Analysis, Modeling, and Stability of Fractional Order Differential Systems 2: The Infinite State Approach 9781786304551, 1786304554, 9781119686842, 1119686849

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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 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: 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 . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Initialization of an integer order differential system . . 1.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Response of a linear system . . . . . . . . . . . . . 1.2.3. Input/output solution . . . . . . . . . . . . . . . . . 1.2.4. State space solution . . . . . . . . . . . . . . . . . . 1.2.5. First-order system example . . . . . . . . . . . . . 1.3. Initialization of a fractional differential equation . . . . 1.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Free response of a simple FDE . . . . . . . . . . . 1.4. Initialization of a fractional differential system . . . . . 1.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.4.2. State space representation . . . . . . . . . . . . . . 1.4.3. Input/output formulation . . . . . . . . . . . . . . . 1.5. Some initialization examples . . . . . . . . . . . . . . . 1.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.5.2. Initialization of the fractional integrator . . . . . . 1.5.3. Initialization of the Riemann–Liouville derivative . 1.5.4. Initialization of an elementary FDS . . . . . . . . . 1.5.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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3 4 4 4 6 7 8 10 10 10 14 14 14 15 17 17 17 19 21 33

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Chapter 2. Observability and Controllability of FDEs/FDSs . . . . . . . . . . 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. A survey of classical approaches to the observability and controllability of fractional differential systems. . . . . . . . 2.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Definition of observability and controllability . . . . . . 2.2.3. Observability and controllability criteria for a linear integer order system . . . . . . . . . . . . . 2.2.4. Observability and controllability of FDS . . . . . . . . . 2.3. Pseudo-observability and pseudo-controllability of an FDS . 2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Elementary approach . . . . . . . . . . . . . . . . . . . . 2.3.3. Cayley–Hamilton approach . . . . . . . . . . . . . . . . 2.3.4. Gramian approach . . . . . . . . . . . . . . . . . . . . . 2.3.5. Gilbert’s approach . . . . . . . . . . . . . . . . . . . . . 2.3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7. Pseudo-controllability example . . . . . . . . . . . . . . 2.4. Observability and controllability of the distributed state . . . 2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Observability of the distributed state . . . . . . . . . . . 2.4.3. Controllability of the distributed state. . . . . . . . . . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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37 39 40 40 41 45 49 52 57 58 60 60 62 64 65

Chapter 3. Improved Initialization of Fractional Order Systems . . . . . . .

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3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Initialization: problem statement . . . . . . . . . . . . . . . . 3.3. Initialization with a fractional observer . . . . . . . . . . . . 3.3.1. Fractional observer definition . . . . . . . . . . . . . . . 3.3.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Convergence analysis . . . . . . . . . . . . . . . . . . . . 3.3.4. Numerical example 1: one-derivative system . . . . . . 3.3.5. Numerical example 2: non-commensurate order system 3.4. Improved initialization . . . . . . . . . . . . . . . . . . . . . 3.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Non-commensurate order principle . . . . . . . . . . . . 3.4.3. Gradient algorithm . . . . . . . . . . . . . . . . . . . . . 3.4.4. One-derivative FDE example . . . . . . . . . . . . . . . 3.4.5. Two-derivative FDE example . . . . . . . . . . . . . . . A.3. Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1. Convergence of gradient algorithm . . . . . . . . . . . . A.3.2. Stability and limit value of λ . . . . . . . . . . . . . . .

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67 68 71 71 72 74 76 78 81 81 82 84 87 91 95 95 98

Contents

Chapter 4. State Control of Fractional Differential Systems . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2. Pseudo-state control of an FDS . . . . . . . . . . . . 4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.2.2. Numerical simulation example . . . . . . . . . 4.3. State control of the elementary FDE . . . . . . . . . 4.3.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.3.2. State control of a fractional integrator. . . . . . 4.4. State control of an FDS . . . . . . . . . . . . . . . . 4.4.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.4.2. Principle of state control . . . . . . . . . . . . . 4.4.3. State control of two integrators in series . . . . 4.4.4. Numerical example . . . . . . . . . . . . . . . . 4.4.5. State control of a two-derivative FDE. . . . . . 4.4.6. Pseudo-state control of the two-derivative FDE 4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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99 100 100 101 103 103 104 121 121 122 124 126 129 130 131

Chapter 5. Fractional Model-based Control of the Diffusive RC Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Stability of Linear FDEs Using the Nyquist Criterion . . . . . . . . . . . .

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Part 2. Stability of Fractional Differential Equations and Systems . . . . . .

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133 134 134 134 139 149 154 154 155 156 158 159 164 165

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6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Simulation and stability of fractional differential equations . 6.2.1. Simulation of an FDE. . . . . . . . . . . . . . . . . . . . 6.2.2. Stability of the simulation scheme . . . . . . . . . . . . . 6.2.3. Stability analysis of FDEs using the Nyquist criterion .

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5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Identification of the RC line using a fractional model . . . . . . 5.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. An identification algorithm dedicated to fractional models . 5.2.3. Simulation of the diffusive RC line . . . . . . . . . . . . . . 5.2.4. Experimental identification . . . . . . . . . . . . . . . . . . 5.3. Reset of the RC line . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Natural relaxation . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Principle of the reset technique . . . . . . . . . . . . . . . . 5.3.4. Proposed reset procedure . . . . . . . . . . . . . . . . . . . . 5.3.5. Experimental results . . . . . . . . . . . . . . . . . . . . . . 5.3.6. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6.3. Stability of ordinary differential equations . . . . . . . 6.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Open-loop transfer function . . . . . . . . . . . . . 6.3.3. Drawing of H OL ( jω ) graph in the complex plane. 6.3.4. Stability of the third-order ODE . . . . . . . . . . . 6.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 6.4. Stability analysis of FDEs. . . . . . . . . . . . . . . . . 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 6.4.2. Drawing of H OL ( jω ) graph in the complex plane. 6.4.3. Stability of the one-derivative FDE . . . . . . . . . 6.4.4. Stability of the two-derivative FDE . . . . . . . . . 6.4.5. Stability of the N-derivative FDE . . . . . . . . . . 6.4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 6.5. Stability analysis of ODEs with time delays. . . . . . . 6.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6.5.3. Stability analysis . . . . . . . . . . . . . . . . . . . 6.5.4. Application to an example . . . . . . . . . . . . . . 6.6. Stability analysis of FDEs with time delays . . . . . . . 6.6.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6.6.2. Stability . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3. Application to an example . . . . . . . . . . . . . .

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175 175 176 177 178 182 182 182 182 184 187 194 195 195 195 196 196 198 200 200 201 202

Chapter 7. Fractional Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Pseudo-energy stored in a fractional integrator . . . . . . . . 7.3. Energy stored and dissipated in a fractional integrator . . . . 7.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Electrical distributed network . . . . . . . . . . . . . . . 7.3.3. Stored energy . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4. Power dissipated in the fractional integrator . . . . . . . 7.3.5. Energy storage . . . . . . . . . . . . . . . . . . . . . . . 7.3.6. Integer order and fractional order integrators . . . . . . . 7.3.7. Characterization of fractional energy and its dissipation 7.3.8. Fractional energy invariance . . . . . . . . . . . . . . . . 7.4. Closed-loop and open-loop fractional energies . . . . . . . . 7.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Energy of the closed-loop model . . . . . . . . . . . . . 7.4.3. Energy of the open-loop model . . . . . . . . . . . . . . 7.4.4. Stored energies with a step input excitation . . . . . . .

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Contents

Chapter 8. Lyapunov Stability of Commensurate Order Fractional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Lyapunov stability of a one-derivative FDE. . . . . . . . . 8.2.1. Problem statement . . . . . . . . . . . . . . . . . . . . 8.2.2. Numerical simulation . . . . . . . . . . . . . . . . . . . 8.2.3. Physical interpretation . . . . . . . . . . . . . . . . . . 8.2.4. Theoretical interpretation. . . . . . . . . . . . . . . . . 8.3. Lyapunov stability of an N-derivative FDE . . . . . . . . . 8.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. The integer order case . . . . . . . . . . . . . . . . . . 8.3.3. Lyapunov function of N-derivative systems . . . . . . 8.3.4. Stability condition . . . . . . . . . . . . . . . . . . . . 8.4. Lyapunov stability of a two-derivative commensurate order FDE . . . . . . . . . . . . . . . . . . . . . 8.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. State space model of the open-loop representation. . . 8.4.3. State space models of the closed-loop representation . 8.4.4. Energy and stability of the open-loop representation . 8.4.5. Energy and stability of the closed-loop representation 8.4.6. Definition of a stability test for a > 0 . . . . . . . . . 8.5. Lyapunov stability of an N-derivative FDE ( N > 2 ) . . . . 8.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Problem statement . . . . . . . . . . . . . . . . . . . . 8.5.3. LMI generalization for N = 3 . . . . . . . . . . . . . . 8.5.4. Application example . . . . . . . . . . . . . . . . . . . A.8. Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8.1. Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . A.8.2. Matignon’s criterion . . . . . . . . . . . . . . . . . . .

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269 269 270 271 272 274 276 281 281 282 283 289 290 290 291

Chapter 9. Lyapunov Stability of Non-commensurate Order Fractional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Stored energy, dissipation and energy balance in fractional electrical devices . . . . . . . . . . . . . . . . . . . 9.2.1. Usual capacitor and inductor devices . . . . . . . . . 9.2.2. Fractional capacitor and inductor . . . . . . . . . . . 9.2.3. Energy storage and dissipation in fractional devices . 9.2.4. Reversibility of energy and energy balance. . . . . . 9.3. The usual series RLC circuit . . . . . . . . . . . . . . . . 9.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Analysis of the series RLC circuit . . . . . . . . . . .

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

9.3.3. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. The series RLC* fractional circuit . . . . . . . . . . . . . . . . . . 9.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Analysis of the series RLC* circuit . . . . . . . . . . . . . . . 9.4.3. Experimental stability analysis . . . . . . . . . . . . . . . . . 9.4.4. Theoretical stability analysis . . . . . . . . . . . . . . . . . . . 9.4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. The series RLL*C* circuit . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Circuit modeling . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. The series RL*C* fractional circuit . . . . . . . . . . . . . . . . . 9.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2. Analysis of the series RL*C* circuit . . . . . . . . . . . . . . 9.6.3. Theoretical stability analysis . . . . . . . . . . . . . . . . . . . 9.7. Stability of a commensurate order FDE: energy balance approach 9.7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2. Analysis of the commensurate order FDE . . . . . . . . . . . 9.7.3. Application to stability . . . . . . . . . . . . . . . . . . . . . . 9.8. Stability of a commensurate order FDE: physical interpretation of the usual approach . . . . . . . . . . . . . . . . . . . . 9.8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2. Commensurate order system . . . . . . . . . . . . . . . . . . . 9.8.3. Lyapunov function of a fractional differential system . . . . . 9.8.4. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9. Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9.1. The infinite length LG line . . . . . . . . . . . . . . . . . . . A.9.2. Energy storage and dissipation in the fractional capacitor . . A.9.3. Some integrals . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

304 306 306 306 307 310 314 315 315 317 320 320 320 322 325 325 325 327

. . . . . . . . . .

. . . . . . . . . .

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. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

328 328 329 329 331 334 335 335 339 341

Chapter 10. An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems . . . . . . . . . . . . . . . . . . . . . . . . .

343

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Indirect Lyapunov method. . . . . . . . . . . . . . . . . . . . . . . 10.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3. Nonlinear system analysis . . . . . . . . . . . . . . . . . . . . 10.2.4. Local stability of a one-derivative nonlinear fractional system 10.3. Lyapunov direct method . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2. The variable gradient method. . . . . . . . . . . . . . . . . . . 10.3.3. Nonlinear system with one derivative . . . . . . . . . . . . . . 10.3.4. Nonlinear system with two fractional derivatives . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

343 344 344 344 345 349 353 353 353 354 357

Contents

10.4. The Van der Pol oscillator . . . . . . . . . . . . . . . 10.4.1. Electrical nonlinear system . . . . . . . . . . . . 10.4.2. Van der Pol oscillator . . . . . . . . . . . . . . . 10.4.3. Simulation of the nonlinear system . . . . . . . 10.4.4. Limit cycle . . . . . . . . . . . . . . . . . . . . . 10.5. Analysis of local stability . . . . . . . . . . . . . . . 10.5.1. Linearization . . . . . . . . . . . . . . . . . . . . 10.5.2. Local stability . . . . . . . . . . . . . . . . . . . 10.5.3. Validation of stability results . . . . . . . . . . . 10.6. Large signal analysis . . . . . . . . . . . . . . . . . . 10.6.1. Introduction . . . . . . . . . . . . . . . . . . . . 10.6.2. Approximation of the first harmonic [MUL 09] 10.6.3. Lyapunov function and oscillation frequency . . 10.6.4. Amplitude of the limit cycle . . . . . . . . . . . 10.6.5. Prediction of the limit cycle . . . . . . . . . . .

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

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xi

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

363 363 364 364 365 366 366 367 369 371 371 371 372 372 374

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

377

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

395

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 2

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 2

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 of Volume 1. 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 2

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 September 2019

PART 1

Initialization, State Observation and Control

Analysis, Modeling and Stability of Fractional Order Differential Systems 2: 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 Initialization of Fractional Order Systems

1.1. Introduction The initialization problem, or the initial value problem, is considered an elementary topic in the integer order case. On the contrary, for a long time it has been considered a very complex problem in the fractional order case; certainly because fractional derivatives, mainly Caputo’s derivative, have perpetuated a deep confusion about initial conditions (see Chapter 8 of Volume 1). The initialization of an ODE is an obvious problem. Initial conditions are well defined because this concept directly refers (or indirectly) to the system state X (t ) [KAI 80, CHE 84, ZAD 08]. In the fractional order case, the definition of system state has been confused for a long time because it was not possible to directly generalize the concepts of integer order state space to fractional systems. As demonstrated previously, the variable x(t ) , output of the fractional order integrator, is only a pseudo-state variable, unable to represent the true internal state of the fractional system (see Chapter 7 of Volume 1). Moreover, since the so-called initial conditions of Caputo’s derivative were considered as the value of the system state at t = 0 , it was impossible to predict the future system dynamics based on these pseudo-initial conditions, as in the integer order case (see Chapter 8 of Volume 1). Thus, the initialization of fractional systems seemed an inextricable problem. There are many publications related to the initialization of fractional systems; the reader can refer to [FUK 03, ORT 03, FUK 04, ORT 08, SAB 10a] and to the overview [TEN 14].

However, Lorenzo and Hartley, aware of these difficulties, proposed a methodology to avoid explicit reference to the system state [LOR 01, HAR 02, LOR 08a, LOR 08b, GAM 11]. This technique, known as the history function approach [HAR 09b, HAR 11], can be considered as an input/output method,

4

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

independent of the true internal system state. More recently, the infinite state approach has given a solution to the initialization problem in terms of the system state [TRI 09b, TRI 11a, TRI 11d, TRI 12a, TRI 12b, TRI 12c, TAR 16a, TAR 16c]. As stated previously, the pseudo state x(t ) is replaced by z (ω , t ) , which is the internal distributed state of any fractional order system; therefore, initialization can be treated as in the integer order case. Moreover, a collaboration between the two research teams exhibited the equivalence and the complementarity of the two approaches [HAR 13], which will be mentioned later. Some papers rely on these two approaches [DU 11, MAO 15, YUA 18, ZHA 18]. However, practical initialization remains a difficult problem. Despite the attractiveness of the history function approach, it only provides tractable solutions in a limited number of theoretical cases [HAR 11]. Similarly, although the infinite state approach makes it possible to use the same techniques as in the integer order case, such as state observers [TRI 11a], accurate estimation of the initial state remains a difficult topic, mainly caused by the infinite dimension of this state. Therefore, this section is only dedicated to the fundamentals of initialization; a practical solution will be proposed in Chapter 3. In fact, as demonstrated in Chapters 6, 7 and 8 of Volume 1, theoretical fractional system initialization can be considered as a solved problem. However, a comparative analysis of the two approaches is useful to extract the fundamentals of initialization of fractional systems. 1.2. Initialization of an integer order differential system 1.2.1. Introduction

There are two distinct problems: the initial values of an ordinary differential equation (known as the initial value problem) [KOR 68] and the response of this ODE to any excitation u (t ) that takes into account initial conditions at t = t0 , summarizing the past behavior of the system. As the solution of the first problem is straightforward, only the second problem will be discussed. 1.2.2. Response of a linear system

Consider the linear system characterized by the transfer function H ( s ) = L {h ( t )} (where h ( t ) is the impulse response) with input u ( t ) and output

y ( t ) . The model H ( s ) corresponds to an input/output representation, i.e. y(t ) = h ( t ) * u ( t ) or Y ( s) = H ( s ) U ( s )

[1.1]

Initialization of Fractional Order Systems

5

A state space model can be associated with this system  X (t ) = AX (t ) + B u (t )  y (t ) = C X (t ) 

dim ( X (t ) ) = N

[1.2]

where X ( t ) is the system state. Assume that we consider the behavior of this system since t = −a (the system is assumed to be initially at rest at t = −a ). The objective is to express the system response y ( t ) for t ≥ 0 without the explicit use of past behavior, corresponding to −a < t < 0 . According to the linear system theory, y ( t ) must take into account this past

behavior using initial conditions X ( 0 ) at t = 0 , thanks to a free response or an initialization function ψ ( t ) , and a forced response caused by the excitation u ( t ) for t > 0 [KAI 80]; therefore

y(t ) = y free ( t ) + y forced ( t ) = ψ ( t ) + y forced ( t ) for t ≥ 0

[1.3]

This problem can be summarized by the graphs presented in Figure 1.1. Let us call history functions (or past behavior, or pre-history) the graphs of u ( t )

and y ( t ) for −a < t < 0 .

