N  dt N 

[1.49]

Then, if we consider that all the initial conditions are equal to 0, we obtain  d N f (t )  N L  = s F ( s ) with N dt  

N > 0 integer

[1.50]

Therefore, as a generalization of the previous result for n = N real, if f (t ) is 1 , its input has to be the nth-order fractional sn derivative of f (t ) : D n ( f (t )) (see Figure 1.7).

the output of the fractional integrator

Dn (f (t))

sn

f(t)

Figure 1.7. Implicit fractional order differentiation

Then f (t ) = hn (t ) * D n ( f (t ))

[1.51]

1 L { D n ( f (t ))} n s

[1.52]

or F (s) =

Therefore, if the initial conditions are equal to 0 (this complex concept will be defined correctly in Chapter 8), we obtain L { D n ( f (t ))} = s n F ( s )

[1.53]

The Fractional Integrator

17

Note that s n can be interpreted as sn =

1 s−n

[1.54]

Therefore s n = L {h− n (t )}

[1.55]

D n ( f (t )) = h− n (t ) ∗ f (t )

[1.56]

and

1.5. Simulation of FDEs with a fractional integrator 1.5.1. Simulation of a one-derivative FDE

Consider the elementary FDE: D n ( x(t )) + a0 x(t ) = u (t )

[1.57]

We can write: D n ( x(t )) = v(t ) = u (t ) − a0 x(t )

[1.58]

where v(t ) is the input of the fractional integration operator I n ( s ) = its output. I.C.

u(t)

x(t)

v(t) s

n

a0

Figure 1.8. Simulation of a FDE

1 and x(t ) is sn

18

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore, the simulation of this FDE is performed as in the integer order case, 1 1 according to Figure 1.1, where integrator is replaced by n , which leads to s s Figure 1.8. 1.5.2. FDE

Consider the general linear FDE [POI 03, TRI 09b]: D mN ( y (t )) + aN −1 D mN −1 ( y (t )) +  + a1 D m1 ( y (t )) + a0 y (t ) = bM D mM (u (t )) +  + b1 D m1 (u (t )) + b0 u (t )

[1.59]

whose transfer function is: H ( s) =

b0 + b1 s m1 +  + bM s mM Y ( s) B(s) = = mN −1 mN m1 U ( s ) a0 + a1 s +  + aN −1 s A( s ) +s

[1.60]

The fractional differentiation orders: m1 < m2 <  < mN

[1.61]

are real positive numbers; they are called external or explicit orders. It is necessary to define internal or implicit differentiation orders such as: n1 = m1  ni = mi − mi −1  nN = mN − mN −1

1.5.3. Simulation of the general linear FDE

Consider the previous FDE [1.59].

[1.62]

The Fractional Integrator

19

Define: X (s) =

1 U ( s) A( s )

[1.63]

and Y (s) = B( s) X ( s)

[1.64]

which allow the introduction of the classical controller canonical state space form [WIB 71, KAI 80, ZAD 08]: x1 (t ) = x(t ) x2 (t ) = D n1 ( x1 (t ) )  xi (t ) = D ni−1 ( xi −1 (t ) )

[1.65]

 xN (t ) = D nN −1 ( xN −1 (t ) ) D nN ( xN (t ) ) = −a0 x1 (t )  − aN −1 xN (t ) + u (t ) = ε (t )

and x1 (t ) = I n1 ( x2 (t ) )  xi −1 (t ) = I ni−1 ( xi (t ) ) 

[1.66]

xN −1 (t ) = I nN −1 ( xN (t ) ) xN (t ) = I nN ( ε (t ) )

Finally, y (t ) is obtained using the relation: Y (s) = B( s) X ( s)

[1.67]

Corresponding to: M

y (t ) =  bi xi +1 (t ) i =0

[1.68]

20

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

This simulation scheme is based on a state space model which requires N fractional integration operators, whose transfer functions are respectively I nN ( s ), I nN −1 ( s ), , I n1 ( s ) and connected according to the analog simulation

{

}

diagram of Figure 1.9, which is the closed-loop representation of system [1.59]. The modeling of FDEs is analyzed in Chapter 7.

y(t) bM

ε(t)

xN(t)

InN (s)

b0

x1(t)

xM+1(t)

InM+1 (s)

In1 (s)

u(t)

aN aM a0

Figure 1.9. Simulation of an FDE with fractional integrators

A.1. Appendix A.1.1. Lord Kelvin’s principle

Lord Kelvin (Sir W. Thomson) proposed in 1876 [THO 76] a general principle for mechanical integration of the differential equation: dy n ( x) dy ( x) dy n −1 ( x) ,..., , u ( x)) = f ( x, y ( x), n dx dx dx n −1

[1.69]

This integration was performed using mechanical integrators, which were connected according to Figure 1.10 (in fact, this figure was proposed by Kailath [KAI 80], who interpreted the original paper).

The Fractional Integrator

yn – 1(0)

y 1 (0) 1

n

y (x)

yn – 1(x)

∫

21

y1 (x)

dx

y(x)

∫

dx

f (x,y, y1... yn – 1,u)

u(x)

Figure 1.10. Lord Kelvin’s scheme for simulation of the nth-order ODE

Thanks to the principle of mechanical integrators, the independent variable x is not necessarily time t . The main idea is that the chain of integrators avoids the use dy ( x ) dy n −1 ( x) ,..., are implicit. Although Lord of differentiators, i.e. the derivatives dx dx n −1 Kelvin can be considered as the father of ODE simulation, he was not able to perform it because friction problems prevented the use of a chain of integrators. A.1.2. A brief history of analog computing

A good introduction to the history of analog computing is presented in the special issue of IEEE Control Systems Magazine (June 2005 Volume 25 Number 3) with many references therein. In 1931, Vannever Bush [OWE 86] of the MIT used Lord Kelvin’s principle and performed the ODE simulation using mechanical integrators. The friction problem was solved by torque amplifiers. This first analog computer was called the differential analyzer. It was regularly improved until the end of World War II. In 1932, D. Hartree (Manchester University G.B.) visited the MIT and proposed a simplified version of the differential analyzer, using Meccano components. Another computer was built in Cambridge University. These first computers were not used for automatic control simulation. In fact, they have been applied to scientific computing, as required by quantum mechanics.

22

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Since 1945, these mechanical computers have been abandoned and replaced by electronic computers, themselves very complex, but more easily duplicated. The heart of these analog computers remained the integrator: it was based on the operational amplifier principle [RAG 47]. The first operational amplifiers, based on a chopper technique, were built with vacuum tubes: they exhibited serious drift problems. An essential improvement was the use of DC differential amplifiers provided by integrated devices in the 1960s. These analog computers have been used in many fields of scientific computing, but particularly for the simulation of automatic control problems arising, for instance, in aeronautics [LEV 64]. Finally, they were supplanted by digital computers and numerical algorithms at the beginning of the 1970s. A.1.3. Interpretation of the RK2 algorithm

Consider the ODE: dy (t ) = f (t , y (t )) dt

[1.70]

If we know a solution y (t ) at the instant t , we can write: y (t + h) = y (t ) + 

t +h

t

f (τ , y (τ ))dτ

[1.71]

The practical problem is to approximate the integral. If we use the rectangular rule [HAR 98], we can write:



t +h

t

f (τ , y (τ ))dτ ≈ hf (t )

[1.72]

which corresponds to Euler’s algorithm: y (t + h) ≈ y (t ) + hf (t )

[1.73]

The integral can also be approximated using the trapezoidal rule [HAR 98]:



t +h

t

f (τ , y (τ )) dτ ≈ h

f (t ) + f (t + h) 2

[1.74]

The Fractional Integrator

where

f (t + h) is not available at instant t . The curve

23

f (τ , y (τ )) can be

approximated by a parabola in the interval [t , t + h ] ; thus, the mean of y , (t ) and h y , (t + h) is equal to y , (t + ) . 2

Therefore:  h  h   h  f (t ) + f (t + h) y,  t +  = = f t + , yt +  2  2  2  2 

[1.75]

h y (t + ) is not available, but it can be approximated by Euler’s algorithm: 2 h  h y  t +  ≈ y (t ) + f (t ) 2  2

[1.76]

and finally: h  h  y (t + h) ≈ y (t ) + hf  t + , y (t ) + f (t )  2  2 

[1.77]

which is known as the Runge–Kutta 2 algorithm [HAR 98] (also known as Heun’s method). Therefore, we can conclude that the RK2 algorithm is equivalent to explicit integration based on the trapezoidal rule. A.1.4. The gamma function

The gamma function Γ(α ) is defined by the integral [ANG 65, POD 99]: ∞

Γ(α ) =  e − x xα −1dx

[1.78]

0

If α is an integer, the gamma function corresponds to the factorial function: Γ(α + 1) = α !

[1.79]

Some important relations: Γ(α + 1) = αΓ(α )

[1.80]

24

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Γ(α )Γ(α + 1) =

π

[1.81]

sin(nπ )

1  1.3..(2 N − 1)  Γ N +  = π 2 2N 

N integer

[1.82]

Integral relation: ∞

y

α − ay

e

dy =

0

Γ(α + 1) aα +1

[1.83]

Remarkable values: Γ(0) = +∞

π 3 Γ  = 2 2

1 Γ  = π 2 Γ(2) = 1

Γ(1) = 1

[1.84]

2 Frequency Approach to the Synthesis of the Fractional Integrator

2.1. Introduction In Chapter 1, we defined fractional integration as the key tool for modeling and simulation of FDEs. Unfortunately, the definition of Riemann–Liouville integration

x (t ) = I n (v (t )) = hn (t ) *v (t )

[2.1]

where

hn (t ) =

t n −1 H (t ) Γ(n )

[2.2]

does not provide a suitable technique for the numerical simulation of fractional integration, which is not a classical integral, but in fact a convolution integral. Therefore, in the first step, we relate this operation to a frequency approach based on the Laplace transform of hn (t ) , thus:

I n (s ) = L { hn (t ) } =

1

sn

[2.3]

In the second step (Chapter 6), we will relate fractional integration to a time approach, called the infinite state approach, moreover demonstrating that these two approaches are equivalent and complementary.

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

26

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

The synthesis of the fractional integrator thanks to a frequency methodology is in fact based on Oustaloup’s technique [OUS 00] for the synthesis of the fractional differentiator D n (s ) = s n . Therefore, in this chapter, we present a frequency approximation of I n (s ) which will be used to derive a modal formulation, basis of the numerical integration algorithm and of the fractional integrator state variables. 2.2. Frequency synthesis of the fractional derivator The first researcher who proposed a constant phase template, i.e.

θ = − ( n + 1)

π

0 < n 0  θ =θˆi

−1

θˆi +1 = θˆi −   J ′' (θ ) + μ I  J ′ (θ )  

[5.24]

The stability of the gradient method depends on the parameter λ (if λ ≈ 0 , stability is ensured, but convergence is very slow). If θˆi is close to θ opt , criterion J is a quadratic function of θ and the optimum

θ opt is theoretically obtained in one iteration with the Gauss–Newton algorithm; otherwise, divergence of this algorithm can occur. Levenberg–Marquardt is a compromise between the two previous algorithms. If θˆinit is far from θ opt , we must choose a large value of μ , thus

θˆi +1 ≈ θˆi −

′ J (θ ) ) ( μ

1

θ =θˆ

(

′ ≡ θˆi − λ J (θ ) i

)

θ =θˆi

with λ ≈ 0

[5.25]

Therefore, the algorithm behaves like the gradient one and convergence is slow but ensured. On the contrary, if μ → 0

 

−1

  θ =θˆi

θˆi +1 ≈ θˆi −   J ′' (θ )  J ′ (θ )  

[5.26]

Therefore, the algorithm behaves like the Gauss–Newton one and fast convergence is ensured. Of course, the value of μ should be automatically monitored to ensure convergence [MAR 63, TRI 01].

122

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

5.4.4. Simulation of yˆ k and σ k

(

)

yˆ ( t ) is a simulation of f θˆ, u ( t ) , i.e. a simulation of a differential system,

which is composed of an integer order part and a fractional order one with Park’s model. yˆ ( t ) should be resimulated at each iteration i of the nonlinear optimization algorithm.

σ k results also of the simulation of a differential system (see [TRI 01]); it should also be simulated at each iteration i . As fractional order n (or n1 ) is unknown, it has to be estimated [VIC 13]. ∂yˆ Derivation of σ k related to n is a complex task [POI 04]. Fortunately, k can be ∂nˆ approximated by ∂yˆ k yˆ k ( ni + Δn ) − yˆ k ( ni ) ≈ Δn → 0 ∂nˆ Δn

[5.27]

Note that Δn can be chosen a priori [DJA 08, JAL 12] because nˆ is necessarily in the interval [ 0,1] . The approximate computation of

∂yˆ k is also possible, but it is more complex ∂θ j

because Δθ j must be adapted at each iteration [HIM 72].

5.4.5. Comments

The main advantage of the output error method is related to its nice statistical properties. If the noise b ( t ) disturbing y * ( t ) is zero mean and as yˆ ( t ) does not depend on b ( t ) , then θ opt can be considered as a confident estimation [WAL 97, TRI 01] of θ , i.e. θ opt → θ if K → ∞ (number of data points). Note that θ MC (estimation of θ provided by the least-squares method) is a biased estimate of θ , i.e. θ MC → θ + Δθ if K → ∞ :

θ MC → θ + Δθ if K → ∞

[5.28]

Modeling of Physical Systems with Fractional Models: an Illustrative Example

123

Δθ is called an estimation bias; it depends on the statistical characteristics of b ( t ) [EYK 74, LJU 87].

Therefore, θ MC is not a confident estimation of θ , but it can be used to initialize a nonlinear optimization technique: θ init = θ MC . Note that it is impossible to know the exact structure of a system model. Therefore, modeling errors are always present, which can be interpreted as a disturbing noise by the identification algorithm. Unfortunately, this modeling noise is not zero mean and θ opt is necessarily biased by modeling errors. Globally, in spite of modeling errors, the output error identification algorithm is a relevant technique. However, simulation of model and sensitivity functions are difficult to implement with complex nonlinear differential systems. Moreover, it is essential to note that it requires large computation time if we compare it to the more simple least-squares techniques [LJU 87]. 5.4.6. Application to the identification of fractional Park’s model

Data measurements u s ( t ) and is ( t ) (we present only the beginning of data files) are shown in Figures 5.9 and 5.10.

Figure 5.9. uS(t) voltage

124

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Figure 5.10. iS(t) current

u s ( t ) is a PMW voltage modulated by a pseudo-random binary sequence. The

sampling period Te is equal to 3.6 ms and K = 9000 . Three model structures are identified and compared: – H ( s ) usual Park’s model; – H n ( s ) fractional Park’s model with Z R ( s ) = Z n ( s ) ; – H n1 , n2 ( s ) fractional Park’s model with Z R ( s ) = Z n1 , n2 ( s ) . The corresponding results are presented in Tables 5.1–5.3. RS (Ω)

Lm ( H )

RR (Ω)

lR ( H )

J

9.96

0.40

4.01

0.08

9.08

Table 5.1. Parameter estimations H(s)

RS (Ω)

Lm ( H )

a0

b0

n

J

9.89

0.43

27.50

7.66

0.89

8.01

Table 5.2. Parameter estimations Hn(s)

Modeling of Physical Systems with Fractional Models: an Illustrative Example

RS (Ω)

Lm ( H )

a0

a1

b0

b1

n1

J

9.93

0.40

42.70

1.93

10.24

0.25

0.48

7.88

125

Table 5.3. Parameter estimations Hn1,n2(s)

Compared to usual Park’s model, there is an improvement with the two fractional models corresponding to a decrease in criterion values. Moreover, H n1 , n2 ( s) provides a better approximation of the induction motor than H n ( s) . Note that the values of Rs and Lm are approximately the same for the three models. H n1 , n2 ( s) is more complex than H n ( s) , accompanied by a low decrease of J: it would seem reasonable to prefer H n ( s) instead of H n1 , n2 ( s) , in contrast to the conclusions of Chapter 4. In fact, a discrimination technique presented in [JAL 13] demonstrates that H n ( s ) is unable to provide a confident behavior model in comparison with H n1 , n2 ( s) , in accordance with the conclusions of Chapter 4. We can conclude that fractional gray box modeling and identification are relevant techniques, but they must be carried out with much care using all the possibilities provided by physics, statistics and advanced identification techniques.

PART 2

The Infinite State Approach

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

6 The Distributed Model of the Fractional Integrator

6.1. Introduction In Part 1, and more specifically in Chapters 1 and 2, the Riemann–Liouville integration was defined, with a particular focus on the fractional order integrator. This operator was synthesized according to a frequency approach. Moreover, thanks to a partial fraction expansion, a modal representation was introduced. This result was not straightforward, considering the classical form of the integrator impulse response: hn (t ) =

t n −1 H (t ) Γ (n)

[6.1]

However, an important question is related to the limit of this modal model when the number of terms is increased to infinity. The answer to this question is provided by the frequency distributed model of the 1 fractional integrator, as a result of the inverse Laplace transform of n . This model s provides answers to fundamental questions, for example, on the relation between frequency and modal representations, the expression of the Riemann–Liouville integral at any initial instant t0 and more globally on the internal distributed state

z (ω , t ) of the integrator. Fundamentally, it will be demonstrated that the Riemann–Liouville integral of any function v(t ) is a convolution between the impulse response hn (t ) and v(t ) ,

130

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

i.e. this integral can be interpreted as the response of an infinite dimension system. This convolution interpretation demonstrates that initialization of the fractional integrator obeys the same laws as any linear system [TRI 12a]. The last part of this chapter is dedicated to a physical interpretation of the distributed model thanks to the comparison of the fractional integrator with an infinite length RC diffusive line [MAA 14]. 6.2. Origin of the frequency distributed model The fractional integrator I n (s) is defined as the Laplace transform of its impulse response hn (t ) :

I n ( s) = L {hn (t )} =

+∞

e 0

with hn (t ) =

− st

hn (t )dt =

1 sn

[6.2]

t n −1 H (t ) . Γ (n)

1 provides sn hn (t ) . However, hn (t ) is not obtained directly: we obtain an intermediary model, which is the frequency distributed model. Reciprocally, the inverse Laplace transform [SPI 65, LEP 80] of

Let:

1 hn (t ) = L−1  n  s 

[6.3]

γ + jω  1 I n ( s)e st ds for t > 0 hn (t ) = 2 jπ γ −jω   hn (t ) = 0 for t < 0 

[6.4]

with:

This integral is calculated with a Bromwich contour [LEP 80]. As the function I n (s) is multiform with a branch point in s = 0 , we use a cut in the complex plane, which leads to a modified Bromwich contour (for more details, see Appendix A6).

The Distributed Model of the Fractional Integrator

131

Therefore, we obtain:

hn ( t ) = lim

sin nπ

R →∞ ε →0

π

R

ε ω

− n −ω t

e

dω

[6.5]

Let us define a weighting function:

sin nπ

μn (ω ) =

π

ω −n 0 < n < 1

[6.6]

Then: ∞

hn (t ) =  μn (ω )e −ωt d ω

[6.7]

0

Note that this equation is similar to the discrete one obtained in Chapter 2: J

hn (t ) ≈  c j e

−ω j t

[6.8]

j =0

Equation [6.7] is the limit of [6.8] when the interval between two consecutive modes ω j and ω j +1 tends to 0. Equation [6.7] provides a new interpretation of hn (t ) , but does it correspond to definition [6.1]? Let us define u = ω t , then: hn (t ) =

∞

−n

∞

sin nπ  u  − u du sin nπ n −1 − n − u = e t  u e du π 0  t  π t 0

[6.9]

Let us recall (Chapter 1, Appendix A.1.4): ∞

u 0

− n −u

e du = Γ(1 − n)

[6.10]

132

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As

sin nπ

π

=

1 Γ(n)Γ(1 − n)

[6.11]

we obtain finally: hn (t ) =

t n −1 H (t ) Γ (n)

[6.12]

Therefore, we verify whether the two expressions of hn (t ) are equivalent, which can be summarized by: ∞

L 1 L t n −1 H (t ) → n →  μn (ω )e −ωt d ω → hn (t ) s Γ ( n) 0 −1

hn (t ) =

[6.13]

An essential interest of [6.7] is to highlight that hn (t ) is the result of an infinite dimension sum of e − ω t modes, weighted by μn (ω ) , where ω varies from 0 to ∞ . Equation [6.1] shows that hn (t ) is characterized by long memory (when t → ∞ ), due to t n −1 , equivalently to the action of very low frequency modes ( ω → 0 ). On the contrary, for very short times ( t → 0 ), it is the action of high frequency modes ( ω → ∞ ) that is predominant, leading to lim ( h n (t ) ) → ∞ . t →0

Let us apply the Laplace transform to equation [6.7], then: ∞ ∞ ∞  L {hn (t )} =  e − st hn (t )dt =  e − st   μn (ω ) e −ωt d ω  dt 0 0 0 

[6.14]

As ∞

e

− st −ω t

e

dω =

0

1 s +ω

[6.15]

we obtain a fundamental result:

L {hn (t )} =

μ (ω ) 1 = n dω n s s +ω 0 ∞

[6.16]

The Distributed Model of the Fractional Integrator

133

6.3. Frequency distributed model

Let us recall that the integrator I n (s) is defined by the convolution equation:

x(t ) = hn ( t ) ∗ v(t )

[6.17]

or equivalently

x( s ) = I n ( s ) v ( s )

[6.18]

where hn (t ) is defined as: ∞

hn ( t ) =  μn (ω ) e−ωt d ω

[6.19]

0

1 , s +ω corresponding to the impulsive input v(t ) = δ (t ) . On the contrary, equation [6.19] means that x(t ) = hn (t ) is the weighted integral of all of these elementary impulse responses e − ω t . Note that e − ω t is the impulse response of the elementary transfer function

1 to any s +ω input v(t ) . Therefore, z (ω , t ) is the solution of the differential equation: Let z (ω , t ) be the elementary response of the transfer function

∂z (ω , t ) = −ω z (ω , t ) + v(t ) ∂t Note that

ω and t .

[6.20]

d ∂ is replaced by since z (ω , t ) is a function of two variables, dt ∂t

Thus, the convolution equation x(t ) = hn (t ) ∗ v(t ) can be interpreted as a frequency distributed differential system:

134

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

 ∂z (ω , t )  ∂t = −ω z (ω , t ) + v(t )ω ∈ [ 0, +∞[  ∞   x(t ) = 0 μn (ω ) z (ω , t )d ω   μ (ω ) = sin nπ ω − n 0 < n < 1  n π

[6.21]

REMARK 1.– The integer order integrator corresponds to n = 1 . This integer order operator is characterized by only one mode ω = 0 . SO

∂z (0, t ) dz (t ) = = v(t ) dt ∂t

Moreover, the weighting function

μn ( ω )

becomes a Dirac impulse

μ1 (ω ) = δ (ω ) and: ∞

x(t ) =  δ (ω ) z (t )d ω = z (t ) 0

REMARK 2.– The frequency distributed model of the fractional integrator is also called the diffusive representation model [MON 98]. The “diffusive” term can create a confusion with the diffusive equation. We will show in the last part of this chapter that there is some equivalence between the distributed model and the diffusive equation, but it is another problem. We prefer to qualify [6.21] as the frequency distributed model of

1 , where the sn

variables z (ω , t ) are distributed state variables. Moreover, because this infinite dimension distributed model is the basis of a methodology for the modeling of FDSs, we characterize this methodology as the infinite state approach. 6.4. Finite dimension approximation of the fractional integrator

Equations [6.21] represent the theoretical distributed model of model is not adapted to the numerical simulation of the operator.

1 . However, this sn

The Distributed Model of the Fractional Integrator

135

This numerical simulation requires the discretization of the weighting function

μn (ω ) . The simplest numerical approximation is based on a stepwise discretization,

according to Figure 6.1, i.e. the distributed differential equation [6.21] is replaced, for frequency ω j , by:

dz (ω j , t )

=

dt

dz j (t ) dt

= −ω j z j (t ) + v(t )

[6.22]

Moreover, the integral: ∞

x(t ) =  μ (ω ) z (ω , t )d ω 0

μn (ω ) =

sin nπ

π

ω −n 0 < n < 1

becomes: J

x(t ) ≈  μ n (ω j )z j (t )Δω j

[6.23]

j =1

Let us define:

α j = μn (ω j )Δω j

[6.24]

which is the area of the step j . Therefore: J

x (t ) ≈  α j z j (t ) j =1

Figure 6.1. Discretization of the weighting function

[6.25]

136

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

6.5. Frequency synthesis and distributed model

Let us recall remember that thanks to the partial fraction expansion used in Chapter 2, section 2.5, it is possible to define a modal model: J

x(t ) ≈  c j z j (t )

[6.26]

j =0

Equations [6.25] and [6.26] are equivalent, except for the first frequency ( ω0 = 0 ). Moreover, it is important to note that frequencies ω j should not be chosen according to an arithmetic law, due to the very slow decrease of ω − n . Any frequency discretization can be used; however, the previous geometric distribution defined in Chapter 2 is a powerful guide. Therefore, using this geometric distribution for equation [6.24], frequencies ω j verify:

Δω j = ω 'j +1 − ω 'j

[6.27]

 ' ωj  ωj = α  ω 'j +1 = ηω j 

[6.28]

with

Nevertheless, it is important to remind that [6.25] starts at j = 1 , whereas [6.26] starts at j = 0 , with ω0 = 0 . Because lim (ω − n ) = ∞ , it is not possible to include ω0 = 0 in equation [6.25]. ω →0

Nevertheless, this inclusion is necessary, as demonstrated thereafter. Consider the unit step response of the system

a0 , so s + a0 n

The Distributed Model of the Fractional Integrator

x( s ) =

1 a0 s s n + a0

137

[6.29]

Recall that x(∞)th = lim sx( s) = 1 . s →0

1 Note that x( s ) can be written as x( s ) = s

a0

1 sn

1 + a0

1 sn

[6.30]

Thus, if we use approximation [6.25]: J αj 1 ≈  n s j =1 s + ω j

[6.31]

we obtain

αj j =1 ω j x (∞ ) = < 1 = x (∞ )th J α j 1 + a0  j =1 ω j J

a0 

[6.32]

which means that there is a static error, regardless of the discretization procedure. On the contrary, if we use approximation [6.26]: J cj 1 ≈ n ωj + s s j =0

J c  c j a0  0 +    s j =1 ω j  = 1 = x(∞) we obtain x(∞) = lim th s →0  c0 J c j  1 + a0  +    s j =1 ω j 

[6.33]

[6.34]

Thus, it is equation [6.26] starting at j = 0 , with ω0 = 0 , that provides the best approximation of the fractional integrator (see Chapter 2).

138

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore, we can conclude that the frequency distributed model [6.21] provides the theoretical justification of equation [6.26], without modification of the conclusions presented in Chapter 2. Moreover, we can use equation [6.16] to understand the connection between the frequency response and the modal representation of the fractional integrator. Let us recall that the frequency response of I n ( s ) is defined as:

1 I n ( ju ) =  n   s s = ju

[6.35]

where u varies from 0 to +∞ . The modal representation is defined by the distribution μn (ω ) of modes ω , verifying equation [6.16]:

μ (ω ) 1 = n dω n s s +ω 0 ∞

where ω varies from 0 to +∞ . Thus, the frequency response of the modal model is defined by:

I n ( ju ) =

( ju )

μn (ω ) dω ju + ω 0

∞

1 n

=

[6.36]

where ω and u vary from 0 to +∞ . Note that it is essential to avoid confusion between variables u (for the frequency response) and ω (for the modal distribution). 6.6. Numerical validation of the distributed model

In order to verify the equivalence between the frequency model and the distributed model, we propose the following two numerical computations. 6.6.1. Reconstruction of the weighting function

Consider the frequency model defined in Chapter 2.

The Distributed Model of the Fractional Integrator

s  1+ ω' G j I dn ( s ) = n ∏  s  s j =1 1+ ω j  j=J

  J cj  = c0 +   s j =1 s + ω j  

139

[6.37]

The equivalence between equations [6.25] and [6.26] implies: J

α j =1

J

j

z j (t ) ≡  c j z j (t )

[6.38]

j =0

i.e. α j ≡ c j where α j = μn (ω j )Δω j . Therefore, the μn (ω j ) weighting function can be reconstructed using

μ n (ω j ) =

cj

[6.39]

Δω j

with

ω j = αηω j −1   ' ωj ω j = α  ω 'j +1 = ηω j 

[6.40]

Δω j = ω 'j +1 − ω 'j (see Chapter 2). For this numerical computation, we use the following values: n = 0.5; ωb = 10−5 rad/s; ωh = 10+5 rad/s; J = N cel = 30

We present in Figure 6.2 the theoretical curve μn (ω ) =

[6.41]

sin ( nπ )

π

ω − n and the

numerical values obtained with equation [6.39], using log/log coordinates.

