Table Of Content

Analysis, Modeling and Stability of Fractional Order Differential Systems 2 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 2 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 John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK 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: 2019947403 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-455-1 Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Part 1. Initialization, State Observation and Control . . . . . . . . . . . . . . . 1 Chapter 1. Initialization of Fractional Order Systems . . . . . . . . . . . . . . 3 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Initialization of an integer order differential system . . . . . . . . . . . . . . . 4 1.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2. Response of a linear system . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3. Input/output solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4. State space solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.5. First-order system example . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3. Initialization of a fractional differential equation . . . . . . . . . . . . . . . . . 10 1.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2. Free response of a simple FDE . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4. Initialization of a fractional differential system . . . . . . . . . . . . . . . . . . 14 1.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2. State space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.3. Input/output formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5. Some initialization examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.2. Initialization of the fractional integrator . . . . . . . . . . . . . . . . . . . 17 1.5.3. Initialization of the Riemann–Liouville derivative . . . . . . . . . . . . . . 19 1.5.4. Initialization of an elementary FDS . . . . . . . . . . . . . . . . . . . . . . 21 1.5.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 vi Analysis, Modeling and Stability of Fractional Order Differential Systems 2 Chapter 2. Observability and Controllability of FDEs/FDSs . . . . . . . . . . 35 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2. A survey of classical approaches to the observability and controllability of fractional differential systems . . . . . . . . . . . . . . . . . . 37 2.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.2. Definition of observability and controllability . . . . . . . . . . . . . . . . 37 2.2.3. Observability and controllability criteria for a linear integer order system . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.4. Observability and controllability of FDS . . . . . . . . . . . . . . . . . . . 39 2.3. Pseudo-observability and pseudo-controllability of an FDS . . . . . . . . . . . 40 2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.2. Elementary approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.3. Cayley–Hamilton approach . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.4. Gramian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.5. Gilbert’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.3.7. Pseudo-controllability example . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4. Observability and controllability of the distributed state . . . . . . . . . . . . . 60 2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.2. Observability of the distributed state . . . . . . . . . . . . . . . . . . . . . 62 2.4.3. Controllability of the distributed state . . . . . . . . . . . . . . . . . . . . . 64 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 3. Improved Initialization of Fractional Order Systems . . . . . . . 67 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2. Initialization: problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3. Initialization with a fractional observer . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1. Fractional observer definition . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.3. Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.4. Numerical example 1: one-derivative system . . . . . . . . . . . . . . . . 76 3.3.5. Numerical example 2: non-commensurate order system . . . . . . . . . . 78 3.4. Improved initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4.2. Non-commensurate order principle . . . . . . . . . . . . . . . . . . . . . . 82 3.4.3. Gradient algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.4. One-derivative FDE example . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4.5. Two-derivative FDE example . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.3. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 A.3.1. Convergence of gradient algorithm . . . . . . . . . . . . . . . . . . . . . . 95 A.3.2. Stability and limit value of λ . . . . . . . . . . . . . . . . . . . . . . . . . 98 Contents vii Chapter 4. State Control of Fractional Differential Systems . . . . . . . . . . 99 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2. Pseudo-state control of an FDS . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.2. Numerical simulation example . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3. State control of the elementary FDE . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.2. State control of a fractional integrator . . . . . . . . . . . . . . . . . . . . . 104 4.4. State control of an FDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4.2. Principle of state control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.3. State control of two integrators in series . . . . . . . . . . . . . . . . . . . 124 4.4.4. Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4.5. State control of a two-derivative FDE . . . . . . . . . . . . . . . . . . . . . 129 4.4.6. Pseudo-state control of the two-derivative FDE . . . . . . . . . . . . . . . 130 4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Chapter 5. Fractional Model-based Control of the Diffusive RC Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2. Identification of the RC line using a fractional model . . . . . . . . . . . . . . 134 5.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.2. An identification algorithm dedicated to fractional models . . . . . . . . . 134 5.2.3. Simulation of the diffusive RC line . . . . . . . . . . . . . . . . . . . . . . 139 5.2.4. Experimental identification . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3. Reset of the RC line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.3.2. Natural relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.3.3. Principle of the reset technique . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3.4. Proposed reset procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3.5. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.6. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Part 2. Stability of Fractional Differential Equations and Systems . . . . . . 167 Chapter 6. Stability of Linear FDEs Using the Nyquist Criterion . . . . . . . 169 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2. Simulation and stability of fractional differential equations . . . . . . . . . . . 171 6.2.1. Simulation of an FDE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2.2. Stability of the simulation scheme . . . . . . . . . . . . . . . . . . . . . . . 172 6.2.3. Stability analysis of FDEs using the Nyquist criterion . . . . . . . . . . . 174 viii Analysis, Modeling and Stability of Fractional Order Differential Systems 2 6.3. Stability of ordinary differential equations . . . . . . . . . . . . . . . . . . . . 175 6.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.3.2. Open-loop transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.3.3. Drawing of H (jω) graph in the complex plane . . . . . . . . . . . . . . 177 OL 6.3.4. Stability of the third-order ODE . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.4. Stability analysis of FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.4.2. Drawing of H (jω) graph in the complex plane . . . . . . . . . . . . . . 182 OL 6.4.3. Stability of the one-derivative FDE . . . . . . . . . . . . . . . . . . . . . . 184 6.4.4. Stability of the two-derivative FDE . . . . . . . . . . . . . . . . . . . . . . 187 6.4.5. Stability of the N-derivative FDE . . . . . . . . . . . . . . . . . . . . . . . 194 6.4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.5. Stability analysis of ODEs with time delays. . . . . . . . . . . . . . . . . . . . 195 6.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.5.2. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.5.3. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.5.4. Application to an example . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.6. Stability analysis of FDEs with time delays . . . . . . . . . . . . . . . . . . . . 200 6.6.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.6.2. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.6.3. Application to an example . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter 7. Fractional Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.2. Pseudo-energy stored in a fractional integrator . . . . . . . . . . . . . . . . . . 206 7.3. Energy stored and dissipated in a fractional integrator . . . . . . . . . . . . . . 211 7.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.2. Electrical distributed network . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.3. Stored energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 7.3.4. Power dissipated in the fractional integrator . . . . . . . . . . . . . . . . . 215 7.3.5. Energy storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3.6. Integer order and fractional order integrators . . . . . . . . . . . . . . . . . 219 7.3.7. Characterization of fractional energy and its dissipation . . . . . . . . . . 226 7.3.8. Fractional energy invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.4. Closed-loop and open-loop fractional energies . . . . . . . . . . . . . . . . . . 234 7.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.4.2. Energy of the closed-loop model . . . . . . . . . . . . . . . . . . . . . . . 234 7.4.3. Energy of the open-loop model . . . . . . . . . . . . . . . . . . . . . . . . 237 7.4.4. Stored energies with a step input excitation . . . . . . . . . . . . . . . . . 239

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