Usually, the solution

y (t )

( t > 0 ) is expressed using a state space

representation, with the explicit use of X ( 0 ) . However, it is also possible to express y ( t ) with a convolution integral (i.e. with an input/output approach), without the explicit use of the initial state X ( 0 ) . This second solution is used by Lorenzo and Hartley in the history function approach, whereas the infinite state approach refers to a state space representation, i.e. to the first solution.

6

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 1.1. The initialization problem

1.2.3. Input/output solution

Let us express the convolution y(t ) = h ( t ) * u ( t ) since t = − a : t

y (t ) =  h ( t − τ ) u (τ ) dτ

[1.4]

−a

y ( t ) can be expressed as the sum of two integrals, from τ = −a to 0 and from τ = 0 to t . Therefore 0

t

−a

0

y (t ) =  h ( t − τ ) u (τ ) dτ +  h ( t − τ ) u (τ ) dτ

[1.5]

Initialization of Fractional Order Systems

7

Then, let us define 0

ψ (t ) =  h ( t − τ ) u (τ )dτ for t > 0

[1.6]

−a

and t

y forced (t ) =  h ( t − τ ) u (τ ) dτ

[1.7]

0

1.2.4. State space solution

Let X ( 0 ) be the system state at t = 0 , summarizing the past behavior from

τ = −a to τ = 0 .

Then, on the one hand X free (t ) = e At X (0)   At  y free (t ) = ψ (t ) = Ce X (0)

[1.8]

and on the other hand t  A( t −τ ) Bu (τ ) dτ  X forced (t ) =  e  0  t A( t −τ ) y ( t ) C = forced 0 e Bu (τ ) dτ  

[1.9]

REMARK 1.– The input/output solution, expressed with the impulse response, refers only to the observable and controllable part of the system state [KAI 80, ZAD 08]. On the contrary, the state space solution takes into account all state components; therefore, it represents the more complete solution. REMARK 2.– As X ( t ) = 0 for t ≤ −a , we can write t

X (t ) =  e −a

0

=

e

−a

A( t −τ )

A( t −τ )

B u (τ ) dτ t

Bu (τ ) dτ +  e 0

[1.10] A( t −τ )

B u (τ ) dτ

8

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Therefore 0

ψ (t ) = C  e A(t −τ ) Bu (τ ) dτ

[1.11]

−a

Using [1.10], we obtain 0

X (0) =

e

− Aτ

−a

Bu (τ ) dτ

[1.12]

i.e. 0

ψ (t ) = Ce At  e − Aτ Bu (τ ) dτ = Ce At X ( 0 )

[1.13]

−a

Thus, we can verify the fundamental equality 0

ψ (t ) =

 h ( t − τ ) u (τ ) dτ = Ce

At

−a

X (0)

[1.14]

1.2.5. First-order system example

Consider the first-order system H (s) =

X (s) 1 = s + α U (s)

y (t ) = x (t )

[1.15]

and its impulse response h ( t ) = L−1 { H ( s )} = e −α t H ( t )

[1.16]

Let us apply a step input U at instant t = −a . Therefore

u ( t ) = UH ( t + a )

[1.17]

Initialization of Fractional Order Systems

9

Then (see Figure 1.2) t

x(t ) =  e −α ( t −τ ) Udτ = Ue −α t −a

t

e

−α τ

dτ =

−a

U 1 − e −α (t + a )  t > −a α

[1.18]

Figure 1.2. Step response of a first-order system

Obviously x (0) =

U

 1− e α

−α a



[1.19]

Then, the free response is expressed as x free (t ) = e −α t x ( 0 ) = e −α t

U

 1− e α

−α a



[1.20]

We can also write 0

0

−a

−a

 1 − e −α a    α 

ψ (t ) =  e −α ( t −τ )Udτ = Ue −α t  eα τ dτ = Ue −α t 

[1.21]

10

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

i.e.

ψ (t ) = e−α t x ( 0 )

[1.22]

Thus, we verify again that the input/output and the state space representations are equivalent:

ψ (t ) = x free ( t )

[1.23]

1.3. Initialization of a fractional differential equation 1.3.1. Introduction

As mentioned in the introduction, the initialization of an FDE cannot be performed as in the integer order case, using the initial value of the pseudo-state variable X ( t ) . It has been proved that the knowledge of X ( 0 ) is unable to give information on the future behavior of the FDE (see Chapters 7 and 8 of Volume 1). In fact, the future behavior depends on the initial value of the internal distributed state variables zi (ω ,0 ) , with ∞

xi (0) =  μ ni (ω ) zi (ω , 0 ) d ω

[1.24]

0

Let us remember that an infinity of different dynamics (each one depending on a particular frequency distribution zi (ω ,0 ) ) can correspond to the same pseudo-initial value xi ( 0 ) [TAR 16a, TAR 16c]. As the general problem was treated previously (see Chapter 7 of Volume 1), we only present thereafter some simulation examples for a one-derivative FDE, in order to illustrate the dependence of dynamics on the initial distributed state z (ω,0 ) . 1.3.2. Free response of a simple FDE

Consider the elementary FDE D n ( x ( t ) ) + a0 x (t ) = 0 0 < n < 1

[1.25]

Initialization of Fractional Order Systems

11

The following distributed differential system is associated with this FDE:  ∂z (ω , t )  ∂t = − ω z (ω , t ) − a0 x(t )   ∞  x (t ) = μ n (ω ) z (ω , t ) d ω 0 

[1.26]

The distributed initial condition is z (ω,0 ) , which verifies the initial value ∞

x(0) =  μn (ω ) z (ω , 0 ) d ω

[1.27]

0

In order to emphasize the dependence of x ( t ) ( t > 0 ) on z (ω,0 ) , we consider a

numerical simulation of [1.25, 1.26] with three different distributions z (ω,0 ) corresponding to the same value x ( 0 ) [1.27].

Therefore, system [1.25] is frequency discretized (see Chapter 2 of Volume 1):  dz j (t ) = −ω j z j ( t ) − a0 x(t )   dt  J  x(t ) =  c j z j ( t )  j =0

[1.28]

The simulation parameters are

J = 20 ; ωb = 10 −3 rd / s ; ωh = 10 +3 rd / s ; Te = 10 −3 s

[1.29]

Each initial condition z (ω,0 ) corresponds to the discretized distribution

{ z (0)

 z j ( 0)  zJ ( 0) }

0

[1.30]

with the constraint J

x(0) =  c j z j ( 0 ) j =0

[1.31]

12

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Three distributions are considered: they are designed to correspond to the same initial value x(0) : 1) z j ( 0 ) = 0 for j = 1 to J , with z0 ( 0 ) =

2) z j ( 0 ) =

x ( 0)

[1.32]

c0

α

( j + 1)

2

1

J

x0 ( 0 ) = α  j =0

3) z j ( 0 ) =

α j +1 J

x (0) = α  j =0

with

( j +1) 2

[1.33]

with 1 ( j + 1)

[1.34]

The distribution [1.32] corresponds to the pseudo-initial condition of Caputo, requiring x ( t ) = cte = x ( 0 ) for t = −∞ to 0 (see Chapter 3 of Volume 1). Therefore, x ( 0 ) depends only on z0 ( 0 ) ( z j ( 0 ) = 0 ). It corresponds to the

slower decrease of x ( t ) , as observed in Figure 1.3.

Distributions [1.33] and [1.34] have components at high frequency. Their amplitude decrease with frequency; however, the high-frequency components of distribution [1.33] are relatively more important than those of distribution [1.34] (see Figure 1.4). Consequently, the decrease of x ( t ) with distribution [1.33] is faster than with distribution [1.34] (see Figure 1.3). Although the considered distributions are essentially theoretical, the main conclusion is that the corresponding transients characterized by the same initial value x ( 0 ) are completely different. Obviously, the knowledge of x ( 0 ) is of no

help to explain the different dynamics of x ( t ) , because these dynamics depend

essentially on z (ω,0 ) .

Initialization of Fractional Order Systems

Figure 1.3. Free responses with the same initial value. For a color version of the figures in this chapter see www.iste.co.uk/trigeassou/analysis2.zip

Figure 1.4. Comparison of distributions 2 and 3

13

14

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

1.4. Initialization of a fractional differential system 1.4.1. Introduction

The initialization of an FDS is closely related to the initialization of its integer order counterpart (see section 1.2). There are two approaches: one deals with the input/output formulation, and the other deals with the state space representation. However, the analysis of this problem is more complex, mainly caused by the duality between the pseudo-state variables and the distributed internal state variables. Moreover, in contrast with the integer order case, analytical solutions can be formulated only in a limited number of particular cases. The principles of the two approaches are presented later in a theoretical framework. Some examples are used to illustrate these two methodologies in the next section.

1.4.2. State space representation

The presentation is limited to the case of an N-derivative commensurate order FDS. Consider  D n ( X ( t ) ) = AX (t ) + B u ( t ) 0 < n < 1 and dim ( X ( t ) ) = N  y (t ) = C X (t ) 

[1.35]

A distributed integer order differential system is associated with [1.35]:  ∂ Z (ω , t ) = −ω Z (ω , t ) + A X (t ) + B u ( t )  ∂t   +∞  X (t ) =  μ n (ω ) Z (ω , t ) d ω  0

[1.36]

Let Z (ω, 0 ) be the distributed initial state at t = 0 . As demonstrated in Chapter 9 of Volume 1, the general response of [1.35, 1.36] to the excitation u ( t ) with the initial state Z (ω, 0 ) can be expressed using

Initialization of Fractional Order Systems

15

the Mittag-Leffler technique as X (t ) =

d En ,1 ( At n ) * X 0 ( t ) + En ,1 ( At n ) * B u ( t ) dt

{

}

[1.37]

where X 0 (t ) =

+∞

ω  μ (ω ) Z (ω , 0 ) e d ω − t

n

[1.38]

0

X 0 ( t ) represents the free responses of the N integrators

1 sn

associated

with [1.35, 1.36]. u (t ) is an equivalent input [MON 10] expressed as u (t ) = 0 Dt1− n ( u ( t ) )

[1.39]

En ,1 ( At n ) is the matrix Mittag-Leffler function.

The first term of [1.37] represents the free response or the initialization function of the pseudo-state X ( t ) . Thus, the initialization function of y ( t ) is expressed as

ψ (t ) = C

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

{

}

[1.40]

Obviously, it will be very difficult to analytically express ψ ( t ) in the general case. 1.4.3. Input/output formulation

Let H n ( s ) be the transfer function of [1.35]: H n (s) =

b0 + b1 s n +  + bM s Mn Y (s) B(s) M≤N = = N −1) n ( n Nn U ( s ) a0 + a1 s +  + aN −1 s A( s ) +s

[1.41]

16

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Let hn ( t ) be its impulse response: hn ( t ) = L−1 { H n ( s )}

[1.42]

Therefore, we can write

y ( t ) = hn ( t ) * u ( t )

[1.43]

As noted previously, the convolution integral is calculated since t = −a , and the system is assumed to be at rest for t < −a . Therefore t

y (t ) =

 h ( t − τ ) u (τ ) dτ

[1.44]

n

−a

which corresponds to y (t ) =

0

t

−a

0

 hn ( t − τ ) u (τ ) dτ +  hn ( t − τ ) u (τ ) dτ

[1.45]

Thus, the general expression of the initialization function is 0

ψ (t ) =

 h ( t − τ ) u (τ ) dτ n

for t > 0

[1.46]

−a

and the forced response is expressed as t

y forced (t ) =  hn ( t − τ ) u (τ ) dτ

[1.47]

0

The history function approach of Lorenzo and Hartley corresponds to the application of this input/output methodology [LOR 01, HAR 11]. The authors’ main objective was to express ψ (t ) using the Mittag-Leffler function and the incomplete Gamma function [HAR 07]. Moreover, they were forced to use simplifying assumptions, such as constant and ramp history functions, in order to formulate analytical expressions of ψ (t ) [HAR 09b, HAR 09c].

Initialization of Fractional Order Systems

17

1.5. Some initialization examples 1.5.1. Introduction

A comparative analysis of the two approaches was published in [HAR 13]. Two simple examples were used: initialization of the fractional integrator and the fractional differentiator. They are presented later. A third example that deals with an elementary FDS is also presented to illustrate the complexity of analytical fractional order initialization. 1.5.2. Initialization of the fractional integrator

Consider the elementary integrator X (s)

U (s)

=

1 sn

0 < n 0

z ( ω , t ) = z ( ω , 0 ) e −ω t

[1.54]

and x(t ) = ψ ( t ) =

+∞



μ n ( ω ) z ( ω , 0 ) e −ω t d ω =

sin ( nπ ) +∞ e − ω (t + a )

π

0

 0

ωn

dω

[1.55]

As (Appendix A.1. Chapter 1 Volume 1) +∞

x

α

e − β x dx =

0

Γ (α + 1)

[1.56]

β (α + 1)

we obtain

ψ (t ) =

sin ( nπ ) Γ (1 − n )

( t + a ) 1− n

π

=

( t + a ) n −1 Γ (n)

[1.57]

1.5.2.2. Input/output approach

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

[1.58]

Then

ψ (t ) =

( t − τ ) n −1  Γ ( n ) δ (τ + a ) dτ −a 0

[1.59]

Let

ψ (s) = L {ψ ( t ) } =

+∞

−st  e ψ ( t ) dt =

0

0

1 e −τ s Γ ( n, −τ s ) δ (τ + a ) dτ s n Γ ( n ) −a

[1.60]

Initialization of Fractional Order Systems

19

Therefore, we obtain [HAR 13]

ψ ( s) =

1 e a s Γ ( n, as ) sn Γ ( n)

[1.61]

where Γ ( n, as ) is the incomplete gamma function [HAR 07]. As

ψ ( s) =

eas Γ (ν + 1, as ) sν +1

= L{ (t + a ) ν }

[1.62]

We finally obtain

ψ (t ) =

as n −1 1 −1  e Γ ( n, as )  ( t + a ) L  =  Γ (n) Γ (n) sn  

[1.63]

REMARK 3.– We can directly obtain ψ ( t ) without using its Laplace transform

ψ ( s) .

Since +∞

 f ( μ ) δ ( μ + a ) d μ = f ( −a )

[1.64]

−∞

Using [1.59], we directly obtain

ψ (t ) =

( t + a ) n −1 Γ (n)

1.5.3. Initialization of the Riemann–Liouville derivative

The Riemann–Liouville derivative is defined by the relation x (t ) =

RL −a

Dtn ( u ( t ) ) =

d dt

{

−a

}

I t1− n ( u ( t ) )

[1.65]

As stated previously, we apply a Dirac impulse at t = −a , i.e. u ( t ) = δ ( t + a ) .