140

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Figure 6.2. Comparison of weighting functions

Inside the frequency interval

{10

−3

rad/s;10+3 rad/s} , we can note a perfect

equivalence between the theoretical and computed values. Differences appear at: – high frequencies, for ω > 103 rad/s : frequency synthesis is not valid for

ω > ωh , so there is a necessary difference for ω > 103 rad/s ; – low frequencies, for ω < 10 −3 rad/s : because we have introduced an integer integrator, in order to obtain an infinite static gain for ω → 0 , there is a necessary difference for ω < ωb . 6.6.2. Reconstruction of the impulse response

The frequency distributed model corresponds to: dz j (t ) dt

= −ω j z j (t ) + v(t ) j = 0 to J J

x(t ) =  c j z j (t ) j =0

[6.42]

The Distributed Model of the Fractional Integrator

141

If v(t ) = δ (t ) , the response x(t) corresponds to the impulse response hn (t ) of the frequency discretized model. As v(t ) = δ (t ) , we obtain z j (t ) = e

−ω j t

J

−ω t so hn (t ) =  c j e j

[6.43]

j =0

whereas [6.1] shows that hn (t ) =

t n −1 H (t ) . Γ( n)

The main difference between hn (t ) and hn (t ) corresponds to t = 0: J

hn (0) → ∞ , whereas hn (0) =  c j is a finite value. j =0

In Figure 6.3, we compare hn (t ) and hn (t ) with the previous values [6.41]. The two impulse responses are compared on the interval −3

Te = 10 s .

Figure 6.3. Comparison of impulse responses

[0;100s]

with

142

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

We note a perfect equivalence in the interval [ 0;100s ] . We can conclude that either the reconstruction of the weighting function or the reconstruction of the impulse response confirms that the distributed frequency model defined in Chapter 2 converges asymptotically to the ideal model provided by the 1 inverse Laplace transform of n 0 < n < 1 . s 6.7. Riemann–Liouville integration and convolution

In Chapter 1, we introduced a simplified definition of the Riemann–Liouville integral:

I n (v ) = 

t

(t −τ )

n −1

v(τ )dτ

Γ( n)

0

[6.44]

where the lower bound of the integral is equal to 0. Remember that this simplified formulation of the Riemann–Liouville integral can be used to define the fractional integrator. Now, we use a more general definition of this integral. Let a I tn ( v ) be the Riemann–Liouville integral where the lower bound is equal to a , with a ∈ R , i.e.:

I (v ) = 

t

(t −τ )

n −1

v(τ )dτ

[6.45]

This expression, where hn ( t ) =

t n −1 represents the convolution between hn ( t ) Γ (n)

n a t

a

Γ ( n)

and v ( t ) , is not adapted to a general system theory formulation. Obviously, we can refer to the previous distributed definition [6.19]: +∞

hn (t ) =  μn (ω ) e−ω t dω 0

The Distributed Model of the Fractional Integrator

Let v ( t ) be an input defined for t ∈ ]−∞; +∞[ and consider x ( t ) = −∞ I

143

( v ) , i.e. v (t ) .

n t

with a lower bound equal to −∞ . Thus, we take into account all the past of Let us recall the distributed differential equation:

∂z (ω , t ) ∂t

= −ω z (ω , t ) + v(t )

where z (ω , t ) is the result of a convolution [ZEM 65, KRA 92, SCH 98] between e −ω t and v ( t ) (where e −ω t is the impulse response of the elementary system

1 ). s +ω

Therefore t

z (ω, t ) = v(t ) ∗ e−ωt =  e

−ω ( t −τ )

−∞

v(τ )dτ

[6.46]

and +∞

+∞

t

0

0

−∞

x(t ) =  μn (ω ) z (ω, t )dω =  μn (ω )  e

−ω ( t −τ )

v(τ )dτ dω

[6.47]

Consider an instant t0 ∈ ]−∞; t [ . We propose to express x ( t ) using t0 I tn ( v ) . We can write [6.47] as: x (t ) =



+∞

0

μ n (ω )

(

t0

−∞

e

− ω ( t −τ )

t

v (τ ) dτ +  e t0

− ω ( t −τ )

)

v (τ ) dτ d ω

[6.48]

where

I tn (v) = 

t0

+∞

0

t

μn (ω ) e−ω ( t −τ ) v(τ )dτ d ω

[6.49]

t0

As:



t0

−∞

e

−ω ( t −τ )

v(τ )dτ = e

−ω ( t − t0 )



t0

−∞

e

−ω ( t0 −τ )

v(τ )dτ = e

−ω ( t −t0 )

z ( ω , t0 )

[6.50]

144

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

We obtain: +∞

x(t ) =  μn (ω )e

−ω ( t − t0 )

0

z (ω, t0 )dω + t0 I tn (v)

[6.51]

where z (ω , t0 ) is the elementary distributed state of the fractional integrator at

instant t0 : z (ω , t0 ) summarizes the past of the integrator for the elementary

frequency ω , i.e. the contribution of v ( t ) at frequency ω for t < t0 (since t = −∞ ).

Therefore, in equation [6.51], the first term represents the free response of the integrator for t ≥ t0 , based on the distributed initial condition z (ω , t0 ) , whereas the second term corresponds to the forced response of the integrator for t ≥ t0 . We recognize the general expression of a linear system response [ZAD 08, WIB 71, KAI 80], the sum of a free response and a forced response, where the initial condition (here z (ω , t0 ) , with ω varying from 0 to ∞ ) summarizes the past (or pre-history) of the system at t = t0 . Obviously, any instant t0 can verify [6.51]: so we can choose t0 = 0 , which makes it possible to use the Laplace transform. Let z (ω , 0 ) be the initial condition at t0 = 0 . Then, the Laplace transform applied to the distributed differential equation yields:

sz (ω , s ) − z (ω , 0 ) = −ω z (ω , s ) + v( s)

[6.52]

so

z (ω , s ) =

z ( ω , 0 ) + v( s )

[6.53]

s +ω

and +∞

+∞

0

0

x( s) =  μn (ω ) z (ω, s)d ω =  μn (ω )

+∞ μ (ω ) z (ω ,0) d ω +  v( s ) n dω [6.54] 0 s +ω s +ω

The Distributed Model of the Fractional Integrator

145

As (equation [6.16]):



+∞

0

μn (ω ) 1 dω = n s +ω s

we obtain finally: +∞

x( s ) =  μn ( ω ) 0

z (ω , 0) 1 d ω + n v( s ) s +ω s

[6.55]

corresponding in the time domain to: +∞

x(t ) =  μn (ω )e−ωt z (ω,0)dω + 0 Itn (v)

[6.56]

0

REMARK 3.– Let us recall that the integer order integration is a particular case of the fractional order integration with n = 1 ( μn (ω ) is replaced by δ (ω ) for n = 1 ): ∞

h1 ( t ) =  δ (ω )e −ωt d ω = H ( t )

[6.57]

0

Therefore t

x1 (t ) = h1 (t ) ∗ v(t ) =



−∞

H ( t − τ )v (τ ) dτ =

t

 v (τ ) dτ

[6.58]

−∞

as H ( t − τ ) = 1 for τ ≤ t . The convolution integral is in fact a simple integration in the integer order case. Thus, with an instant t0 ∈ ]−∞; t [ , we can write:

x1 (t ) =

t0

t

t

−∞

t0

t0

 v (τ ) dτ +  v (τ ) dτ = x1 (t0 ) +  v (τ ) dτ

[6.59]

where x1 ( t0 ) is the initial condition of the integrator at t = t0 . Therefore:

x1 (t ) = x1 (t0 ) + t0 I t1 (v)

[6.60]

146

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

If we use hn ( t ) =

t n −1 H (t ) , we can write Γ (n)

(t −τ ) v(τ )dτ −∞ Γ ( n) n −1 n −1 t (t −τ ) t (t −τ ) = v(τ )dτ +  v(τ )dτ −∞ t Γ ( n) Γ (n) n −∞ I t (v ) = x (t ) = 

n −1

t

[6.61]

0

0

As a generalization of the integer order case, it would be tempting to write:

(t −τ ) −∞ Γ ( n)

x(t0 ) =  i.e.

n −1

t0

v(τ )dτ

[6.62]

n −∞ t

I (v) = x(t ) = x(t0 ) +t0 I tn (v)

[6.63]

where x ( t0 ) would be the initial condition of the Riemann–Liouville integral at t0 . In fact:

(t −τ ) −∞ Γ ( n )

n −1

t0

v(τ )dτ = x(t , t0 )

[6.64]

where x ( t , t0 ) depends on t and t0 . Obviously, it is wrong to use equations [6.62] and [6.63] because x ( t , t0 ) is not a constant. Note that: +∞

x(t0 ) = { x(t )}t =t = 0 I 0n (v) +  μn (ω )e− 0 z (ω, t0 )dω 0

Therefore: +∞

x(t0 ) =  μn (ω )z (ω, t0 )dω 0

0

[6.65]

The Distributed Model of the Fractional Integrator

147

In other words, equation [6.61] is correct only at t = t0 , but it is not able to predict x ( t ) for t > t0 . This misuse of the convolution integral [6.61] is unfortunately at the origin of many errors in the definition of the initial conditions of fractional derivatives. 6.7.1. Conclusion

We have established a fundamental result: fractional integration is a convolution process which can be expressed using the linear system theory, thanks to the distributed model of the fractional integrator. Note that the Riemann–Liouville integral of f ( t ) can be formulated as

t0

I tn ( f )

using any instant t0 ( t0 = 0 in practice), if we take into account the past ( t < t0 ) using the distributed initial condition z (ω , t0 ) . Moreover, this fundamental result has many consequences: it is used to characterize the transients of FDEs or FDSs (Chapter 7) and those of the Riemann–Liouville and Caputo fractional derivatives (Chapter 8). 6.8. Physical interpretation of the frequency distributed model 6.8.1. The infinite RC transmission line

Electrical lines were used during the 19th Century across the Atlantic Ocean to transmit telegraph messages. Theory of these transmission lines was established by Thomson and particularly by Heaviside who formulated the telegrapher’s equations, which were an adaptation of Maxwell’s equations [BEN 79]. When the inductor effect is negligible, the transmission line is equivalent to an RC line. Heaviside demonstrated that the RC line behaves as a fractional integrator [OLD 74, MIL 93]. Based on this equivalence, we propose to provide a physical interpretation of the distributed infinite state model and its distributed state variable z (ω , t ) . Consider the RC line elementary cell shown in Figure 6.4 [OLD 74, FAR 82].

148

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Figure 6.4. RC line elementary cell

C=

∂C ( x) ∂x

R=

∂R( x) ∂x

[6.66]

are capacity and resistance densities. This elementary cell is governed by the voltage/current equations:

∂v( x, t ) ∂i( x, t ) ∂v( x, t ) = − Ri( x, t ) and = −C ∂x ∂x ∂t

[6.67]

corresponding to the well-known partial derivative diffusion equation [KOR 68, FAR 82]:

1 ∂v( x, t ) ∂ 2 v ( x, t ) ∂v( x, t ) i ( x, t ) = − = RC 2 ∂x R ∂ t ∂x

[6.68]

Consider an infinite length transmission line. The general solution of [6.68] is easily obtained using the Laplace transform, as in Chapter 4. Let us define:

V ( x, s) = L {v( x, t )}

[6.69]

  ∂v( x, t )   L  ∂t  = sV ( x, s) − v( x, 0)     2 2  L  ∂ v ( x, t )  = d V ( x, s )   2  dx 2  ∂x  

[6.70]

Assuming that v( x, 0) = 0 ∀x , [6.68] is equivalent to the differential equation:

The Distributed Model of the Fractional Integrator

d 2V ( x, s) − RCsV ( x, s ) = 0 dx 2

149

[6.71]

whose general solution is:

V ( x, s) = Ae

Γx

+ Be−

Γx

Γ = sRC

lim v ( x, t ) = 0 A = 0 because  so V ( x, s ) = Be −  x→∞

[6.72] Γx

[6.73]

where B depends on the excitation at the terminal x = 0 of the transmission line. Using [6.68], we obtain:

I ( x, s ) =

B Γ − e R

Γx

[6.74]

Let us define the impedance of the RC line as Z ( x, s ) =

V ( x, s ) R 1 so Z ( x, s ) = I ( x, s ) C s 0.5

[6.75]

Then: V ( x, s ) =

R 1 I ( x, s ) C s 0.5

[6.76]

This result means that the infinite RC line behaves like a fractional integrator, with a fractional order equal to 0.5. 6.8.2. RC line and spatial Fourier transform

The infinite dimension RC line is characterized by the variables v( x, t ), i( x, t ) with the condition:

v ( x, t ) = 0   for x < 0 i ( x, t ) = 0  The RC line variables at terminal x = 0 are v(0, t ) and i(0, t ) .

[6.77]

150

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Let us define +∞

+∞

−∞

0

Vx ( j χ , t ) =  v( x, t )e− j χ x dx =  v( x, t )e− j χ x dx

[6.78]

which is the spatial Fourier transform [KRA 92] of v( x, t ) .

Vx ( j χ , t ) = Re ( χ , t ) + jI m ( χ , t )

[6.79]

where χ ∈ ]−∞; +∞[ is a spatial frequency

[6.80]

The Fourier transform of

∂v( x, t ) is defined as: ∂x

∞ ∂v ( x, t )  ∂v( x, t )  F e − jν x dx  = 0 ∂x  ∂x 

[6.81]

with the condition:

 ∂v( x, t )  F  = j χVx ( j χ , t ) − v(0, t )  ∂x  Thus, the Fourier transform of

[6.82]

∂ 2 v ( x, t ) is equal to: ∂x 2

 ∂ 2 v ( x, t )  2 F  = − χ Vx ( j χ , t ) − j χ v(0, t ) + Ri (0, t ) 2 x ∂  

[6.83]

Therefore, the Fourier transform of equation [6.67] is:

dVx ( j χ , t ) χ2 jχ 1 =− Vx ( j χ , t ) − v(0, t ) + i (0, t ) dt RC RC C

[6.84]

Thanks to [6.79], this equation is equivalent to two differential systems:

dRe ( χ , t ) χ2 1 =− Re ( χ , t ) + i (0, t ) dt RC C

[6.85]

The Distributed Model of the Fractional Integrator

dI m ( χ , t ) χ2 χ =− Im (χ , t) − v(0, t ) dt RC RC

151

[6.86]

The solutions Re ( χ , t ) and I m ( χ , t ) are characterized by:

Re (− χ , t ) = Re ( χ , t ) and I m (− χ , t ) = − I m ( χ , t )

[6.87]

Reciprocally, if Vx ( j χ , t ) is known, we obtain v( x, t ) using the inverse Fourier transform:

v ( x, t ) =

1 2π



+∞

−∞

Vx ( j χ , t )e j χ x d χ

[6.88]

6.8.3. Impulse response of the RC line

We are interested in v(0, t ) when i(0, t ) = δ (t ) . According to [6.88]:

v(0, t ) =

1 2π



+∞

−∞

Vx ( j χ , t )d χ

[6.89]

Therefore

v(0, t ) =

+∞ 1  +∞ Re ( χ , t )d χ + j  I m ( χ , t )d χ  −∞  2π  −∞

[6.90]

According to [6.87]

v(0, t ) =

1

π

+∞

0

Re ( χ , t )d χ

[6.91]

where Re ( χ , t ) is the solution of the differential equation [6.85], with χ ∈ ]−∞; +∞[ . 2

χ t 1 − RC e As As i (0, t ) = δ (t ), C

is the solution of [6.85] for χ > 0 and also for χ < 0 .

[6.92]

152

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Hence: 2

χ t 2 − RC e C

Re ( χ , t ) =

[6.93]

and:

v(0, t ) =

1

π



∞

0

2

χ t 2 − RC e dχ C

As [KOR 68]:



∞

0

2 2

e − a x dx =

[6.94]

π a

[6.95]

We finally obtain:

v(0, t ) =

1

π

R 1 C t

Let us recall that the impulse response of a fractional integrator is hn (t ) =

[6.96]

t n −1 . Γ ( n)

Therefore, for n = 0.5:

h0.5 (t ) =

1

π t

with Γ(0.5) = π

[6.97]

Thus: v (0, t ) =

R h0.5 (t ) C

[6.98]

Obviously, this result expressed in the time domain is equivalent to [6.75] expressed in the s variable Laplace transform domain. On the contrary, because v(0, t ) is obtained using [6.85] and [6.91], we can compare the following two differential systems: – for the fractional integrator, the output y (t ) is the solution of the distributed system:

The Distributed Model of the Fractional Integrator

∂z (ω , t ) = −ω z (ω , t ) + u (t )ω ∈ [ 0; +∞[ ∂t ∞

y(t ) =  μn (ω ) z(ω, t )dω 0

153

[6.99]

[6.100]

– for the transmission line, the output v(0, t ) is the solution of the distributed system:

dRe ( χ , t ) χ2 1 =− Re ( χ , t ) + i (0, t ) χ ∈ ]−∞; +∞[ dt RC C

v(0, t ) =

1

π

+∞

0

Re ( χ , t )d χ

[6.101]

[6.102]

Equations [6.99], [6.100] and [6.101], [6.102] are analogous, so:

ω≡

χ2 RC

[6.103]

Note that [ω ] = sec−1 [ χ ] = m−1 [ R ] = Ωm −1 [C ] = Fm−1 . Therefore:

 χ2  1 = sec −1  =  RC  ΩF

[6.104]

Consequently, the two expressions have the same physical dimension. The main conclusion is that according to [6.103], the distributed model of the fractional integrator, where the time frequency ω ranges from 0 to ∞ , corresponds to the Fourier transform model of the RC line, where the spatial frequency χ ranges from −∞ to +∞ . 6.8.4. General solution

The previous result can be generalized to any excitation i(0, t ) according to the equivalence:

154

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

ω≡

χ2 RC

or χ = ± RCω

[6.105]

Equation [6.101] can be expressed as:

dRe

(

RCω , t

) = −ω R

e

dt

(

)

RCω , t +

1 i (0, t ) C

[6.106]

and according to [6.93]: Re ( χ , t ) = 2 Re

(

RCω , t

Re

(

)

[6.107]

Therefore:

v(0, t ) =

2

π



+∞

0

)

RCω , t d χ

[6.108]

Note that:

dχ =

RC d ω 2 ω

RC dω = 2χ

[6.109]

hence:

v(0, t ) =

RC

π



+∞

0

Re

(

RCω , t

ω

)d ω

[6.110]

On the contrary, using system [6.99] with v(t ) = i (0, t ) and comparing the two differential systems, we obtain:

Re

(

)

RCω , t ≡

1 z (ω , t ) C

[6.111]

Hence, using z (ω , t ) in [6.102], we obtain: v (0, t ) =

R ∞ 1 −0.5 ω z (ω , t )d ω C 0 π

[6.112]

The Distributed Model of the Fractional Integrator

As: sin nπ = sin 0.5π = 1

155

[6.113]

[6.112] can be expressed as: v(0, t ) =

R ∞ sin 0.5π −0.5 ω z (ω , t )d ω C 0 π

[6.114]

which corresponds to the distributed model of the fractional integrator with:

μ0.5 (ω ) =

sin 0.5π

π

ω −0.5

[6.115]

Conclusion

The fractional integrator, with order equal to 0.5, is exactly equivalent to an infinite dimension RC line, i.e. to an infinite dimensional partial differential diffusive equation. The frequency distributed model of the integrator corresponds to the real part of the spatial Fourier transform of the diffusive system. The modes ω of the integrator distributed model correspond to the spatial frequencies χ of the harmonic diffusive response. 6.8.5. Initialization in the time and spatial domains

It is well known [KOR 68, FAR 82] that the initial condition of the diffusive RC line corresponds to v( x, 0) , x ranging from 0 to ∞ . On the contrary, the initial condition of the fractional integrator distributed model is z (ω , 0) , ω ranging from 0 to ∞ . According to the equivalence:

z (ω , t ) ≡ Re ( χ , t )

[6.116]

and the definition:

Re ( χ , 0) = Re  F {v( x, 0)}

[6.117]

the frequency distributed initial condition z (ω , 0) is directly equivalent to the spatial distributed initial condition v( x, 0) :

z (ω , 0) ≡ v( x, 0)

[6.118]

156

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Initialization of the RC line with v( x, 0) requires knowledge of the initial voltage, particularly for large values of x . Recall that due to the Fourier transform, large values of x correspond to low frequency values of χ . Therefore, initialization of the integrator with z (ω , 0) requires knowledge of ω for any value of the frequency spectrum, particularly for very low values of ω , exhibiting the specificity of the long range memory of a fractional system. This equivalence between the fractional integrator and the RC line will be used in Chapter 5 Volume 2 to derive a technique for the control of the distributed state v( x, t ) of the RC line. A.6. Appendix: inverse Laplace transform of the fractional integrator

Let H ( s) = L {h(t )} . Reciprocally, for a given transfer function H ( s) , h(t ) is provided by the inverse Laplace transform h(t ) = L−1 { H ( s )} [SPI 65]:

 1  h(t ) =  2 jπ  

γ + j∞



H ( s )e st ds

for t > 0

0

for t < 0

γ − j∞

The integral is calculated using a Bromwich contour [LEP 80].

Figure 6.5. Modified Bromwich contour C

[6.119]

The Distributed Model of the Fractional Integrator

157

1 0n < 1 is a multiform function, a cut is necessary in the complex sn plane, which leads to the modified Bromwich contour presented in Figure 6.5: As H ( s) =

Thus, we can write:

1 2 jπ

1

 s

n

e st ds =

C

 1   +  + +  + +   2 jπ  AB BDE EH HJK KL LNA 

[6.120]

Referring to Cauchy’s theorem [KOR 68, LEP 80]: 1 2 jπ



=0

1 2 jπ



+

[6.121]

C

As

BDE

1 2 jπ



= 0 and

LNA

1 2 jπ



=0

[6.122]

HJK

then:

 1    +  2 jπ  EH KL  AB R R − xt − xt  1  e e dx +  dx  = lim − − jπ n jπ n 2π j  ε ( xe ) ) ε ( xe  1 2 jπ



= hn (t ) = lim( R → ∞, ε → 0) −

[6.123]

Finally, we obtain ∞

hn (t ) =  0

sin nπ

π

x − n e − xt dx

[6.124]

Physical considerations indicate that x corresponds to a frequency ω , so let us define ω = x . Note that e − ω t is the impulse response ( z (ω , t ) = e −ωt ) of the elementary system

1 when its input is v(t ) = δ (t ) . s +ω

158

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Finally, equation [6.124] means that hn (t ) is the weighted contribution of all these elementary systems, with ω ranging from ω = 0 to ∞ , where the frequency weight is

μn (ω ) =

sin nπ

π

ω −n

[6.125]

Therefore ∞

hn (t ) =  μn (ω )e −ωt d ω 0

[6.126]

7 Modeling of FDEs and FDSs

7.1. Introduction Chapters 1–3 were dedicated to the numerical simulation of FDEs using fractional integrators. As in the integer order case, the integration operator is the key tool for the modeling of FDSs. In Chapter 6, a theoretical framework was provided to the analysis of this operator. Moreover, it was proved that the Riemann–Liouville integration corresponds basically to a convolution process, obeying the well-known principles of the system theory [KAI 80]. This convolution interpretation provides a natural solution to the initialization of FDSs, recognized as a major problem of fractional calculus for a long time [LOR 01, FUK 04, TEN 14]. Numerical simulations highlight the role of the distributed state variable z (ω , t ) in the interpretation of system transients and reciprocally the inability of pseudo-state variables x(t ) to predict them [SAB 10a, TRI 09b, TAR 16a]. This type of modeling, based on fractional integrators, can be qualified as the closed-loop or internal representation [TRI 12b, TRI 12c].

Nevertheless, another representation of linear commensurate order fractional systems exists, which is not based on fractional integrators. The inverse Laplace 1 transform of the elementary transfer function n provides another distributed s +a model of this system, known as the diffusive representation [MON 98, HEL 98, MON 05a, MON 05b], which can be qualified as the open-loop or external representation.

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

160

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

These two representations of FDSs exhibit complementary features, which will be used for the analysis of fractional Lyapunov stability in Chapters 7 and 8, Volume 2. An application of the open-loop representation is the computation of the Mittag-Leffler function (section 7.8). 7.2. Closed-loop modeling of an elementary FDS

Consider the one-derivative FDE: D n (x ) + a0 x(t ) = u (t )

[7.1]

which can also be presented as an elementary FDS: D n (x ) = −a0 x(t ) + u (t )

[7.2]

where v(t ) is the input of the fractional integrator used to simulate or model the FDE and x(t ) is its output (see Figure 7.1).

Figure 7.1. Analog simulation of the one-derivative FDE

Modeling of FDEs and FDSs

Using the distributed model of

1 sn

161

, modeling of the FDE is based on the

equations:  ∂z (ω , t )  ∂t = −ω z (ω , t ) + v(t )  +∞   x(t ) = μ n (ω ) z (ω , t )dω  0  = v ( t ) u (t ) − a0 x(t )   sin (nπ ) − n μ n (ω ) = ω π 



where z (ω , t ) is the distributed internal state of

[7.3]

1

and also the state of the FDE, as

sn

in the integer order case. Using the Laplace transform and defining the initial state at t = 0 , i.e. z (ω ,0 ) , we obtain: sz (ω , s ) − z (ω ,0) = −ω z (ω , s ) + u ( s) − a0 X ( s)

[7.4]

Therefore z (ω , s ) =

z (ω ,0) + u ( s ) − a 0 X ( s)

(s + ω )

[7.5]

 μ (ω ) z(ω, s )dω

[7.6]

and +∞

X (s) =

n

0

Thus +∞

X (s) =

 0

μ n (ω ) z (ω ,0) dω + [u ( s ) − a0 x( s )] (s + ω )

+∞

μ n (ω )

 (s + ω ) dω 0

[7.7]

162

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Recall equation [6.16], thus: X ( s) =

sn n s + a0

+∞

 0

μ n (ω ) z (ω ,0) 1 dω + n U (s) (s + ω ) s + a0

[7.8]

The first term of [7.8] corresponds to the system’s free response initialized by z (ω ,0) , whereas the second term represents the forced response. This means that the FDE response x(t ) is the sum of a free response and a forced response, as in the integer order case [KAI 80, ZAD 08]. The distributed state z (ω , t ) is the state of the 1 FDE and also the state of its associated integrator . sn The analytical expression of x(t ) will be formulated in Chapter 9.

7.3. Closed-loop modeling of an FDS 7.3.1. Modeling of an N-derivative FDS

As any linear FDE can be written as a FDSs, we present the general case of an N -derivative FDS. Let D n ( X (t ) ) = A X (t ) + Bu (t )

dim( X (t ) ) = dim(n ) = N

X = [x1 (t )  xi (t )  x N (t )] T

n = [n1  ni  n N ] T

[

0 < ni ≤ 1

[7.9]

]

Dtn ( X (t ) ) = Dtn1 (x1 (t ) )  Dtni (xi (t ) )  Dtn N (x N (t ) )

T

Note that we directly consider the non-commensurate order case. More specifically, the order n i can be equal to 1 , as in the induction motor example presented in Chapter 5. Note also that it is possible to consider 1 < ni' < 2 , with ni' = 1 + ni (so 0 < ni < 1 ). Thus, it is another reason to introduce integer orders in the model of [7.9].

Modeling of FDEs and FDSs

163

Obviously, this general model also corresponds to the commensurate order case with ni = n ∀i . The fractional integrator

1

s ni

associated with the pseudo-state xi (t ) corresponds

to the equation:  ∂zi (ω , t ) i = 1,2,..., N = −ω zi (ω , t ) + vi (t )  ∂t  +∞   xi (t ) = μ ni (ω ) zi (ω , t )dω  0  n vi (t ) = D i (xi (t ) )  μ n (ω ) = sin (niπ ) ω − ni 0 < ni < 1  i π



[7.10]

For ni = 1 , these equations are replaced by:  ∂zi (0, t ) = vi (t )  ∂t   xi (t ) = zi (0, t )  v (t ) = D1 (x (t ) ) i  i μ ni (ω ) = δ (ω )

[7.11]

Using the Laplace transform, we can write: z (ω ,0) + vi ( s) zi (ω , s ) = i (s + ω )

[7.12]

Let us define

Z (ω , t ) = [z (ω , t )  z (ω , t )  z (ω , t )]T 1 i N  T V (t ) = [v1 (t )  vi (t )  v N (t )] and the matrix

[7.13]

164

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

 μ n (ω ) 0   1      μ n (ω ) =  μ ni (ω )       0 μ n N (ω ) 

[

]

[7.14]

Then +∞   X (t ) = μ n (ω ) Z (ω , t )dω  0  V (t ) = D n ( X (t ) ) = B u (t ) + A X (t )

[

]

[7.15]

In the Laplace domain, we obtain: X (s) =

+∞



[μ

0

n

+∞ [μ n (ω )] dω [BU (s) + A X (s)] (ω )]Z (ω ,0) dω +  (s + ω ) (s + ω ) 0

[7.16]

Note that

+∞

 0

 1  s n ,1   ( ) μn ω 1 dω =  n  =  (s + ω ) s      0 

[

]

        1  n, N  s 

0

 1 s n, i

[7.17]

and  s n1  −1  1  n   n = s =  s    0

[ ]

       nN  s 

0

 s ni

[7.18]

Modeling of FDEs and FDSs

165

Thus, we can write +∞

X (s) =



[μn (ω )] Z (ω,0)dω +  1  [B U (s) + A X (s)]  sn   

(s + ω )

0

[7.19]

[ ]

Left multiplication of [7.19] by s n and simple calculations yield +∞

[ [ ] ] [s ]  (Zs(ω+ ω,0)) dω + [ [s ]− A] −1

n

X ( s) = s − A

n

n

−1

B U ( s)

[7.20]

0

As noted previously, the first term corresponds to the free response initialized by Z (ω ,0) , whereas the second term represents the forced response. Obviously, equation [7.20] is more simple in the commensurate order case (i.e. ni = n ∀i ). Then

[μn (ω )] = μn (ω ) I

[ ]

and s n = s n I

[7.21]

Therefore

[

X (s) = s n I − A

+∞

] [s ]  (Zs(ω+ ω,0)) dω + [s I − A] −1

n

n

−1

B U ( s)

[7.22]

0

The analytical expression of X (t ) will be formulated in Chapter 9. Previous modeling can be qualified as closed-loop modeling, thanks to the closed-loop nature shown in Figure 7.1. However, we can also call it the internal modeling of the FDS, since the state variables zi (ω , t ) are those of associated 1

“inside” the system. The external (or diffusive) representation of s ni FDS will be presented in section 7.7. integrators

7.3.2. Distributed state

Equations [7.8] and [7.22] can be used to express the distributed state.

166

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

For example, consider again the elementary system [7.1] and its step response x(t ) caused by u (t ) = U H (t ) with initial conditions equal to 0, i.e. z (ω ,0) = 0 ∀ ω . Then, X ( s) =

1 U s + a0 s

[7.23]

n

As v(t ) = Dn ( x(t )) and z (ω , s ) =

we obtain: z (ω , s ) =

1 V ( s) s +ω

sn 1 U n s + a0 s + ω s

[7.24]

[7.25]

Therefore, we can calculate the components of z (ω , t ) at the steady state, i.e. z (ω , ∞) : z (ω , ∞) =

lim s→0

s z (ω , s ) =

sn U n s → 0 s + a0 s + ω lim

[7.26]

Thus:

z (ω , ∞) = 0

∀ω ≠ 0

[7.27]

and z (0, ∞) =

sn U =∞ s → 0 s n + a0 s lim

[7.28]

Therefore, z (0, ∞) is not defined. Nevertheless, x(∞) is perfectly defined because x(∞) =

U a

Hence, z (0, ∞) is indirectly characterized by the integral equation:

[7.29]

Modeling of FDEs and FDSs

∞

U

 μ n (ω ) z (0, ∞) dω = a0

167

[7.30]

0

REMARK 1.– With the frequency discretized model (Chapter 2): J

x(t ) =

c

j

z j (t )

j =0

[7.31]

and:  z j (∞ ) = 0 ∀ j ≠ 0  . U   z0 ( ∞ ) = a c 0 0 

[7.32]

We can also express the components of z (ω , t ) when the system is at rest. Consider the free response caused by z (ω ,0) ≠ 0 , then sn s s + a0

X (s) =

∞

 0

μ n (ω ) z (ω ,0) dω s +ω

[7.33]

which represents the contribution of all the elementary z (ω ,0) frequency components. Therefore, z (ω , s ) =

sn z (ω ,0) n s + a0 s + ω

[7.34]

Of course, the system is at rest for t → ∞ and: z (ω , ∞) =

lim lim s s n z (ω ,0) s Z (ω , s ) = =0 n s→0 s → 0 s + a0 s + ω

[7.35]

168

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

so we obtain z (ω , ∞) = 0 ∀ ω

[7.36]

7.4. Transients of the one-derivative FDS 7.4.1. Numerical simulation

Consider again the elementary system [7.1]:

D n ( x(t )) + a0 x(t ) = u (t ) with a0 = 1 and n = 0.5. Simulation is performed using equations [7.3]. Practically, a numerical algorithm is used to perform the simulation, where the distributed state variable z (ω , t ) is frequency discretized into components z j (t ) : dz j (t ) dt x(t ) =

= − ω j z j (t ) + v(t ) for j = 0 to J J

 c j z j (t )

[7.37]

j =0

v(t ) = u (t ) − a x(t )

Finally, using the Z-transform technique, the complete numerical algorithm is: vk = uk − a xk

t = k Te

z j , k +1 = α j z j , k + β j

vk

xk +1 =

[7.38]

J

c j =0

j

z j , k +1

where α j , β j and c j have been defined in Chapter 2 (equations [2.37]–[2.40]). We consider the following input:

u (t ) = U  u (t ) = − U

for 0 < t < T for T < t < 3 T

with U = 1 T = 5 s

[7.39]

Modeling of FDEs and FDSs

169

The reference output, using the Mittag-Leffler function (Appendix A.3), is:

[ (t ) = U H (t ) [1 − E

] )]

xmit (t ) = U H (t ) 1 − En ,1 (−a t n ) for 0 < t < T xmit

n ,1 (−a t

[

n

[7.40]

]

− 2 U H (t − T ) 1 − En ,1 (−a (t − T ) n ) for T < t < 3 T

where En , 1(− at n ) is the Mittag-Leffler function. The system is simulated with the following parameters:

ωb = 0.001 rd / s ωh = 1000 rd / s J = 20 Te = 5 ms

[7.41]

The reference and simulated outputs are shown in Figure 7.2: as expected and in accordance with Chapter 3, the two curves fit exactly.

Figure 7.2. Comparison of Mittag-Leffler and simulated responses. For a color version of the figures in this chapter, see www.iste.co.uk/trigeassou/analysis1.zip

7.4.2. Initialization at t = t1

In Figure 7.2, we note that x(t ) is equal to 0 at the instant t1 = 5.25 s . Therefore, we propose to initialize the system at t = t1 u (t ) = 0 for t > t1 .

by imposing

170

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Thus, we modify the computation of xmit (t ) for t > t1 , according to: for t1 < t < 3 T :

[

]

[

x mit (t ) = U H (t ) 1 − E n , 1 (− a t n ) − 2 U H (t − T ) 1 − E n , 1 (−a(t − T ) n )

[

+ U H (t − t1 ) 1 − E n , 1 (−a (t − t1 ) n

]

]

[7.42]

which corresponds to the equivalent input: u (t ) = U  u (t ) = − U u (t ) = 0 

for 0 < t < T for T < t < t1

[7.43]

for t1 < t < 3 T

As u (t ) = 0 for t > t1 , the reference xmit (t ) represents the free response of the system for t > t1 , starting from x(t1 ) = 0 . Hence, we create a reference free response as presented in Figure 7.3.

Figure 7.3. Comparison of reference and initialized responses

Moreover, at t = t1 , we measure the state z (ω , t1 ) of the fractional integrator (see Figure 7.5). Therefore, it is possible to initialize the system, with u (t ) = 0 for t > t1 .

z (ω , t1 )

and

Modeling of FDEs and FDSs

171

The initialized response is plotted in Figure 7.3: we can note that the two responses fit perfectly. The conclusions are as follows: – the initial state z (ω , t1 ) perfectly summarizes the past behavior for t < t1 , so the initialized response fits exactly the reference free response for t > t1 ; – the initial state z (ω , t1 ) corresponds to x(t1 ) = 0 : therefore, we verify that the only knowledge of the pseudo-state variable does not allow the prediction of the fractional system’s free response; it is necessary to refer to its internal state z (ω , t1 ) . 7.4.3. Initialization at different instants

Finally, we consider the following modified input: u (t ) = U  u (t ) = − U u (t ) = U 

for

0 π and the constraint [7.56] cannot be respected: therefore, there is no pole n for a stable FDE;

so

– on the contrary, if a < 0 , then λ > 0 so θ = 0 + 2kπ and ψ = 0 +

2kπ n

1

which implies r = (− a) n : therefore, there is a positive unstable pole α .  1  Then, let us calculate the impulse response hn, a (t ) = L−1   using the sn + a  inverse Laplace transform (see Appendix A.7). We obtain the general expression: hn, a (t ) = e α t +

+∞

μ

n, a

(ω ) e −ω t dω

[7.57]

0

The first term corresponds to the response of the integer order unstable mode e , whereas the second term is a stable aperiodic multimode. αt

Because there is no pole if a > 0 , the impulse response hn, a (t ) is only composed of the stable multimode:

178

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

+∞

μ

hn, a (t ) =

n, a

(ω ) e −ω t dω

[7.58]

0

where sin(nπ )

μ n, a (ω ) =

π

ωn

[7.59]

a 2 + 2a (cos nπ )ω n + ω 2 n

REMARK 3.– If a = 0 , then H n, a ( s) =

1 sn

, i.e. we recover the fractional integrator.

It is easy to verify that:

μ n,0 (ω ) =

sin (nπ )

π

ω − n = μ n (ω )

Expressions [7.57] and [7.58] are the external impulse responses [ORT 15] of the FDE; they are also known as the diffusive representation of the FDE [MON 98, HEL 98]. Now, consider the response x(t ) to any input u (t ) : x(t ) = hn, a (t ) * u (t )

[7.60]

Let ξ (ω , t ) be the distributed variable associated with hn, a (t ) . Then, as with the fractional integrator, we can write:  ∂ξ (ω , t ) = −ω ξ (ω , t ) + u (t )  ∂t  +∞   x(t ) = μ n, a (ω )ξ (ω , t ) dω  0 

[7.61]



Let ξ (ω ,0) be the initial condition at t = 0 of system [7.61]. Then +∞

x(t ) =

 0

μ n, a (ω )ξ (ω ,0) e −ωt dω +

t +∞

 μ 0

0

n, a

(ω ) e −ω (t −τ ) u (τ ) dω dτ

[7.62]

Modeling of FDEs and FDSs

179

The first term represents the free response, whereas the second term corresponds to the forced response. Model [7.61] is called external or open-loop because it is not referred to an internal integrator. Obviously, the distributed variables z (ω , t ) (internal state variable) and ξ (ω , t ) (external state variable) are completely different. 7.6.3. External representation of a two-derivative FDE

Consider the system H n, a 0 , a1 ( s ) =

s

2n

1 + a1s n + a0

0 < n 0 ∀ a1 and the roots λ1 and λ2 are real with opposite signs; – if a 0 > 0 , then Δ > 0 if a1 > 2a0 , so λ1 and λ2 are real with the same signs; – if a 0 > 0 , then Δ < 0 if −2a0 < a1 < 2a0 . Then the roots λ1 and λ2 are complex conjugate numbers:

λ1 = a + jb , λ2 = a − jb with a = −

a1 2

and b =

a0 −

a12 4

[7.64]

We can also write b

λ1 = ρe jθ , λ2 = ρe − jθ with ρ = a0 and θ = arctan  a

[7.65]

180

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

With λ1 and λ2 , we can write H n, a 0 , a1 ( s) =

A1 A + n 2 s − λ1 s − λ2

[7.66]

n

Now consider the possible poles of H n, a0 , a1 ( s) . If λ1 and λ2 are real, it is the same case as noted previously (section 7.2.2). However, if λ1 and λ2 are complex, there are two conjugate poles:

s1 = re jψ and s 2 = re − jψ if θ < nπ = θ lim .

[7.67]

These poles can be expressed as s1 = α + jβ α = cos(ψ ) and β = sin(ψ ) .

and s 2 = α − jβ

They have a negative real part (i.e. they are stable poles) if nπ

< θ < nπ or

π 2

with

< ψ < π , i.e. if

θ lim

< θ < θ lim . Therefore, we can express the impulse response 2 2 hn, a0 , a1 (t ) using the inverse Laplace transform (see Appendix A.7): hn, a 0 , a1 (t ) = c1e (α + jβ )t + c2 e (α − jβ )t +

+∞

μ

n , a 0 , , a1

(ω ) e −ωt dω

[7.68]

0

This important case will be used in Chapter 8 Volume 2 to analyze system stability. 7.6.4. External representation of an N-derivative FDE

Consider the N -derivative commensurate order FDE (or FDS): H n ( s) =

N (s) D( s)

[7.69]

In this case, the characteristic equation D (s ) has N roots λi that can be real or complex and

Modeling of FDEs and FDSs

H n ( s) =

A A1 A + n 2 + ... + n N s − λ1 s − λ2 s − λN n

181

[7.70]

These roots λi can correspond to possible poles si . The impulse response hn, H (t ) is the sum of elementary impulse responses as defined in sections 7.6.2 and 7.6.3. Therefore: N

hn, H (t ) =



ci e s i t +

i =1

+∞

N

 μ i =1

n , λi

(ω ) e −ωt dω

[7.71]

0

The sum of multimodes gives an equivalent aperiodic multimode, with an equivalent weighting function μ n, H (ω ) . Therefore, we can write: N

hn, H (t ) =

c e i

i =1

si t

+∞

+

μ

n, H

(ω ) e −ωt dω

[7.72]

0

Let ξ (ω , t ) be the distributed state variable associated with the equivalent multimode. Then, for any excitation u (t ) :  ∂ξ (ω , t ) = −ω ξ (ω , t ) + u (t )  ∂t  +∞   xmulti (t ) = μ n, H (ω )ξ (ω , t ) dω  0 



[7.73]

Moreover, a response ci e si t * u (t ) is associated with each possible integer order pole si . Therefore, the global forced response x(t ) is expressed as: N

x(t ) =

 i =1

ci e s i t * u (t ) +

+∞

μ 0

n, H

(ω ) ξ (ω , t ) dω

[7.74]

182

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

7.7. External and internal representations of an FDS

We have defined two representations of a fractional system, closed-loop and open-loop (or diffusive). These two representations are equivalent from an input/output point of view. The open-loop (or external) representation explains the fractional dynamics with the aperiodic multimode, whereas dumped oscillating dynamics are explained by possible integer order conjugate complex poles (for a stable system). The interest of external representation is to allow direct analysis of system dynamics, particularly to characterize stability with the nature of poles (see Chapter 8, Volume 2). The closed-loop (or internal) representation does not make it possible to directly characterize system dynamics: it requires numerical simulation or the calculation of the system response with the distributed exponential technique (see Chapter 9). The external representation requires prior calculation of the roots of D(s ) and

possible poles, and also of the weighting function μ n, H (ω ) of the multimode.

On the contrary, the closed-loop representation requires no prior calculation. Moreover, the external representation requires linearity and commensurability of the FDS. On the contrary, the closed-loop representation is more general because it applies to any FDS, as there is no requirement related to linearity or commensurability. REMARK 4.– The integer order case corresponds to the commensurate order n = 1 . Then, the poles of H (s ) coincide with the roots of D(s ) . Moreover, the multimode vanishes for n = 1 . Then, hH (t ) =

N

 ci e

i =1

λit

for n = 1 . This means that the external

(or diffusive) representation is nothing else than the impulse response representation in the integer order case. REMARK 5.– The closed-loop representation, i.e. modeling using fractional integrators, was defined originally by Trigeassou in a European Control Conference paper in 1999 [TRI 99]. At the same time, and independently, this representation was investigated by Heleschewitz in his PhD [HEL 00]. However, this representation was not developed because the author lacked an efficient approximation technique (infinite gain for

Modeling of FDEs and FDSs

183

ω → 0 ) for the fractional integrator. Therefore, Heleschewitz focused his works mainly on the analysis of the external representation, called the diffusive representation in his PhD (for more details, see [MON 98, MON 05a, MON 05b, HEL 00]). It was also used by Sabatier [SAB 08b, SAB 10a, SAB 12]. REMARK 6.– In his first monograph [OUS 83], Oustaloup analyzed the time 1 performances of the reference system n 0 < n < 2 step response with a s +a technique based on the inverse Laplace transform, comparable to the diffusive representation, but without the same theoretical background. 7.8. Computation of the Mittag-Leffler function 7.8.1. Introduction

We propose a technique for the computation of the Mittag-Leffler function:

En,1 (−a t n )

a>0

0 < n 0

Let us recall the definition of this function (see Appendix A.3): En,1 (− a t n ) =

∞



k =O

(− a t n ) k Γ(n k + 1)

[7.75]

Truncation of this sum to a large value K > > 1 enables direct computation of this function, as with the exponential function: (−at ) k k =0 k! K

e− a t = 

[7.76]

However, whereas this computation is independent of t for the exponential function, we note a divergence of the sum for the Mittag-Leffler function occurring for large values of t . Thus, it is a challenge to propose a computation technique independent of t values. Several techniques have been proposed, for example approaches based on an integral formulation [GOR 02, ORT 17].

184

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

We propose to use the equation

x(t ) = (1− En,1 (−a t n )) H (t )

[7.77]

where x(t ) is the unit step response of the elementary system: H (s) =

a sn + a

a>0

[7.78]

This technique was already proposed by Heleschewitz [HEL 00]; with the same principle, we propose an improved formulation. 7.8.2. Divergence of direct computation

Direct computation is compared to a technique based on the erf function

erf (− a t ) [KOR 68] for n = 0.5 [HEL 00]: EO.5,1 ( − a t ) = a 2 t (1 + erf ( − a t ))

Figure 7.9 presents the graphs of the two functions for

[7.79]

a =1 .

Figure 7.9. Divergence of the Mittag-Leffler function for t > tmax

We verify whether t is limited to a value t max depending on a (see Table 7.1).

Modeling of FDEs and FDSs

a t max

(s)

0.1

1

10

3000

30

0.3

185

Table 7.1. Divergence of the Mittag-Leffler function versus a , n = 0.5

Note that the computation with the erf function is also divergent for t > tmax . 7.8.3. Step response approach

We first use the computation of x(t ) based on the closed-loop representation for the simulation of H ( s ) =

a sn + a

. This technique is very easy to implement;

however, there is an obvious drawback: in order to obtain En,1 (−a t n ) for the desired value t , it is necessary to recursively compute all the previous values x(τ ) 0 t max . 7.8.4. Improved step response approach

We use the same step response technique, but with another formulation of H (s ) . With the open-loop (or diffusive) representation, we can express H (s ) as: ∞

H (s) = a  0

μ n, a (ω ) dω (s + ω )

[7.80]

Let u (t ) = H (t ) , then: ∞

H ( s) = a  0

As

μ n,a (ω ) s (s + ω )

dω

[7.81]

1 1 1 = − , we obtain: s (s + ω ) ω s ω (s + ω ) ∞  ∞ μ n, a (ω )  μ n, a (ω ) x( s ) = a   dω −  dω  ωs ω (s + ω ) 0  0 

[7.82]

Let us recall that x(∞) = lim s x( s ) = 1 , thus: s →0

∞  ∞ μ n, a (ω )  ∞ μ n, a (ω ) μ n, a (ω ) lim a  s  dω − s  dω  = a  dω s →0 ωs ω (s + ω ) ω 0 0  0 

[7.83]

Therefore, ∞

x(t ) = L−1 [ x( s )]=1− a  0

μ n, a (ω ) − ω t e dω ω

[7.84]

Modeling of FDEs and FDSs

187

Thus, ∞

En ,1 (−a t n ) = a  0

μ n, a (ω ) − ω t e dω ω

[7.85]

with sin(nπ )

μ n, a (ω ) =

π

ωn

a 2 + 2a (cos nπ )ωn + ω 2 n

0 < n 0 , 0 < n < 1 and ∀ t > 0 . A.7. Appendix: inverse Laplace transform of

The impulse response of

1 n

s +a transform [SPI 65, LEP 80, POD 99]: h(t ) =

1 s +a n

is calculated using the inverse Laplace

1 γ + jω

1 e st ds  2 jπ γ − jω s n + a

1  with a Bromwich contour, in the same way as L −1  n  (see Appendix A.6). s 

If a > 0 , there is no pole.

[7.88]

190

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore: 1 2 jπ



C

e st ds = 0 sn + a

[7.89]

and R→∞  1   h(t ) = lim  −  +  ε → 0  2 jπ    EH KL  R R→∞ e − xt 1  = lim  ε → 0 2 π j  a + ( x e − jπ ) n ε

     R

dx −  ε

 e − xt dx  jπ n a + (x e ) 

[7.90]

Finally, we obtain ∞

h(t ) =

μ

n, a ( x) e

−xt

dx

[7.91]

0

where sin( nπ )

μ n, a (x ) =

π

xn

[7.92]

a 2 + 2a (cos nπ ) x n + x 2 n 1

On the contrary, if a < 0 , there is a pole α = (−a) n inside the Bromwich contour in the positive complex half plane and 1 2 jπ



e st ds = eαt Cs +a n

[7.93]

which yields: h(t ) = e α t +

+∞

μ

n, a

(x ) e − x t dx

[7.94]

0

Note that if a < 0 , then e α t is an unstable mode, whereas the multimode remains stable.

Modeling of FDEs and FDSs

191

REMARK 7.– These results also apply if a is a complex number. For example, consider 1

H (s ) =

[7.95]

a0 + a1s n + s 2 n

Therefore H (s ) =

1 ( s − λ1 ) ( s n − λ2 )

[7.96]

n

where λ1 and λ2 are the roots of H (s ) such that: λ1, 2 = a ± jb = ρe ± jθ   2 2  ρ = a + b = a0 

b a

θ = arctan 

[7.97]

For each root λ1 or λ2 , we can use the previous results, with two complex weighting functions μ n, λ1 (ω ) and μ n, λ1 (ω ) . Combining these weighting functions, we obtain a real weighting function μ n, a0 , a1 (ω ) . Finally, using the poles α ± jβ corresponding to λ1 and λ2 , we can express the impulse response as: +∞

h(t ) =

μ

n , a 0 , a1 (ω ) e

−ω t

dω + c1 e (α + jβ )t + c2 e (α − jβ )t

[7.98]

0

Therefore, the impulse response is composed of a stable aperiodic multimode and an integer order oscillatory mode, which is stable if α < 0 or unstable if α > 0 .

8 Fractional Differentiation

8.1. Introduction Fractional calculus is generally referred to the fractional derivatives of Riemann– Liouville and Caputo [DIE 10]. Paradoxically, from the beginning of this book, we have addressed fractional calculus only with the definition of the Riemann–Liouville 1 integral and the fractional integrator n . Fractional differentiation is mentioned s only through its Laplace transform s n and with the Grünwald–Letnikov derivative. This surprising approach is motivated by the definition of fractional differentiation which requires the integer order differentiation and the fractional order integration [TRI 13d]. Moreover, the use of Riemann–Liouville or Caputo derivatives requires us to specify their initial conditions [POD 99]. However, as we will demonstrate it, these initial conditions require us to introduce the initial state of the associated Riemann–Liouville integral. Consequently, usual initial conditions of these derivatives have to be revisited [HAR 09a, LOR 11], as highlighted by many research works [SAB 10a, TRI 10b, TRI 13a, TRI 15]. Let us finally recall that the usual approach to the analysis of fractional systems dynamics is based on the properties of these derivatives, particularly on their initial conditions [MAT 96, BET 06, MON 10]. As demonstrated in Chapter 7, FDS transients can be expressed with the initial conditions of internal integrators. Therefore, we intend to demonstrate that FDS transients are independent of the type of fractional differentiation used to analyze them. Finally, numerical simulations will illustrate the transients of Riemann–Liouville and Caputo derivatives.

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

194

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

8.2. Implicit fractional differentiation 1 , its input sn v (t ) is equal to the implicit fractional derivative I Dtn ( x) ( I for implicit).

Let us consider Figure 8.1: as x(t ) is the output of the operator

z(ω,t)

u(t)

v(t)

x(t) sn

a0

Figure 8.1. Analog simulation of the one-derivative FDS

Of course, for initial conditions equal to 0 at t = 0 , we can write: n 0 t

I

(

I 0

Dtn ( x) ) = x(t )

[8.1]

which can also be interpreted as:

x(t ) = hn (t ) *

I 0

Dtn ( x)

[8.2]

or using the Laplace transform: x( s ) =

1 L{ sn

I 0

}

[8.3]

x( s)

[8.4]

Dtn ( x)

i.e. L{

I 0

Dtn ( x)

}=s

n

Fractional Differentiation

195

Let us define: I 0

Dtn ( x) = L−1 { s n x( s)

} = L {s } −1

n

*x(t )

[8.5]

where: L

−1

{

sn

}=h

−n

(t )

[8.6]

Then, equation [8.1] can be interpreted as: n 0 t

I

(

I 0

Dtn ( x) ) = hn (t ) * h− n (t ) * x(t ) = δ (t ) * x(t ) = x(t )

[8.7]

where h− n (t ) is called the convolution inverse of hn (t ) [MAT 94]. It is important to note that the fractional integration process of Figure 8.1 is not a reversible operation: we obtain 0I Dtn ( x) only inside the closed loop of Figure 8.1, used for the simulation of: D n ( x(t ) ) + a0 x(t ) = u (t )

[8.8]

We can conclude that the implicit fractional derivative: – corresponds to the input of the fractional integrator; – can be interpreted as the convolution of x(t ) with the impulse response h− n (t ). As demonstrated in the previous chapters, implicit derivative plays a fundamental role in the analysis of FDS dynamics. 8.3. Explicit Riemann–Liouville and Caputo fractional derivatives 8.3.1. Definitions

The modeling of FDSs was presented in Chapter 7, with fractional orders ni verifying 0 < ni ≤ 1 . Therefore, we are interested principally in the properties of Riemann–Liouville and Caputo derivatives for 0 < n < 1 .

196

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

The fractional derivative of a function f (t ) was defined as Dn ( f (t )) in Chapter 1, using the Laplace transform and for initial conditions equal to 0: L { D n ( f (t ))} = s n F ( s)

[8.9]

Equation [8.4] can be written as: s n F (s) =

1 1− n

s

[8.10]

s F ( s)

or s n F (s) = s

1 F ( s) s1− n

[8.11]

Equation [8.10] corresponds to: C 0

 1  Dtn ( f (t ) ) = L−1  1− n  * L s 

−1

{s F ( s)}

=

1− n 0 t

I

 df (t )     dt 

[8.12]

where C0 Dtn ( f (t )) is the Caputo derivative, whereas equation [8.11] corresponds to: RL 0

where

RL 0

Dtn ( f (t ) ) = L−1 {s} * L

−1

 1  d 1− n  1− n F ( s )  = ( 0 I t ( f (t ) ) ) s  dt

[8.13]

Dtn ( f (t )) is the Riemann–Liouville derivative.

For each derivative, 0 It1− n ( ) is the Riemann–Liouville integral of the order 1 − n associated with integer order differentiation. REMARK 1.– These definitions are easily generalized to any value n : N − 1 < n < N , where N is an integer.

As we can write: L { D n ( f (t ) )

}=s

n

F ( s) =

1 s

N −n

s N F (s) = s N

1 s

N −n

F ( s)

[8.14]

Fractional Differentiation

197

therefore C 0

 d N f (t )   1  Dtn ( f (t ) ) = L−1  N − n  * L−1 {s N F ( s )} = hN − n (t ) *   N s   dt 

(t −τ ) = 0 Γ ( N − n) t

N − n −1

d N f (τ ) dτ dt N

[8.15]

and RL 0

N  1  d  N − n F ( s )  = N ( hN − n (t ) * f (t ) ) s  dt  f (τ ) dτ   

Dtn ( f (t ) ) = L−1 {s N } * L

dN = N dt

 t ( t − τ ) N − n −1   0 Γ ( N − n) 

−1

[8.16]

We can note that these two derivatives are based on the fractional integration of the order N − n = 1 − n' , where 0 < n ' < 1 . Thus, as noted previously, the fractional integration is of the order 1 − n' , whereas associated integer order differentiation is of the order N . 8.3.2. Theoretical prerequisites

The thereafter analysis of the different fractional derivatives relies on a fundamental assumption: we assume the existence of Dtn ( f (t )) , which is the exact fractional derivative of f (t ) . We define three derivatives, I Dtn ( f (t )) , C Dtn ( f (t )) and

RL

Dtn ( f (t )) in order to

compute or calculate Dtn ( f (t )) . Due to initial conditions at t = 0 which will be specified afterwards, implicit and explicit fractional derivatives provide different values of Dtn ( f (t )) , which is one of the major problems of fractional calculus [ORT 08, ORT 11]. In order to get rid of these initial conditions at t = 0 (or at t = t0 ), we define these derivatives since t = −∞ , i.e. the associated integrals are defined since t = −∞.