20

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

1.5.3.1. State space approach

Let zRL (ω , t ) be the distributed variable associated with ∂z RL (ω , t ) = −ω z RL (ω , t ) + δ ( t + a ) ∂t

1 : s1− n

[1.66]

Therefore zRL (ω , t ) = e

−ω ( t + a )

and zRL (ω,0 ) = e−ω a

[1.67]

Then, for t > 0

zRL (ω , t ) = zRL (ω , 0 ) e−ω t

[1.68]

Therefore x(t ) = ψ ( t ) =

sin ( (1 − n ) π ) d  +∞ e −ω (t + a )  dω   1− n π dt  0 ω 

[1.69]

As +∞

 0

e −ω (t + a )

ω

1− n

dω =

Γ (n)

(t + a ) n

[1.70]

we obtain

ψ (t ) =

sin ( (1 − n ) π ) d  Γ ( n )    π dt  ( t + a ) n 

[1.71]

ψ (t ) =

−n − n +1 (t + a ) ( ) Γ (1 − n )

[1.72]

thus

Initialization of Fractional Order Systems

21

1.5.3.2. Input/output approach

Let us define α = 1 − n and q = n ; Lorenzo and Hartley demonstrate [HAR 13] that sα s =  e −τ s Γ ( q, −τ s ) δ (τ + a ) dτ = q e a s Γ ( q, as ) Γ ( q ) −a s Γ(q) 0

ψ (s) =

[1.73]

Thus

ψ (t ) = L−1{ψ ( s )} =

1 d Γ (q ) dt

{(t + a ) ( ) }= Γ(1−−n n ) (t + a ) (

− n +1)

q −1

[1.74]

REMARK 4.– Again, we can directly express ψ ( t ) without the Laplace transform

ψ (s) . Using the definition of x ( t ) x (t ) = −RLa Dtn (δ ( t + a ) ) =

d dt

{

−a

}

I t1− n (δ ( t + a ) )

[1.75]

Then 0  d  1 d  1 −n  ( t − τ ) − nδ (τ + a ) dτ  =  ( t + a )    dt  Γ (1 − n ) − a  dt  Γ (1 − n )  −n − n +1 = (t + a ) ( ) Γ (1 − n )

ψ (t ) =

[1.76]

1.5.4. Initialization of an elementary FDS 1.5.4.1. Introduction

Consider the initialization of the elementary FDS at t = 0 : H n ( s) =

1 s +α n

[1.77]

excited by a step input U at t = −a , so u ( t ) = UH ( t + a ) ; this system is assumed to be at rest at t = −a .

22

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This example is used to illustrate the complexity of the initialization problem for a system less basic than the integrator or the differentiator. Two fractional system representations will be used: the closed-loop formulation and the open-loop formulation (see Chapter 7 of Volume 1). 1.5.4.2. Open-loop model formulation

Consider again H n (s) =

X (s) 1 = s + α U (s)

[1.78]

n

Using the inverse Laplace transform, we can express its impulse response hn (t ) as hn (t ) =

+∞

ω  μ α (ω ) e d ω − t

n,

[1.79]

0

with (see Chapter 7 of Volume 1)

μn ,α (ω ) =

sin ( nπ )

π

ωn

α + 2α cos ( nπ ) ω n + ω 2 n 2

[1.80]

The open-loop model is characterized by the distributed variable ξ (ω , t ) :  ∂ξ (ω , t ) = −ω ξ (ω , t ) + u (t )  ∂t   +∞  x (t ) = μ n ,α (ω ) ξ (ω , t ) d ω 0 

[1.81]

u (t ) = UH (t + a )

[1.82]

where

Our objective is to initialize this system at t = 0 using ξ (ω , t ) and ξ (ω , 0) .

Initialization of Fractional Order Systems

23

1) State space approach Then t

ξ (ω , t ) = e −ω t *UH ( t + a ) = U  e −ω (t −τ ) dτ

[1.83]

−a

and 0

0

−a

−a

ξ (ω , 0) = U  e −ω ( 0 −τ ) dτ = U  eωτ dτ

[1.84]

Therefore

ξ free (ω, t ) == e−ω t ξ (ω,0) for t > 0

[1.85]

and

ψ (t ) =

+∞

ω  μ α (ω ) ξ (ω , 0 ) e d ω − t

n,

[1.86]

0

2) Input/output approach

x ( t ) is defined as x (t ) =

t

 h ( t − τ ) u (τ ) dτ n

−a

t  +∞  = U    μn ,α (ω ) e−ω (t −τ ) d ω  dτ −a  0  ∞  t −ω (t −τ )  dτ  d ω = U  μn ,α (ω )   e 0  −a 

[1.87]

Therefore ∞



0

 −a

0



∞





0

 −a

0



ψ ( t ) = U  μn ,α (ω )   e−ω (t −τ ) dτ  d ω = U  μn ,α (ω ) e −ω t   eωτ dτ  d ω ∞

=  μn,α (ω ) ξ (ω , 0 ) e

−ω t

dω

0

Obviously, equation [1.88] exactly corresponds to [1.86].



[1.88]

24

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

1.5.4.3. Closed-loop model, Mittag-Leffler formulation

The system [1.78] corresponds to the elementary FDS: D n ( x ( t ) ) = −α x (t ) + u ( t ) 0 < n < 1

[1.89]

and to the closed-loop distributed model (recall that ξ (ω , t ) ≠ z (ω , t ) ):  ∂z (ω , t )  ∂t = −ω z (ω , t ) − α x(t ) + u ( t )   +∞  x(t ) =  μ n (ω ) z (ω , t ) d ω  0

[1.90]

1) Input/output approach Let us recall that if u ( t ) = U H ( t ) or u ( s ) = x (s) =

U 1 s sn + α

U , we can express s

[1.91]

As (see Chapter 3 of Volume 1)

α sn + α

=

sn + α − sn sn 1 = − sn + α sn + α

[1.92]

we can write x ( s ) ==

 U1 α U 1 sn = −  n n α s s + α α  s s ( s + α ) 

[1.93]

As   sn n L−1   = En ,1 ( −α t ) H ( t ) n s s ( + ) α  

[1.94]

we obtain x (t ) =

U

1 − En ,1 ( −α t )  H ( t ) α



n



[1.95]

Initialization of Fractional Order Systems

25

If we use a step input applied at t = −a , it is necessary to translate the time scale; thus x (t ) =

(

)

U n 1 − En ,1 −α ( t + a )  H ( t + a )  α

[1.96]

This result corresponds to x (t ) =

t

 h ( t − τ ) u (τ ) dτ

[1.97]

n

−a

and

ψ (t ) =

0

 h ( t − τ ) u (τ ) dτ

[1.98]

n

−a

Note that if we use the fictive excitation u ( t )

u ( t ) = UH ( t + a ) − UH ( t ) i.e. u ( t ) = 0 for t > 0

[1.99]

we can write

ψ (t ) =

0

 h ( t − τ ) u (τ ) dτ

[1.100]

n

−a

therefore

ψ (t ) =

t

t

 h ( t − τ ) U H (τ + a ) dτ −  h ( t − τ ) U H (τ ) dτ n

n

−a

[1.101]

0

Using [1.99], we obtain the expression of ψ ( t )

ψ (t ) =

(

)

U U n 1 − En ,1 −α ( t + a )  H ( t + a ) −  1 − En ,1 ( −α t n )  H ( t )   α α

[1.102]

i.e. for t > 0

ψ (t ) =

U

E ( −α t α  n ,1 

n

) − E ( − α ( t + a ) )  n

n ,1

[1.103]

26

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

We can conclude that the history function approach (input/output approach) provides an analytic formulation of the initialization function ψ ( t ) . 2) State space approach With u ( t ) = UH ( t ) and no initial condition, the Laplace transform of [1.90] gives

z (ω , s ) =

−α x ( s ) + s +ω

U s

[1.104]

As

x (s) =

U s (s + α )

[1.105]

n

we obtain z (ω , s ) =

U sn ( s + ω ) s (sn + α )

[1.106]

Therefore, as [1.94]   sn n L−1   = En ,1 ( −α t ) H (t ) n  s (s + α ) 

we obtain

{

z (ω , t ) = L−1 { z (ω , s )} = U e −ω t * En ,1 ( −α t n )

}

[1.107]

For u ( t ) = U H ( t + a ) (i.e. a step input at t = −a ), we can write t

z (ω , t ) = U  e −ω ( t −τ ) En ,1 ( −ατ n ) dτ −a

[1.108]

Initialization of Fractional Order Systems

27

If we want to initialize the system at t = 0 , we obtain 0

z (ω , 0 ) = U  e ωτ En ,1 ( −ατ n ) dτ

[1.109]

−a

Thus, the free response of [1.89] for t ≥ 0 , using z (ω,0 ) , is expressed as

ψ (t ) =

dx ( t ) d En,1 ( −α t n ) * x0 ( t ) = En ,1 ( −α t n ) * 0 dt dt

(

)

1 : sn

where x0 ( t ) is the free response of

x0 (t ) =

+∞

[1.110]

ω  μ (ω ) z (ω , 0 ) e d ω − t

n

[1.111]

0

Therefore t

ψ (t ) =  0

d ( x0 ( t − τ ) ) En,1 ( −ατ n ) dτ dt

[1.112]

An expression of ψ ( t ) can be formulated as +∞

t



0

0



ψ ( t ) = −  ω μ n (ω ) z (ω , 0 )   e −ω (t −τ ) En ,1 ( − ατ n ) dτ  d ω

[1.113]

where 0

z (ω , 0 ) = U  e ωτ En ,1 ( −ατ n ) dτ

[1.114]

−a

Obviously, it will be difficult to obtain an analytical formulation of ψ ( t ) in order to compare it to [1.103]. However, we can verify the identity of equations [1.103] and [1.113] that perform a numerical simulation.

28

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Therefore, we use the frequency discretized model of [1.90], i.e.  d z j (t ) = −ω j z j ( t ) − α x ( t ) + U H (t + a )   dt  J  x(t ) =  c j z j ( t )  j =0

[1.115]

The following simulation parameters are used:

J = 20 ; ωb = 10 −3 rd/s; ωh = 10 +3 rd/s; Te = 10 −3 s; a = 2.5 s ; U = 1 ; α = 1 The simulation of [1.115] provides z (ω,0 ) at t = 0 . Then, the initialization function ψ ( t ) is provided by the distributed model [1.90] initialized by z (ω,0 ) with u ( t ) = 0 :  d z j (t ) = −ω j z j ( t ) − αψ ( t )   dt  J  ψ (t ) =  c j z j ( t )  j =0

Figure 1.5. Comparison of initialization functions

[1.116]

Initialization of Fractional Order Systems

29

Figure 1.5 presents x ( t ) provided by equation [1.115], ψ ( t ) provided

by [1.116] and ψ ( t ) provided by [1.103].

Thus, we numerically verify that the two expressions of ψ ( t ) are identical. 1.5.4.4. Closed-loop model and distributed exponential formulation

In Chapter 9 of Volume 1, we expressed FDS transients using two approaches: the Mittag-Leffler technique and the distributed exponential formulation. Thus, we intend to express ψ ( t ) with this new formulation, in order to highlight its potentialities. Consider again the elementary FDS [1.89]: D n ( x ( t ) ) = −α x (t ) + u ( t ) 0 < n < 1

corresponding to the distributed differential system [1.90]. The step response of the distributed state z (ω , t ) with z (ω , − a ) = 0 ∀ ω and u (t ) = U H (t + a) is expressed as t ∞   z (ω , t ) =  exp  (t − τ )  f (ω , ξ ) d ξ U dτ   0 −a 0 ∞   =  exp  (t − τ )  f (ω , ξ ) d ξ U dτ 0 −a   t ∞  +  exp  (t − τ )  f (ω , ξ ) d ξ 0 0 

[1.117]

 U dτ 

Therefore, we can define

ψ z (ω , t ) = U

0



∞

−a



0

 exp (t − τ ) 

 f (ω , ξ ) dξ  dτ 

[1.118]

where

 ∞  exp t  f (ω , ξ ) dξ   0 

[1.119]

30

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

is the distributed exponential with

f (ω , ξ ) = − ω δ (ξ − ω ) − α μn (ξ )

[1.120]

(see the definitions of Chapter 9 of Volume 1). Finally ∞

ψ (t ) =  μ n (ω )ψ z (ω , t ) d ω

[1.121]

0

1) State space formulation Let us define ∞

g (ω , α ) =  f (ω , ξ ) d ξ

[1.122]

0

We can express t

z (ω , t ) = U exp { t g (ω , α )}  exp { − τ g (ω , α )} dτ −a

 −1  t = U exp { t g (ω , α )}    exp { −τ g (ω , α )} − a  g (ω , α ) 

[1.123]

Therefore

z (ω , t ) =

U ( exp { (t + a) g (ω, α )} −1) g (ω , α )

[1.124]

U ( exp { a g (ω, α )} −1) g (ω , α )

[1.125]

Note that

z (ω , 0) = Thus

ψ z (ω, t ) = exp { t g (ω, α )} z (ω,0)

[1.126]

Initialization of Fractional Order Systems

31

and ∞

ψ (t ) =  μ n (ω ) exp { t g (ω , α ) } z (ω , 0) d ω

[1.127]

0

REMARK 5.– If n =1 , then μ1 (ω ) = δ (ω ) ; therefore, f (ω , ξ ) = − α δ (ω ) . ∞

Then,



f (ω , ξ ) d ξ ) = − α = g (0, α ) and exp { t g (0, α ) } = e− α t .

0

Therefore, x (t ) = z (0, t ) = − and x (0) = z (0, 0) =

U

α

Thus, ψ (t ) = e

( 1− e

−α t

U

α

−α a

e − α t ( e − α a − 1)

).

x (0)

[1.128]

This result corresponds of course to [1.22]. 2) Input/output formulation As mentioned previously, using the fictive input u (t ) [1.99]

u (t ) = U H (t + a) − U H (t ) we can write t

x(t ) =  hn (t − τ ) u (τ ) dτ

[1.129]

−a

where ∞

hn (t ) =  μ n (ω ) hn , z (ω , t ) d ω

[1.130]

hn, z (ω, t ) = exp { t g (ω, α ) } z (ω,0) = exp { t g (ω, α ) }

[1.131]

0

and

since z (ω , 0) =1 ∀ ω when system [1.89] is excited by a Dirac impulse δ (t ) .

32

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Then ∞

t

x(t ) = x1 (t ) − x2 (t ) =  μ n (ω )  exp {(t − τ ) g (ω , α )}UH (τ + a ) dτ d ω −a

0

∞

t

0

−a

[1.132]

−  μ n (ω )  exp {(t − τ ) g (ω , α )}UH (τ ) dτ d ω

ψ (t ) = x1 (t ) − x2 (t ) for t > 0

[1.133]

therefore ∞

ψ (t ) =  μ n (ω ) 0

U exp { t g (ω , α )}  exp { a g (ω , α )} − 1 d ω g (ω , α )

[1.134]

Since

z (ω , 0) =

U ( exp { a g (ω, α )} −1) g (ω , α )

[1.135]

we obtain ∞

ψ (t ) =  μ n (ω ) exp { t g (ω , α )} z (ω , 0) d ω

[1.136]

0

i.e. the same result [1.127] as obtained previously. REMARK 6.– Using the Mittag-Leffler formulation: x(t ) =

∞

U

( 1− E α

n ,1

(−α t n ) ) H (t ) = U  0

μ n (ω )  exp { t g (ω , α )} −1 d ω g (ω , α ) 

[1.137]

therefore ∞

En ,1 (−α t n ) = 1 + α  0

μ n (ω ) (1 − exp { t g (ω , α )}) dω g (ω , α )

[1.138]

Initialization of Fractional Order Systems

33

If n =1 , the right side of [1.138] is equal to ∞

δ (ω ) (1 − e−α t ) dω =1 −1 + e−α t − α 0

1+ α 

(

i.e. e−α t = En,1 (−α t n )

)

n =1

= E1,1 (−α t )

[1.139]

[1.140]

REMARK 7.– Using the distributed exponential formulation, we obtain results that are equivalent to those of the integer order case (section 1.2.5). This means that this formulation makes it possible to generalize the integer order case: this is obviously interesting to derive mathematical proofs. However, we have to keep in mind that the apparent simplicity of the distributed exponential formulation hides, in fact, major computational complexity (see Chapter 9 of Volume 1). 1.5.5. Conclusion

The previous comparative analysis has made it possible to formally verify the equivalence of the history function and infinite state approaches, each of which provides a solution to the initialization of linear fractional differential systems. However, these theoretical approaches cannot provide an analytical solution for systems more complex than the one-derivative FDS. Moreover, the theoretical assumptions of this comparative analysis are too restrictive, particularly the assumption that the system is at rest at instant t = −a is not realistic. In fact, this analysis has proved the influence of past behavior on initialization (or long memory phenomenon). Theoretically, we have to take into account all the dynamical history of the system, since t = − ∞. Practically, it will be necessary to truncate past dynamical behavior (or pre-history). Therefore, the system cannot be considered at rest at instant t = −a. Thus, the practical initialization based on an arbitrary truncated dynamical past behavior is studied in Chapter 3.

2 Observability and Controllability of FDEs/FDSs

2.1. Introduction In contrast with the initialization problem, the observability and controllability problems of a linear FDS are considered as solved by the fractional calculus community, at least in the commensurate order case. The authors of these works [MAT 96, BET 08, MON 10, BAL 13] have directly generalized the results related to integer order differential systems to the fractional case, assimilating the fractional order pseudo-state to the integer order state. As demonstrated previously in Chapter 8 of Volume 1, any proof based on this postulate is questionable. However, as it will be demonstrated later, the result is paradoxical: even with an erroneous proof, the conclusion is right, i.e. usual observability and controllability criteria apply to the commensurate order FDS. In fact, this paradox has a logical explanation, since integer order systems are a particular class of commensurate order fractional systems. The objective of this section is to provide proofs based on the internal distributed state z (ω , t ) , instead of the pseudo-state x (t ) . First, it is necessary to recall the different approaches used in the integer order case (Kalman [KAL 60], Gilbert [GIL 63], gramian [KAI 80, CHE 84, ZAD 08]) and their fractional order counterparts, based on the pseudo-state [MAT 96].

Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

36

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Fundamentally, all these techniques are based directly or indirectly on the concept of the system state X (t ) such as

 X (t ) = AX (t ) + B u (t ) dim ( X (t ) ) = N  y (t ) = C X (t ) 

[2.1]

The observability and controllability criteria are based on the properties of the triplet { A, B, C} , deduced from the fundamental equations: for an integer order system  X (t ) = e A t X (0) + e A t * B u (t )  y (t ) = C X (t ) 

[2.2]

and for a commensurate order FDS n  D ( X (t ) ) = AX (t ) + B u (t ) dim ( X (t ) ) = N 0 < n < 1  y (t ) = C X (t ) 

[2.3]

The different proofs are mainly based on the fundamental equations established in Chapter 9 of Volume 1, i.e. d dt

X (t ) =

{ E ( At ) } * X n

n ,1

0

( t ) + En,1 ( At n ) * B

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

}

u ( t )

+ En ,1 ( At ) * B u ( t )

[2.4]

n

where X 0 (t ) =

+∞

 μ (ω ) n

Z (ω , 0 ) e −ω t d ω

[2.5]

0

and

u (t ) =

0

Dt1− n ( u ( t ) )

[2.6]

Observability and Controllability of FDEs/FDSs

37

It will be necessary to make a distinction between pseudo-observability and pseudo-controllability related to the pseudo-state X (t ) and exact observability and controllability related to the infinite dimension internal state Z (ω , t ) [TAR 16b]. Finally, the frequency discretization of Z (ω , t ) will enable the definition of approximate observability and controllability conditions in finite time. 2.2. A survey of classical approaches to the observability and controllability of fractional differential systems 2.2.1. Introduction

The observability and controllability conditions for linear FDSs are directly deduced from those of their integer order counterparts. Therefore, it is useful to briefly recall the definitions and results of the integer order case. Then, the different techniques related to FDSs will be surveyed. 2.2.2. Definition of observability and controllability

Consider the integer order differential system [2.1], with dim ( X (t ) ) = N . Let X (0) be the initial state of this system at t = 0 . Consider a finite time interval t ∈  0, t f  . The system is observable [KAI 80] if the knowledge of y (t ) on this interval makes it possible to estimate its initial state X (0) . Let u (t ) be an input signal on the finite time interval  0, t f  . The system is controllable [KAI 80] if a signal u (t ) exists on this interval, making it possible to transfer the system state from any initial state X (0) to any final state X (t f ) .

2.2.3. Observability and controllability criteria for a linear integer order system

The observability and controllability concepts were defined in 1960 by Kalman [KAL 60], in order to study the optimal control of linear systems, such as [2.1].

38

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Fundamentally, they are based on equations [2.2]. This system is observable if its observability matrix

 C   CA       [ϑ ] =  CAi  dim [ϑ ] = N × N      N −1  CA 

[2.7]

has rank N . Similarly, this system is controllable if its controllability matrix

AB  Ai B  AN −1 B  dim [

C

C

[ ] =  B

]= N×N

[2.8]

has rank N . Different techniques can be used to prove these results. Let us briefly recall their principles, because they will be further analyzed to apply them to an FDS with the internal state Z (ω , t ) . For example, Kailath proposes two elementary techniques [WIB 71, KAI 80]. d N −1 y ( t ) , and Observability is based on the successive differentiation of y (t ) until dt N −1 controllability is based on an input u (t ) composed of a sum of Dirac impulses and their derivatives until

d N −1δ ( t ) dt N −1

.

Although these techniques are completely unrealistic in a practical context, they provide a simple way to establish conditions [2.7] and [2.8]. More realistic is the technique based on the Cayley–Hamilton theorem [KAI 80, ZAD 08], which is applied on the interval 0, t f  . A more complex approach is based on observability and controllability gramians [KAI 80, CHE 84, ZAD 08].

Observability and Controllability of FDEs/FDSs

39

Consider the observability gramian Wϑ ( t f ) =

T

tf

 (e ) Aτ

0

(

C C e A τ dτ dim Wϑ ( t f T

)) = N × N

[2.9]

The system [2.1] is observable on 0, t f  if the rank of matrix Wϑ ( t f to N .

)

is equal

Consider the controllability gramian tf

0

T

(e ) Aτ

T

(

dτ dim W ( t f C

C

W ( t f ) =  e A τ BB

)) = N × N

[2.10] C

The system [2.1] is controllable on 0, t f  if the rank of matrix W ( t f to N .

) is equal

Finally, it is important to remember the technique proposed by Gilbert [GIL 63], which is historically the natural complement of Kalman’s approach [KAL 60]. Gilbert’s approach is not based on equations [2.2]. It uses the algebraic properties of the triplet { A, B, C} and the diagonal representation of system [2.1] to demonstrate the conditions [2.7] and [2.8] without the explicit use of the system state X (t ) . 2.2.4. Observability and controllability of FDS

Consider the linear commensurate order FDS [2.3]. All the techniques used to analyze observability and controllability of this FDS are based on the assimilation of the pseudo-state X (t ) to the true state X (t ) of an integer order differential system. Consequently, the initial state of the FDS is assumed to be X (0) , whereas it is perfectly established that X (0) is unable to predict X (t ) (for t > 0 ) [TAR 16a]. Moreover, the exponential matrix e At is replaced by the matrix Mittag-Leffler

(

)

function En,1 At n [MON 10]. Therefore, equations [2.4] become  X ( t ) = En ,1 ( At n ) X ( 0 ) + En ,1 ( At n ) * B u ( t ) 0 < n 0 ! X (t ) is only the pseudo-state: in order to obtain X (t ) = 0 ∀ t > 0 , it is required that Z (ω , t ) = 0 ∀ t > 0 and ∀ ω , which is a more drastic condition! This state control problem is discussed in Chapter 4. 2.3.3. Cayley–Hamilton approach 2.3.3.1. Introduction

The previous approach is completely unrealistic from a practical point of view, either for observability or controllability. On the contrary, the Cayley– Hamilton approach allows a realistic application (at least theoretically) of the pseudo-observability and pseudo-controllability principles in a finite time domain 0, t f  . Let us first recall the Sylvester/Cayley–Hamilton theorem [GAN 66, KOR 68, DEN 69]. Let

A

be

a

{λ0 , λ1 , , λi , , λN −1}

square

matrix

( dim( A) = N × N )

whose

eigenvalues

are distinct. Then, a matrix function of A can be expressed

with a finite number of terms such as Ai ( i = 0 to N − 1 ). Therefore, we can write for the matrix Mittag-Leffler function [MON 10] N −1

En ,1 ( At n ) =  ai (t ) Ai

[2.38]

i =0

2.3.3.2. Pseudo-observability

The objective is to estimate the components of the initial pseudo-state

 +∞  X (0) =   μn (ω ) Z (ω , 0) e−ω t d ω  0 t =0

[2.39]

46

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

using observations of the output y (t ) on the finite interval  0, t f  with u (t ) = 0 . Then

y(t ) = C En,1 ( At n ) ∗ g 0 (t )

[2.40]

Using [2.38], we can write  N −1  y (t ) = C   ai (t ) Ai  ∗ g 0 (t ) = C  i =0 

t

N −1

0

i =0

  a (t − τ ) i

Ai g 0 (τ ) dτ

[2.41]

Let us define

ξ T (t − τ ) = [ a0 (t − τ ) 

ai (t − τ )  aN −1 (t − τ )]

[2.42]

CAi g 0 (τ )  CAN −1 g 0 (τ ) 

[2.43]

and

φ T (τ ) =  C g 0 (τ )  Therefore  C   CA       φ (τ ) =  i   CA     N −1  CA 

g 0 (τ ) = [ϑ ] g 0 (τ )

[2.44]

[ϑ ]

[2.45]

Consequently t

y (t ) =  ξ (t − τ ) 0

T

g 0 (τ ) dτ

The information on Z (ω , 0) , i.e. on X (0) , is included in the N components of g 0 (τ ) . In order to access the N components of X (0) , it is necessary to have N

Observability and Controllability of FDEs/FDSs

47

equations. We can obtain these N equations by multiplying y (t ) by ξ (t ) and integrating on  0, t f  [MON 10]. Therefore tf

tf

t

T  ξ ( t ) y ( t ) dt =  ξ ( t )   ξ ( t − τ ) 0

=

0

0

tf

 T  ξ (t ) ξ (t −τ ) 0 t

 0

[ϑ ]

g 0 (τ )

 g 0 (τ ) dτ  dt 

[ϑ ]  dτ  dt 

[2.46]

Note that tf

t  0  0 f ( t − τ ) h (τ ) dτ dt  =

tf



tf

f ( t ) dt

 h (t )

dt

[2.47]

T  ξ ( t ) y ( t ) dt =  ξ ( t ) ξ ( t ) dt [ϑ ]  g 0 ( t ) dt

[2.48]

0

0

Therefore tf

tf

tf

0

0

0

Let tf

M =  ξ (t ) ξ

T

(t )

dt

[2.49]

0

Note that tf

 g (t ) 0

tf

dt =

0

 0

d X 0 (t ) dt

dt = X 0 ( t f ) − X 0 ( 0 )

[2.50]

Therefore tf

 ξ (t ) y (t ) 0

dt = M

[ϑ ]

( X (t ) − X 0

f

0

( 0))

with X 0 ( 0 ) = X ( 0 )

[2.51]

48

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

and

(

)

X 0 ( t f ) − X 0 ( 0 ) = ( M [ϑ ])

−1

tf

 ξ (t ) y (t )

dt

[2.52]

0

Consequently, it is possible to estimate the N components of the pseudo-state X (0) using the observations of y (t ) on the finite time interval  0, t f  if the matrix [ϑ ] is invertible, i.e. if rank ([ϑ ]) = N . REMARK 3.– The estimation of X ( 0 ) using [2.52] is not a straightforward problem, but it is more realistic than the elementary approach using the successive fractional derivatives of y (t ) . 2.3.3.3. Pseudo-controllability

Our objective is to transfer the pseudo-state X (t ) from X (0) to X ( t f ) = 0 using the input u (t ) defined on  0, t f  . Using equation [2.4], we can write

X (t f ) = 0 =

{

tf

}

En,1 ( At n ) * g 0 ( t )

t =t f

(

+  En ,1 A ( t f − τ ) 0

n

)

B u (τ ) dτ

[2.53]

Using [2.38], we obtain

{

}

En,1 ( At n ) * g 0 ( t )

t =t f

N −1

tf

i =0

0

= − Ai B

 a (t i

f

− τ ) u (τ ) dτ

[2.54]

Let us define

ψ T ( t f ) =  ψ 0 ( t f )  ψ i ( t f )  ψ N −1 ( t f ) 

[2.55]

where tf

ψ j ( t f ) =  a j ( t f − τ ) u (τ ) dτ 0

[2.56]

Observability and Controllability of FDEs/FDSs

49

Thus, we obtain

{ E ( At ) * g n

n ,1

0

(t )

}

t =t f

=[

=  B  Ai B  AN −1 B  ψ ( t f

C

−

]

ψ (t f

)

[2.57]

)

The objective is to calculate the N components of ψ ( t f ) , i.e. u (t ) (equation [2.56]).

[ ]

is invertible, i.e. if rank ([

C

C

This is only possible if the matrix

]) = N .

REMARK 4.– Again, the calculation of u (t ) on  0, t f  using the N components of

ψ ( t f ) is not an easy task. However, it is more realistic than the elementary

approach where the input is a sum of Dirac impulses! 2.3.4. Gramian approach 2.3.4.1. Controllability gramian

The objective is to define the controllability gramian of fractional system [2.3] based on equations [2.4], [2.5] and [2.6]. This gramian will be used to define the input u (t ) on the time interval 0, t f  , allowing the transfer of the pseudo-state

X (t ) from X (0) to X ( t f ) .

Using [2.4], we can write tf

(

 En,1 A ( t f − τ ) 0

n

)B

tf

(

u (τ ) dτ = X ( t f ) −  En,1 A ( t f − τ ) 0

n

)g

0

(τ ) dτ

= Δ (t f

)

[2.58] Assume that u (t ) can be expressed as

(

T u (τ ) = B En ,1 A ( t f − τ )

n

)

T

η

[2.59]

50

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Then tf

(

tf

)

(

)

(

T  En,1 A ( t f − τ ) B u (τ ) dτ =  En,1 A ( t f − τ ) B B En,1 A ( t f − τ ) 0

n

0

n

) dτ η

n T

[2.60] Let us define the controllability gramian of X (t ) tf

(

WC ( t f ) =  En ,1 A ( t f − τ ) 0

n

(

)

B B T En ,1 A ( t f − τ )

n

)

T

dτ

[2.61]

Then, equation [2.58] can be expressed as

WC ( t f ) η = Δ ( t f

)

[2.62]

i.e.

η = Wc ( t f ) Δ ( t f ) −1

[2.63]

Consequently, the input u (t ) , on interval  0, t f  , is expressed as

(

T u (τ ) = B En ,1 A ( t f − τ )

)

n T

Wc ( t f

)

−1

u (t ) can be calculated only if Wc ( t f

(

rank WC ( t f

) )=N

tf  n  X ( t f ) −  En,1 A ( t f − τ )  0 

(

)

)

 g 0 (τ ) dτ    [2.64]

is invertible, i.e. if [2.65]

2.3.4.2. Observability gramian

The objective is to define the observability gramian of [2.3] based on equations [2.4], [2.5] and [2.6] for u (t ) = 0 on the time interval 0, t f  .

Observability and Controllability of FDEs/FDSs

51

We can write

y ( t ) = C En,1 ( At n ) ∗ g 0 ( t )

as the “energy” of y (t ) on the time interval 0, t f  .

2 0 →t f

y (t )

Let us define

[2.66]

Then 2

y (t )

tf

 ( C E ( At ) ∗ g ( t ) ) ( C E ( At ) ∗ g ( t ) ) n

=

T

n ,1

n

n ,1

0

0

tf

=  g 0 ( t ) ∗ En,1 ( At n ) T

T

0

0

dt

[2.67]

C T C En ,1 ( At n ) ∗ g 0 ( t ) dt

Thus, we can write using the convolution property [2.47] y (t )

2

tf

tf

=  g 0 ( t ) dt  En,1 ( At T

0

0

)

n T

C C En ,1 ( At T

tf

n

) dt  g ( t ) 0

0

dt

[2.68]

Recall that tf

tf

0

0

 g 0 ( t ) dt =



d X 0 (t ) dt

dt = X 0 ( t f ) − X 0 ( 0 )

[2.69]

Let us define the observability gramian as tf

(

Wϑ ( t f ) =  En,1 A ( t ) 0

)

n T

(

C C En ,1 A ( t ) T

n

)

dt

[2.70]

Then y (t )

2 0→t f

(

)

== X 0 ( t f ) − X 0 ( 0 ) Wϑ ( t f T

) ( X (t ) − X (0)) 0

0

f

This quadratic form is positive definite if the matrix Wϑ ( t f

)

[2.71]

is positive definite,

i.e. if

(

rank Wϑ ( t f

)) = N

[2.72]

52

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

REMARK 5.– A matrix M is positive definite if its N eigenvalues are strictly positive [GAN 66]:

λi > 0 for i = 1 to N Recall that [DEN 69] N

det ( M ) = ∏ λi i =1

Therefore, if one of the eigenvalues is equal to 0, then det( M ) = 0 and M is not invertible. Thus, a positive definite matrix M verifies the condition rank ( M ) = N . 2.3.5. Gilbert’s approach 2.3.5.1. Introduction

Gilbert proposed [GIL 63] a dual approach to Kalman’s one [KAL 60], in order to demonstrate the observability and controllability conditions. Gilbert’s approach is based on the algebraic properties of the triplet { A, B, C} . Moreover, it provides an interpretation of the observability and controllability conditions in terms of the modal representation of the differential system. 2.3.5.2. Pseudo-observability and pseudo-controllability

Consider

again system [2.3] of A are distinct.