198

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Let us recall that hα (t ) =

t α −1 . Γ(α )

Therefore, for the implicit derivative: I −∞

note that

t

Dtn ( f (t ) ) =

I −∞

h

−n

(t − τ ) f (τ ) dτ

[8.17]

−∞

Dtn ( f ) is the input of the fractional integrator

1 delivering sn

n −∞ t

I (f)

since t = −∞ with: n −∞ I t ( f (t ) ) =

t

 h (t − τ ) f (τ )dτ

[8.18]

n

−∞

For the Caputo derivative: C −∞

Dtn ( f (t ) ) =

t

h

1− n

(t − τ )

−∞

df (τ ) dτ dτ

[8.19]

and for the Riemann–Liouville derivative: RL −∞

Dtn ( f (t ) ) =

t  d   h1− n (t − τ ) f (τ ) dτ  dt  −∞ 

[8.20]

Because these three expressions of Dn ( f ) are calculated since t = −∞ , the possible transients caused by initial conditions at t = −∞ have completely vanished at instant t . Therefore, these derivatives are equal at instant t to the fractional derivative Dn ( f ) : I −∞

Dtn ( f ) =

c −∞

Dtn ( f ) =

RL −∞

Dtn ( f ) = Dtn ( f )

Of course, this equality is not verified for 0I Dtn ( f ) , 0c Dtn ( f ) and

[8.21] RL 0

Dtn ( f ) .

8.3.3. Comments

Fractional differentiation obeys different definitions, unlike the integer order case, where the definition is unique. Moreover, it is important to note that each of

Fractional Differentiation

199

these definitions is related to fractional integration, of the order n for the implicit derivative and of the order 1 − n for the Caputo and Riemann–Liouville derivatives. The role of this fractional integration in the usual approach to fractional differentiation is untold. On the contrary, this fractional integration plays an essential role in the analysis of transients. It is one of the paradoxes of fractional calculus where focus is put on fractional differentiation, whereas it will be necessary to concentrate on fractional integration. Moreover, it is the cause of wrong formulations of fractional derivative initial conditions, as we will prove it in the next section. 8.4. Initial conditions of fractional derivatives 8.4.1. Introduction I −∞

It has been demonstrated that

Dtn ( f ) ,

c −∞

Dtn ( f ) and

RL −∞

Dtn ( f ) provide the

same value Dtn ( f ) at any instant t . Practically, these derivatives are calculated since an instant t0 ( t0 = 0 afterwards), so we will consider

I t0

Dtn ( f ) ,

c t0

Dtn ( f ) and

RL t0

Dtn ( f ) . Because these

derivatives depend on a specific Riemann–Liouville integral, this integration process introduces distributed states z I (ω , t0 ) , zC (ω , t0 ) or z RL (ω , t0 ) . Equality of the three derivatives at any instant t is necessarily related to these distributed states. Practically, because the Laplace transform is defined for t ≥ 0 , the equality of the derivative Laplace transforms will be expressed at initial instant t0 = 0 . Moreover, because the initial conditions of the derivatives are formulated in the framework of FDS, we first consider the one-derivative case, which we will generalize to N -derivative FDSs in the next section. Let Dtn ( x ) + a0 x ( t ) = u ( t ) 0 < n < 1

be the one-derivative FDS (or FDE). We assume that u (t ) is applied to the elementary system since t = −∞ .

[8.22]

200

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

8.4.2. Implicit derivative 1 (see Figure 8.1). The input sn of the integrator provides the implicit derivative and its internal state z (ω , 0) = z I (ω , 0 ) at instant t0 = 0 . z I (ω , 0 ) summarizes the past of the system,

Let x(t ) be the output of the fractional integrator

i.e. the influence at t = 0 of u (t ) since t = −∞ . Now, if we start the integration process at t0 = 0 , we obtain 0I Dtn ( x) . The dynamics of the fractional integrator verify the distributed differential system: ∂z I (ω , t ) ∂t x(t ) =

= −ω z I (ω , t ) + 0I Dtn ( f (t ) )

[8.23]

+∞

 μ (ω ) n

z I (ω , t ) d ω

0

where z I (ω , 0 ) is the distributed initial state. If the fractional integrator is initialized with z I (ω , 0 ) , we obtain the initialized n implicit derivative 0I Dinit ,t ( x) .

Using the Laplace transform, we can write: z I (ω , s ) =

n z I (ω , 0 ) + L { 0I Dinit , t ( x )}

[8.24]

(s +ω)

Therefore x (s) =

+∞

 0

μn (ω ) zI (ω , 0 ) n d ω + L { 0I Dinit ,t ( x) (s +ω)

+∞

}  (μs +(ωω)) dω n

[8.25]

0

Thus, we finally obtain: n n n L { 0I Dinit , t ( x )} = s x ( s ) − s

+∞

 0

μn (ω ) zI (ω , 0 ) dω (s +ω)

[8.26]

Fractional Differentiation

201

8.4.3. Caputo derivative

The Caputo derivative, calculated since t = −∞ , i.e. −∞I Dtn ( x) , is provided by a fractional integrator of the order (1 − n) excited by the integer order derivative dx(t ) , according to Figure 8.2. dt

Figure 8.2. Caputo derivative principle

zc (ω , t ) is the distributed state associated with

1 1− n

s

; note that it is different from

z I (ω , t ) (or z RL (ω , t ) ).

Let zc (ω , 0) be the value at t = 0 , summarizing the past behavior of the integrator ( t < 0 ). The dynamics of ∂zc (ω , t ) ∂t c 0

= −ω zc (ω , t ) +

n Dinit ,t ( x ) =

+∞



1 1− n

s

verify the distributed differential system:

dx(t ) dt

[8.27]

μ1− n (ω ) zc (ω , t ) d ω

0

n Let 0c Dinit , t ( x ) be the initialized Caputo derivative calculated since t = 0 , where

the integrator

1 1− n

s

has been initialized by zc (ω , 0) .

202

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

The Laplace transform provides:  dx(t )  zc ( ω , 0 ) + L    dt  zc ( ω , s ) = (s + ω)

[8.28]

 dx(t )  As s x( s ) − x(0) = L    dt 

[8.29]

we obtain:

{

L

c 0

n Dinit , t ( x )} = s x ( s )

+∞

μ1− n (ω )

 ( s + ω ) dω 0

+∞ μ (ω ) zc (ω , 0 ) μ (ω ) − x(0)  1− n d ω +  1− n dω s +ω) (s + ω) 0 ( 0 +∞

As

1 1−n

s

[8.30]

∞

μ1− n (ω ) d ω , we finally obtain: s +ω 0

=

n n L { 0c Dinit , t ( x )} = s x ( s ) −

x(0) + s1− n

+∞

 0

μ1− n (ω ) zc (ω , 0 ) dω (s + ω)

[8.31]

REMARK 2.– The commonly accepted definition of Caputo’s initial conditions [POD 99] is expressed as: L ( 0c Dtn ( x ) ) = s x ( s ) −

x (0) s1− n

[8.32]

There is a serious problem with this definition because it does not take into account 1 the distributed state of the fractional integrator 1− n at t = 0 . s x (0) cannot be interpreted as the initial condition of the Caputo derivative: equation

[8.31] clearly exhibits that these initial conditions are x (0) and zc (ω , 0) . The same problem exists with the Riemann–Liouville derivative.

Fractional Differentiation

203

8.4.4. Riemann–Liouville derivative RL n Dt ( x) , is The Riemann–Liouville derivative, calculated since t = −∞ , i.e. −∞ provided by the integer order derivative of the output of the fractional integrator 1 , according to Figure 8.3. 1− n s

RL (ω,t)

I

x(t)

1– –n

(x(t))

RL

Dn (x(t))

d

I 1–n (s)

dt

Figure 8.3. Riemann–Liouville derivative principle

z RL (ω , t ) is the distributed state associated with

1 1− n

s

, and z RL (ω , 0) summarizes

the past behavior of the integrator (for t < 0 ). The dynamics of

∂z RL (ω , t ) ∂t RL 0

Let

verify the distributed differential system:

s

= −ω z RL (ω , t ) + x(t )

D(t )tn ( x) =

RL 0

1 1− n

d  dt 

+∞





[8.33]

μ1− n (ω ) zRL (ω , t ) d ω  

0

n Dinit , t ( x ) be the initialized Riemann–Liouville derivative calculated since

t = 0 , where the integrator

1 is initialized by z RL (ω , 0) . s1− n

The Laplace transform provides: z RL (ω , s ) =

z RL (ω , 0 ) + x( s )

(s +ω)

[8.34]

204

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

 +∞  n L { RL0 D(t )init ( x ) = s }   μ1− n (ω ) zRL (ω, t ) dω  − { −∞ It1− n ( x)}t =0 ,t   0

where

{

1− n −∞ t

I

( x)}

t =0

=

1− n −∞ 0

I

[8.35]

( x) .

Therefore, we finally obtain: n n L { RL0 D(t )init , t ( x )} = s x ( s ) −

 +∞ μ1− n (ω ) z RL (ω , 0 )  1− n + I ( x ) s dω   −∞ 0 (s + ω)  0 

[8.36] REMARK 3.– The initial conditions of Riemann–Liouville and Caputo derivatives have motivated several interpretations (see, for example, [POD 02a, TRI 10b]). 8.4.5. Relations between initial conditions

The previous three equations correspond to the initialized fractional derivatives. These initialized derivatives provide the exact value of Dtn ( x) , since the influence of the past ( t < 0 ) is summarized by the initial conditions. Therefore, by equality of the Laplace transforms of the three derivatives, we can formulate the relations between the different initial conditions. 8.4.5.1. Relation between the implicit and Caputo initial conditions s n x( s) − s n

+∞

 0

+∞ μ n (ω ) z I (ω , 0 ) μ ( ω ) zc ( ω , 0 ) x(0) d ω = s n x ( s ) − 1− n +  1− n dω s (s + ω) (s + ω) 0 [8.37]

Therefore, we obtain +∞

 0

μn (ω ) zI (ω , 0 ) x(0) 1 dω = − n s s (s +ω)

+∞

 0

μ1− n (ω ) zc (ω , 0 ) dω (s + ω)

[8.38]

8.4.5.2. Relation between the implicit and Riemann–Liouville initial conditions

Equality of Laplace transforms provides:

Fractional Differentiation

+∞

 0

205

+∞ μ n ( ω ) z I (ω , 0 ) μ (ω ) z RL (ω , 0 ) 1 d ω = n ( −∞ I 01− n ( x) ) − s1− n  1− n dω s (s + ω) (s + ω) 0

[8.39] 8.5. Initial conditions in the general case 8.5.1. Introduction

Let us consider the N -derivative FDS: Dtn ( X (t ) ) = AX (t ) + Bu (t ) dim ( X (t ) ) = N X = [ x1 (t )  xi (t )  xN (t ) ]

T

n = [ n1  ni  nN ]

T

[8.40]

0 < ni ≤ 1

Dtn ( X (t ) ) =  Dtn1 ( x1 (t ) )  Dtni ( xi (t ) )  DtnN ( xN (t ) ) 

T

We use the same procedure as with N = 1 . The three derivatives are calculated since t = −∞ , and we use the Laplace transform to infer relations between the initial conditions at t = 0 . 8.5.2. Implicit derivatives

The implicit derivative integrator

I 0

Dtn ( xi ) corresponds to the input of the fractional

1 ( i = 1 to N ) operating since t = 0 . s ni

Let Z I (ω , t ) be the distributed state variables of the integrators and Z I (ω , 0) their initial conditions at t0 = 0 . Then: ∂ Z I (ω , t ) ∂t X (t ) =

+∞

 0

= −ω Z I (ω , t ) + 0I Dtn ( X )

[8.41]  μ n (ω )  Z I (ω , t ) d ω

206

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore Z I (ω , s ) =

Z I (ω , 0 ) + L

{

I 0

(s + ω)

Dtn ( X )}

[8.42]

Thus, we obtain: L

{

I 0

Dtn ( X )} =  s n  X ( s ) −  s n 

+∞

 0

[ μn (ω ) ] Z I (ω , 0 ) dω (s +ω)

[8.43]

 μn (ω )  and  s n  were defined in Chapter 7. 8.5.3. Caputo derivatives c 0

Dtn ( X ) is defined as:

c 0

  dx (t )  Dtn ( X (t ) ) =  0 I t1− n1  1    dt  

1− ni 0 t

I

 dxi (t )      dt 

1− nN 0 t

I

 dxN (t )       dt  

T

[8.44] Z C (ω , t )

Let

be the distributed state variable of the integrators

1 s1− ni

( i = 1 to N ) and Z C (ω , 0) be their initial conditions at t0 = 0 . Then: ∂ Z C (ω , t ) ∂t C 0

n t

D (X ) =

= −ω Z C (ω , t ) + +∞

 0

d X (t ) dt

[8.45]

 μ1− n (ω )  Z C (ω , t ) d ω

Therefore: z C (ω , s ) =

z C (ω , 0 ) + s X ( s ) − X (0)

(s + ω)

[8.46]

Fractional Differentiation

207

Thus:  1  L { C0 Dtn ( X )} =  s n  X ( s ) −  1− n  X ( 0 ) + s 

+∞

 0

[ μ1− n (ω ) ] Z C (ω , 0 ) dω (s + ω)

[8.47]

8.5.4. Riemann–Liouville derivatives RL 0

Dtn ( X ) is defined as: RL 0

d  dt 

(

( x1 (t ) ) )

1− n1 0 t

I



d dt

(

Dtn ( X (t ) ) = 1− ni 0 t

I

(x

ni

(t )

))



d dt

(

1− nN 0 t

I

(x

nN

 (t )  

))

Let Z RL (ω , t ) be the distributed state variable of the integrators

[8.48]

T

1 1− ni

s

( i = 1 to

N ) and Z RL (ω , 0) be their initial conditions at t0 = 0 .

Then: ∂ Z RL (ω , t ) ∂t RL 0

= −ω Z RL (ω , t ) + X (t )

+∞  d  D (X ) =    μ1− n (ω )  Z RL (ω , t ) d ω  dt  0 

[8.49]

n t

Therefore: Z RL (ω , s ) =

Z RL (ω , 0 ) + X ( s )

[8.50]

(s + ω)

Thus: +∞  μ1− n (ω )  Z RL (ω , 0 ) dω +s  t =0 (s + ω) 0

L { RL0 Dtn ( X )} =  s n  X ( s ) − { −∞ I t1− n ( X )}

[8.51] 8.5.5. Relations between initial conditions

As in the one-derivative case, we obtain the following.

208

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

8.5.5.1. Relation between the implicit and Caputo initial conditions +∞

 0

+∞  μn (ω )  Z I (ω , 0 ) 1  1   μ1− n (ω )  Z C (ω , 0 ) dω = X ( 0 ) −  n    dω s (s +ω) (s +ω) s  0

[8.52]

8.5.5.2. Relation between the implicit and Riemann–Liouville initial conditions +∞

 0

 μn (ω )  Z I (ω , 0 ) dω (s +ω)

1 = n s 

{

1− n −∞ t

I

( X )}

t =0

−  s1− n 

+∞

 0

 μ1− n (ω )  Z RL (ω , 0 ) dω (s +ω)

[8.53]

In the commensurate order case (i.e. n1 = ... = ni = ... = nN ), these relations are simplified since:

[8.54]

8.6. Unicity of FDS transients 8.6.1. Transients of the elementary FDE

Consider the elementary FDE: Dtn ( x ) + a0 x(t ) = u (t ) 0 < n < 1

[8.55]

We can express the free response using each of the fractional derivatives and their respective initial conditions. Therefore, we obtain three expressions of x(t ) : xI (t ), xC (t ), xRL (t ) : 1) For the implicit derivative: Dtn ( x ) =

I 0

Dtn ( x ) the initial condition is

z I (ω , 0).

Then, we obtain for the free response ( u (t ) = 0 ): L { 0I Dtn ( xI )} + a0 xI ( s ) = 0 0 < n < 1

[8.56]

Fractional Differentiation

209

Using equation [8.26], we finally obtain: xI ( s ) =

sn s n + a0

+∞

 0

μn (ω ) zI (ω , 0 ) dω (s + ω)

[8.57]

Note that this result corresponds to equation [7.8] because z I (ω , 0) = z (ω , 0) . 2) For the Caputo derivative: Dtn ( x ) = C0 Dtn ( xC ) the initial conditions are xC (0) and zC (ω , 0) . Then, we obtain for the free response: L { C0 Dtn ( xC )} + a0 xC ( s ) = 0 0 < n < 1

[8.58]

Using [8.31], we finally obtain: xC ( s ) =

(s

sn − 1 n

+ a0 )

xC (0) −

(s

1 n

+ a0

+∞

) 0

μ1− n (ω ) zC (ω , 0 ) dω (s +ω)

3) For the Riemann–Liouville derivative: Dtn ( x ) = conditions are

1− n −∞ 0

I

RL 0

[8.59]

Dtn ( xRL ) and the initial

( x) and zRL (ω , 0) . Then, we obtain for the free response:

L { RL0 Dtn ( xRL )} + a0 xRL ( s ) = 0 0 < n < 1

[8.60]

Using [8.36], we finally obtain: xRL ( s ) =

+∞ μ1− n (ω ) z RL (ω , 0 ) ( x) s − n dω  ( s + a0 ) ( s + a0 ) 0 (s +ω)

1− n −∞ 0 n

I

[8.61]

8.6.2. Unicity of transients

The true “physical” free response of the FDE depends only on its initial condition and obviously cannot depend on the choice of the fractional derivative used to analyze the FDE. However, the three responses xI (t ) , xC (t ) and xRL (t ) seem to be different because of specific initial conditions. In fact, we have determined relations between these initial conditions.

210

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Consider, for example, xI (t ) and xC (t ) : xI ( s ) =

sn s n + a0

+∞

 0

μn (ω ) zI (ω , 0 ) dω (s + ω)

[8.62]

Using relation [8.38] between the implicit and Caputo initial conditions, we obtain: xI ( s ) =

 xC (0) 1 sn − n  n ( s + a0 )  s s

n −1

xC (0) 1 = n − n ( s + a0 ) s ( s + a0 ) s

+∞

 0

+∞

 0

 μ1− n (ω ) zC (ω , 0 ) dω  (s + ω) 

μ1− n (ω ) zC (ω , 0 ) d ω = xC ( s ) (s +ω)

[8.63]

This means that xI (t ) = xC (t ) = x(t )

[8.64]

where x(t ) is the free response expressed in Chapter 7. Applying the same approach to xRL (t ) , it is straightforward to obtain xI (t ) = xRL (t ) = x(t )

Therefore, we can conclude that: xI (t ) = xC (t ) = xRL (t ) = x(t ). i.e. there is unicity of the free responses calculated with the different fractional derivatives. Obviously, this result applies directly to an N -derivative FDS, using relations [8.52] and [8.53]. 8.6.3. Conclusion

We can use either I Dtn ( x ) , C0 Dtn ( xC ) or

RL 0

Dtn ( xRL ) to express the free response

of: Dtn ( X (t ) ) = AX (t ) + Bu (t ) dim ( X (t ) ) = N

since we have proved the equivalence of X I (t ) , X C (t ) or X RL (t ) .

[8.65]

Fractional Differentiation

211

Nevertheless, there is a fundamental difficulty to express X C (t ) or X RL (t ) . Note that Z I (ω , 0) = Z (ω , 0) is a “natural” initial condition because it is intrinsically linked to fractional integrators

1 , i.e. Z I (ω , 0) is directly available s ni

and its interpretation is straightforward. On the contrary, Z C (ω , 0) and Z RL (ω , 0) are not directly available because they require the calculation of the corresponding fractional derivatives with integrators 1 1 . Moreover, their interpretation in terms of ni dynamics, i.e. of system 1− ni s s dynamics, is not straightforward (see section 8.7). It is necessary to remember that the Caputo derivative has motivated an unjustified interest due to the apparent simplicity of equation [8.32]. It has been interpreted as a strong similitude between the integer order derivative and the Caputo derivative, where x (0) has been interpreted as a physical initial condition of this derivative. Consequently, the free response of

xI ( s) =

1 is expressed as: s + a0 n

sn x(0) s ( s + a0 ) n

[8.66]

This expression is obviously wrong, as it is illustrated by the numerical simulation of section 8.7. Expression [8.26] using Z I (ω , 0) = Z (ω , 0) is not as simple as its erroneous homolog [8.32], but it is the price to pay in fractional calculus, where dynamical transients are more complex than in the integer order case. Finally, let us highlight that Z I (ω , t ) = Z (ω , t ) generalizes the concept of the integer order state variable to the fractional order case: – X (t ) is the state vector for integer order systems; – X (t ) is the pseudo-state vector, whereas Z (ω , t ) is the distributed state vector for fractional order systems.

212

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

8.7. Numerical simulation of Caputo and Riemann–Liouville transients 8.7.1. Introduction

Unicity of FDS transients was proved in the previous section. Nevertheless, it is interesting to verify this fundamental result by numerical simulations. Therefore, the purpose is to illustrate that the transients of an elementary onederivative FDE, expressed either with the Caputo derivative or the Riemann– Liouville derivative, are identical to the transients expressed with the implicit derivative, i.e. with the distributed variable z (ω , t ) . 8.7.2. Simulation of Caputo derivative initialization

Consider the elementary system [8.22]:

Dn ( x(t )) + a0 x(t ) = u(t ) The input of the system is:

u (t ) = 1  u (t ) = 0

0 < t < t0 t > t0

with t0 = 10 s

[8.67]

The system internal state z I (ω , t ) is equal to 0 at t = 0 . Then, at t = t0 , this internal state is z I (ω , t0 ) . As u (t ) = 0 for t > t0 , we observe the free response of the system, initialized by z I (ω , t0 ) : the response x(t ) is presented in Figure 8.5. In order to initialize this system with the Caputo derivative, we must compute dx(t ) dx(t ) and C Dtn ( x(t )) = I 1− n ( ) for 0 < t < t0 dt dt

At t = t0 , we obtain x(t0 ) and zC (ω , t0 ) (internal state of I 1− n ( s) ).

[8.68]

Fractional Differentiation

213

Let us recall equation [8.59] with t0 = 0 :

X ( s) =

s n −1 x(0) 1 − n n s + a0 s + a0

∞

μ1− n (ω ) zC (ω ,0) dω s +ω 0



[8.69]

Let us define:

X Cap ( s) =

s n −1 x(0) s n + a0

[8.70]

and

X z ,Cap ( s) =

1 s n + a0

∞

μ1− n (ω ) zC (ω , 0) dω s +ω 0



[8.71]

Then, for t ≥ t0 : xCap (t ) = x(t0 ) En , 1 (−a0 (t − t0 ) n )

[8.72]

This term is usually considered as the free response of the system, initialized by x(t0 ) [LES 11]. Of course, this solution does not work because there is a transient, even though x(t0 ) = 0 ! Therefore, it is necessary to use the supplementary term: xz ,Cap (t ) = L−1 { X z , Cap ( s )}

[8.73]

∞

μ1− n (ω ) zC (ω , 0) d ω is the free response of the I 1− n ( s) integrator, with the s +ω 0



initial condition zC (ω , 0) , and xz ,Cap (t ) is the response of response, according to Figure 8.4.

1 to this free s n + a0

214

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Figure 8.4. Analysis of the Caputo supplementary term

Finally

x(t ) = xCap (t ) − xz , Cap (t ) These two signals are plotted in Figure 8.5.

Figure 8.5. Simulation of Caputo derivative initialization. For a color version of this figure, see www.iste.co.uk/trigeassou/analysis1.zip

[8.74]

Fractional Differentiation

Obviously,

215

xCap (t ) is different from x(t ) , although they share the same starting

point x(t0 ) . Finally, if we add − xz , Cap (t ) to xCap (t ) , we exactly obtain x(t ) . Therefore, it is possible to correctly initialize the FDS with the Caputo derivative, but the procedure is more complex than the direct one, based only on z I (ω , t0 ) = z (ω , t0 ) . 8.7.3. Simulation of Riemann–Liouville initialization

We consider the same system [8.22] and use the same procedure as described previously. First, we must compute I 1− n ( x(t )) and differentiate this integral to obtain RL

d 1− n [ I ( x(t ))] for 0 < t < t0 dt

Dtn ( x(t )) =

{

}

At t = t0 , we obtain I 1− n ( x(t ))

t0

[8.75]

and z RL (ω , t0 ) (state of I 1− n ( s) ).

Let us recall equation [8.61] with t0 = 0 :

{I X (s) =

1− n

( x(t ))}

s n s + a0

−

0

n

s + a0

∞

μ1− n (ω ) z RL (ω , 0) dω s +ω 0



[8.76]

Let us define: X RL ( s) =

{I

1− n

( x(t ))}

0

n

s + a0

[8.77]

and

X z , RL ( s) =

s s n + a0

∞

μ1− n (ω ) zRL (ω , 0) dω s +ω 0



[8.78]

216

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Then xRL (t ) = { I1− n ( x(t ))}t

0

1 En , n (− a0 (t − t0 ) n ) (t − t0 ) n −1

[8.79]

This term is called the Riemann–Liouville free response [LES 11]. Of course, it cannot fit the true free response because xRL (t ) → ∞ when t → t0 . Therefore, it is necessary to use the supplementary term: xz , RL (t ) = L−1 { X z , RL ( s )}

[8.80]

∞

μ1− n (ω ) zRL (ω , 0) d ω is the free response of the I 1− n ( s) integrator, with the s ω + 0 initial condition z RL (ω , 0) , and xz , RL (t ) is the integer derivative of the response of



1 to this free response, according to Figure 8.6. s + a0 n

Figure 8.6. Analysis of the Riemann–Liouville supplementary term

Note that xz , RL (t ) → ∞ when t → t0 because of the integer derivative action. These two signals are plotted in Figure 8.7 with x(t ) .

Fractional Differentiation

217

Figure 8.7. Simulation of Riemann–Liouville derivative initialization

First, we note an important change in the amplitude scale in comparison with Figure 8.5: because of the integer derivative action, the signals xRL (t ) and xz , RL (t ) are larger than x(t ) , particularly for t → t0+ . Nevertheless, when we add

− xz , RL (t ) to xRL (t ) , we exactly obtain x(t ) . Therefore, it is possible to initialize the FDS with the Riemann–Liouville derivative, with a similar procedure to the Caputo derivative and the conclusion is the same as summarized previously. We can use either the Caputo derivative or the Riemann–Liouville derivative to initialize the FDS, but the two procedures are obviously more complex than the direct one, based only on z I (ω , t0 ) = z (ω , t0 ) !

9 Analytical Expressions of FDS Transients

9.1. Introduction The determination of the analytical expression X (t ) formulating the transients of the FDS: D n ( X (t ) ) = AX (t ) + Bu (t ) dim ( X (t ) ) = N

[9.1]

is considered as a trivial problem because many researchers think that it has already been solved for several years with the initial condition X (0) , where X (t ) is the pseudo-state vector [MAT 96, BET 08]. Unfortunately, we have previously demonstrated that X (t ) is unable to predict future system behavior, which must be replaced by the distributed state vector Z (ω , t ) , in the theoretical framework of the closed-loop representation. Moreover, it is generally admitted that X (t ) expressions depend on the choice of the fractional derivative. We also demonstrated in Chapter 8 that it is a false problem and transients have a unique expression, which can be based on the distributed initial condition Z (ω , 0) . Therefore, the objective of this chapter is to express X (t ) , i.e. the free and forced responses of [9.1] using the distributed state Z (ω , t ) . Two solutions are proposed: the first solution is based on the Mittag-Leffler approach [MON 10, ORT 18] and the second solution is based on the new concept of distributed exponential, i.e. using all the potentialities of the infinite state approach. Moreover, beyond these theoretical expressions, another practical objective is to formulate computable solutions derived from these expressions. Basically, the two theoretical solutions will be derived from Picard’s method; let us briefly recall its principle [KOR 68, POD 99].

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

220

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

This method is used to derive the solution of the nonlinear system: dy ( x)  = f ( x, y ( x ) )  dx   y ( x) = y ( x0 ) for x ∈ [ x0 , X ] 

[9.2]

The solution y ( x ) verifies the integral equation: y ( x) =

x0

 dy ( x)  I 1x  =  dx 

x0

I 1x ( f ( x, y ( x ) ) + y ( x0 )

[9.3]

As y ( x ) on the right side is unknown, it must be replaced by iterative estimates yi ( x) . At the first step, y0 ( x) = y ( x0 ) . Therefore x

y1 ( x) = y( x0 ) +  f ( μ , y0 ( μ ) )d μ

[9.4]

x0

Then, at the second step, with estimate y1 ( x) , we can calculate: x

y2 ( x) = y( x0 ) +  f ( μ , y1 ( μ ) )d μ

[9.5]

x0

Therefore, at iteration k, we obtain x

yk ( x) = y ( x0 ) +  f ( μ , yk ( μ ) )d μ

[9.6]

x0

This method makes it possible to obtain y ( x ) as:

y( x) = lim yk ( x) k →∞

[9.7]

This technique is essentially used in the nonlinear case [KOR 68], but obviously it can be used in the linear case.

Analytical Expressions of FDS Transients

221

9.2. Mittag-Leffler approach 9.2.1. Free response of the elementary FDS Consider the elementary system D n ( x(t ) ) = ax(t ) 0 < n < 1

[9.8]

We have previously demonstrated that z (ω , 0) is its initial condition, i.e. the initial condition of the associated fractional integrator

1 with the closed-loop sn

representation. Let us recall that:

 ∂z (ω , t ) = −ω z (ω , t ) + v(t )   ∂t +∞  = ( ) x t  0 μn (ω ) z (ω, t ) dω   sin(nπ ) − n ω  μn (ω ) = π 

[9.9]

The free response x0 (t ) of the integrator, with the initial condition z (ω , 0) , corresponds to v (t ) = 0 . Therefore

z(ω, t ) = z (ω,0)e−ω t

[9.10]

and x0 (t ) =

+∞

ω  μ (ω ) z (ω , 0 ) e d ω n

− t

[9.11]

0

We look for the solution x(t ) of system [9.8] using Picard’s method. Basically, x(t ) is the solution of the integral equation: x(t ) = 0 I tn ( D n ( x(t ) ) = 0 I tn ( ax(τ ) ) + x0 (t )

[9.12]

222

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As x(t ) on the right side is unknown, it must be replaced iteratively by x0 (t ) , x1 (t ) , …, xk (t ) . The first estimate is x0 (t ) . Therefore x1 (t ) = 0 I tn ( ax0 (τ ) ) + x0 (t )

[9.13]

and using the Laplace transform technique:

x1 ( s ) = L { 0 I tn ( ax0 (τ ) )} + x0 ( s ) =

a a  x ( s ) + x0 ( s) = 1 + n n 0 s  s

  x0 ( s ) 

[9.14]

The second estimate x2 (t ) corresponds to x2 (t ) = 0 I tn ( ax1 (τ ) ) + x0 (t )

[9.15]

Therefore

x2 ( s) =

a sn

 a  1 + n s  

2   a a   x ( s ) + x ( s ) = 1 + + x ( s)   0  0  s n  s n   0    

[9.16]

Consequently, at iteration k: k −1 k  a a a  xk ( s) = 1 + n +…+  n  +  n   x0 ( s )  s s   s   

[9.17]

Therefore j

k a x( s ) = lim   n  x0 ( s ) k →∞ j =0  s  ∞  aj   ak  = s lim   nj x0 ( s ) = s   n ( k +1) x0 ( s ) k →∞ j = 0  ss  k =0  s  k

[9.18]

Analytical Expressions of FDS Transients

223

Let us recall Appendix A.3:

 ak  n  nk +1  = L { En,1 (at )}  k =0  s  ∞

[9.19]

where En,1 (at n ) is the Mittag-Leffler function defined as  ( at n )k   En ,1 (at ) =   k = 0  Γ( nk + 1)    ∞

n

[9.20]

Since x(s) = s L {En ,1 (at n )} x0 ( s)

[9.21]

we obtain x(t) = L-1 {sEn ,1 (at n )} * L-1 { x0 ( s)}

[9.22]

Therefore x(t) =

d En ,1 (at n ) * x0 (t ) dt

[9.23]

9.2.2. Free response of the N-derivative FDS

Consider

 D n ( X (t ) ) = AX (t ) 0 < n < 1  dim ( X (t ) ) = N

[9.24]

The initial condition of the integrators is Z (ω , 0) , and the free response X 0 (t ) of the integrators is defined as X 0 (t ) =

+∞

ω  μ (ω ) Z ( ω , 0 ) e d ω n

0

− t

[9.25]

224

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Basically, X (t ) is the solution of the integral equation: X (t ) = 0 I tn ( AX (τ ) ) + X 0 (t )

[9.26]

As noted previously, the first estimate is X 1 (t ) = 0 I tn ( AX 0 (τ ) ) + X 0 (t )

[9.27]

which is expressed as X 1 ( s ) with the Laplace transform technique: X 1 (s) =

1 1   A X 0 ( s) + X 0 ( s) =  I + n A  X 0 ( s) sn s  

[9.28]

Then X 2 (t ) = 0 I tn ( AX 1 (τ ) ) + X 0 (t )

[9.29]

and 1  1  A  I + n A  X 0 (s) + X 0 (s) n s  s  1 1   =  I + n A + 2 n A2   X 0 ( s ) s s  

X 2 (s) =

[9.30]

Therefore, at iteration k 1 1 1   X k ( s ) =  I + n A +…+ ( k −1) n A( k −1) + kn Ak  X 0 ( s ) s s s  

[9.31]

and X ( s ) is the limit of this procedure: k  1  X ( s ) = lim   jn A j  X 0 ( s ) k →∞  j =0  s

 1  = s lim   jn +1 A j  X 0 ( s ) k →∞  j =0  s k

[9.32]

En ,1 ( At n ) is the matrix Mittag-Leffler function [MON 10] defined by the

Laplace transform:

Analytical Expressions of FDS Transients

∞  Ak  L { En ,1 ( At n )} =   nk +1  k =0  s 

225

[9.33]

Then X ( s ) = sL { En ,1 ( At n )} X 0 ( s )

[9.34]

and X (t ) =

d {En,1 ( At n )} * X 0 (t ) dt

[9.35]

where  ( At n )k   En ,1 ( At ) =   k = 0  Γ ( nk + 1)    n

∞

[9.36]

is the matrix Mittag-Leffler function. 9.2.3. Complete solution of the N-derivative FDS

An excitation u (t ) is introduced in the previous FDS, so

 D n ( X (t ) ) = AX (t ) + Bu (t ) 0 < n < 1  dim ( X (t ) ) = N Z (ω , 0) is the initial condition of integrators

[9.37]

1 and X 0 (t ) is their free sn

response [9.25]. Basically, X (t ) is the solution of the integral equation: X (t ) = 0 I tn ( A X (τ ) + Bu (τ ) ) + X 0 (t )

[9.38]

As noted previously, the first estimate is: X 1 (t ) = 0 I tn ( AX 0 (τ ) + Bu (τ ) ) + X 0 (t )

[9.39]

226

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore 1 ( AX 0 (s) + Bu ( s) ) + X 0 ( s) sn 1  1  =  I + n A  X 0 ( s ) + n Bu ( s ) s s  

[9.40]

X 2 (t ) = 0 I tn ( AX 1 (τ ) + Bu (τ ) ) + X 0 (t )

[9.41]

X 1 (s) =

Then

Therefore 1 1 1   X 2 ( s ) =  I + n A + 2 n A2  X 0 ( s ) + n s s s  

1    I + n A  Bu ( s) s  

[9.42]

At iteration k , we obtain 1 1 1   X k ( s ) =  I + n A +…+ ( k −1) n A( k −1) + kn Ak  X 0 ( s ) s s s   1 1 1  + n  I + n A +…+ ( k −1) n A( k −1)  Bu ( s ) s  s s 

[9.43]

Consequently, X ( s ) is the limit of this procedure: k 1  1  X ( s ) = lim   jn A j  X 0 ( s ) + lim n k →∞ k →∞ s s   j =0

k −1

 1

   s j =0

jn

 A j  Bu ( s ) 

∞

1 ∞  1  1   = s   kn +1 Ak  X 0 ( s ) + n s   kn +1 Ak  Bu ( s ) s k =0  s   k =0  s

[9.44]

Therefore X ( s ) = sL { En ,1 ( At n )} X 0 ( s ) + s1− n L { En,1 ( At n )} Bu ( s )

[9.45]

Let us define [MON 10]

u (s) = s1− n

u(s)

[9.46]

Analytical Expressions of FDS Transients

227

Then X(t) =

d {En,1 ( At n )} * X0 (t) + En,1 ( At n ) * Bu (t ) dt

[9.47]

where u (t ) = 0 Dt1− n ( u (t ) )

[9.48]

It would be possible to generalize this methodology to the non-commensurate order case. However, it would be of limited interest because even in the commensurate order case, practical computation of X (t ) is difficult.

9.3. Distributed exponential approach 9.3.1. Introduction

The Mittag-Leffler approach is very popular among fractional calculus users [POD 99]. We demonstrated in the previous section that it can be adapted to the infinite state approach. Nevertheless, the expression X(t) =

d {En,1 ( At n )} * X0 (t) dt

[9.49]

where X 0 (t ) =

+∞

ω  μ (ω ) Z (ω , 0 ) e n

− t

dω

[9.50]

0

does not provide a straightforward access to the components of X (t ) because it is based on a convolution integral. Therefore, it seems more appropriate to determine another expression of X (t ) using all the properties of the infinite state method. 9.3.2. Solution of D n ( x(t ) ) = ax(t ) using frequency discretization

Consider the elementary system

228

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

D n ( x(t ) ) = ax(t ) 0 < n < 1

[9.51]

which can be expressed with the internal distributed variable z (ω , t ) :  ∂z (ω , t ) = −ω z (ω , t ) + ax(t )   ∂t +∞   x(t ) =  μn (ω ) z (ω , t ) d ω 0   sin(nπ ) − n ω  μn (ω ) = π 

[9.52]

Frequency discretization of this distributed differential system leads to (see Chapter 2):  ∂z j ( t ) = −ω j z j ( t ) + ax (t )   ∂t J   x (t ) =  c j z j ( t ) j =0  c = μ (ω )Δω n j j  j 

j = 0,1, …, J

[9.53]

Let us define

Z (t ) =  z0 (t ) … z j (t ) … zJ (t ) 

T

[9.54]

Equation [9.54] can be written as x(t ) = C I Z (t )

[9.55]

with C I = c0

c1 … c j

… cJ 

[9.56]

Analytical Expressions of FDS Transients

229

Let us define 0  0  −ω  1      AI =   −ω j        −ωJ  0

1 1    BI =   1    1

[9.57]

Thus, equation [9.53] can be expressed as d Z (t ) dt

= AI Z ( t ) + aB I C I Z (t ) = Asyst Z(t)

[9.58]

with Asyst = AI + aB I C I

[9.59]

The differential equation [9.58] corresponds to the frequency discretization of the distributed differential system [9.52]. Let Z (0) = {Z (t )}t =0

[9.60]

Then, the solution of [9.58] with the initial condition [9.60] is expressed with the matrix exponential eA t [KAI 80], i.e. Asyst t

Z (t ) = e

Z (0)

[9.61]

This technique is completely trivial in the context of integer order differential systems. However, what is the limit of [9.61] if J → ∞ , ωb → 0 , ωh → ∞ , Δω → 0 ? Asyst t

This means that Z (t ) → z (ω , t ) and Z (0) → z (ω , 0) , which implies that e becomes a matrix exponential of infinite dimension. Therefore, our objective is to express this limit.

230

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

9.3.3. Solution of D n ( x(t ) ) = ax(t ) using a continuous approach

Consider again the distributed differential system ∂z (ω , t ) ∂t

= −ω z (ω , t ) + ax(t )

[9.62]

Using the equation defining x(t ) , we obtain ∂z (ω , t ) ∂t

+∞

= −ω z (ω , t ) + a  μn (ω ) z (ω , t ) d ω

[9.63]

0

In fact, equation [9.63] is not correct because it is necessary to separate the frequency ω (of z (ω , t ) ) and the frequency ξ used in the calculus of the integral defining x(t ) . Indeed, we must write ∂z (ω , t ) ∂t

+∞

= −ω z (ω , t ) + a  μn (ξ ) z (ξ , t ) d ω

[9.64]

0

Consequently, there are two frequency variables ω and ξ . It is possible to use only one variable ξ thanks to a mathematical trick. Consider the frequency Dirac impulse δ (ξ ) [ZEM 65], defined as +∞

 δ (ξ ) d ξ = 1

[9.65]

0

Therefore, we can write ∞

∞

0

0

 ωδ (ξ − ω ) z (ξ , t )dξ = ω  δ (ξ − ω ) z (ξ , t )dξ = ω z (ω , t )

[9.66]

Let us define f (ω , ξ ) = −ωδ (ξ − ω ) + a μ n (ξ )

[9.67]

Analytical Expressions of FDS Transients

231

Then ∂z (ω , t ) ∂t

+∞

= −ω z (ω , t ) + a  μ n (ξ ) z (ξ , t ) d ξ 0

+∞

+∞

0

0

= −ω  δ (ξ − ω ) z (ξ , t ) d ξ + a  μ n (ξ ) z (ξ , t ) d ξ

[9.68]

∞

=  f ( ω , ξ ) z (ξ , t ) d ξ 0

This means that the limit of the discretized system d Z (t ) dt

= Asyst Z(t)

[9.69]

is the distributed differential system ∂z (ω , t ) ∂t

∞

=  f (ω , ξ ) z (ξ , t ) d ξ

[9.70]

0

∞

Therefore,

 f (ω , ξ ) z (ξ , t ) d ξ

is the continuous expression equivalent to

0

Asyst Z(t) .

REMARK 1.– We can verify equation [9.70]. The discrete equivalent of ∞

−  ωδ (ξ − ω ) z (ξ , t )d ξ is −δ (ξ k − ω j ) z (ξk , t ) = −ω j z j (t ) , 0

i.e.

dz j (t ) dt

J

= −ω j z j (t ) + a  ck zk (t )

which corresponds to

for

j = 0 to

J,

k =0

d Z (t ) dt

= AI Z ( t ) + aB I C I Z (t ) = Asyst Z(t) .

Note that two indexes j and k are necessary in order to separate ω j and ξk .

232

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

9.3.4. Solution of D n ( x(t ) ) = ax(t ) using Picard’s method

Consider the distributed differential system: ∂z (ω , t ) ∂t

∞

=  f ( ω , ξ ) z (ξ , t ) d ξ

ω ∈ [ 0; +∞[

[9.71]

0

with the initial condition z (ω , 0 ) . We use Picard’s method. The solution z (ω , t ) verifies the integral equation:

∞  z (ω , t ) = 0 I t1   f (ω , ξ ) z (ω , t ) d ξ  + z (ω , 0 ) 0 

[9.72]

At the first iteration, z0 (ω , t ) = z (ω , 0 ) t ∞   Therefore, z 1 (ω , t ) = z (ω , 0 ) +    f (ω , ξ ) z (ω , 0 ) d ξ dτ 0 0  ∞

As

 f (ω , ξ ) z (ω , 0 ) d ξ

does not depend on τ and ω

0

we obtain ∞

z 1 (ω , t ) = z (ω , 0 ) + tz (ω , 0)  f (ω , ξ ) d ξ

[9.73]

0

Then, at the second iteration: t ∞

z 2 (ω , t ) = z (ω , 0 ) +   f (ω , ξ ) z1 (ω , t ) d ξ dτ 0 0

t ∞

= z (ω , 0) +   0 0

∞   f (ω , ξ )  τ z (ω , 0)  f (ω , ξ ) d ξ + z (ω , 0)  d ξ dτ 0  

[9.74]

Analytical Expressions of FDS Transients

233

Thus ∞  t2 z 2 (ω , t ) = z (ω ,0 ) + t z (ω ,0)  f (ω , ξ ) dξ + z (ω ,0)   f (ω , ξ ) dξ  2  0  0 ∞

2

[9.75]

Consequently, at kth iteration: j

∞  tj  zk (ω , t ) =    f (ω , ξ )d ξ  z (ω , 0) j =0 j !  0 

k

[9.76]

j

∞  tj  f (ω , ξ )d ξ  z (ω , 0)   k →∞ ! j j =0 0  k

z (ω , t ) = lim  so

k  ∞ t k ∞   =     f (ω , ξ )d ξ   z (ω , 0)  k = 0 k !  0  

[9.77]

Let us define the distributed exponential as: ∞ k ∞    ∞ t  exp t  f (ω , ξ )d ξ  =    f (ω , ξ )d ξ  k !  0  k = 0  0 

Thus, the solution of

∂z (ω , t ) ∂t

k

[9.78]

∞

=  f (ω , ξ ) z (ξ , t ) d ξ with the initial condition 0

z (ω , 0 ) is expressed as:

∞  z (ω , t ) = exp t  f (ω , ξ )d ξ  z (ω , 0) ω ∈ [ 0; +∞[  0  Asyst t

which is the continuous limit of Z (t ) = e

Z (0) .

REMARK 2.– 1) The elementary ODE dx ( t ) dt

= ax (t )

with the initial condition x(0) verifies the general solution:

[9.79]

234

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

x(t ) = ea t x(0) The distributed elementary ODE ∂z (ω , t ) ∂t

∞

=  f (ω , ξ ) z (ξ , t ) d ξ 0

with the initial condition z (ω , 0) verifies the general solution:

∞  z (ω , t ) = exp t  f (ω , ξ )d ξ  z (ω , 0) ω ∈ [ 0; +∞[  0  ∞  Therefore, the distributed exponential exp t  f (ω , ξ )dξ  is the generalization  0  of the elementary exponential e at of the integer order case. 2) Consider n = 1 , then ω = 0 and μ1 (ξ ) = δ (ξ ) so f (ω , ξ ) = −ωδ (ξ − ω ) + a μ1 (ξ ) = 0 + aδ (ξ ) and

∞

∞

0

0

 f (ω, ξ )d (ξ ) =  aδ (ξ )dξ = a

∞  at Thus, exp t  f (ω , ξ )d ξ  = exp {a t} = e  0  i.e. we recover the integer order case. 3) The main interest of the distributed exponential, in fact of its discretized version, is to provide a straightforward numerical solution expressed as: Asyst t

Z (t ) = e

Z0

for the free response of the elementary FDE D n ( x(t ) ) = a x(t )

Analytical Expressions of FDS Transients

235

with the initial condition z (ω , 0 ) . On the contrary, the Mittag-Leffler approach relies on a convolution x(t ) =

d En,1 ( at n ) * x0 (t ) dt ∞

with x0 (t ) =  μn (ω ) z (ω , 0 ) e −ωt d ω 0

Obviously, the computation of this convolution is a complex operation. Thus, the distributed exponential approach provides a tractable solution, generalization of the exponential in the integer order case. 9.3.5. Solution of D n ( X (t ) ) = AX (t )

Consider the non-commensurate order system: D n ( X (t ) ) = AX (t ) dim

( X (t ) ) = N

n = [ n1  ni  nN ]

T

[9.80]

Let Z (ω , t ) be the vector of distributed states zi (ω , t ) associated with each integrator

1 s ni

i = 1 to N . Then, system [9.80] is equivalent to

 ∂ Z (ω , t ) = −ω  Z (ω , t ) + AX (t )   ∂t  +∞  X (t ) =  μ (ξ )  Z (ξ , t ) d ξ 0  n  

[9.81]

where  μ n (ξ )  was defined in Chapter 7. As in the one-derivative case, we can write ∂ Z (ω , t ) ∂t

+∞

= −ω Z (ω , t ) + A   μn (ξ )  Z (ξ , t ) d ξ 0

[9.82]

236

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Note that

ω Z (ω , t ) =

+∞

 ωδ (ξ − ω ) Z (ξ , t ) dξ

[9.83]

0

Therefore, we can define the matrix F (ω , ξ ) : F (ω , ξ ) = −ωδ (ξ − ω ) I + A  μn (ξ ) 

[9.84]

Then, system [9.81] can be expressed as ∂ Z (ω , t ) ∂t

=

+∞

 F ( ξ , ω ) Z (ξ , t ) d ξ

[9.85]

0

which is the generalization to the N -derivative case of ∂z (ω , t ) ∂t

=

+∞

 f (ξ , ω ) z (ξ , t ) d ξ

[9.86]

0

where f (ξ , ω ) = −ωδ (ξ − ω ) + a μ n (ξ ) .

∞  Therefore, the distributed exponential exp t  f (ω , ξ )dξ  is replaced by the  0  ∞  distributed matrix exponential exp t  F (ω, ξ )d ξ  , i.e.  0  ∞  Z (ω , t ) = exp t  F (ω , ξ )d ξ  Z (ω ,0) ω ∈ [ 0; +∞[  0 

[9.87]

is the solution of the non-commensurate order FDS D n ( X (t ) ) = AX (t ) with the initial condition Z (ω , 0) . Finally, we obtain X (t ) , the solution of [9.80] with: ∞  ∞  X ( t ) =   μn (ω )  exp t  F (ω , ξ ) d ξ  Z (ω , 0) d ω 0  0 

[9.88]

Analytical Expressions of FDS Transients

9.3.6. Solution of D

n

237

( X (t ) ) = AX (t ) + Bu(t )

Let us again consider the N -derivative FDS:

D

n

( X (t ) ) = AX (t ) + Bu(t )

dim ( X (t ) ) = N

[9.89]

Z (ω , t ) is the sum of the previous free response and the forced response due to

the input u (t ) . Therefore, as in the integer order case [KAI 80, ZAD 08]: ∞  Z f (ω , t ) = exp t  F (ω , ξ )d ξ  * Bu (t )  0  t ∞   =  exp (t − τ )  F (ω , ξ ) d ξ  Bu (τ ) dτ 0 0  

[9.90]

ω ∈ [ 0, +∞[

Thus, the complete solution is: ∞  Z (ω , t ) = exp t  F (ω , ξ )d ξ  Z (ω , 0)  0  t ∞   +  exp (t − τ )  F (ω , ξ )d ξ  Bu (τ )dτ 0 0  

[9.91]

ω ∈ [ 0, +∞[

and the response X (t ) of the system [9.80] is ∞ X ( t ) =   μn (ω )  Z (ω , t )d ω 0

[9.92]

9.4. Numerical computation of analytical transients 9.4.1. Introduction ∞

The convolution of easy task; moreover, as

d  En ,1 (at n )  with x0 (t ) =  μ n (ω ) z (ω , 0)e −ωt d ω is not an dt  0 En ,1 (at n ) is not convergent for large values of t (see

Chapter 7), computation of the free response with the Mittag-Leffler approach, even

238

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

with an elementary fractional system, is a complex problem without practical interest. On the contrary, computation of the free response based on the distributed exponential, in fact with its discretized expression, is straightforward, since it is the generalization of the integer order case which benefits many numerical tools [MOK 97]. However, computation of the forced response is not as simple as in the integer order case, i.e. a dedicated approach is necessary. Two examples are proposed to highlight the interest of the discrete distributed exponential technique. 9.4.2. Computation of the forced response

Consider the elementary system:

Dtn ( x(t )) = ax(t ) + bu(t )

[9.93]

Its frequency discretization leads to the equivalent integer order differential system: d Z (t ) = Asyst Z (t ) + BI u (t ) dt

[9.94]

with Asyst = AI + aBI CI . Therefore, the global solution of [9.94] with the discrete initial condition Z (t0 ) is:

Z (t ) = e

Asyst ( t − t0 )

t

Z (t0 ) +  e

Asyst ( t −τ )

BI u (τ )dτ

[9.95]

t0

Numerical computation of the free response is straightforward, as in the integer order case. In particular, we obtain the value of Z (t ) directly, i.e. without recursive computation of intermediary values since t0 . On the contrary, there is a problem with the forced response.

Analytical Expressions of FDS Transients

239

Let t = kTe , where Te is the sampling period. u (t ) must be replaced between instants (k − 1)Te and k Te by a constant value uk −1 (zero-holder [KAI 80, KRA 92]). Therefore, the convolution integral becomes: t



e

Asyst ( t −τ )

t −Te

BI uk −1dτ = ( Asyst )−1 e

Asyst Te

− I  BI uk −1

[9.96]

Let us recall that dim( Asyst ) = ( J + 1)( J + 1) , where J + 1 is the number of frequency modes, with the constraint J >> 1 . Moreover, as the frequency modes range from ωb to ωh , with ωh >> ωb , the matrix Asyst is ill-conditioned and cannot be inverted [FRA 68, DEN 69, MAR 93]. Therefore, it is necessary to numerically compute the convolution integral. t

Let M =



e

Asyst ( t −τ )

dτ and μ = t − τ

[9.97]

t −Te Te

Then, M =  e

Asyst μ

dμ

[9.98]

0

t

Therefore,  e

Asyst ( t −τ )

BI u (τ )dτ corresponds to the recursive equation:

t0

Z f (k ) = F Z f (k − 1) + Gu(k − 1) where F = e

Asyst Te

and G = M BI

[9.99] [9.100]

M is precomputed thanks to a numerical integration algorithm [HAR 98, NOU 91].

For example, we can compute M using the simplest elementary algorithm: I −1

M = e i =0

Asyst Δt

with Δt =

Te I

[9.101]

240

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

In order to appreciate the accuracy of this technique, we must compute the forced b with u (t ) = UH (t ) and compare it to the response x(t ) = CI Z f (t ) of n s +a reference output xref (t ) = U (1 − En ,1 (− at n )) (see Appendix A.3). The simulation is performed with

ωb = 10− 3 rad/s, ωh = 103 rad/s, J = 30, I = 1000, a = b = 1 Figure 9.1 presents the computation error ε(t) = x(t) – xref (t) for different values of Te.

Figure 9.1. Computation error. For a color version of the figures in this chapter see www.iste.co.uk/trigeassou/analysis1.zip

This error is very low for Te = 10−3 s and Te = 10−2 s . As expected, it is more important for Te = 0.1s , but nevertheless acceptable. 9.4.3. Step response of a three-derivative FDS

Consider the non-commensurate order FDS:

Dtn ( X (t )) = AX (t ) + Bu(t )

[9.102]

with n = [ n1n2 n3 ] X (t )T = [ x1 (t ) x2 (t ) x3 (t ) ]

[9.103]

T

Analytical Expressions of FDS Transients

 0 A =  0  −a0

1 0 −a1

0  0  1  B = 0 1  −a2 

241

[9.104]

n1 = 0.4 n2 = 0.5 n3 = 0.6 a0 = 1 a1 = 0.1 a2 = 0.1

Asyst , B syst and C syst correspond to:

Asyst

AI 1   = 0  −a0 B I 3 C I 1

BI 2 C I 2 AI 2 −a1 B I 3 C I 2

  BI 2 C I 3  AI 3 − a2 B I 3 C 2 0

B syst = [ 0 0 B I 3 ] C syst = [C I 1 C I 2 T

[9.105]

CI3 ]

Let u (t ) = U H (t ) U = 1 . X (t ) is computed with the recursive equation: X (k ) = F X (k − 1) + GU

[9.106]

G is precomputed with the previous integration technique ( I = 2000 ).

The graphs of x1 (t ) , x2 (t ) and x3 (t ) are plotted in Figure 9.2.

xi(t)

Step responses Te = 0.1s

time (s)

Figure 9.2. Step responses of pseudo-state variables

242

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Note that dim ( Asyst ) = (153) (153) , which is obviously a large value for an integer order system. Thus, this example perfectly illustrates the numerical robustness of the discrete distributed exponential algorithm.

10 Infinite State and Fractional Differentiation of Functions

10.1. Introduction Fractional calculus has concerned the calculation of fractional integrals and derivatives for any kind of functions for a long time (see [OLD 72, MIL 93] and the references therein). Of course, integration of FDEs has also been an important issue, stimulated by physical and engineering applications [SAM 87, POD 99]. In the previous chapters, the infinite state approach was applied to the modeling and analysis of FDEs. In fact, this methodology also applies to other domains of fractional calculus, and particularly to the calculation of the fractional derivative of a function g (t ) , i.e. to Dtn (g ) . This derivative depends on the differentiation technique used: Riemann–Liouville, Caputo or Grünwald–Letnikov.