{λ1 , , λi , , λN }

and

assume

that

the

N

eigenvalues

Let M be the matrix whose columns are the eigenvectors of A. Consider the transformation X = MW Recall that [KOR 68]

[2.73]

Observability and Controllability of FDEs/FDSs

 λ1        M −1 AM = Ad =  λ2       λN 

53

[2.74]

Then  D n (W (t ) ) = Ad W (t ) + β u (t ) dim (W (t ) ) = N  y (t ) = γ W (t ) 

[2.75]

with

β = M −1 B

and

γ = CM

[2.76]

The system [2.75] is the modal representation of system [2.3]. Therefore, they share the same properties. Observability and controllability conditions are obvious with the modal representation: – system [2.75] is observable if all the components of γ are different from 0; – system [2.75] is controllable if all the components of β are different from 0. REMARK 6.– Consider the non-commensurate order FDS  D n ( X (t ) ) = AX (t ) + B u (t ) dim ( X (t ) ) = N  y (t ) = C X (t )  where

n = [ n1  n i  nN ] 0 < ni ≤ 1 T

If we use transformation [2.73], we obtain D n ( X (t ) ) = D n ( MW (t ) ) ≠ MD n (W (t ) )

54

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This means that the non-commensurate order system does not possess a modal representation [TRI 10c]. Consequently, according to Gilbert’s approach, it is not possible to establish the observability and controllability conditions for a non-commensurate order system. 2.3.5.3. Observability condition

Consider the observability matrix [ϑ ] given in [2.7]. As γ = C M , we have

C = γ M −1 . Note that

Ai = ( M Ad M −1 ) = M Adi M −1 i

[2.77]

Then

CAi = C M Adi M −1 = γ M −1 M Adi M −1 = γ Adi M −1

[2.78]

Therefore  γ   γA  d      [ϑ ] =  γ A i  M −1   d     γ Ad N −1   

[2.79]

Moreover

γ Adi = [γ 1  γ i  γ N ]

= γ 1λ1

i

 γ i λi

i

λ1i 0       i   λi       i λN   0

i  γ N λN 

[2.80]

Observability and Controllability of FDEs/FDSs

 γi  γ1      i [ϑ ] =  γ 1λ1  γ i λi i      N −1 N −1  γ i λi γ 1λ1   1  1       i i =  λ1  λi         N −1 N −1  λi   λ1

γN

    i  γ N λ N  M −1     N −1   γ N λN  1  γ 1 0         M −1 λN i   γi          γ N  λN N −1   0 

55

[2.81]

Let us define  1    [V ] =  λ1i    N −1  λ1



1









 

i

 

N −1



λi

 λi



1    λN i      λN N −1 

[2.82]

and  γ1 0 γ 1  γ    2       and γ =  γi [γ ] =     γi      0  γ N  γ N −1

        

[2.83]

Therefore

[ϑ ] = [V ] [γ ] [V ]

M −1

is the Vandermonde matrix [GIL 63, KOR 68] verifying

[2.84]

56

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

det ([V ]) ≠ 0

[2.85]

Therefore, we can write det ([ϑ ]) = det ([V ]) det ([γ ]) det ( M )

−1

[2.86]

with N

det ([γ ]) = ∏ γ i

[2.87]

i =1

As [V ] and M −1 are invertible, det ([ϑ ]) = 0 if only one of the components of

γ is equal to 0. Thus, if γ verifies Gilbert’s condition, we have

rank ([ϑ ]) = N

[2.88]

Consider the controllability matrix

Ai B = ( M Ad M −1 )

i

C

2.3.5.4. Controllability condition

[ ]

given in [2.8] and the fact that

B = M Adi M −1 B = M Adi M −1 B = M Adi β

[2.89]

Therefore

C

[ ] =  β

i

Ad β  Ad β  Ad

N −1

β 

[2.90]

Let us define  β1  β  2   β =  βi    β N −1

 0   β1          βi  and [ β ] =          0  β N  

[2.91]

Observability and Controllability of FDEs/FDSs

57

We can write as previously using the Vandermonde matrix

C

[ ] = M [ β ] [V ]

[2.92]

As

C

det ([

]) = det ( M )

det ([ β ]) det ([V ])

[2.93]

and N

det ([ β ]) = ∏ β i

[2.94]

i =1

C

we obtain det ([

]) = 0

if only one of the components of β is equal to 0. Thus, if

β verifies Gilbert’s condition, we have

C

rank ([

])

= N.

2.3.6. Conclusion

The approaches based on the pseudo-state X (t ) or on the algebraic properties of the triplet { A, B, C} are equivalent for a commensurate order fractional system. They lead to the same pseudo-observability and pseudo-controllability conditions, i.e.

rank ([ϑ ]) = N

[2.95]

and

C

rank ([

]) = N

[2.96]

These properties are independent of the fractional order n . They are also right for n = 1 , which is a particular case of a commensurate order fractional differential system.

58

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

On the contrary, they do not apply to non-commensurate order fractional systems. To the best of our knowledge, it is not possible to demonstrate the pseudoobservability and pseudo-controllability conditions for these systems. 2.3.7. Pseudo-controllability example

Consider the two-derivative FDS:  D n ( X (t ) ) = AX (t ) + B u (t )  x1 (t )  X (t ) =    y (t ) = C X (t )  x2 ( t )  

[2.97]

 −2 1  1 A= B =   n = 0.5  1  α −4 

[2.98]

with

and

C

[ ] = [B

1 −1  AB ] =   1 α − 4 

[2.99]

thus

C

det ([

]) = α − 3

[2.100]

Therefore

C

det ([

]) = 0 for α = 3

[2.101]

According to Gilbert’s approach, with the modal model, one component of β is equal to 0 for α = 3 , i.e. one component of the system pseudo-state is not influenced by u (t ) . This means that it is not possible to distinguish the dynamics of x1 (t ) and

x2 (t ) for α = 3 . Consequently, we simulate the system responses x1 (t ) and x2 (t ) to the step input u (t ) = H (t ) with the following parameters: J = 20 Te = 10 −3 s ω h = 103 rd / s ωb = 10 −3 rd / s

[2.102]

Observability and Controllability of FDEs/FDSs

59

The graphs of x1 (t ) and x2 (t ) are presented in Figure 2.1 for α = 1 , α = 3 and α = 5.

Figure 2.1. Pseudo-controllability of the commensurate order case. For a color version of the figures in this chapter see, www.iste.co.uk/trigeassou/analysis2.zip

We can verify that the graphs of x1 (t ) and x2 (t ) are different for α = 1 and α = 5 . On the contrary, for α = 3 , these graphs coincide perfectly. Then, we consider the non-commensurate order FDS   D n1 ( x1 (t ) )    n2  = AX (t ) + Bu (t ) .   D ( x2 (t ) )   y (t ) = C X (t ) 

This system is simulated with the same step input; matrices A and B and parameters are given in the previous example. Obviously, for n1 = n2 = 0.5 , the graphs of x1 (t ) and x2 (t ) perfectly coincide for α = 3 .

60

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

For n1 = 0.55 and n2 = 0.45 , we experimentally determine the value of α corresponding to the closest graphs for x1 (t ) and x2 (t ) : α = 3.085 and J = 0.153, where J =  ( x1 (k ) − x2 (k )) 2 .

For n1 = 0.6 and n2 = 04 , we obtain α = 3.17 corresponding graphs are presented in Figure 2.2.

and

J = 0.603 . The

Figure 2.2. Pseudo-controllability of the non-commensurate order case

Thus, we can conclude that the pseudo-controllability of the pseudo-state can only be approximate for a non-commensurate order fractional system. 2.4. Observability and controllability of the distributed state 2.4.1. Introduction

Pseudo-observability and pseudo-controllability of the pseudo-state X (t ) have been investigated previously. In fact, the final objective is to characterize the observability and controllability of the internal distributed state, i.e. z (ω , t ) for the closed-loop form or ξ (ω , t ) for the open-loop form (see Chapter 7 of Volume 1).

Observability and Controllability of FDEs/FDSs

61

In Chapter 9 of Volume 1, we proposed an equation expressing the transients of system [2.3] based on the distributed state variable Z (ω , t ) :  ∞  Z (ω , t ) = exp t  F (ω , ξ )d ξ  Z (ω , 0)  0  t ∞   +  exp (t − τ )  F (ω , ξ )d ξ  B u (τ ) dτ ω ∈ [ 0, +∞[ 0 0  

[2.103]

for the closed-loop form. However, it is difficult to use [2.103], and the problem can be treated more simply. 1 associated with the pseudo-state xi (t ) , sn characterized by its internal distributed variable zi (ω , t ) : Consider each fractional integrator

 ∂zi (ω , t ) = −ω zi (ω , t ) + vi ( t )   ∂t  +∞  x (t ) = μ (ω ) z (ω , t ) d ω i i 0 n 

[2.104]

An obvious and necessary condition for the observability of zi (ω , t ) is that the pseudo-state xi (t ) is observable. Similarly, a necessary condition for controllability is that the input vi (t ) controls the output xi (t ) , i.e. the pseudo-state xi (t ) is controllable. 1 is composed of sn an infinity of variables zi (ω , t ) connected in parallel, excited by the same input vi (t ) , where the output xi (t ) is the weighted sum of all the variables zi (ω , t ) , the model of each integrator is a modal form, i.e. the distributed variable zi (ω , t ) is perfectly observable and controllable. Moreover, according to equation [2.104], as each integrator

Consequently, a necessary and sufficient condition for observability and controllability of Z (ω , t ) is that the pseudo-state X (t ) is pseudo-observable and pseudo-controllable. Thus, at first sight, observability and controllability of Z (ω , t ) is a solved problem.

62

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

In fact, as the integrator is an infinite dimensional system [CUR 95], it is necessary to provide a more precise definition of the observability and controllability of Z (ω , t ) . 2.4.2. Observability of the distributed state

Let us recall that any system is observable if its initial state z (ω , 0) can be estimated using observations of its output x (t ) on the finite time interval 0, t f  . 2.4.2.1. Approximate observability of the distributed state

Recall that the infinite dimensional distributed system [2.104] must be frequency discretized for practical use, i.e. the distributed variable z (ω , t ) should be replaced by the finite dimensional approximation:

Z d ( t ) =  z0 ( t ) z1 ( t )  z j ( t )  z J ( t ) 

T

[2.105]

corresponding to the state model  d Z d (t ) = AI Z d ( t ) + B I v ( t )  dim ( Z d ( t ) ) = J + 1  dt  x(t ) = C I Z d ( t ) 

[2.106]

0  0  c0  1  −ω  c  1 1    1        T B I =   AI =   CI =   −ω j c j    1             −ω J  1 0 cJ 

[2.107]

with

Therefore, for v(t ) = 0 , we can express x (t ) as x ( t ) = C I e AI t Z d ( 0 )

[2.108]

Observability and Controllability of FDEs/FDSs

63

Let us divide the observation interval 0, t f  into J elementary intervals Δt such as Δt =

tf

[2.109]

J

 t0 = 0 t j = j Δt for j = 1,..., j ,...J   t J = t f

[2.110]

and define

X d =  x ( 0 ) x ( t1 )  x ( t j )  x ( t J = t f

)

 

T

[2.111]

Therefore, according to [2.108], we can express X d as  C I e AI 0   A I t1  C I e     X d =  AI t j  Z d ( 0 ) = [ϑI ] Z d ( 0 )  Ce        Ce AI t f   

[2.112]

Independently of numerical problems, the matrix [ϑI ] is invertible; thus, it is possible to approximately estimate the initial state Z d (0) using J + 1 observations of x (t ) on the finite time interval 0, t f  . 2.4.2.2. Exact observability of the distributed state

Now, let us consider z (ω , 0) , where

z (ω , 0 ) = lim Z d ( 0 )

[2.113]

J →∞

ω J →∞ ω1 → 0

Practically, a minimum value Δtmin is imposed by numerical constraints. As t f = J Δtmin , t f has to be infinite since J → ∞ .

64

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This means that the exact observability of the distributed state z (ω , t ) requires an infinite time interval 0, t f  . 2.4.3. Controllability of the distributed state

Any system is controllable if it is possible to define an input v(t ) on a finite time interval 0, t f  , allowing the transfer of the system state z (ω , t ) from any initial state z (ω , 0) to any final state z (ω , t f ) . 2.4.3.1. Approximate controllability of the distributed state

Consider the frequency discretized model [2.106]. Then, we can express t

Z d ( t ) = e AI t Z d ( 0 ) +  e

AI ( t −τ )

B I v (τ ) dτ

0

[2.114]

Note that [2.114] is the finite dimension approximation of [2.104] for a simple integrator. If we consider that input v(t ) is composed of constant elementary inputs v j on the time interval Δt =

tf J

, we can write:

Z d ( j + 1) = F Z d ( j ) + G v ( j )

[2.115]

where Δt

F = e AI Δt and G =  e AI μ d μ B I

[2.116]

0

Then, we can express Z d ( j + 1) from Z d (0) using the different increments v j of v(t ) . Therefore

CI

Z d ( j + 1) = F J +1 Z d ( 0 ) + [

]

V

[2.117]

Observability and Controllability of FDEs/FDSs

65

where

V = v0

v1  v j  vJ 

T

[2.118]

is the control input. Thus, the control input V allowing the transfer of Z d (t ) from any initial state Z d (0) to any final state Z d ( J + 1) is expressed as

CI

V =[

]

−1

(

Z d ( j + 1) − F J +1 Z d ( 0 )

)

[2.119]

CI

As mentioned previously, independently of numerical problems (see Chapter 4), the matrix [ ] is invertible, and it is possible to define an input v(t ) on the finite time interval 0, t f  verifying the controllability requirements. Note that the practical control of the distributed state is analyzed in Chapter 4. 2.4.3.2. Exact controllability of the distributed state

Let us consider z (ω , t f ) defined as z (ω , t f ) = lim Z d ( J + 1)

[2.120]

J →∞

ω J →∞ ω1 → 0

As mentioned previously, consider the minimum time interval Δtmin : as t f = J Δtmin , t f has to be infinite since J → ∞ . This means that exact controllability of z (ω , t ) requires an infinite time interval

0, t f  . 2.5. Conclusion

Consider the commensurate order system  D n ( X (t ) ) = AX (t ) + B u (t ) dim ( X (t ) ) = N  y (t ) = C X (t ) 0 < n n

π 2

∀ i = 1, 2, , N

73

[3.15]

For a non-commensurate order system, the stability condition is more complex to derive. For this purpose, we can use the technique proposed in Chapter 6, which is derived from the Nyquist criterion [NYQ 32, TRI 09c]. For example, consider the two-derivative system:  0 A=  − a0

1  n  K   0  T b  B =   C =  0  n =  1  and K =  1   − a1  1   b1   n2  K2 

[3.16]

Let us define the transfer function H ( s ) =

1  D (s)

[3.17]

where

(

D ( s ) = det  s n  − [ A − KC ]

)

[3.18]

and  s n1 0   s n  =  n2  0 s 

[3.19]

D ( s ) is the well-known characteristic polynomial [KAI 80, ZAD 08] corresponding to the system [3.16, 3.17] which is of the form

D ( s ) = α 0 + α1s n1 + s n1 + n2 Thus, the estimation error will be stable if H ( s ) =

[3.20] 1 is stable, i.e. if α 0 > 0  D (s)

and α1 > 0 (according to the Nyquist stability criterion in Chapter 6).

74

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

3.3.3. Convergence analysis

Since ε x (t ) is only the pseudo-state error, it is necessary to associate a distributed state variable error ξi (ω , t ) with each component ε x ,i (t ) of ε x (t ) . Thus, equivalent to [3.12] and [3.14], and taking into account [3.11], the distributed state vector error ξ (ω , t ) verifies the following equation (note that

ξ (ω , t ) does not correspond to the open-loop distributed state in this section):  ∂ξ (ω , t ) = − ω ξ (ω , t ) + [ A − K C ] ε x (t )   ∂ t ∞  ε x (t ) = 0  μn (ω )  ξ (ω , t ) d ω  dim(ε x (t )) = N  

[3.21]

Using [3.14] and [3.21] leads to +∞

 0

 μ n (ω )  ξ (ω , t ) d ω = X (ω , t ) − Xˆ (ω , t ) ∀ ω ∈ [0, +∞[

[3.22]

or

ξ (ω , t ) = Z (ω , t ) − Zˆ (ω , t ) ∀ ω ∈ [0, +∞[

[3.23]

where ξ (ω , t ) is the state vector error. The observer starts at t = t p , and it is supposed to be at rest at this instant because we have no information on Z (ω , t p ) . Therefore

Zˆ (ω, t p ) = 0

∀ ω ∈ [0, +∞[

[3.24]

Thus, [3.23] leads to

ξ (ω , t p ) = Z (ω , t p ) ∀ω ∈ [0, +∞[

[3.25]

Improved Initialization of Fractional Order Systems

75

In order to simplify the notations, let us consider the reduced time

t = t − t p  ξ (ω , t p ) = ξ (ω , 0) ∀ω

[3.26]

Then, the Laplace transform of [3.21] leads to

(

ε x ( s) =  s n  − ( A − K C )

)

−1

 s n 

+∞

 0

 μ n (ω )  ξ (ω , 0)   dω s +ω

[3.27]

with

0

 s n1    s n  =      0

 s ni

      nN  s 

[3.28]

Therefore

ξ (ω , s ) =

(

 s n  − ( A − K C )

)

−1

ξ (ω , 0)  s n  s + ω

[3.29]

and

(

lim ξ (ω , t ) = lim sξ (ω , s ) = lim s  s n  − ( A − K C ) t →∞ s →0 s →0

)

−1

ξ (ω , 0)  s n  s +ω

[3.30]

This result means that

lim ξ (ω, t ) = 0 ∀ ω ∈ [0, +∞[ t →∞

[3.31]

Thus, we can conclude that all the components of Zˆ (ω , t ) converge to those of Z (ω , t ) as t → ∞ . Nevertheless, the dynamics of ξ (ω , t ) are imposed by 1/( s + ω ) (see [3.30]). Thus, the lower frequency modes of Zˆ (ω , t ) require an infinite time to converge. Consequently, the requirement for convergence ∀ ω is that t0 − t p → ∞ .