Let us consider the calculation of Dtn (g ) with the Caputo derivative. In the first step, we demonstrate that the calculation of the Caputo derivative on an infinite interval ]− ∞;+∞[ makes it possible to express the fractional derivative of the function g (t ) . In the second step, we analyze the calculation of the Caputo derivative on a truncated interval [t0 ;+∞[ and the initial condition problem arising at t = t0 . Thus, we demonstrate that knowledge of these initial conditions makes it possible to correct the calculation on a truncated interval and thus to recover Dtn (g ) [TRI 11d, TRI 11e]. We consider some generic functions and formulate their fractional derivatives and the corresponding initial conditions.

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

244

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Moreover, we demonstrate that the implicit derivative is also able to express the fractional derivative Dtn (g ) . Finally, numerical simulations are used to illustrate previous results for the particular case of the sine function. 10.2. Calculation of the Caputo derivative

Let us consider a function g (t ) defined for t ∈ ]− ∞;+∞[ , and we propose to analytically determine its fractional derivative − ∞C Dtn (g ) for 0 < n < 1 . Let us recall (Chapter 8) that the Caputo derivative calculated since t = −∞ is equal to the fractional derivative of f (t ) because all the possible transients have vanished, so C n − ∞ Dt ( g )

= Dtn ( g )

[10.1]

where Dtn (g ) is the theoretical fractional derivative of g (t ) . Let us recall that the Caputo derivative is defined as: C n 1− n   − ∞ Dt ( g ) = − ∞ I t

dg (t )    dt 

[10.2]

i.e. the Caputo derivative is the Riemann–Liouville integral, with order 1 − n , of the dg (t ) integer order derivative . dt Let zc (ω , t ) be the distributed state variable of the integrator

1 1− n

s

. Thus,

zc (ω , t ) and Dtn (g ) are solutions of the distributed differential system: ∂zc (ω , t ) dg (t ) = −ω zc (ω , t ) + ∂t dt

[10.3]

+∞ C n n − ∞ Dt ( g ) = Dt ( g ) =  μ1− n (ω ) zc (ω , t ) dω 0

[10.4]

μ1− n (ω ) =

sin((1 − n)π ) − (1− n)

π

ω

[10.5]

Infinite State and Fractional Differentiation of Functions

Equation [10.3] means that zc (ω , t ) is the convolution of impulse response hω (t ) of a first-order elementary system hω (t ) = e

−ω t

1 s+ω

H (t )

dg (t ) dt

245

with the

, where [10.6]

Therefore zc (ω , t ) = e

t −ω (t −τ ) dg (τ ) −ω t dg (t ) * dτ = e dt dτ −∞

[10.7]

With zc (ω , t ) , we finally obtain the fractional derivative as the weighted integral of all the frequency components, using equation [10.4]. Then, we apply this methodology to some generic functions (see [POD 99]). 10.2.1. Fractional derivative of the Heaviside function

Consider the Heaviside function: 1 g (t ) = H (t + T ) =  0

for t ≥ −T

[10.8]

for t < −T

We note that dg (t ) dt

= δ (t + T )

[10.9]

Therefore, using equation [10.3]: zc (ω , t ) = e

−ω t

* δ (t + T ) = e

−ω (t +T )

H (t + T )

[10.10]

Then, using equation [10.4]: +∞ −ω (t +T ) dω Dtn ( g ) =  μ1− n (ω ) e 0

[10.11]

246

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As +∞ −ωt h 1 − n (t ) =  μ 1 − n (ω ) e dω 0

[10.12]

we can write +∞ −ω (t +T ) h1− n (t + T ) =  μ1− n (ω ) e dω 0

[10.13]

Let us recall that h1− n (t ) =

t −n Γ(1 − n)

[10.14]

Therefore h1− n (t + T ) =

(t + T ) − n Γ(1 − n)

[10.15]

and Dtn ( H (t + T )) =

(t + T ) − n H (t + T ) Γ (1 − n)

[10.16]

10.2.2. Fractional derivative of the power function

Let us define the power function: (t + T )α g (t ) = (t + T )α H (t + T ) =   0

for t ≥ −T for t < −T

[10.17]

Then α (t + T )α −1 for t ≥ −T dg (t ) = α (t + T )α −1 H (t + T ) =  dt  0 for t < −T

[10.18]

Infinite State and Fractional Differentiation of Functions

247

Consider the variable change:

τ =t +T

[10.19]

Therefore ατ α −1 dg (τ ) = ατ α −1H (τ ) =  dτ 0

for τ ≥ 0 for τ < 0

[10.20]

Using equation [10.3] zc (ω , τ ) = e −ωτ * ατ α −1

[10.21]

and with equation [10.4] +∞ −ω τ Dτn ( g ) =  μ1− n (ω ) e * ατ α −1 dω 0 + ∞ −ω τ  =   μ1− n (ω ) e dω  * ατ α −1  0 

[10.22]

+∞ τ −n −ωτ dω = h  μ1− n (ω ) e 1− n (τ ) = Γ(1 − n) 0

[10.23]

As

we obtain

Dτn ( g ) =

α Γ(1 − n)

τ − n *τ α −1

[10.24]

Using the convolution lemma (Appendix A.10):

τ − n *τ α −1 =

Γ(α + 1) Γ(1 − n) α − n τ Γ(α − n + 1)

[10.25]

We finally obtain

(

)

Dtn (t + T ) α H (t + T ) =

α Γ(α + 1) (t + T ) α −n H (α + T ) Γ(α − n + 1)

[10.26]

248

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

10.2.3. Fractional derivative of the exponential function

Consider the exponential function: g (t ) = e

λt

t ∈] − ∞ ; + ∞ [

[10.27]

Then dg (t ) dt

=λe

λt

[10.28]

Using equation [10.3] t −ω (t −τ ) λτ * λ e λt = λ  e e dτ −∞ λ e −ω t  τ (ω + λ )  t λe λ t −ω t t τ (ω + λ ) = λe dτ = e = e  − ∞ λ + ω λ + ω  −∞ zc (ω , t ) = e

−ω t

[10.29]

Therefore, with equation [10.4]

∞ λ eλ t Dtn ( g ) =  μ1− n (ω ) dω λ +ω 0

[10.30]

Let us recall that ∞μ 1− n (ω ) dω = 1  s1− n 0 s +ω

[10.31]

Therefore, with s = λ , we obtain ∞μ 1− n (ω ) dω = 1  0 λ +ω λ 1− n

[10.32]

λ eλ t λt =λ ne Dtn ( g ) = n 1 − λ

[10.33]

and

Infinite State and Fractional Differentiation of Functions

249

10.2.4. Fractional derivative of the sine function

Consider the sine function

]

g (t ) = sin (Ωt + θ )

t ∈ −∞;+ ∞

[

[10.34]

and its derivative dg (t ) dt

= Ω cos ( Ω t + θ

)

[10.35]

As cos ( Ω t + θ ) =

e

j (Ωt +θ )

+e

− j (Ωt +θ )

[10.36]

2

Equation [10.3] can be written as ∂zc (ω , t ) ∂t

= − ω zc (ω , t ) +

Ω 2

[

e

j (Ωt +θ )

+e

− j (Ωt +θ )

]

[10.37]

Therefore, this equation corresponds to −ω t * Ω cos (Ωt + θ ) zc (ω , t ) = e t Ω −ω (t −τ )  j (Ωτ +θ ) − j (Ωτ +θ )  e dτ = +e  e   2 −∞ t τ (ω − jΩ ) − jθ  Ω −ω t  t τ (ω + jΩ ) jθ e dτ +  e e dτ  = e e  2 −∞ − ∞  =

[10.38]

Ω −ω t  1 1 t (ω + jΩ ) jθ t (ω − jΩ ) − jθ  e e e + e e   2 ω − jΩ  ω + jΩ 

and using equation [10.40]: j (Ωt +θ ) − j (Ωt +θ )  ∞ e e Ω ∞ Dtn ( g ) =   μ1− n (ω ) dω +  μ1− n (ω ) dω  2 0 ω + jΩ ω − jΩ  0  

[10.39]

250

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As (equation [10.32]), ∞μ 1−n (ω ) dω = 1  0 x +ω x1−n

[10.40]

we obtain, with x = j Ω : j ∞μ 1 1− n (ω ) dω = =e  0 jΩ+ω ( j Ω )1− n

 nπ   2

  − e

j  π   2  Ω n −1

[10.41]

Finally nπ π    − j  Ωt + nπ − π +θ   Ω j  Ωt + 2 − 2 +θ  2 2  +e  Dtn (g ) = e   2  

[10.42]

nπ π   = Ω n cos  Ωt + − +θ  2 2  

which yields: nπ   Dtn (sin (Ωt + θ )) = Ω n sin  Ωt + θ +  2  

[10.43]

10.3. Initial conditions of the Caputo derivative

We have previously defined the initial conditions of the Caputo derivative in the context of FDS transients. Our objective is to revisit these definitions, in the context of the calculation of the Caputo derivative applied to a function g (t ) . In order to express the fractional derivative Dtn (g ) , we have calculated the Caputo derivative on an infinite interval ] − ∞ ; + ∞ [ . Practically, the derivative is

calculated on a truncated interval [t0 ;+∞[ .

Consequently, the function g (t ) is truncated. Let g * (t ) be this new function: g * (t ) = g (t ) H (t − t0 )

[10.44]

Infinite State and Fractional Differentiation of Functions

251

Therefore, we calculate * 1− n  C n  t 0 Dt ( g ) = t 0 I t 

dg * (t )    dt 

[10.45]

On the other hand dH (t − t0 ) dg * (t ) d (g (t ) H (t − t0 ) ) dg (t ) = = H (t − t0 ) + g (t ) dt dt dt dt

[10.46]

dH (t − t0 ) = δ (t − t0 ) dt

[10.47]

As

We obtain dg * (t ) dg (t ) H (t − t0 ) + g (t0 )δ (t − t0 ) = dt dt

[10.48]

Then: C n * 1− n   t 0 Dt ( g ) = t 0 I t

dg (t )  H (t − t0 ) + g (t0 ) t 0 I t1− n (δ (t − t0 ))  dt 

[10.49]

REMARK 1.– Let x(t )=t 0 I t1− n (δ (t − t0 )) , and define ∂z (ω , t ) ∂t

= −ω z (ω , t ) + δ (t − t0 )

Then z (ω , t ) = e−ωt * δ (t − t0 ) = e −ω (t −t0 ) H (t − t0 ) thus x(t ) =

+∞

+∞

0

0

 μ1− n (ω )z(ω , t ) dω =

−ω (t − t )  μ1− n (ω )e 0 dω =

(t − t0 )− n Γ(1 − n)

[10.50]

252

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Therefore, [10.49] is expressed as

(t − t0 )− n dg (t )  H (t − t0 ) + g (t0 ) Γ(1 − n)  dt 

* 1− n  C n  t 0 Dt ( g ) = t 0 I t

[10.51]

We have previously defined [10.1]:  dg (t )  Dtn ( g )= − ∞ I t1− n    dt 

Therefore, using the convolution relation (see Chapter 6), we can write 1− n  dg (t )  1− n  dg (t )    =t 0 I t  + −∞ It dt    dt 

+∞

 μ1− n (ω )zc (ω, t0 )e

−ω (t − t 0 )

dω

[10.52]

0

where zc (ω , t0 ) is the value of the distributed variable zc (ω , t ) of the integrator 1 1− n

s

at instant t0 (when the integrator acts since t = −∞ ).

Note that

dg (t )  dg (t )   H (t − t0 ) = t 0 I t1− n    dt   dt 

1− n  t0 It 

so equation [10.51] can be written as

(t − t0 )− n dg (t )   dg (t )  C n * H (t − t0 ) = t 0 I t1− n  = t 0 Dt ( g ) − g (t0 ) Γ(1 − n)  dt   dt 

1− n   t0 It

[10.53]

Similarly, [10.52] can be written as 1− n   t0 It

dg (t )  1− n  dg (t )   =−∞ It  −  dt   dt 

Equaling these two expressions of

c n t0 Dt

(g ) = D *

n t (g)

+ g (t0 )

+∞

 μ1− n (ω )zc (ω , t0 )e

−ω (t − t 0 )

dω

[10.54]

0

1− n   t0 It

(t − t0 )− n Γ(1 − n)

dg (t )   , we obtain:  dt 

−

+∞

 μ1−n (ω )zc (ω , t0 )e 0

−ω (t −t0 )

dω

[10.55]

Infinite State and Fractional Differentiation of Functions

253

The two “initial conditions” are g (t0 ) and zc (ω , t0 ) . The distributed variable zc (ω , t0 ) is a true initial condition, corresponding to the 1 , acting since t = −∞ . distributed variable zc (ω , t ) at t = t0 of the integrator 1− n s On the contrary, g (t0 ) is a pseudo-initial condition [ORT 11]: it corresponds to the jump of the truncated function g * (t ) from g * (t ) = 0 for t < t0 to g (t ) at t = t0 . The derivative of this jump provides the Dirac g (t0 )δ (t − t0 ) . Moreover, note that lim g (t0 )

t → +∞

(t − t0 )− n Γ(1 − n)

=0

[10.56]

and +∞

lim

t → +∞

 μ1− n (ω )zc (ω, t0 )e

−ω (t − t 0 )

dω = 0

[10.57]

0

Therefore lim

t → +∞

c n t 0 Dt

(g (t )) = D (g (t )) *

n t

[10.58]

This means that the Caputo derivative, calculated since any instant t0 , tends asymptotically towards the fractional derivative of g (t ) , with a long memory transient. 10.4. Transients of fractional derivatives 10.4.1. Introduction

We have demonstrated that the Caputo derivative of the truncated function g (t ) = g (t ) H (t − t0 ) converges asymptotically towards the exact value Dtn (g ) , with a long memory transient. *

254

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

This convergence is characterized by a transient d t 0 (t ) , depending on the “initial conditions” g (t0 ) and zc (ω , t0 ) . This transient is defined as

(

)

Dtn (g (t ) )=t 0cDtn g * (t ) + dt 0 (t )

[10.59]

i.e. d t 0 (t ) =

+∞

−ω (t − t )  μ1− n (ω )zc (ω, t0 )e 0 dω − g (t0 ) 0

(t − t0 )− n Γ(1 − n)

[10.60]

Let us define these “initial conditions” for the previous generic functions; in order to simplify calculations, let t0 = 0 .

10.4.2. Heaviside function

Consider g (t ) = H (t + T ) with T > 0 . Then 1 for t ≥ 0 g * (t ) = H (t + T )H (t ) = H (t ) =  0 for t < 0

[10.61]

We have previously calculated zc (ω , t ) = e −ω (t +T )H (t + T )

[10.62]

Therefore zc (ω ,0 ) = e −ωT

[10.63]

g (0 ) = H (0) = 1

[10.64]

and

Infinite State and Fractional Differentiation of Functions

255

Then c n 0 Dt

(g (t ))= D (g (t )) c n 0 t +∞

*

= Dtn (g (t ) ) − = Dtn (g (t ) ) −

 μ1− n (ω )zc (ω ,0)e

− ωt

0 +∞

− ωT − ωt  μ1− n (ω )e e dω + 0

= Dtn (g (t ) ) −

dω + g ( 0)

t −n Γ(1 − n)

[10.65]

t −n Γ(1 − n)

(t + T ) − n t −n + Γ(1 − n) Γ(1 − n)

Thus d t 0 (t ) = −

(t + T ) − n t −n + Γ(1 − n) Γ(1 − n)

[10.66]

10.4.3. Power function

g (t ) = (t + T )α H (t + T )

Consider

with

T >0

and

its

truncation

α

*

g (t ) = (t + T ) H (t ) .

We have previously calculated zc (ω , t ) = e −ωt * α t α −1

[10.67]

0  α −1 −ωτ ( ) z , 0 ω α =  c  e (0 − τ ) dτ  −T  g ( 0) = T α 

[10.68]

and

Therefore d t 0 (t ) =

+∞

 μ1− n (ω ) zc (ω ,0) dω − 0

T α t −n Γ(1 − n)

[10.69]

256

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

10.4.4. Exponential function

Consider g (t ) = e λ t and its truncation g * (t ) = e λ t H (t ) . We have previously calculated zc (ω , t ) =

λ eλ t λ +ω

[10.70]

Therefore

λ   zc (ω ,0 ) =  λ +ω  g ( 0) = 1

[10.71]

Consequently d t 0 (t ) =

+∞



μ1− n (ω )

0

t −n λ e−ω t dω − λ +ω Γ(1 − n)

[10.72]

10.4.5. Sine function

Consider g (t ) = sin (Ωt + θ ) and its truncation g * (t ) = sin (Ωt + θ )H (t ) . We have previously calculated

zc (ω , t ) =

 Ω −ωt  1 1 e  et (ω + jΩ )e jθ + et (ω − jΩ )e − jθ  ω − jΩ 2  ω + jΩ 

[10.73]

Therefore   1 Ω 1  ω cos(θ ) + Ω sin(θ )  e jθ + e − jθ  = Ω   zc (ω ,0 ) =    2  ω + jΩ ω − jΩ ω 2 + Ω2       g (0) = sin(θ )

[10.74]

Infinite State and Fractional Differentiation of Functions

257

Consider the particular value θ = 0 , and then Ωω   z c (ω ,0 ) = ω 2 + Ω 2    g ( 0) = 0

[10.75]

Therefore d t 0 (t ) =

+∞



Ωω



 μ1− n (ω ) ω 2 + Ω 2  e

−ω t

dω

[10.76]

0

10.5. Calculation of fractional derivatives with the implicit derivative 10.5.1. Introduction

We have demonstrated the interest of the infinite state approach for the calculation of the fractional derivative of a function g (t ) , i.e. the distributed model 1 zc (ω , t ) of the fractional integrator makes it possible to calculate this 1− n s fractional derivative using the Caputo derivative definition. Similarly, we can calculate fractional derivatives with the distributed model associated with the Riemann–Liouville derivative. More surprisingly, we can also use the implicit derivative. Of course, it is not possible to use the closed-loop formulation of the implicit derivative. Let us recall that the analytical expression of this derivative (see Chapter 8) corresponds to: I n − ∞ Dt ( g )

= h− n (t ) * g (t )

[10.77]

where h− n (t ) =

+∞

 0

μ − n (ω ) e −ω t dω =

t − ( n +1) Γ( − n)

[10.78]

258

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As L{ hα (t )} =

1 , we obtain, for α = −n sα 1

L{ h− n (t )} =

s−n

= sn

[10.79]

Let z I (ω , t ) be the distributed variable associated with the implicit derivative:  ∂z I (ω , t ) = −ω z I (ω , t ) + g (t )  ∂t  +∞   − ∞I Dtn ( g ) = Dtn ( g ) = μ − n (ω ) z I (ω , t ) dω   0 

[10.80]

10.5.2. Fractional derivative of the Heaviside function

Consider the Heaviside function g (t ) = H (t + T ) . Then

∂z I (ω , t ) = −ω z I (ω , t ) + H (t + T ) ∂t

[10.81]

Therefore z I (ω , t ) = e −ω t * H (t + T )

[10.82]

and Dtn (g (t ) ) =

+∞

 μ − n (ω ) e 0 +∞

= H (t + T ) *

 μ− n (ω ) e 0

= H (t + T ) *

−ω t

t − n −1 Γ ( − n)

* H (t + T ) dω

−ω t

dω

[10.83]

Infinite State and Fractional Differentiation of Functions

259

Thus Dtn (g (t ) ) =

t

1 (t − τ )− ( n +1) dτ Γ(− n) −T

1  (t − τ )− n  =   Γ(− n)  (− n) 

t

−T

1 (t + T )− n = Γ ( − n) ( − n)

[10.84]

As Γ( x + 1) = xΓ( x) (see Appendix A.1.4), we obtain Γ (1 − n) = − nΓ ( − n) . Therefore Dtn (g (t ) ) =

(t + T ) − n H (t + T ) Γ(1 − n)

[10.85]

10.5.3. Fractional derivative of the power function

Consider the power function g (t ) = (t + T )α H (t + T ) . Then ∂z I (ω , t ) ∂t

= −ωz I (ω , t ) + (t + T )α H (t + T )

[10.86]

Consider the variable change τ = (t + T ) Therefore

∂z I (ω , t ) = −ω z I (ω , t ) + (τ )α H (t + T ) ∂t

[10.87]

z I (ω , τ ) = e −ωτ * τ α

[10.88]

and

Then Dτn (g ) =

+∞

+∞

0

0

−ω τ − ωτ α α  μ − n (ω ) e *τ dω = τ *  μ− n (ω ) e dω

[10.89]

260

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

As +∞

 μ − n (ω )e

− ωτ

dω =

0

τ − n −1

Γ(− n )

[10.90]

we obtain Dτn (g ) =

τ −n −1

Γ(− n )

*τα

[10.91]

Using the convolution lemma (Appendix A.10): Dτn (g ) =

Γ(− n ) Γ(α + 1) α − n τ Γ(− n ) Γ(α − n + 1)

[10.92]

Finally Dtn (g ) =

Γ(α + 1)

Γ(α − n + 1)

(t + T )α − n H (t + T )

[10.93]

10.5.4. Fractional derivative of the exponential function

Consider the exponential function g (t ) = e λ t . Therefore

∂z I (ω , t ) = −ω z I (ω , t ) + e λ t ∂t

[10.94]

z I (ω , t ) = e −ωt * eλt

[10.95]

and

Using the results of section 10.2.3, we obtain z I (ω , t ) =

e λt

λ +ω

[10.96]

Infinite State and Fractional Differentiation of Functions

261

Therefore +∞

Dtn (g ) =  μ − n (ω ) 0

+∞ μ (ω ) dω = eλt  − n dω λ +ω 0 λ +ω

e λt

[10.97]

As +∞

 0

μ − n (ω ) dω = s n s +ω

[10.98]

With s = λ , we obtain Dtn (g ) = λn e λ t

[10.99]

10.5.5. Fractional derivative of the sine function

Consider the sine function g (t ) = sin(Ωt + θ ) =

e j (Ωt +θ ) + e − j (Ωt +θ ) 2j

.

Therefore

∂z I (ω , t ) = −ω z I (ω , t ) + g (t ) ∂t

[10.100]

and  e j (Ωt +θ ) + e − j (Ωt +θ )   z I (ω , t ) = e −ωt *   2j   =

t  1  −ω (t −τ ) j (Ωτ +θ ) − j (Ωτ +θ ) +e e dτ   e 2 j − ∞ 

=

e − ωt 2j

t  t (ω + jΩ )τ jθ  e e d τ e (ω − jΩ )τ e − jθ dτ  +   − ∞  −∞

=

e − ωt 2j

 1  1 (ω + jΩ )t e jθ + e (ω − jΩ )t e − jθ   ω + jΩ e ω − jΩ  

=

1 2j

(

 e j (Ωt +θ ) e − j ( Ωt +θ )  +   ω − jΩ   ω + jΩ

)

[10.101]

262

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Then Dtn (g ) =

1 2j

∞ ∞ e − j (Ωt +θ )  e j ( Ωt +θ ) dω  dω +  μ − n (ω )   μ − n (ω ) ω + jΩ ω − jΩ  0  0

[10.102]

As +∞

μ − n (ω )

0

ω+x



dω = x n

n

[10.103]

n

where x = ± jΩ and x = Ω e

±j

nπ 2

, we obtain

nπ   j (Ωt +θ + nπ ) − j (Ωt +θ + ) e 2 +e 2  n n  Dt (g ) = Ω   2j    

[10.104]

Therefore

Dtn (g ) = Ω n sin(Ω t + θ +

nπ ) 2

[10.105]

10.5.6. Conclusion

Obviously, we have obtained the same results for Dtn (g ) using either the Caputo derivative or the implicit derivative. As discussed previously, it would be straightforward to define the initial conditions of the implicit derivative and the corresponding transients of I t0

Dtn ( g * (t ) ) , where g * (t ) is the truncation of function g (t ) .

10.6. Numerical validation of Caputo derivative transients 10.6.1. Introduction

Previous results are essentially theoretical. They do not provide a practical understanding on the computation of the Caputo derivative and particularly of

Infinite State and Fractional Differentiation of Functions

263

transients caused by initial conditions. Thus, we hereafter illustrate this problem with the sine function. The objective is to compare the exact fractional derivative Dtn (g ) and the Caputo derivative C0 Dtn ( g ) computed since t = 0 . Thanks to the theoretical values of the corresponding initial conditions, theoretical transients can be computed and compared to the observed ones. Let us recall: Sine function: g (t ) = sin (Ωt + θ ) .

nπ   n n Fractional derivative: Dt (sin (Ωt + θ )) = Ω sin  Ωt + θ + . 2   Computed Caputo derivative:

C 0

Dtn ( g ) .

Transients: +∞

( ) μ

Dtn ( g ) = t0cDtn g * +

1−n

(ω ) zc (ω , t0 )e −ω (t −t ) dω − g (t0 ) (t − t0 )

−n

0

0

Γ(1 − n)

[10.106]

Let us note that the term related to g (t0 ) is a false problem: it represents the amplitude of the jump caused by the truncation of g (t ) . The corresponding transient dg (t ) for t > 0 + . Thus, we will only dt consider the influence of the distributed initial condition zC (ω ,0) .

can be easily removed by the computation of

 ω cos(θ ) + Ω sin(θ )  We have demonstrated: z c (ω ,0) = Ω  . ω 2 + Ω2  

The corresponding transient is defined as: ∞

 ω cos(θ ) + Ω sin(θ )  dth (t ) =  μ1− n (ω ) Ω   dω ω 2 + Ω2   0

264

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

It is compared to d (t ) = Dtn ( g ) − C0 Dtn ( g )

10.6.2. Simulation results

Numerical simulations are performed with: Riemann–Liouville integration: 1 − n = 0.5

ωb =10 − 4 rd / s ωh = 10 4 rd / s Sine function: Ω = 2 rd / s

−

π 2

J = 30

0

[10.107]

Let I (t ) = t α * t β =

t

α β  (t − τ ) τ dτ

[10.108]

0

and

{ }{ }

L{I (t )} = L t α L t β

[10.109]

Recall that

{ }

L tn =

Γ(n + 1) for t > 0 s n+1

[10.110]

Infinite State and Fractional Differentiation of Functions

267

Then L{I (t )} =

Γ(α + 1) Γ( β + 1) Γ(α + 1)Γ( β + 1) = s α +1 s β +1 sα + β + 2

[10.111]

As t α + β +1  1  L−1  α + β + 2  = s  Γ(α + β + 2)

[10.112]

we obtain t α * t β = L−1{I ( s )} =

Γ(α + 1)Γ( β + 1)t α + β +1 for t > 0 Γ(α + β + 2)

[10.113]

References

[AGA 53] AGARWAL R.P., “A propos d'une note de M. Pierre Humbert”, Comptes Rendus de l’Académie des Sciences, Paris, vol. 236, no. 21, pp. 2031–2032, 1953. [AGR 04] AGRAWAL O.P., “A general formulation and solution scheme for fractional optimal control problems”, Nonlinear Dynamics, vol. 38, pp. 323–337, 2004. [AGR 07] AGRAWAL O.P., KUMAR P., “Comparison of five numerical schemes for fractional differential equations”, in SABATIER J. et al. (eds), Advances in Fractional Calculus, pp. 43–60, Springer, 2007. [AGU 14] AGUILA-CAMACHO N., DUARTE-MERMOUD M.A., GALLEGOS J.A., “Lyapunov functions for fractional order systems”, Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 2951–2957, 2014. [ALG 95] ALGER P.L., Induction Machines. Their Behavior and Uses, Taylor & Francis, 1995. [ANG 65] ANGOT A., Compléments de mathématiques, Editions de la Revue d’Optique, 1965. [BAG 85] BAGLEY R.L., TORVIK P.J., “Fractional calculus in the transient analysis of viscoelastically damped structures”, AIAA Journal, vol. 23, no. 6, pp. 918–925, 1985. [BAL 13] BALACHANDRAN K., GOVINDARAJ K., ORTIGUEIRA M.D. et al., “Observability and controllability of fractional linear dynamical systems”, IFAC Workshop FDA’13, vol. 6 Part 1, Grenoble, France, pp. 893–898, 2013. [BAL 06] BALEANU D., “Fractional Hamiltonian analysis of irregular systems”, Signal Processing, vol. 26, pp. 2632–2636, 2006. [BAR 04] BARBOSA R.S., MACHADO J.A.R., FERREIRA I.M., “Tuning of PID controllers based on Bode’s ideal transfer function”, Non Linear Dynamics, vol. 38, pp. 305–321, 2004. [BAR 07] BARBOSA R.S., TENREIRO MACHADO J.A., VINAGRE B.M. et al., “Analysis of the Van der Pol oscillator containing derivatives of fractional order”, Journal of Vibration and Control, vol. 13, nos 9–10, pp. 1291–1301, 2007. [BAT 01] BATTAGLIA J.L., COIS O., PUISEGUR L. et al., “Solving an inverse heat conduction problem using a non-integer identified model”, International Journal of Heat and Mass Transfer, vol. 44, no. 14, pp. 2671–2680, 2001. Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

270

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[BAT 02] BATTAGLIA J.L., Méthodes d’identification de modèles à dérivées d’ordre non entier et de réduction modale. Application à la résolution de problèmes thermiques inverses dans les systèmes industriels, Habilitation à Diriger des Recherches, Université Bordeaux 1, 2002. [BEN 04] BENCHELLAL A., BACHIR S., POINOT T. et al., “Identification of a non-integer model of induction machines”, IFAC Workshop FDA’2004, Bordeaux, France, pp. 400–407, 2004. [BEN 06] BENCHELLAL A., POINOT T., TRIGEASSOU J.C., “Approximation and identification of diffusive interface by fractional models”, Signal Processing, vol. 86, no. 10, pp. 2712–2727, 2006. [BEN 08a] BENCHELLAL A., Modélisation des interfaces de diffusion à l’aide d’opérateurs d’intégration fractionnaires, PhD Thesis, University of Poitiers, France, 2008. [BEN 08b] BENCHELLAL A., POINOT T., TRIGEASSOU J.C., “Fractional modeling and identification of a thermal process”, Journal of Vibration and Control, vol. 14, nos 9–10, pp. 1403–1414, 2008. [BEN 93] BENNET S., “A history of control engineering”, vol. 1: 1800–1930 and vol. 2: 1930–1955, IEE Control Engineering Series, Peter Peregrinus, 1979 and 1993. [BET 06] BETTAYEB M., DJENNOUNE S., “A note on the controllability and the observability of fractional dynamic systems”, IFAC FDA’06 Workshop, Porto, Portugal, 2006. [BET 08] BETTAYEB M., DJENNOUNE S., “New results on the controllability and observability of fractional dynamical systems”, Journal of Vibration and Control, vol. 14, pp. 1531–1541, 2008. [BOD 45] BODE H.W., Network Analysis and Feedback Amplifier Design, Van Nostrand, New York, 1945. [BON 00] BONNET C., PARTINGTON J.-R., “Coprime factorization and stability of fractional differential systems”, Systems and Control Letters, vol. 41, pp. 167–174, 2000. [BOS 86] BOSE B.K., Power Electronics and AC Drives, Prentice-Hall, 1986. [BOU 67] BOUDAREL R., DELMAS J., GUICHET P., Commande optimale des processus, vols 1–4, Dunod, 1967. [BOY 94] BOYD S., EL GHAOUI L., FERON E. et al., Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [CAN 05a] CANAT S., FAUCHER J., “Modeling, identification and simulation of induction machine with fractional derivative”, in LE MEHAUTE A. et al. (eds), Fractional Differentiation and its Applications, Ubooks Verlag, 2005. [CAN 05b] CANAT S., Contribution à la modélisation dynamique d'ordre non entier de la machine asynchrone à cage, Thesis, INPT, France, 2005. [CAP 69] CAPUTO M., Elasticita e Dissipazione, Zanichelli, Bologna, 1969.