76

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

3.3.4. Numerical example 1: one-derivative system

Consider the transfer function H n ( s) =

b0 a0 = 1 ; b0 = 2 ; n = 0.5 s n + a0

[3.32]

Therefore, A = −a0 , B = 1 and C = b0 . At t = 0 , the system is at rest, i.e. z (ω , 0) = 0∀ω , x(0) = 0 and y (0) = 0 . The input of the system is a unity step on [ 0, t0 ] : 1 for t ∈ [ 0, t0 ] u (t ) =   0 for t > t0

[3.33]

The past history interval corresponds to t p = 0.5s and t0 = 2s . We simulate the system response y (t ) with

J = Ncel = 20

ωb = 0.001 rd / s ωh = 1000

rd / s

Te = 10−3 s

The observer starts at t = t p with K = 20 and zˆ(ω , t p ) = 0 ∀ω . At t = t0 , the initial state is z (ω , t0 ) , whereas the initial state of the observer is zˆ(ω , t0 ) . Then, at t = t0 , we switch off the observer (K = 0). Consequently, its response represents the free response of the system initialized with the estimated initial condition zˆ(ω , t0 ) , which we call the initialized response yinit (t ) . The input u (t ) , the theoretical system response y (t ) , the observer response

yˆ(t ) and the initialized response yinit (t ) are plotted on [0,3 t0 ] in Figure 3.1. We can note that the observer response yˆ(t ) is quickly close to y (t ) . However on [t0 ,3t0 ] , yinit (t ) is progressively different from y (t ) . The first conclusion is that the equality of the pseudo-states ( x (t ) and xˆ(t ) ) at t = t0 is not a guarantee for a good initialization.

Improved Initialization of Fractional Order Systems

77

Figure 3.1. Input u (t ) and outputs y (t ) , yˆ(t ) and yinit (t ) . For a color version of the figures in this chapter see www.iste.co.uk/trigeassou/analysis2.zip

The comparison between the components z j (t0 ) and zˆ j (t0 ) (for j = 0, , 20 ) provides the explanation of this difference (see Figure 3.2): the high frequency modes are correctly estimated, whereas there is a poor estimation of the very low frequency modes. Consequently, the initialization of the fractional system is poor for long time behaviour, as highlighted on [t0 ,3 t0 ] (Figure 3.1). REMARK 1.– The increase in the observer gain K shows that there is some improvement of the convergence of low frequency modes. Therefore, very large values of K (because stability is not theoretically affected) would theoretically improve convergence. In fact, a practical value K max is imposed by the numerical computation of the observer: note that the fractional integrator is approximated by a finite dimension model, and each mode is time discretized. Thus, observer stability has to respect a more restrictive condition K < K max . Consequently, there is no simple solution to the improvement of the convergence of low frequency modes.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 3.2. Comparison between modes z j (t0 ) and zˆ j (t0 ) for j = 0,1, 2, , 20

3.3.5. Numerical example 2: non-commensurate order system

Consider the transfer function H ( s) =

b0 + b1 s n1 a0 = 1 ; a1 = 2 ; b0 = b1 = 1 a0 + a1 s n1 + s n1 + n2

n1 = 0.5 ; n2 = 0.3 ;

[3.34]

corresponding to the observer canonical form [KAI 80]:  n1   K1   0 − a0  b0  T  0  A=  B =  b  C =  1  n =  n  and K =  K  1 − a     2  2 1  1

[3.35]

The values of the gain K have to be chosen to respect observer stability. Using the stability criterion of section 3.3.2, theoretical stability is ensured for K1 ≥ 0 and K2 ≥ 0 .

Improved Initialization of Fractional Order Systems

79

The simulation protocol is the same as previously with

J = 20

ωb = 10−4 rd / s ωh = 10+4

rd / s

Te = 10−4 s

t p = 0.4s t0 = 1s

[3.36]

The graphs of u ( t ) , y ( t ) and yˆ ( t ) for different values of K are presented in Figure 3.3. Figures 3.4 and 3.5 present the graphs of the corresponding modes of Zˆ 1 ( t0 ) and Zˆ 2 ( t0 ) . 10  Obviously, the best estimation of Z ( t0 ) is provided by K =   . Nevertheless, 10  the low frequency modes exhibit a poor convergence, as in the previous example.

Figure 3.3. u (t ) , y (t ) , yˆ(t ) and yinit (t )

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 3.4. Comparison between modes z1, j (t0 ) and zˆ1, j (t0 ) for j = 0,1, 2, , 20

Figure 3.5. Comparison between modes z2, j (t0 ) and zˆ2, j (t0 ) for j = 0,1, 2, , 20

Improved Initialization of Fractional Order Systems

81

3.4. Improved initialization 3.4.1. Introduction

The previous convergence analysis demonstrated that the lower frequency modes of Z (ω , t0 ) are not correctly estimated because they require an infinite history interval [ t p , t0 ] for complete convergence. Obviously, this requirement is not acceptable in practice. A straightforward approach to improve Z (ω , t0 ) estimation would be to artificially broaden the interval

[ t p , t0 ]

by repeated

forward/backward observations. This technique has been successfully used for the initialization of a PDE (see [RAM 10] and the references therein) where the direct model X (t ) = A X ( t ) is used to perform forward observation, whereas the backward model X (t ) = − AT X ( t ) is used to perform backward observation. Unfortunately, the application of this methodology to a fractional system is forbidden by numerical problems due to the fractional backward model [BOU 67]. This model is very sensitive to numerical errors that affect the high frequency modes. Apparently, this iterative procedure cannot be used with a fractional system. However, this approach demonstrates that the improvement of Z (ω , t0 ) estimation depends on the quality of the initial value Zˆ (ω , t ) of the observer. p

Thus, we propose a solution to estimate Z (ω , t p ) using a fixed history interval

[ t p , t0 ] and the free response of the system starting at t = t p . As demonstrated thereafter, this estimation makes it possible to initialize the observer and then to improve the estimation of Z (ω , t0 ) . The estimation of Z (ω , t p ) is based on a gradient approach using the open-loop responses X OL (t ) of the fractional integrators

1 (closed-loop representation). s ni

As demonstrated thereafter, X OL (t ) is deduced from the free response of the

system y free (t ) . Since y (t ) = y free (t ) + y forced (t ) , the free response y free (t ) is calculated from the knowledge of y (t ) on [ t p , t0 ] and on the simulation of

y forced (t ) based on the knowledge of u (t ) on [ t p , t0 ] and of the system parameters.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

3.4.2. Non-commensurate order principle

Let us consider the Laplace transform of the system response [3.3, 3.4] (see Chapter 7 of Volume 1): −1 −1  +∞ Z (ω , 0)  y ( s) = C   s n  − A  s n     μn (ω )  d ω  + C   s n  − A Bu ( s) [3.37] s +ω 0 

where the first term represents the free response of the system initialized by Z (ω , 0) , and the second term represents the forced response depending only on the input u (t ) . Let us define −1  +∞ Z (ω , 0) y free ( s ) = C   s n  − A  s n     μn (ω )  s +ω 0

 dω  

[3.38]

and  +∞ X OL ( s) =    μn (ω )  0

Z (ω , 0) s +ω

 dω  

[3.39]

Then

y free ( s ) = C   s n  − A 

−1

 s n 

X OL ( s )

[3.40]

Since we consider the free response initialized by Z (ω , t p ) , we obtain X OL (t ) =

+∞

 0

 μ n (ω )  e −ω ( t − tp ) Z (ω , t p ) d ω for t > t p

[3.41]

As noted previously, the simplification of notations is based on the reduced time variable t − t p = t ; therefore, Z (ω , t p ) = Z (ω, 0) . Then X OL (t ) =

+∞

 0

 μ n (ω )  e −ω t

Z (ω , 0)

d ω for t > 0

[3.42]

Improved Initialization of Fractional Order Systems

83

Practically, we use the frequency discretized model (see Chapter 2 of Volume 1); thus, we replace Z (ω , 0) (in fact, Z (ω , t p ) ) with

θ T =  Z 1 (0)T  Z i (0)T  Z N (0)T  dim ( Z i ( 0 ) ) = ( J i + 1) θ iT = Z i (0)T =  zi ,0 (0)...zi , j (0)...zi , J (0) 

[3.43]

and X OL (t ) with X OL (t ) = Φ (t ) θ

[3.44]

where C    Φ (t ) =     

1, I

e

( A 1, I t )

       t ) ( A e N ,I  

0

 C

i,I

e

( A i ,I t )

 0

C N ,I

[3.45]

Therefore, X OL (t ) is linear in the parameter vector θ . Note that in [3.45], Ai , I is diagonal; thus, the matrix Φ(t ) is easily computed using Ai , I and Ci , I . As demonstrated thereafter, we can compute the free response of the system y free (t ) for t ∈ [t p , t0 ] (see sections 3.4.3.1 and 3.4.3.2). Therefore, expressing X OL (t ) in terms of y free (t ) on the history interval, we can estimate θ as follows. Let us define T e(t ) = f (t ) − ϕ (t ) θˆ

[3.46]

where θˆ is an estimation of θ . f (t ) and ϕ (t ) are known functions on [t p , t0 ] (see the illustrative examples given in sections 3.4.3.1 and 3.4.3.2).

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Note that the least-squares technique [EYK 74, NOR 86], based on the quadratic criterion

t0

 e (t )

2

, requires a matrix inversion that cannot be performed since the

t =t p

eigenvalues of this matrix are distributed from 0 to + ∞ , due to the wide range of

ω j modes (see Chapter 4 for an analysis of this problem). Consequently, the gradient technique [RIC 71, LJU 87, TRI 88] is more appropriate as it does not require matrix inversion. 3.4.3. Gradient algorithm

Consider the quadratic criterion

J (t ) = e2 (t )

[3.47]

Let t − t p = k Te (with Te being the sampling time, k ∈ Ν ) and θˆ k be the estimation at t = k Te . Thus, e k = f − ϕ T θˆ k k k  2 = J e  k k

[3.48]

Using the online gradient algorithm [LJU 87], the new estimation θˆ k +1 at (k + 1) Te corresponds to

 dJ 

θˆk +1 = θˆ k − λ   λ > 0  dθˆ k

[3.49]

θˆ k +1 = θˆ k + 2λ ek ϕ k

[3.50]

or

The gradient algorithm presents two known drawbacks [RIC 71, TRI 88]: its stability depends on λ and it is highly sensitive to measurement noise, which imposes a low value of λ . On the contrary, in a deterministic context, convergence is relatively fast (with a high value of λ respecting λ < λmax ). Moreover, it provides a confident estimation of Z (ω , t p ) without performing matrix inversion [TAR 16c, MAA 17].

Improved Initialization of Fractional Order Systems

85

Practically, several sequences of the gradient algorithm are necessary (one sequence corresponding to t p → t0 ), initialized at the first sequence by θˆ = 0 (see Appendix A.3. for convergence and stability of the gradient algorithm). 3.4.3.1. Example 1

Consider the transfer function system H n (s) =

b0 a0 + s n

[3.51]

Thus, dim ( X OL ) = N = 1 . Relation [3.51] leads to a0

1 y free ( s)+ y free ( s ) = b0 X OL ( s ) sn

[3.52]

Let us define  y free ( s)  I n ( y free (t )) = L−1   n  s 

[3.53]

Then y free (t ) + a0 I n ( y free (t )) = b0 X OL (t )

[3.54]

and using [3.44] T

y free (t )+ a0 I n ( y free (t )) = b0 ϕ (t ) θ

[3.55]

Then, we can express T e(t ) = f (t ) − ϕ (t ) θˆ

[3.56]

 f (t ) = y free (t ) + a0 I n ( y free (t ))  T ( AI t ) ϕ (t ) = b0 C I e

[3.57]

with

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

I n ( y free (t ))

corresponds to the fractional integration of y free ( t ) , which can be

easily computed (see Chapter 2 of Volume 1). Then, using the gradient algorithm [3.42], we can estimate θ . 3.4.3.2. Example 2

Consider the transfer function system H n ( s) =

a0

b0 + b1 s n1 + a1 s n1 + s n1 + n2

[3.58]

Thus, dim ( X OL ) = N = 2 . The calculation of y free ( t ) and X OL ( t ) is based on the observer canonical form [KAI 80]:  0 − a0  b  n  A= B =  0  C = [ 0 1] n =  1   1 − a1  b 1   n2 

[3.59]

After simple calculations, relation [3.40] provides y free

(s)

 1 + 

a1 s n2

+

a

s

0 n1 + n2

  1  =  n2  s 

 1  

X OL ( s )

[3.60]

which leads, in the time domain, to

(

y free (t ) + a1 I n2 ( y free (t )) + a0 I n1 I  n2 ( y free (t )) = [ 0 1] X OL (t ) + [1 0 ] I ( X OL (t )) n2

)

[3.61]

Thus, from [3.44], we obtain

(

y free (t ) + a1 I n2 ( y free (t )) + a0 I n1 I n2 ( y free (t ))

(

)

= [ 0 1] Φ (t ) + [1 0 ] I (Φ (t )) θ n2

)

[3.62]

Improved Initialization of Fractional Order Systems

87

or

(

y free (t ) + a1 I n2 ( y free (t )) + a0 I n1 I n2 ( y free (t ))

)

[3.63]

=  I (Φ1 (t )) Φ 2 (t )  θ n2

where 0  C1, I e( A1,I t ) Φ (t ) Φ (t ) =  1 = Φ 2 (t )   0  0

0 C 2, I e

( A2 I t )

 θ 1   and θ =   θ 2  

[3.64]

θ 1 corresponds to the first fractional integrator (with order n1 ), and θ 2 corresponds to the second integrator (order n2 ). Then, we obtain

(

 f (t ) = y free (t ) + a1 I n2 ( y free (t )) + a0 I n1 I n2 ( y free (t ))   T n ϕ ( t ) =  I 2 (Φ1 (t )) Φ 2 (t ) 

(

)

[3.65]

)

The fractional integrals I n2 (.) and I n1 I n2 (.) are easily computed using the frequency discretized model of the fractional integrators (see Chapter 2 of Volume 1). Thus, θ can be estimated by the gradient algorithm [3.49]. T

Of course, the regressor ϕ (t ) is more complex than in the first example, but it does not introduce numerical difficulties. 3.4.4. One-derivative FDE example 3.4.4.1. Introduction

Let us consider the system [3.51]: H n (s) =

b0 a0 + s n

[3.66]

Since the direct observation of the system state does not provide a confident estimation zˆ(ω , t0 ) (see section 3.3.4), we use the previous gradient technique to estimate z (ω , t p ) .

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

In Figure 3.6, we plot the step response y (t ) starting at t = 0 and the corresponding free response y free (t ) starting at t p = 0.5 s. This simulation makes it possible to provide the required data

{ f k } of [3.55].

Figure 3.6. The different responses of the system on [ 0, t0 ]

3.4.4.2. Estimation tests of z (ω , t p )

The previous gradient algorithm is used to estimate z (ω , t p ) with λ = 0.0025 (which ensures algorithm stability). In Figure 3.7, we plot the frequency discretized components z j (t p ) and zˆ j (t p ) obtained after several sequences. After one sequence, the estimation is poor, particularly for the higher modes. Therefore, it is necessary to perform several sequences of the gradient algorithm to improve this estimation. We note an improvement for five sequences, and an important one at the low and medium frequencies for 10 sequences.