References

271

[CHA 05] CHATTERJEE A., “Statistical origins of fractional derivatives in viscoelasticity”, Journal of Sound and Vibration, vol. 284, pp. 1239–1245, 2005. [CHE 84] CHEN C.T., Linear System Theory and Design, Oxford University Press, 1984. [CHE 14] CHEN D., ZHANG R., LIN X. et al., “Fractional order Lyapunov stability theorem and its application in synchronization of complex dynamical networks”, Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 12, pp. 4105–4121, 2014. [COI 01] COIS O., OUSTALOUP A., POINOT T. et al., “Fractional state variable filter for system identification by fractional model”, European Control Conference ECC’2001, Porto Portugal, 3–5 September, 2001. [COI 02] COIS O., Systèmes linéaires non entier et identification par modèle non entier: application en thermique, PhD Thesis, University of Bordeaux, France, 2002. [COL 33] COLE K.S., “Electric conductance of biological systems”, Proceedings Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116, 1933. [COU 85] COULOMB J.L., SABONNADIERE J.C., CAO en électrotechnique, Hermes, Paris, 1985. [CUR 95] CURTAIN R.F., ZWART H.J., An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, NY, 1995. [DAS 11] DAS S., Functional Fractional Calculus, Springer-Verlag, Berlin, 2011. [DEN 69] DENIS-PAPIN M., KAUFMANN A., Cours de calcul matriciel appliqué, Albin Michel, 1969. [DIE 02] DIETHELM K., FORD N.J., “Numerical solution of the Bagley-Torvik equation”, BIT, vol. 42, no. 3, pp. 490–507, 2002. [DIE 08] DIETHELM K., “An investigation of some non-classical methods for the numerical approximation of Caputo-type fractional derivatives”, Numerical Algorithms, vol. 47, pp. 361–390, 2008. [DIE 10] DIETHELM K., “Appendix C: Numerical solution of fractional equations”, in DIETHELM K. et al., The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer Verlag, pp. 195–225, 2010. [DJA 08] DJAMAH T., DJENNOUNE S., BETTAYEB M., “Fractional order system identification”, JTEA 2008, Hammamet, Tunisia, 2008. [DJE 12] DJENNOUNE S., KATZANSTSIS N., SABATIER J., “Discussion on observability and pseudo-state estimation of fractional order systems”, European Journal of Control, vol. 18, pp. 272–276, 2012. [DJE 13] DJENNOUNE S., BETTAYEB M., “Optimal synergetic control for fractional order systems”, Automatica, vol. 49, pp. 2243–2249, 2013. [DOR 67] DORF R.C., Les variables d'état dans l'analyse et la synthèse des systèmes automatiques, translated from English by NASLIN P., Dunod, 1967.

272

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[DU 11] DU M., WANG Z., “Initialized fractional differential equations with Riemann-Liouville fractional order derivative”, ENOC 2011 Conference, Rome, Italy, July 2011. [DU 16] DU B., WEI Y., LIANG S. et al., “Estimation of exact initial states of fractional order systems”, Nonlinear Dynamics, vol. 86, pp. 2061–2070, 2016. [DUG 97] DUGARD L., VERRIEST E.-I., Stability and Control of Time Delay Systems, Lectures Notes in Control and Information Sciences, vol. 228, Springer-Verlag, 1997. [DZI 06] DZIELINSKI A., SIEROCIUK D., “Observer for discrete fractional order state-space systems”, FDA’06 Workshop, vol. 2, pp. 511–516, 2006. [EYK 74] EYKHOFF P., System Identification, John Wiley and Sons, 1974. [FAH 11] FAHD J. et al., “On the Mittag-Leffler stability of Q-fractional nonlinear dynamical systems”, Proceedings of the Romanian Academy, Series A, vol. 12, no. 4/2011, pp. 309–314, 2011. [FAH 12] FAHD J. et al., “On the stability of some discrete fractional non-autonomous systems”, Abstract an Applied Analysis, vol. 2012, Article ID 476581, 9 p., 2012. doi:10.1155/2012/476581. [FAR 82] FARLOW S.J., Partial Differential Equations for Scientists and Engineers, John Wiley and Sons, 1982. [FRA 68] FRANKLIN J.N., Matrix Theory, Prentice-Hall, 1968. [FRI 86] FRIEDLAND B., Control System Design: An Introduction to State-space Methods, McGraw-Hill, 1986. [FUK 03] FUKUNAGA M., SHIMIZU N., “Initial condition problems of fractional viscoelastic equations”, Proceedings of the ASME Conference DETC’03, Chicago, USA, 2003. [FUK 04] FUKUNAGA M., SHIMIZU N., “Role of pre-histories in the initial value problems of fractional viscoelastic equations”, Nonlinear Dynamics, vol. 38, pp. 207–220, 2004. [GAM 11] GAMBONE T., HARTLEY T.T., ADAMS J. et al., “An experimental validation of the initialization response in fractional order systems”, Proceedings of IDETC/CIE FDTA’2011 Conference, August, Washington, DC, 2011. [GAN 66] GANTMACHER F.R., Théorie des matrices, Volumes 1 and 2, Dunod, 1966. [GIB 63] GIBSON J., Nonlinear Automatic Control, McGraw-Hill, 1963. [GIL 63] GILBERT E.G., “Controllability and observability in multivariable control system”, SIAM Journal on Control, Series A, vol. 2, no. 1, pp. 128–151, 1963. [GIL 90] GILLE J.C., DECAULNE P., PELEGRIN M., Théorie et calcul des asservissements linéaires, Dunod-Bordas, Paris, 1990. [GOR 02] GORENFLO R., LOUTCHKO J., LUCHKO Y., “Computation of the Mittag-Leffler function Eα,β (z) and its derivative”, Journal of Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002.

References

273

[HAN 96] HANUS R., BOGAERTS P., Introduction à l’automatique. Vol. 1: Systèmes continus, De Boeck & Larcier, Brussels, 1996. [HAR 98] HARRIS J.W., STOCKER H., Handbook of Mathematics and Computational Science, Springer-Verlag, New York, 1998. [HAR 95] HARTLEY T.T., LORENZO C.F., QAMMER H.K., “Chaos in fractional order Chuaʼ s system”, IEEE Transactions on Circuits and Systems I. Fundamental Theory and Applications, vol. 42, no. 8, pp. 485–490, 1995. [HAR 02] HARTLEY T.T., LORENZO C.L., “Dynamics and control of initialized fractional order system”, Nonlinear Dynamics, vol. 29, pp. 201–233, 2002. [HAR 09a] HARTLEY T.T., LORENZO C.F., “The error incurred in using the Caputo derivative Laplace transform”, Proceedings of the ASME IDET-CIE Conferences San Diego, California, USA, 2009. [HAR 09b] HARTLEY T.T., LORENZO, C.F., “The initialization response of linear fractional order system with constant History function”, 2009 ASME /IDETC, San Diego, CA, 2009. [HAR 11] HARTLEY T.T., LORENZO C.F., “The initialization response of multi term linear fractional order systems with constant history functions”, Proceedings of IDETC/CIE FDTA’2011 Conference, Washington DC, USA, August 2011. [HAR 13] HARTLEY T.T., LORENZO C.F., TRIGEASSOU J.C. et al., “Equivalence of history function based and infinite dimensional state initializations for fractional order operators”, ASME Journal of Computational and Nonlinear Dynamics, vol. 8, no. 4, 2013. [HAR 15a] HARTLEY T.T., TRIGEASSOU J.C., LORENZO C.F. et al., “Energy storage and loss in fractional order systems”, ASME Journal of Computational and Nonlinear Dynamics CND 14-1113, vol. 10, no. 6, 2015. [HAR 15b] HARTLEY T.T., LORENZO C.F., “Realizations for determining the energy stored in fractional order operators”, ASME IDETC-CIE Conference, Boston, USA, August 2015. [HAR 15c] HARTLEY T.T., TRIGEASSOU J.C., LORENZO C.F. et al., “Initialization energy in fractional order systems”, Proceedings of the ASME Conference IDETC/CIE 2015, Boston, USA, 2015. [HEL 00] HELECHEWITZ D., Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels sous representation diffusive, PhD Thesis, ENST Paris, 18 December 2000. [HEL 98] HELESCHEWITZ D., MATIGNON D., “Diffusive realizations of fractional integro-differential operators: structural analysis under approximation”, Conference IFAC, System, Structure and Control, vol. 2, pp. 243–248, Nantes, France, July 1998. [HIM 72] HIMMELBLAU D.M., Applied Nonlinear Programming, McGraw-Hill, 1972. [HOL 89] HOLMAN J.P., Heat Transfer, McGraw-Hill, 1989. [HOT 98] HOTZEL R., Contributions à la théorie structurelle et à la commande des systèmes linéaires fractionnaires, PhD Thesis, University of Paris XI, Orsay, 1998.

274

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[HUL 15] HU J.B., LU G.P., ZHANG S.H., “Lyapunov stability theorem about fractional system without and with delay”, Communications in Nonlinear Science and Numerical Simulation, vol. 20, no. 3, pp. 905–913, 2015. [HUR 95] HURWITZ A., “Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln niet negativen reelen Theilenbesitzt”, Mathematische Annalen, vol. 46, pp. 273–284, 1895. [HUS 09] HUSSON R., Control Methods for Electrical Machines, John Wiley and Sons, 2009. [HWA 06] HWANG C., CHENG Y.-C., “A numerical algorithm for stability testing of fractional delay systems”, Automatica, vol. 42, pp. 825–831, 2006. [JAL 10] JALLOUL A., JELASSI K., TRIGEASSOU J.C., “Fractional modeling of rotor skin effect in induction machines”, IFAC Workshop FDA '10, Badajoz, Spain, 2010. [JAL 12] JALLOUL A., Modélisation et identification des effets de fréquence dans la machine asynchrone par approche d'ordre non entire, PhD Thesis, University of Tunis, Tunisia, 2012. [JAL 13] JALLOUL A., TRIGEASSOU J.C., JELASSI K. et al., “Fractional modeling of rotor skin effect in induction machines”, Nonlinear Dynamics, vol. 73, nos 1–2, pp. 801–813, 2013. [JOO 86] JOOS G., Theoretical Physics, Dover Publications, 1986. [KAB 97] KABBAJ H., Identification d'un modèle circuit prenant en compte les effets de fréquence dans une machine asynchrone à cage d'écureuil, Thesis, INPT France, 1997. [KAI 80] KAILATH T., Linear Systems, Prentice Hall Inc., Englewood Cliffs, 1980. [KAL 60] KALMAN R.E., “On the general theory of control system”, Proceedings of the First IFAC Congress Automatic Control, vol. 1, pp. 481–492, Moscow, 1960. [KHA 15a] KHADHRAOUI A., JELASSI K., TRIGEASSOU J.C. et al., “Identification of fractional model by least squares method and instrumental variable”, Journal of Computational and Nonlinear Dynamics, vol. 10, no. 5, 2015. [KHA 15b] KHADHRAOUI A., JELASSI K., TRIGEASSOU J.C. et al., “Initialization of identification of fractional model by output-error technique”, Journal of Computational and Nonlinear Dynamics, vol. 11, no. 2, 2015. [KHA 16] KHADHRAOUI A., Contribution à l'identification des systèmes d'ordre non entier: application à l'identification des effets de peau dans les barres rotoriques d'une machine asynchrone, PhD Thesis, University of Tunis, Tunisia, 2016. [KHA 96] KHALIL H.K., Nonlinear Systems, Prentice Hall, 1996. [KHA 01] KHAORAPAPONG T., Modélisation d’ordre non entier des effets de fréquence dans les barres rotoriques d’une machine asyncrone, Thesis, France, 2001. [KOR 68] KORN G.A., KORN T.M., Mathematical Handbook for Scientists and Engineers, McGraw-Hill, 1968. [KRA 92] KRANIANSKAS P., Transforms in Signals and Systems, Addison Wesley, 1992.

References

275

[KRA 63] KRASOVSKII N.N., Stability of Motion, Stanford University Press, 1963. [KUN 86] KUNDERT K.S., SANGIOVANNI-VINCENTELLI A., “Simulation of nonlinear circuits in the frequency domain”, IEEE Transactions on Computer Aided Design of Integrated Circuits, vol. CAD-5, no. 4, October 1986. [LAN 89] LANDAU I.D., Identification et commande de systèmes, Hermès, Paris, 1989. [LAS 61] LA SALLE J., LEFSCHETZ S., Stability by Liapunovʼ s Direct Method with Applications, Academic Press, 1961. [LEM 83] LE MÉHAUTÉ A., CREPY G., “Introduction to transfer and motion in fractal media: the geometry of kinetics”, Solid State Ionics, vols 9–10, pp. 17–30, 1983. [LEP 80] LEPAGE W.R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980. [LES 11] LESZCZYNSKI J.S., An Introduction to Fractional Mechanics, Częstochowa University of Technology, 2011. [LEV 59] LEVY E.C., “Complex curve fitting”, IRE Transactions on Automatic Control, vol. AC-4, pp. 37–43, 1959. [LEV 64] LEVINE L., Methods for Solving Engineering Problems Using Analog Computers, McGraw-Hill, New York, 1964. [LI 09] LI Y., CHEN Y.Q., PODLUBNY I., “Mittag Leffler stability of fractional order nonlinear dynamic systems”, Automatica, vol. 45, pp. 1965–1969, 2009. [LI 10] LI Y., CHEN Y.Q., PODLUBNY I., “Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability”, Computers and Mathematics with Applications, vol. 59, pp. 1810–1821, 2010. [LI 14] LI Y., CHEN Y.Q., “Lyapunov stability of fractional order nonlinear systems: a distributed order approach”, ICFDA’14, 23–25 June 2014. [LIN 00] LIN J., POINOT T., TRIGEASSOU J.C., “Parameter estimation of fractional systems: application to the modeling of a lead-acid battery”, 12th IFAC Symposium on System Identification, SYSID 2000, USA, 2000. [LIN 01a] LIN J., Modélisation et identification des systèmes d’ordre non entier, PhD Thesis, University of Poitiers, France, 2001. [LIN 01b] LIN J., POINOT T., TRIGEASSOU J.C., “Parameter estimation of fractional systems. Application to heat transfer”, European Control Conference ECC 2001, Porto, Portugal, pp. 2644–2649, 2001. [LJU 87] LJUNG L., System Identification: Theory for the User, Englewood Cliffs, NJ, 1987. [LOR 01] LORENZO C.F., HARTLEY T.T., “Initialization in fractional order systems”, Proceedings of the European Control Conference (ECC’01), Porto, Portugal, pp. 1471–1476, 2001.

276

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[LOR 08a] LORENZO, C.F., HARTLEY, T.T., “Initialization of fractional differential equation”, ASME Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, 2008. [LOR 08b] LORENZO C.F., HEARTLEY T.T., “Initialization of fractional order operators and fractional differential equations”, Special issue on Fractional Calculus of ASME Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, 021101, April 2008. [LOR 11] LORENZO C., HARTLEY T., “Time-varying initialization and Laplace transform of the Caputo derivative: with order between zero and one”, Proceedings of IDETC/CIE FDTA’2011 Conference, Washington DC, USA, August 2011. [LYA 07] LYAPUNOV A., “Problème général de la stabilité du mouvement”, Annales de la Faculté des Sciences de Toulouse, vol. 9, 1907. [MAA 91] MAAMRI N., TRIGEASSOU J.C., OUSTALOUP A., “Time moments and CRONE control”, IMACS Congress, Lille, France, 1991. [MAA 09] MAAMRI N., TRIGEASSOU J.C., MEHDI D., “A frequency approach to analyze the stability of delayed fractional differential equations”, Proceedings of the European Control Conference ECC’09, Budapest, Hungary, August 2009. [MAA 10] MAAMRI N., TRIGEASSOU J.C., TENOUTIT M., “On fractional PI and PID controllers”, Proceedings of IFAC Workshop FDA'10, Badajoz, Spain. [MAA 14] MAAMRI N., TARI M., TRIGEASSOU J.C., “Physical interpretation and initialization of the fractional integrator”, ICFDA’14 23–25 June 2014. [MAA 15] MAAMRI N., TARI M., TRIGEASSOU J.C., “On the fractional modeling of the diffusive interface”, Proceedings of the ASME Conference IDETC/CIE 2015, Boston, USA, 2015. [MAA 16] MAAMRI N., TRIGEASSOU J.C., “Lyapunov stability of nonlinear fractional systems: the Van der Pol oscillator”, ICFDA'16 Serbia 2016. [MAA 17] MAAMRI N., TARI M., TRIGEASSOU J.C., “Improved initialization of fractional order systems”, 20th World IFAC Congress, Toulouse, France, pp. 8567–8573, 2017. [MAG 06] MAGIN R.L., Fractional Calculus in Bioengineering, Begel House, 2006. [MAN 60] MANABE S., “The noninteger integral and its application to control systems”, Journal of the Institute of Electrical Engineers of Japan, pp. 589–597, May 1960. [MAO 15] MAOLIN D., ZAIHUA W., “Correcting the initialization of models with fractional derivatives via history dependent conditions”, Acta Mechanica Sinica, August 2015. doi: 10.1007/s10409-015-0469-7. [MAR 63] MARQUARDT D.W., “An algorithm for least squares estimation of nonlinear parameters”, Journal of the Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431–441, 1963. [MAR 93] MARCUS M., Matrices and MATLABTM: A Tutorial, Prentice Hall, 1993. [MAT 94] MATIGNON D., Représentations en variables d’état de modèles de guides d’ondes avec dérivation fractionnaire, PhD Thesis, University of Paris XI, Orsay, 1994.

References

277

[MAT 96] MATIGNON D., D’ANDRÉA-NOVEL B., “Some results on controllability and observability of finite dimensional fractional differential systems”, Computational Engineering in Systems Applications Lille France, vol. 2, pp. 952–956, 1996. [MAT 97] MATIGNON D., D’ANDRÉA-NOVEL B., “Observer based controllers for fractional differential systems”, Proceedings of the CDC’97, San Diego, CA, pp. 4967–4972, December 1997. [MAT 98] MATIGNON D., “Stability properties for generalized fractional differential systems”, Proceedings of the ESSAIM, vol. 5, pp. 145–158, 1998. [MAT 10] MATIGNON D., “Optimal control of fractional systems: a diffusive formulation”, Proceedings of the MTNS 2010, Budapest, Hungary, July 2010. [MEN 73] MENDEL J.M., Discrete Techniques of Parameter Estimation, Marcel Dekker, 1973. [MIL 93] MILLER K.S., ROSS B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. [MIT 03] MITTAG-LEFFLER G.M., “Sur la nouvelle fonction Eα (x)”, Comptes Rendus de l'Académie des Sciences, Paris, vol. 137, pp. 554–558, 1903. [MOK 97] MOKHTARI M., MESBAH A., Apprendre et maîtriser MATLABTM, Springer, 1997. [MOM 04] MOMANI S., HADID S., “Lyapunov stability solutions of fractional integro-differential equations”, International Journal of Mathematics and Mathematical Sciences, vol. 47, pp. 2503–2507, 2004. [MON 05a] MONJE C.A., VINAGRE B.M., CHEN Y.Q. et al., “Optimal tunings for fractional fractional PID controllers”, in LE MEHAUTE A. et al. (eds), Fractional Differentiation and Its Applications, pp. 675–686, 2005. [MON 05b] MONTSENY G., Représentation Diffusive, Hermès Lavoisier, 2005. [MON 98] MONTSENY G., “Diffusive representation of pseudo differential time operators”, Proceedings of ESSAIM, vol. 5, pp. 159–175, 1998. [MON 10] MONJE C.A., CHEN Y.Q., VINAGRE B.M. et al., Fractional Order Systems and Control, Sringer-Verlag, London, 2010. [MUL 09] MULLHAUPT P., Introduction à l'analyse et à la commande des systèmes non linéaires, Presses Polytechniques et Universitaires Romandes, 2009. [NAS 68] NASLIN P., Théorie de la commande et variables d’état, Ecole Supérieure d'Electricité, 1968. [NIC 97] NICULESCU S.I., Systèmes à retard, Diderot Editeur Arts et Sciences, Paris, 1997. [NOR 86] NORTON J.P., An Introduction to Identification, Academic Press, 1986. [NOU 91] NOUGIER J.P., Méthodes de calcul numérique, Masson, 1991. [NYQ 32] NYQUIST H., “Regeneration theory”, Bell System Technical Journal, vol. 11, pp. 126–147, 1932.

278

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[OLD 70] OLDHAM K.B., SPANIER J., “The replacement of Fick’s law by a formulation involving semi-differentiation”, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, vol. 26, pp. 331–341, 1970. [OLD 72] OLDHAM K.B., “A signal independent electro-analytical method”, Analytical Chemistry, vol. 44, no. 1, pp. 196–198, 1972. [OLD 74] OLDHAM K.B., SPANIER J., The Fractional Calculus, Academic Press, New York, 1974. [ORT 98] ORTEGA R., LORIA A., NICKLASSON P.J. et al., Passivity Based Control of Euler Lagrange Systems, Springer-Verlag, Berlin, 1998. [ORT 03] ORTIGUEIRA M.D., “On the initial conditions in continuous-time fractional linear systems”, Signal Processing, vol. 83, pp. 2301–2309, 2003. [ORT 08] ORTIGUEIRA M.D., COITO F.J., “Initial conditions: what are we talking about?”, 3rd IFAC Workshop, FDA’08, Ankara, Turkey, 5–7 November 2008. [ORT 11] ORTIGUEIRA M.D., Fractional Calculus for Scientists and Engineers, Springer Science, 2011. [ORT 15] ORTIGUEIRA M.D., TENREIRO M.J., RIVERO M. et al., “Integer/fractional decomposition of the impulse response of fractional linear systems”, Signal Processing, vol. 114, pp. 85–88, 2015. [ORT 18] ORTIGUEIRA M.D., LOPES A.M., TENREIRO M.J., “On the computation of the multi-dimensional Mittag-Leffler function”, Communications in Nonlinear Science and Numerical Simulation, vol. 53, pp. 278–287, 2018. [OUS 81] OUSTALOUP A., “Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation”, IEEE Transactions on Circuits and Systems, vol. 28, no. 10, pp. 1007–1009, 1981. [OUS 83] OUSTALOUP A., Systèmes asservis linéaires d’ordre fractionnaire, Masson, Paris, 1983. [OUS 91] OUSTALOUP A., La commande CRONE, Hermès, Paris, 1991. [OUS 95a] OUSTALOUP A., La dérivation non entière: théorie, synthèse et applications, Hermès, Paris, 1995. [OUS 95b] OUSTALOUP A., MATHIEU B., La commande CRONE, du scalaire au multivariable, Hermès, Paris, 1995. [OUS 00] OUSTALOUP A., LEVRON F., MATHIEU B. et al., “Frequency-band complex noninteger differentiator: characterization and synthesis”, Transactions on Circuits and Systems I; Fundamental Theory and Applications, vol. 47, no. 1, pp. 25–39, 2000. [OUS 05] OUSTALOUP A., COIS O., LE LAY L., Représentation et identification par modèle non entier, Hermes Lavoisier, 2005. [OUS 15] OUSTALOUP A., Diversity and Non-integer Differentiation for System Dynamics, ISTE Ltd, London and John Wiley & Sons, New York, 2015.

References

279

[OWE 86] OWENS L., “Vannevar Bush and the differential analyzer: the text and context of an early computer”, Technology and Culture, vol. 27, no. 1, pp. 63–95, January 1986. [PAR 29] PARK R.H., “Two reaction theory of synchrone machines”, Transaction on AIEE, pp. 27–31, 1929. [PET 11] PETRAS I., Fractional Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Verlag, 2011. [POD 97] PODLUBNY I., DORCAK L., KOSTIAL I., “On fractional derivatives, fractional order systems and PID control”, Proceedings of the Conference on Decision and Control, San Diego, 1997. [POD 99] PODLUBNY I., Fractional Differential Equations, Academic Press, San Diego, 1999. [POD 00] PODLUBNY I., “Matrix approach to discrete fractional calculus”, Journal of Fractional Calculus and Applied Analysis, vol. 3, no. 4, pp. 359–386, 2000. [POD 02a] PODLUBNY I., “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. [POD 02b] PODLUBNY I., PETRAS I., VINAGRE B.M. et al., “Analogue realizations of fractional order controllers”, Nonlinear Dynamics, vol. 29, nos 1–4, pp. 281–296, 2002. [POI 82] POINCARÉ H., “Mémoire sur les courbes limites définies par une équation différentielle”, Journal de Mathématiques, vol. 3, no. 8, pp. 251–296, 1882. [POI 02] POINOT T., TRIGEASSOU J.C., “Parameter estimation of fractional models: application to the modelling of diffusive systems”, 15th IFAC World Congress, Barcelona, Spain, 2002. [POI 03] POINOT T., TRIGEASSOU J.C., “A method for modeling and simulation of fractional systems”, Signal Processing, vol. 83, pp. 2319–2333, 2003. [POI 04] POINOT T., TRIGEASSOU J.C., “Identification of fractional systems using an output-error technique”, Nonlinear Dynamics, vol. 38, nos 1–4, pp. 133–154, 2004. [POL 87] POLOUJADOFF M., “The theory of three phase induction squirrel cage rotors”, Electrical Machines and Power Supplies, vol. 13, pp. 245–264, 1987. [RAG 47] RAGAZZINI J.R., RANDALL R.H., RUSSELL F.A., “Analysis of problems in dynamics by electronic circuits”, Proceedings of the IRE, vol. 35, no. 5, pp. 444–452, May 1947. [RAM 10] RAMDANI K., TUCSNAK M., WEISS G., “Recovering the initial state of an infinite dimensional system using observers”, Automatica, vol. 46, pp. 1616–162, 2010. [RAY 97] RAYNAUD H.F., ZERGAINOH A., “State-space representation of fractional linear filters”, Proceedings of the European Control Conference ECC 97, Brussels, Belgium, 1997. [RET 99] RETIERE N., IVANES M., “An introduction to electrical machine modeling by systems of non-integer order. Application to double-cage induction machines”, IEEE Transactions on Energy Conversion, vol. 14, no. 4, pp. 1026–1032, December 1999.