Improved Initialization of Fractional Order Systems

89

Figure 3.7. Gradient technique estimation of modes zˆ j (t p ) for different sequences

3.4.4.3. Estimation of z (ω , t0 )

The estimation zˆ(ω , t p ) obtained after 10 sequences is selected to initialize the observer, and we keep the same gain K = 20 as with the previous direct approach. In Figure 3.8, we plot z j (t0 ) (exact), zˆ j (t0 ) (direct) and zˆ j (t0 ) initialized by

zˆ j (t p ) (improved). The improvement of z (ω , t0 ) estimation is now significant, whereas zˆ(ω , t0 ) (direct) is far from z (ω , t0 ) at low frequencies, zˆ(ω , t0 ) (initialized by zˆ(ω , t p ) ) is excellent at all frequencies. Thus, the initialization of the one-derivative fractional order model is now excellent (Figure 3.9). As we cannot objectively appreciate the initialization improvement, we compute the difference between the true response y (t ) and its initialization yinit (t ) , i.e.

ε (t ) = y(t ) − yinit (t )

[3.67]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

In Figure 3.10, we plot the direct initialization error and the improved initialization error. Obviously, the proposed methodology provides an important improvement to the initialization problem.

Figure 3.8. Direct and improved estimation zˆ j (t0 ) for j = 0,1, 2, , 20

Figure 3.9. u (t ) , yˆ(t ) (direct), yˆ(t ) (improved) and yinit (t )

Improved Initialization of Fractional Order Systems

91

Figure 3.10. Comparison of initialization errors

3.4.5. Two-derivative FDE example

Let us consider again the non-commensurate example [3.58]: H ( s) =

b0 + b1 s n1 a0 + a1 s n1 + s n1 + n2

[3.68]

In Figure 3.11, we present the step response y (t ) starting at t = 0 and the corresponding forced response y forced (t ) and free response y free (t ) starting at

t p = 0.4s . This simulation makes it possible to provide the required data [3.61].

{ f k } of

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 3.11. The different responses of the system on [ 0, t0 ]

3.4.5.1. Estimation tests of z (ω , t p )

 Z 1 (ω , t p )   The previous gradient algorithm was used to estimate Z (ω , t p ) =   Z 2 (ω , t p )    with λ1 = 10 −04 and λ2 = 5.10 −05 (which ensure algorithm stability).

In Figures 3.12 and 3.13, we respectively plot z1, j (t p ) and z2, j (t p ) estimates for several sequences.

Figure 3.12. Gradient estimation of the modes zˆ1, j (t p ) for different sequences

Improved Initialization of Fractional Order Systems

93

Figure 3.13. Gradient estimation of the modes zˆ2, j (t p ) for different sequences

3.4.5.2. Estimation of Z (ω , t0 )

The observer is initialized with the estimation Zˆ (ω , t p ) obtained after 10  10 sequences; moreover, it operates with the gain K =   on the history interval 10 

t p , t0  . In Figures 3.14 and 3.15, we plot Z (t0 ) exact, Zˆ (t0 ) direct and Zˆ (t0 ) initialized by Zˆ j (t p ) respectively for Z 1 (t0 ) and Z 2 (t0 ) .

Figure 3.14. Direct and improved estimations zˆ1, j (t0 )

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 3.15. Direct and improved estimations zˆ2, j (t0 )

As shown in these figures, Zˆ (ω , t0 ) improved by the gradient technique is now closer to Z (ω , t0 ) exact, as in the previous example. Finally, in Figure 3.16, we compare the system response initialized by Zˆ (ω , t0 ) improved and by Zˆ (ω , t0 ) direct. As shown by the comparison of initialization errors in Figure 3.17, the improved initialized response fits very well with the exact system response, and there is no longer the difference caused by the low frequency modes that characterize the direct initialization.

Figure 3.16. u (t ) , y (t ) , yˆ(t ) (direct) and yˆ(t ) (improved)

Improved Initialization of Fractional Order Systems

95

Figure 3.17. Comparison of initialization errors

A.3. Appendix A.3.1. Convergence of gradient algorithm A.3.1.1. Asymptotic convergence

The gradient algorithm is expressed as

θ k +1 = θ k + 2 λ ek ϕ k T

where ek = f k − ϕ k θ k =

[3.69]

ϕ k T (θ − θ k ) .

Therefore

θ k +1 = θ k + 2 λ

ϕ k ϕ k T (θ − θ k )

[3.70]

Assume that the algorithm converges to a value θ k (k >> 1) , then θ k +1 = θ k . Therefore

2 λ

ϕ k ϕ k T (θ − θ k ) = 0

[3.71]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

T

Since ϕ k ϕ k ≠ 0 , the algorithm converges to

limθ k = θ

[3.72]

i →∞

Of course, this property is classical: the least-squares technique is characterized by lim θˆ = θ because the model is linear in the parameters [LJU 87]. i →∞

A.3.1.2. Convergence rate

In fact, convergence is not sufficient, and the convergence rate is more important in our case. First, consider a one-parameter θ j algorithm, with

e(t ) = (θ j − θ j (t )) c j e

−ω j t

As J (t ) = e2 (t ) , we obtain: dJ d θ j (t )

= −2(θ j − θ j (t )) c j

2

e

−2ω j t

[3.73]

Therefore

θ j , k +1 = θ j , k (1 − 2 λ c j 2 e

−2ω j kTe

2

) + 2λ c j e

−2ω j kTe

θj

[3.74]

This algorithm is stable if 2

1− 2 λ cj e

−2ω j kTe

2

i.e. if 0 < λ c j e

−2ω j kTe

First, consider the case Te ω1 .

, whereas Te > 1 ). Parameters characterized by Te close to

1

ωJ

(high frequency modes) converge

slowly, whereas low frequency modes converge quickly. A.3.2. Stability and limit value of λ

Equations [3.69, 3.70] can be written as

θ k +1 = A(k )θ k + B(k ) θ

[3.79]

Stability of [3.79] is conditioned by the eigenvalues μ j (k ) of A(k ) , with the requirement

μ j (k ) < 1

[3.80]

for N = 1 , we obtain 1 − 2 λ c 2 e

−2ω kTe

tc .

Figure 4.2. FDE pseudo-state control

State Control of Fractional Differential Systems

103

In order to explain this phenomenon, we present in Figure 4.3 the distributions of the internal states z1, j ( t ) and z2, j ( t ) at t = 0 and t = tc .

Figure 4.3. Distributions of FDE internal states

We note that the distributions z1, j (t ) and z 2, j (t ) are modified by the control input; however, we do not obtain z1, j (t c ) = 0 and z 2, j (t c ) = 0 for each value of j (i.e. ∀ ω ), which would be the necessary condition to obtain x1 (t ) = 0 and x2 (t ) = 0 for t > tc .

We can conclude that the true control of fractional system state requires the control of all the frequency modes of the distributed states z1 (ω , t ) and z 2 (ω , t ) . 4.3. State control of the elementary FDE 4.3.1. Introduction

In the previous section, we demonstrated that the state control of an FDE requires the control of z (ω , t ) , which is the distributed state of the closed-loop representation.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This problem was briefly treated in Chapter 2 for controllability objectives. Let us recall that it is not possible to control z (ω , t ) ∀ ω on a finite time interval. On

the contrary, it is possible to control a finite dimension approximation z (t ) on a finite time interval.

Therefore, our objective is to propose a methodology to control the frequency discretized modes z (t ) of an elementary FDE. We demonstrate that the main problem is the state control of the associated fractional integrator. 4.3.2. State control of a fractional integrator 4.3.2.1. Problem statement

Let us consider the fractional integrator

1

sn distributed state z (ω , t ) , excited by an input u (t ) .

, which is characterized by the

More precisely, we consider its finite dimension approximation

z j (t )

(see Chapter 2 of Volume 1) such as  dz j (t ) = −ω j z j (t ) + u (t )   dt  J  x(t ) =  c j z j (t )  j =0

[4.7]

The system [4.7] is in fact a large dimension integer order system (dim Z (t ) = J + 1 ):  d Z (t )  = AI Z (t ) + B I u (t )  dt  x(t ) = C I Z (t )

[

]

Z (t ) = z0 (t ) z1 (t )  z j (t )  z J (t ) T

State Control of Fractional Differential Systems

0  0 1   1  −ω 1       B I =   AI =   −ωj   1        1 − ω J  0

[

C I = c0 c1 ... c j ... c J

105

[4.8]

]

Now consider a time discretization of [4.8] with the sample period Tce . Then

Z k +1 = F Z k + Gu k

[4.9]

REMARK 1.– – j is the frequency index and k is the time index. – For simulation purposes, Te is the time increment, which has to be chosen very small in relation to ωh . On the contrary, the sample period Tce for state control has to be adapted to the time response of the system. – Therefore, in practice, Tce >> Te such as Tce = k cTe with kc >> 1

[ ] transfer of Z (t ) from an initial value Z (0) to any final state Z (t f ) .

Consider the standard problem: calculate u (t ) for t ∈ 0, t f , allowing the

Therefore, it is necessary to compute J + 1 values u j (t ) of the input, i.e.

[

u T = u0

u1  u j  u J

with t f = (J + 1)Tce

]

[4.10]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The different steps of the calculation are  Z (1) = F Z (0 ) + Gu0     Z ( j + 1) = F Z ( j ) + Gu j    Z (J + 1) = F Z (J ) + Gu J

k = 1, t k = Tce  k = j + 1, t k = ( j + 1)Tce 

[4.11]

k = J + 1, t k = (J + 1)Tce

Therefore

[

]

Z (J + 1) = F J +1 Z (0) + F T G  F j G  G u

[4.12]

which corresponds to

]u

[4.13]

C

where

C

Z (J + 1) = F J +1 Z (0) + [

[ ] is the controllability matrix (Chapter 2).

Thus, the control input u is the solution of

C

u =[

]−1 [ Z (J + 1) − F J +1 Z (0) ]

[4.14]

However, let us specify the composition of the matrix

C

The calculation of the ( J + 1 ) components of the finite dimension approximation of z (ω , t ) is not a new problem: fundamentally, it corresponds to the calculation of the input of an integer order differential system, which is a well-known problem.

[ ].

Therefore, recall that system [4.8] is decoupled. Thus, we can express F and G as 0 α 0  β0    β  α1    1       G=  F=  αj   β j           α J   0 β J 

[4.15]

State Control of Fractional Differential Systems

107

with α = e −ω j Tce j  α0 = 1  1 − α j for j = 1,2,...., J and   β 0 = Tce β j = ωj 

C

Therefore, the matrix

[ ] corresponds to

( )

J   α 0 β0   ] =  α j J β 0    α J β J  J

C

[

[4.16]

(α ) β j



0



( ) ( )

0



(α ) β  (α ) β j



j



j

j



J

C

Consider the components of

J

  β0      βj      β J 

[ ] ; note that frequencies ω j

[4.17]

vary from ω j = 0 to

ωj → ∞. If ω j → ∞ , then

α j ≈ e −∞ = 0 and β j ≈

1− 0

→0

ωj

[4.18]

On the contrary, if ω j → 0 , then

α j ≈ e − 0 = 1 and β j → Tce C

[ ] corresponds to

Concretely, the matrix

C

[

 Tce     ] =  α j J β0     0

( )

[4.19]



Tce





  

(α ) β j

j

 0

j

 Tce       βj      0 

[4.20]

108

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

C

Theoretically, the matrix [ ] is invertible since the system [4.8] is diagonal. Practically, this matrix is very ill-conditioned for a conventional numerical technique [NOU 91], due to the very wide spectrum of ω j values. We can conclude that the state control of a fractional integrator is not a conventional problem. It requires a specific approach, as in the case of the initialization problem. 4.3.2.2. Solving a linear system with a recursive least-squares algorithm

Consider the linear system Aθ = B dim(θ ) = N

[4.21]

The solution θ is equal to

θ = A −1 B

[4.22]

if A is invertible. A necessary condition is rank ( A) = N

[4.23]

However, this condition is not sufficient. Let λi ( i = 1 to N) be the eigenvalues of A . The conditioning of the matrix A , i.e. its ability to be inversed, is characterized by [MOK 97] cond ( A) =

λi, max

[4.24]

λi, min

A well-conditioned matrix is characterized by cond ( A) ≈ 1

[4.25]

A high value of cond ( A) indicates bad conditioning.

C

C

[ ] (equations [4.17] and [4.20]), we have cond ( A) → ∞ , i.e. it is impossible to compute [ ]−1 . In the case of

State Control of Fractional Differential Systems

109

C

There are different techniques [MAR 93, MOK 97] adapted to ill-conditioned matrices and linear systems; however, they do not correspond to the problem of [ ] inversion.

C

Thus, we later propose an inversion technique based on the recursive least-squares method, avoiding [ ] inversion. Let θˆ be an estimation of θ , verifying [4.21]. Let us define the estimation error

ε = B − Aθˆ

[4.26]

and consider the quadratic criterion J = εT ε

[4.27]

The value θ opt minimizing J corresponds to  ∂J    =0  ∂θˆ   θˆ =θ opt

[4.28]

∂ε ∂J ∂ε T = −A =2 ε and ˆ ˆ ∂θ ∂θ ∂θˆ

[4.29]

As

we obtain

(

∂J = −2 AT B − Aθˆ ˆ ∂θ

)

[4.30]

As

(

)

 ∂J    = 0  − 2 AT B − Aθˆ opt = 0  ∂θˆ   θˆ =θ opt

[4.31]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Thus

( )

θˆ opt = AT A

−1 T

[4.32]

A B

REMARK 2.– Let us multiply [4.21] by AT . Therefore, we obtain an equivalent solution

(A A)θ = A T

T

( )

B i.e. θ = AT A

−1 T

A B

which is the optimal least-squares solution [4.32]. Thus, the least-squares solution [MEN 73, LJU 87] is also the solution of the linear system [4.21]. Nevertheless, solution [4.32] is not interesting because it also requires a matrix inversion. Recall that there exists a recursive solution to the identification of transfer function models called the recursive least-squares (RLS) algorithm [NOR 86]. Let yk = φ T θ k

[4.33]

be the linear in the parameter model (LP model). Let us define

ε k = yk* − yˆ k = yk* − φ Tk θˆ

[4.34]

and K

J K =  ε k2

[4.35]

k =1

where yk* is the measurement of yk , and yˆ k = φ T θˆ is its estimation based on θˆ k (for k = 1 to K ).

State Control of Fractional Differential Systems

111

The solution θ MC minimizing J K is given by [NOR 86] K  k =1

 

−1

K

θ MC ,K =   φ k φ Tk   φ k y k*

[4.36]

k =1

Then, for K measurements, let us define  K  PK =   φ k φ Tk   k =1  QK =

−1

[4.37]

K

*  φ k yk k =1

Therefore

θ MC , K = PK Q K

[4.38]

Obviously, for K − 1 measurements

θ MC , K −1 = PK −1 Q K −1

[4.39]

Then, replace K with k, i.e. we decide to compute θ MC at each instant k: the RLS algorithm provides the recursive estimation value θˆ based on the previous k

knowledge θˆ k −1 , Pk −1 , as follows [NOR 86, LJU 87]:

θˆ k = θˆ k −1 + K a ε k

[4.40]

k

where K a

k

is an adaptive gain and

 ε k = yk* − φ Tk θˆ k −1  −1  K a = Pk −1φ k 1 + φ Tk Pk −1φ k  k   Pk = Pk −1 − Pk −1φ k φ Tk Pk −1 1 + φ Tk Pk −1φ k  Pk = Pk −1 − K a φ Tk Pk −1  k

(

(

)

[4.41]

)

−1

Normally, the solutions [4.40] and [4.41] appear to be more complex than [4.32]. Note that φ T Pk −1φ k

calculation.

k

is a scalar; hence,

(1 + φ

)

−1 T P φ k k −1 k

is an elementary

112

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Consequently, with algorithms [4.40] and [4.41], we can recursively compute the solution θˆ with K iterations, avoiding matrix inversion required by [4.32]. K

However, the solution θˆ K depends on the initial estimation θˆ 0 and the matrix P0 : how should these initial values be chosen? Note that Pk is related to the covariance matrix of θˆ k , i.e. the confidence related to estimation θˆ [LJU 87]. k

A good confidence in θˆ k corresponds to small values of Pk . On the contrary, no confidence in θˆ is equivalent to large values of P . k

k

At the limit, if there is no information on θˆ 0 and P0 , we have to use θˆ 0 = 0 and P0 = α I with α >> 1 . Finally, note that K >> N corresponds to the case of recursive identification. However, at the limit, we can use K = N , where N = dim(θ ) : of course, this would be a very bad solution for an identification problem with noisy data! However, this is a way to compute the solution of an ill-conditioned problem. Before using [4.40] and [4.41] for the computation of [4.14], we propose to perform numerical simulations to test the influence of the different parameters on the solutions [4.40] and [4.41] when K = N . 4.3.2.3. A numerical example

Consider the elementary example

 e (−0.1t1 ) e (−t1 ) e (−10 t1 )    A = e (− 0.1t 2 ) e (− t 2 ) e (−10 t 2 )  e (− 0.1 t 3 ) e (− t 3 ) e (−10 t 3 )   

[4.42]

t1 = 10 , t 2 = 1 , t3 = 0.1

[4.43]

with

The matrix A is ill-conditioned since cond ( A) = 23.1 .