280

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[RIC 71] RICHALET J., RAULT A., POULIQUEN R., Identification des processus par la méthode du modèle, Gordon and Breach, 1971. [RIC 91] RICHALET J., Pratique de l'identification, Hermès, Paris, 1991. [RIC 03] RICHARD J.P., “Time-delay Systems: an overview of some recent advances and open problems”, Automatica, vol. 39, pp. 1667–1694, 2003. [ROU 77] ROUTH E.J., A Treatise on the Stability of a Given State of Motion, MacMillan London, 1877. [SAB 06] SABATIER J., AOUN M., OUSTALOUP A. et al., “Fractional system identification for lead-acid battery state of charge estimation”, Signal Processing, vol. 86, pp. 2645–2657, 2006. [SAB 08a] SABATIER J. et al., “On stability and performances of fractional order systems”, 3rd IFAC Symposium FDA’08, Ankara, Turkey, 5–7 November 2008. [SAB 08b] SABATIER J. et al., “On a representation of fractional order systems: interests for the initial condition problem”, 3rd IFAC Workshop FDA’08, Ankara, Turkey, 5–7 November 2008. [SAB 09] SABATIER J., MERVEILLAUT M., FENETEAU L. et al., “On observability of fractional order systems”, Proceedings of the ASME IDET-CIE Conferences, San Diego, California, 2009. [SAB 10a] SABATIER J., MERVEILLAUT M., MALTI R. et al., “How to impose physically coherent initial conditions to a fractional system”, Communications in Non Linear Science and Numerical Simulation, vol. 15, no. 5, pp. 1318–1326, 2010. [SAB 10b] SABATIER J., MOZE M., FARGES C., “LMI stability conditions for fractional order systems”, Computers and Mathematics with Applications, vol. 9, pp. 1594–1609, 2010. [SAB 12] SABATIER J., FARGES C., MERVEILLAUT M. et al., “On observability and pseudo state estimation of fractional order system”, European Journal of Control, vol. 18, no. 3, pp. 260–271, 2012. [SAB 13] SABATIER J., FARGES C., TRIGEASSOU J.C., “A stability test for non commensurate fractional order systems”, Systems and Control Letters, vol. 62, no. 9, pp. 739–746, 2013. [SAB 14] SABATIER J., FARGES C., TRIGEASSOU J.C., “Fractional systems state space description: some wrong ideas and proposed solutions”, Journal of Vibration and Control, vol. 20, no. 7, pp. 1076–1084, 2014. [SAB 16] SABATIER J., FARGES C., FADIGA L., “Approximation of a fractional order model by an integer order model: a new approach taking into account approximation error as an uncertainty”, Journal of Vibration and Control, vol. 22, no. 8, pp. 2069–2082, 2016. [SAD 10] SADATI S.J., BALEANU D., RANJBAR A. et al., “Mittag Leffler stability theorem for fractional nonlinear systems with delay”, Abstract and Applied Analysis, vol. 2010, ID 108651, 2010.

References

281

[SAM 93] SAMKO S.G., KILBAS A.A., MARITCHEV O.I., “Fractional integrals and derivatives. Theory and applications”, Translated from Russian, Gordon and Breach, 1993. [SCH 98] SCHWARTZ L., Méthodes mathématiques pour les sciences physiques, Ecole Polytechnique, Hermann, 1998. [SLO 91] SLOTINE J.J.E., LI W., Applied Nonlinear Control, Prentice Hall, 1991. [SPI 65] SPIEGEL M.R., Laplace Transforms, McGraw-Hill, 1965. [STR 15] STROGATZ S.H., Nonlinear Dynamics and Chaos, Westview Press, 2015. [TAR 16a] TARI M., MAAMRI N., TRIGEASSOU J.C., “Initial conditions and initialization of fractional systems”, ASME Journal of Computational and Nonlinear Dynamics, vol. 11, no. 4, 2016. [TAR 16b] TARI M., MAAMRI N., TRIGEASSOU J.C., Observability and Observation of Commensurate Order Fractional Systems, ICFDA’16, Serbia, 2016. [TAR 16c] TARI M., Etat distribué, pseudo-état, observation et initialisation des systèmes fractionnaires, PhD Thesis, University of Poitiers, France, 2016. [TEN 12] TENOUTIT M., MAAMRI N., TRIGEASSOU J.C., “Tuning of a new class of robust fractional-order proportional-integral-derivative controllers”, Journal of Systems and Control Engineering, vol. 226, no. 4, pp. 486–496, 2012. [TEN 13] TENOUTIT M., Définition et réglage de correcteurs robustes d'ordre fractionnaire, PhD Thesis, University of Poitiers, France, 2013. [TEN 14] TENREIRO MACHADO J., “Numerical analysis of the initial conditions of fractional systems”, Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 2935–2941, 2014. [THO 76] THOMSON W., “Mechanical integration of the general linear differential equation of any order with variable coefficients”, Proceedings of the Royal Society, vol. 24, no. 1876, pp. 271–275, 1976. [TRI 09a] TRICAUD C., CHEN Y.Q., “Solution of fractional order optimal control problems using SVD-based rational approximations”, Proceedings of the ACC’2009 Conference, St Louis, Missouri, 2009. [TRI 88] TRIGEASSOU J.C., Recherche de modèles expérimentaux assistée par ordinateur, Tec et Doc Lavoisier, 1988. [TRI 10a] TRICAUD C., CHEN Y.Q., “An approximated method for numerically solving fractional order optimal control problems of general form”, Computers and Mathematics with Applications, vol. 59, pp. 1644–1655, 2010. [TRI 99] TRIGEASSOU J.C., POINOT T., LIN J. et al., “Modeling and identification of a non integer order system”, ECC’99 European Control Conference, Karlsruhe, Germany, 1999. [TRI 01] TRIGEASSOU J.C., POINOT T., “Identification des systèmes à représentation continue. Application à l'estimation de paramètres physiques”, in LANDAU I.D., BESANÇON-VODA A. (eds), Identification des systèmes, Hermès, Paris, pp. 177–211, 2001.

282

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[TRI 09b] TRIGEASSOU J.C., MAAMRI N., “State-space modeling of fractional differential equations and the initial condition problem”, IEEE SSD'09, Djerba, Tunisia, 2009. [TRI 09c] TRIGEASSOU J.C., BENCHELLAL A., MAAMRI N. et al., “A frequency approach to the stability of Fractional Differential Equations”, Transactions on Systems, Signals and Devices, vol. 4, no. 1, pp. 1–26, 2009. [TRI 10b] TRIGEASSOU J.C., MAAMRI N., “The initial conditions of Riemman-Liouville and Caputo derivatives: an integrator interpretation”, FDA’2010 Conference, Badajoz, Spain, October 2010. [TRI 10c] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “The pseudo state space model of linear fractional differential systems”, FDA’2010 Conference, Badajoz, Spain, October 2010. [TRI 11a] TRIGEASSOU J.C, MAAMRI N., “Initial conditions and initialization of linear fractional differential equations”, Signal Processing, vol. 91, no. 3, pp. 427–436, 2011. [TRI 11b] TRIGEASSOU J.C, MAAMRI N., SABATIER J. et al., “A Lyapunov approach to the stability of fractional differentiel equations”, Signal Processing, vol. 91, no. 3, pp. 437–445, 2011. [TRI 11c] TRIGEASSOU J.C., OUSTALOUP A., “Fractional integration: a comparative analysis of fractional integrators”, IEEE SSD'11, Sousse, Tunisia, 2011. [TRI 11d] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Initialization of Riemann-Liouville and Caputo fractional derivatives”, Proceedings of IDETC/CIE FDTA’2011 Conference, Washington, DC, August 2011. [TRI 11e] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Automatic Initialization of the Caputo Fractional Derivative”, CDC-ECC 2011, Orlando, December 2011. [TRI 12a] TRIGEASSOU J.C., MAAMRI N., SABATIER J. et al., “Transients of fractional order integrator and derivatives”, Signal, Image and Video Processing, vol. 6, no. 3, pp. 359–372, 2012. [TRI 12b] TRIGEASSOU J.C., MAAMRI N., SABATIER J. et al., “State variables and transients of fractional order differential systems”, Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3117–3140, 2012. [TRI 12c] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “State variables, initial conditions and transients of fractional order derivatives and systems”, Plenary talk, FDA’12, Nanjing, China, 2012. [TRI 13a] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “The Caputo derivative and the infinite state approach”, 6th Workshop on Fractional Differentiation and its Applications, Grenoble, France, February 2013. [TRI 13b] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Lyapunov stability of linear fractional systems. Part 1: definition of fractional energy”, ASME IDETC-CIE Conference, Portland, Oregon, August 2013.

References

283

[TRI 13c] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Lyapunov stability of linear fractional systems. Part 2: derivation of a stability condition”, ASME IDETC-CIE Conference, Portland, Oregon, August 2013. [TRI 13d] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “The infinite state approach: origin and necessity”, Computers and Mathematics with Applications, vol. 66, pp. 892–907, 2013. [TRI 14] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Lyapunov stability of fractional order systems: the two derivatives case”, ICFDA’14, 23–25 June 2014. [TRI 15] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Analysis of the Caputo derivative and pseudo state representation with the infinite state approach”, in DAOU R.A.Z., MORGAN X. (eds), Fractional Calculus Theory, Nova Editors, 2015. [TRI 16a] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Lyapunov stability of non commensurate fractional order systems: an energy balance approach”, ASME Journal of Computational and Nonlinear Dynamics, vol. 11, no. 4, p. 041007, 2016. [TRI 16b] TRIGEASSOU J.C., MAAMRI N., OUSTALOUP A., “Lyapunov stability of commensurate fractional order systems: a physical interpretation”, ASME Journal of Computational and Nonlinear Dynamics, vol. 11, no. 5, p. 051007, 2016. [TUL 93] TULLEKEN H.J.A.F., “Grey-box modeling and identification using physical knowledge and Bayesian techniques”, Automatica, vol. 29, no. 2, pp. 285–308, 1993. [VAL 07] VALERIO D., DA COSTA J.S., “Tuning rules for fractional PIDs”, in SABATIER J., AGRAWAL O.P., TENREIRO MACHADO J.A. (eds), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. [VAL 08] VALERIO D., ORTIGUEIRA M.D., DA COSTA J.S., “Identifying a transfer function from a frequency response”, ASME Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, 2008. [VER 08] VERGARA V., ZACKER R., “Lyapunov functions and convergence to steady state for differential equations of fractional order”, Mathematische Zeitschrift, vol. 259, pp. 287–309, 2008. [VIC 13] VICTOR S., MALTI R., GARNIER H. et al., “Parameter and differentiation order estimation in fractional models”, Automatica, vol. 49, no. 4, pp. 926–935, 2013. [VIN 00] VINAGRE B.M., PODLUBNY I., DORCAK L. et al., On Fractional PI Controllers: A Frequency Domain Approach, IFAC Workshop, Terrasa, Spain, pp. 53–58, April 2000. [WAL 97] WALTER E., PRONZATO L., Identification of Parametric Models from Experimental Data, Springer, 1997. [WAN 09] WANG X.Y., SONG J.M., “Synchronization of the fractional order hyper-chaos Lorentz systems with activation feedback control”, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3351–3357, 2009. [WES 94] WESTERLUND S., EKSTAM L., “Capacitor theory”, IEEE Trans. on Dielectrics and Electrical Insulation, vol. 1, no. 5, pp. 826–839, 1994.

284

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

[WIB 71] WIBERG D.M., Schaum's Outline of Theory and Problems of State Space and Linear Systems, McGraw-Hill, New York, 1971. [YUA 02] YUAN L., AGRAWAL O.P., “A numerical scheme for dynamic systems containing fractional derivatives”, Journal of Vibration and Acoustics, vol. 124, pp. 321–324, 2002. [YUA 13] YUAN J., SHI B., JI W., “Adaptive sliding mode control of a novel class of fractional chaotic systems”, Advances in Mathematical Physics, vol. 2013, article ID 576709, 2013. [YUA 18] YUAN J., ZHANG Y., LIU J. et al., “Equivalence of initialized fractional integrals and the diffusive model”, ASME Journal of Computational and Nonlinear Dynamics, vol. 13, 2018. [ZAD 08] ZADEH L.A., DESOER C.A., Linear System Theory: The State Space Approach, Dover Publications Inc., 2008. [ZEM 65] ZEMANIAN A.H., Distribution Theory and Transform Analysis, McGraw Hill, 1965. [ZHA 18] ZHAO Y., WEI Y., CHEN Y. et al., “A new look at the fractional initial value problem: the aberration phenomenon”, ASME Journal of Computational and Nonlinear Dynamics, vol. 13, no. 12, 2018.

Index

A, B, C analog computer, 3–5, 8, 21, 22 black box model, 107–109, 116 Bode diagram, 27–32, 49–51 Bromwich contour, 130, 156, 157 Caputo derivative, 193, 195, 196, 198, 201, 202, 204, 206, 209, 211, 212, 214, 215, 217, 243, 244, 250, 253, 257, 262, 263 transients, 262 closed-loop representation, 182, 185 commensurate order model, 82, 88, 89, 91, 96, 97, 101–103 convolution equation, 133 D, E, F diffusive interface, 81–88, 90, 96–98, 101, 102 representation, 134, 159, 176, 178, 183 discrete impulse response (DIR), 70, 71, 74, 78 distributed exponential approach, 235 initial condition, 144, 147, 155 state variable, 159, 168, 172, 181 exponential function, 248, 260 flow diagram, 8, 9

fractional differential equation (FDE), 3, 17, 18, 20 differential system (FDS), 159, 160, 162, 165, 168, 173, 175, 180, 182 differentiation, 193, 194, 199 integrator, 3, 7, 16, 17, 20, 26, 28, 33, 39–41, 45, 47, 49, 129, 130, 134, 137, 138, 142, 144, 147, 149, 152, 153, 155, 156 modeling, 81 order, 3, 6, 7, 9, 15, 16 frequency discretization, 227, 229, 238 distributed model, 129, 130, 134, 138, 140, 147, 155 distribution, 48 synthesis, 28, 41 G, H, I geometric distribution, 28 heat equation, 83 transfer, 81–83, 102 Heaviside function, 245, 254, 258 identification, 107–109, 115, 117–120, 123, 125 implicit derivative, 195, 198–200, 205, 208, 212, 244, 257, 258, 262

Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

286

Analysis, Modeling and Stability of Fractional Order Differential Systems 1

impulse response, 12, 15 induction motor, 107–110, 113, 125 infinite RC line, 149 state approach, 66, 68, 78 initial conditions, 193–195, 197–199, 202, 204–210 initialization, 130, 156, 159, 172 internal state variable, 34, 37, 39 inverse Laplace transform, 129, 130, 142, 156, 159, 175, 177, 180, 183, 189 K, L, M knowledge model, 107, 108, 110, 112 Laplace transform, 7, 9, 13, 14 long memory phenomenon, 92 matrix exponential, 229, 236 Mittag-Leffler approach, 219, 221, 227, 235, 237 function, 72, 75, 78, 160, 168, 169, 183–187, 189 modal model, 37, 40, 41, 129, 136, 138 N, O, P nonlinear optimization, 119, 122, 123 numerical simulation, 56, 72, 244

open-loop representation, 160, 176 ordinary differential equation (ODE), 3–6, 8, 9, 21, 22 output error method (OE), 118, 120, 122 parameter estimation, 98 partial derivative equation (PDE), 81, 82 power function, 246, 259 R, S repeated integration, 10, 12 response forced, 162, 165, 179, 181, 219, 237, 238, 240 free, 162, 165, 167, 170–172, 179, 221, 223, 225, 234, 237, 238 rotor bar, 109–115 short memory principle, 56, 63, 65, 66, 74, 78 sine function, 244, 249, 261, 263 skin effect, 110, 111, 114 spatial Fourier transform, 149, 150, 155 state space model, 9, 20 static simulation error, 66

Other titles from

in Systems and Industrial Engineering – Robotics

2019 ANDRÉ Jean-Claude Industry 4.0: Paradoxes and Conflicts BRIFFAUT Jean-Pierre From Complexity in the Natural Sciences to Complexity in Operations Management Systems (Systems of Systems Complexity Set – Volume 1) BUDINGER Marc, HAZYUK Ion, COÏC Clément Multi-Physics Modeling of Technological Systems FLAUS Jean-Marie Cybersecurity of Industrial Systems KUMAR Kaushik, DAVIM Paulo J. Optimization for Engineering Problems VANDERHAEGEN Frédéric, MAAOUI Choubeila, SALLAK Mohamed, BERDJAG Denis Automation Challenges of Socio-technical Systems

2018 BERRAH Lamia, CLIVILLÉ Vincent, FOULLOY Laurent Industrial Objectives and Industrial Performance: Concepts and Fuzzy Handling GONZALEZ-FELIU Jesus Sustainable Urban Logistics: Planning and Evaluation GROUS Ammar Applied Mechanical Design LEROY Alain Production Availability and Reliability: Use in the Oil and Gas Industry MARÉ Jean-Charles Aerospace Actuators 3: European Commercial Aircraft and Tiltrotor Aircraft MAXA Jean-Aimé, BEN MAHMOUD Mohamed Slim, LARRIEU Nicolas Model-driven Development for Embedded Software: Application to Communications for Drone Swarm MBIHI Jean Analog Automation and Digital Feedback Control Techniques Advanced Techniques and Technology of Computer-Aided Feedback Control MORANA Joëlle Logistics SIMON Christophe, WEBER Philippe, SALLAK Mohamed Data Uncertainty and Important Measures (Systems Dependability Assessment Set – Volume 3) TANIGUCHI Eiichi, THOMPSON Russell G. City Logistics 1: New Opportunities and Challenges City Logistics 2: Modeling and Planning Initiatives City Logistics 3: Towards Sustainable and Liveable Cities

ZELM Martin, JAEKEL Frank-Walter, DOUMEINGTS Guy, WOLLSCHLAEGER Martin Enterprise Interoperability: Smart Services and Business Impact of Enterprise Interoperability

2017 ANDRÉ Jean-Claude From Additive Manufacturing to 3D/4D Printing 1: From Concepts to Achievements From Additive Manufacturing to 3D/4D Printing 2: Current Techniques, Improvements and their Limitations From Additive Manufacturing to 3D/4D Printing 3: Breakthrough Innovations: Programmable Material, 4D Printing and Bio-printing ARCHIMÈDE Bernard, VALLESPIR Bruno Enterprise Interoperability: INTEROP-PGSO Vision CAMMAN Christelle, FIORE Claude, LIVOLSI Laurent, QUERRO Pascal Supply Chain Management and Business Performance: The VASC Model FEYEL Philippe Robust Control, Optimization with Metaheuristics MARÉ Jean-Charles Aerospace Actuators 2: Signal-by-Wire and Power-by-Wire POPESCU Dumitru, AMIRA Gharbi, STEFANOIU Dan, BORNE Pierre Process Control Design for Industrial Applications RÉVEILLAC Jean-Michel Modeling and Simulation of Logistics Flows 1: Theory and Fundamentals Modeling and Simulation of Logistics Flows 2: Dashboards, Traffic Planning and Management Modeling and Simulation of Logistics Flows 3: Discrete and Continuous Flows in 2D/3D

2016 ANDRÉ Michel, SAMARAS Zissis Energy and Environment (Research for Innovative Transports Set - Volume 1) AUBRY Jean-François, BRINZEI Nicolae, MAZOUNI Mohammed-Habib Systems Dependability Assessment: Benefits of Petri Net Models (Systems Dependability Assessment Set - Volume 1) BLANQUART Corinne, CLAUSEN Uwe, JACOB Bernard Towards Innovative Freight and Logistics (Research for Innovative Transports Set - Volume 2) COHEN Simon, YANNIS George Traffic Management (Research for Innovative Transports Set - Volume 3) MARÉ Jean-Charles Aerospace Actuators 1: Needs, Reliability and Hydraulic Power Solutions REZG Nidhal, HAJEJ Zied, BOSCHIAN-CAMPANER Valerio Production and Maintenance Optimization Problems: Logistic Constraints and Leasing Warranty Services TORRENTI Jean-Michel, LA TORRE Francesca Materials and Infrastructures 1 (Research for Innovative Transports Set Volume 5A) Materials and Infrastructures 2 (Research for Innovative Transports Set Volume 5B) WEBER Philippe, SIMON Christophe Benefits of Bayesian Network Models (Systems Dependability Assessment Set – Volume 2) YANNIS George, COHEN Simon Traffic Safety (Research for Innovative Transports Set - Volume 4)

2015 AUBRY Jean-François, BRINZEI Nicolae Systems Dependability Assessment: Modeling with Graphs and Finite State Automata BOULANGER Jean-Louis CENELEC 50128 and IEC 62279 Standards BRIFFAUT Jean-Pierre E-Enabled Operations Management MISSIKOFF Michele, CANDUCCI Massimo, MAIDEN Neil Enterprise Innovation

2014 CHETTO Maryline Real-time Systems Scheduling Volume 1 – Fundamentals Volume 2 – Focuses DAVIM J. Paulo Machinability of Advanced Materials ESTAMPE Dominique Supply Chain Performance and Evaluation Models FAVRE Bernard Introduction to Sustainable Transports GAUTHIER Michaël, ANDREFF Nicolas, DOMBRE Etienne Intracorporeal Robotics: From Milliscale to Nanoscale MICOUIN Patrice Model Based Systems Engineering: Fundamentals and Methods MILLOT Patrick Designing HumanMachine Cooperation Systems NI Zhenjiang, PACORET Céline, BENOSMAN Ryad, RÉGNIER Stéphane Haptic Feedback Teleoperation of Optical Tweezers

OUSTALOUP Alain Diversity and Non-integer Differentiation for System Dynamics REZG Nidhal, DELLAGI Sofien, KHATAD Abdelhakim Joint Optimization of Maintenance and Production Policies STEFANOIU Dan, BORNE Pierre, POPESCU Dumitru, FILIP Florin Gh., EL KAMEL Abdelkader Optimization in Engineering Sciences: Metaheuristics, Stochastic Methods and Decision Support

2013 ALAZARD Daniel Reverse Engineering in Control Design ARIOUI Hichem, NEHAOUA Lamri Driving Simulation CHADLI Mohammed, COPPIER Hervé Command-control for Real-time Systems DAAFOUZ Jamal, TARBOURIECH Sophie, SIGALOTTI Mario Hybrid Systems with Constraints FEYEL Philippe Loop-shaping Robust Control FLAUS Jean-Marie Risk Analysis: Socio-technical and Industrial Systems FRIBOURG Laurent, SOULAT Romain Control of Switching Systems by Invariance Analysis: Application to Power Electronics GROSSARD Mathieu, REGNIER Stéphane, CHAILLET Nicolas Flexible Robotics: Applications to Multiscale Manipulations GRUNN Emmanuel, PHAM Anh Tuan Modeling of Complex Systems: Application to Aeronautical Dynamics

HABIB Maki K., DAVIM J. Paulo Interdisciplinary Mechatronics: Engineering Science and Research Development HAMMADI Slim, KSOURI Mekki Multimodal Transport Systems JARBOUI Bassem, SIARRY Patrick, TEGHEM Jacques Metaheuristics for Production Scheduling KIRILLOV Oleg N., PELINOVSKY Dmitry E. Nonlinear Physical Systems LE Vu Tuan Hieu, STOICA Cristina, ALAMO Teodoro, CAMACHO Eduardo F., DUMUR Didier Zonotopes: From Guaranteed State-estimation to Control MACHADO Carolina, DAVIM J. Paulo Management and Engineering Innovation MORANA Joëlle Sustainable Supply Chain Management SANDOU Guillaume Metaheuristic Optimization for the Design of Automatic Control Laws STOICAN Florin, OLARU Sorin Set-theoretic Fault Detection in Multisensor Systems

2012 AÏT-KADI Daoud, CHOUINARD Marc, MARCOTTE Suzanne, RIOPEL Diane Sustainable Reverse Logistics Network: Engineering and Management BORNE Pierre, POPESCU Dumitru, FILIP Florin G., STEFANOIU Dan Optimization in Engineering Sciences: Exact Methods CHADLI Mohammed, BORNE Pierre Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems DAVIM J. Paulo Lasers in Manufacturing

DECLERCK Philippe Discrete Event Systems in Dioid Algebra and Conventional Algebra DOUMIATI Moustapha, CHARARA Ali, VICTORINO Alessandro, LECHNER Daniel Vehicle Dynamics Estimation using Kalman Filtering: Experimental Validation GUERRERO José A, LOZANO Rogelio Flight Formation Control HAMMADI Slim, KSOURI Mekki Advanced Mobility and Transport Engineering MAILLARD Pierre Competitive Quality Strategies MATTA Nada, VANDENBOOMGAERDE Yves, ARLAT Jean Supervision and Safety of Complex Systems POLER Raul et al. Intelligent Non-hierarchical Manufacturing Networks TROCCAZ Jocelyne Medical Robotics YALAOUI Alice, CHEHADE Hicham, YALAOUI Farouk, AMODEO Lionel Optimization of Logistics ZELM Martin et al. Enterprise Interoperability –I-EASA12 Proceedings

2011 CANTOT Pascal, LUZEAUX Dominique Simulation and Modeling of Systems of Systems DAVIM J. Paulo Mechatronics DAVIM J. Paulo Wood Machining

GROUS Ammar Applied Metrology for Manufacturing Engineering KOLSKI Christophe Human–Computer Interactions in Transport LUZEAUX Dominique, RUAULT Jean-René, WIPPLER Jean-Luc Complex Systems and Systems of Systems Engineering ZELM Martin, et al. Enterprise Interoperability: IWEI2011 Proceedings

2010 BOTTA-GENOULAZ Valérie, CAMPAGNE Jean-Pierre, LLERENA Daniel, PELLEGRIN Claude Supply Chain Performance / Collaboration, Alignement and Coordination BOURLÈS Henri, GODFREY K.C. Kwan Linear Systems BOURRIÈRES Jean-Paul Proceedings of CEISIE’09 CHAILLET Nicolas, REGNIER Stéphane Microrobotics for Micromanipulation DAVIM J. Paulo Sustainable Manufacturing GIORDANO Max, MATHIEU Luc, VILLENEUVE François Product Life-Cycle Management / Geometric Variations LOZANO Rogelio Unmanned Aerial Vehicles / Embedded Control LUZEAUX Dominique, RUAULT Jean-René Systems of Systems VILLENEUVE François, MATHIEU Luc Geometric Tolerancing of Products

2009 DIAZ Michel Petri Nets / Fundamental Models, Verification and Applications OZEL Tugrul, DAVIM J. Paulo Intelligent Machining PITRAT Jacques Artificial Beings

2008 ARTIGUES Christian, DEMASSEY Sophie, NERON Emmanuel Resources–Constrained Project Scheduling BILLAUT Jean-Charles, MOUKRIM Aziz, SANLAVILLE Eric Flexibility and Robustness in Scheduling DOCHAIN Denis Bioprocess Control LOPEZ Pierre, ROUBELLAT François Production Scheduling THIERRY Caroline, THOMAS André, BEL Gérard Supply Chain Simulation and Management

2007 DE LARMINAT

Philippe Analysis and Control of Linear Systems

DOMBRE Etienne, KHALIL Wisama Robot Manipulators LAMNABHI Françoise et al. Taming Heterogeneity and Complexity of Embedded Control LIMNIOS Nikolaos Fault Trees

2006 FRENCH COLLEGE OF METROLOGY Metrology in Industry NAJIM Kaddour Control of Continuous Linear Systems