State Control of Fractional Differential Systems

113

We impose

θ = [1 1 1] T

[4.44]

Thus, the equality Aθ = B leads to

B = [0.3679 1.2728 2.2628] T

[4.45]

Then, using algorithms [4.40] and [4.41], we obtain the successive estimations ˆ θ 1 , θˆ 2 , θˆ 3 . In order to objectively judge the quality of θˆ 3 , we use the quadratic criterion 3

(

J =  bk − φ Tk θˆ 3 k =1

2

)

[4.46]

Therefore, with θˆ 0 = 0 and α = 103 , we obtain θˆ1 = [0.9928 0.0001 0.0000] T ˆ T θ 2 = [1.0101 0.9680 0.0001] with J = 3.61E − 05 θˆ = [0.9884 1.0377 0.9315] T  3

[4.47]

θˆ3 is close to θ , but there are remaining errors. Obviously, we can improve the estimations with a new recursion, initialized by ˆ θ 3 and P3 . Therefore, we obtain θˆ 4 = [0.9939 1.0205 0.9486] T ˆ T θ 5 = [0.9937 1.0205 0.9595] with J = 1.05E − 05 θˆ = [0.9936 1.0206 0.9629] T  6

[4.48]

There is an improvement, but the convergence is too slow. Another solution is to increase P0 , i.e. α .

114

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Let α = 1010 ; we obtain θˆ1 = [1.0001 0.0001 0.0000] T ˆ T θ 2 = [1.0001 1.0001 0.0001] with J = 4.98E − 19 θˆ = [1.0000 1.0000 1.0000] T  3

[4.49]

Thus, with a large value of α , only one recursion is required to obtain θˆ3 ≈ θ . 4.3.2.4. State control of the fractional integrator using the RLS algorithm

We have previously demonstrated that the control input u , which allows the

( )

transfer of the internal state Z from an initial value Z (0) to any final value Z t f , is given by equations [4.40] and [4.41].

( )

Thereafter, we are interested in the standard case Z t f = 0 with t f = (J + 1)Tce . Therefore, the control input u has to satisfy

C

− F J +1 Z (0) = [

]u

[4.50]

This equation can be decomposed into J + 1 lines. For line j , we can write

[

− α Jj +1 z j (0) = α Jj β j

α Jj −1β j  α Jj − j β j  β j

]

 u0  u   1    u j      u J 

[4.51]

which corresponds, with the RLS notation, to y j = φT u j

Therefore, we can recursively compute the control input u with the RLS algorithm, starting from u 0 = 0 with P0 = α I .

State Control of Fractional Differential Systems

115

Index j varies from 1 to J + 1 :

(

u j = u j −1 + K a y j − φ Tj u j −1 j

)

[4.52]

with  J +1  y j = −α j z j (0 )  T J J −1 J− j φ j = α j β j α j β j  α j β j  β j  T θ j = u0, j u1, j  u j , j  u J , j  T −1  K a j = Pj −1φ j 1 + φ j Pj −1φ j   Pj = Pj −1 − K a φ T Pj −1 j j 

[

[

]

)

(

] [4.53]

The quality of the estimation uˆ = u J +1 is characterized by the criterion

J z (α ) =

 (y j =1

)

2

J +1

j

− φ Tj uˆ

[4.54]

This criterion depends on α as uˆ depends on α . Moreover, we can characterize the achieved control input uˆ by its “energy” criterion J u (α ) =

 (u 2j Tce )

J +1

[4.55]

j =1

C

Let us recall that in contrast to the elementary example 4.3.2.3, it is impossible to know the “true” value of u because cond ([ ]) → ∞ . Thus, the criterion J z (α ) is the only way to appreciate the quality of estimation uˆ . 4.3.2.5. Computation of a control input example

Consider the fractional integrator

n = 0.5 and J = 30

1 sn

with [4.56]

116

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The integrator is simulated with the usual parameters

ωb = 10 −03 rd / s , ω h = 10 +03 rd / s and Te = 10 −03 s

[4.57]

The sample period for the control input u is Tce = kcTe with kc = 200

[4.58]

In order to use a realistic initial condition, we perform an initialization simulation with u (t ) = uinit (t )  D 0.5 (xinit (t )) = uinit (t )

[4.59]

The input uinit (t ) and the response xinit (t ) are presented in Figure 4.4.

Figure 4.4. Initialization procedure

The final distributed state Z of this simulation is used as an initial condition for the computation of control input u in the interval [0, Tce (J + 1)] .

State Control of Fractional Differential Systems

117

The corresponding graphs of u (t ) and x(t ) (for α = 106 ) are presented in

( )

Figure 4.5; the corresponding distributions of Z (0) and Z t f

are presented in

Figure 4.6.

Figure 4.5. Fractional integrator state control

Figure 4.6. Initial and final distributions of the internal state

118

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

We obtain J z (α ) = 7.7 E −06 and J u (α ) = 3.18 . A new computation of u is performed with α = 108 . We obtain J z (α ) = 3.2 E −07 and J u (α ) = 14.1 .

( ) is closer to

The conclusion is that Z t f

0 , but at the same time, the control

input u (t ) is more “energetic” than previously. In order to appreciate the influence of α on J z (α ) and J u (α ) , we repeat the computation of u for increasing values of α (see Table 4.1).

α J Z (α ) J u (α )

104

1.7 E

106 −4

0.72

7.7 E

108 −6

3.18

3.2 E

1010 −7

1.6 E

14.1

−8

1012

1.8 E

62.5

−9

188.3

1014

2.7 E

1016

−9

2.1 E −7

774

6350

Table 4.1. Influence of α

J z (α ) is minimum for α = 1012 , while J u (α ) regularly increases with α . A compromise is necessary between precision and control input energy. For example,

α = 106 or α = 108 seem to be an acceptable compromise. There is another parameter with a great influence on uˆ , i.e. the sample period Tce . With Tce varying from 100ms to 400ms , with α = 108 , we obtain the

following values of J z (α ) and J u (α ) (see Table 4.2). Tce (s) J Z (α ) J u (α )

0.1

5.9 E

0.2 −6

81.1

3.2 E

0.3 −7

6.7 E

14.1 Table 4.2. Influence of Tce

5.5

0.4 −8

2.5 E −8 3.0

State Control of Fractional Differential Systems

119

4.3.2.6. Conclusion

This numerical simulation has shown the main features of the proposed methodology for the state control of the fractional integrator. The RLS algorithm makes it possible to compute the control input u (t ) ; however, the result is not unique. It depends on the initialization parameter α : with the criterion J z (α ) , we can appreciate the precision of numerical computation. However, a “better” control input u (t ) corresponds to an increase in control input “energy”. Moreover, for an imposed value of α , J z (α ) and J u (α ) decrease when the sample period Tce is increased. The compromise J z (α ) / J u (α ) depends on α and Tce ; thus, the user is confronted with an optimization problem with a constraint on the “energy” of the control input. 4.3.2.7. State control of a one-derivative FDE

We are interested in the state control of the fractional integrator because this operation is fundamental for the state control of the closed-loop representation of a fractional system. Let us recall the simulation graph for a one-derivative FDE, where u (t ) is the input of the fractional integrator (Figure 4.7) and uc (t ) is the input of the FDE.

Figure 4.7. State control of a one-derivative FDE

120

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This graph corresponds to the equation u (t ) = D n (x(t )) = uc (t ) − a0 x(t )

[4.60]

Previously, we calculated the control input u (t ) , allowing the transfer of Z (t )

( )

from any initial state Z (0) to an arbitrary final state Z t f . When we apply the control input u (t ) to

1 sn

, we can compute Z (t ) and x(t ) , where

J

x(t ) =  c j z j (t ) = C I Z (t )

[4.61]

j =0

Using equation [4.60], with the knowledge of u (t ) and x(t ) , we can compute the control input uc (t ) applied to the FDE, allowing the transfer of its state Z (t ) from

( )

Z (0) to Z t f (since the integrator state is also the FDE state).

Therefore uc (t ) = u (t ) + a0 x(t ) t ∈ [0, Tce (J + 1)]

[4.62]

Consider the control input u (t ) corresponding to Tce = 0.2 s , α = 10 6 and n = 0.5

[4.63]

Then, if we consider the system x(s ) =

1 uc (s ) s n + a0

a0 = 1

[4.64]

we obtain the control input uc (t ) (with [4.62]) presented in Figure 4.8 and the corresponding output x(t ) , which corresponds exactly to x(t ) , presented in Figure 4.5.

State Control of Fractional Differential Systems

121

Figure 4.8. FDE state control

4.4. State control of an FDS 4.4.1. Introduction

The objective of this section is to propose a methodology for the state control of the distributed state Z (ω , t ) (in fact, a finite dimensional discretization Z ) of a non-commensurate order system D n ( X (t )) = A X (t ) + Bu (t )

n T = [n1

n 2  ni

 nN

]

[4.65]

However, we restrict the class of FDSs to the non-commensurate order FDEs represented by the transfer function y (s )

uc (s )

=

b0 a0 + a1s

n1

+ s n1 + n 2 +

[4.66]

corresponding to the simulation graph of Chapter 1 of Volume 1, where the 1 are connected in series, as a generalization of the elementary case integrators s ni considered previously.

122

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

For a single integrator, the system matrix A is diagonal. Thus, it is possible to explicitly formulate the relations between Z (t ) and the coefficients α i and β j . For an integrator chain, the composition of matrix A is more complex due to the couplings between the integrators imposed by the relations ui (t ) = xi +1 (t ) . Therefore, we present thereafter a general formulation of the relation Z k +1 = F Z k + G u k depending on Asys and B sys . Practically, this relation will be used for the two-integrator case. 4.4.2. Principle of state control

Remember that the objective is not the state control of Z (ω , t ) , which would require an infinite control time, but the state control of a finite dimensional approximation Z (t ) defined as Z T (t ) = [Z 1 (t ) Z 2 (t )  Z i (t )  Z N (t )]

[4.67]

where

[

]

Z iT (t ) = zi,0 (t ) zi,1 (t )  zi, j (t )  zi, J (t ) Each integrator

1 s ni

[4.68]

is characterized by the state model:

 d Z i (t ) = AI ,i Z i (t ) + B I ,i vi (t )  i = 1,2,  , N  dt  xi (t ) = C Z (t ) I ,i i 

[4.69]

with the constraint vi (t ) = xi +1 (t )

For the whole system, the state space model is

[4.70]

State Control of Fractional Differential Systems

 d Z (t ) = Asys Z i (t ) + B sys u (t )   dt  X (t ) = C sys Z (t ) 

123

[4.71]

Knowledge of the matrices Asys and B sys enables us to determine the matrices F and G corresponding to the discrete state-space representation [4.71] for the

sample period Tce thanks to standard algorithms available in mathematical solvers [MOK 97]. Then, we can write Z k +1 = F Z k + G u k

[4.72]

We previously demonstrated that the control input {uk } for a single integrator with J + 1 modes must be composed of J + 1 terms, varying from u0 to u J . For an integrator chain composed of N elements, each one with J + 1 modes, the control input {uk } will be composed of N (J + 1) terms, varying from u0 to u J max with J max = N (J + 1) − 1 .

Therefore, the explicit system of equations based on u0 to u J max is Z = F Z + Gu0 0  1   Z j +1 = F Z j + Gu j   Z  J max +1 = F Z J max + Gu J max

[4.73]

corresponding to

[

]

Z J max +1 = F J max +1 Z 0 + F J max G

 F jG  G u

[4.74]

with

[

u T = u0

u1  u j  u J max

]

[4.75]

124

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

i.e.

C

Z J max +1 = F J max +1 Z 0 + [

]u

[4.76]

C

[ ] is the controllability matrix (Chapter 2). The control input u allows the transfer of Z (t ) the final value Z (t f ) with t f = Tce (J max + 1) .

where

from any initial value Z (0) to

The control input u is provided by

C

u =[

]−1 [ Z (t f ) − F J

max +1

]

Z (0)

[4.77]

As mentioned previously, the control input u is recursively computed using the RLS algorithm [4.40] and [4.41]. 4.4.3. State control of two integrators in series

Consider the series system in Figure 4.9.

Figure 4.9. Two fractional integrators in series

The integrators

1 s

n2

and

1 s n1

are characterized by the state equations

 d Z 2 (t ) = AI ,2 Z 2 (t ) + B I ,2u (t )   dt  x2 (t ) = C Z (t ) I ,2 2 

 d Z 1 (t ) = AI ,1 Z 1 (t ) + B I ,1 x2 (t ) = AI ,1 Z 1 (t ) + B I ,1 C I ,2 Z 2 (t )   dt  x1 (t ) = C Z 1 (t ) I ,1 

[4.78]

State Control of Fractional Differential Systems

125

Let us define

Z (t ) = [Z 1 (t ) Z 2 (t )] T

[4.79]

Then  d  Z 1 (t )  AI ,1 B I ,1 C I , 2   Z 1 (t )  0  +    u (t ) = AI , 2   Z 2 (t )  B I , 2   dt  Z 2 (t )  0  0   Z 1 (t )  x1 (t ) C I ,1   X (t ) =  x (t ) =  0 C   Z (t ) I ,2   2   2   

[4.80]

0  0 1   −ω 1 1, i        B I ,i =   AI ,i =   − ω j ,i   1        1 − ω J ,i  0

[4.81]

with

[

C I ,i = c0,i

c1,i  c j ,i  c J ,i

]

Therefore  d Z (t ) = Asys Z (t ) + B sysu (t )   dt  X (t ) = C sys Z (t ) 

[4.82]

The Z -transform [KRA 92], with the sampling time Tce , provides the matrices

{F, G} of the discrete state equations. Thus, we can write

C

( )

Z t f = F J max +1 Z 0 + [

]u

with t f = (J max + 1)Tce

[4.83]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This linear system is composed of J max+1 lines from j = 1 to J max+1 .

( )

Consider the standard case Z t f = 0 .

the line j of the matrix

C

Let us define f j (0) as the component j of the product F J max +1 Z 0 and φ T as j

[ ].

Then, the line j of equation [4.83] corresponds to f j (0 ) + φ Tj u = 0 j = 1,2,  , J max + 1

[4.84]

The control input u is provided by the minimization of the quadratic criterion J max +1



j =1

(− f (0) + φ uˆ ) j

T j

2

[4.85]

4.4.4. Numerical example

Consider the following example a0 = 1 , a1 = 1 , n1 = 0.7 , n2 = 0.5 , Te = 10 −03 s

[4.86]

ω h = 10 +3 rd / s , ωb = 10 −3 rd / s and J = 30 for each integrator and Tce = 300ms .  Z (0 ) The initial state  1  is provided by the simulation of the corresponding  Z 2 (0 ) FDE (see Figure 4.10).

Z 1 (0) and Z 2 (0) are plotted in Figure 4.12.

State Control of Fractional Differential Systems

127

Figure 4.10. Initialization procedure

The control input uˆ is computed using the RLS algorithm with α = 1010 . The graphs of u (t ) , x1 (t ) and x2 (t ) are presented in Figure 4.11; the distributions of Z (0) and Z (T ) are presented in Figure 4.12.

Figure 4.11. State control of two fractional integrators in series

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 4.12. Distribution of initial and final internal states

For α = 1010 , we obtain  J Z (α ) = 3.6 E − 08   J u (α ) = 106

[4.87]

As discussed previously, we test the influence of Tce (see Table 4.3). Tce (s) J Z (α ) J u (α )

0.1

1.8 E

0.2 −5

14000

3.2 E

0.3 −7

3.6 E

670

for α = 1010

0.4 −8

106

1.3 E −8 28

Table 4.3. Influence of Tce

We note that the increase of Tce allows a better precision and a lower “energy” of the control input.

State Control of Fractional Differential Systems

129

4.4.5. State control of a two-derivative FDE

Consider the FDE  D n1 (x1 (t ) ) = x2 (t )  u (t ) = D n 2 (x2 (t ) ) = uc (t ) − a0 x1 (t ) − a1 x2 (t )

[4.88]

Using the previous control input u (t ) , we can express the FDE control input

( )

uc (t ) , allowing the transfer of Z (t ) from Z (0) to Z t f .

Using  x1 (t ) = C I ,1 Z 1 (t )   x2 (t ) = C I ,2 Z 2 (t )

[4.89]

[

]

we can compute uc (t ) for t ∈ 0, t f thanks to the equation uc (t ) = u (t ) + a0 x1 (t ) + a1 x2 (t )

[4.90]

Consider the previous example with the same parameter values ( a0 = 1 a1 = 1 ). The corresponding graphs of u (t ) , x1 (t ) and x2 (t ) are presented in Figure 4.13.

Figure 4.13. Two-derivative FDE state control

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

4.4.6. Pseudo-state control of the two-derivative FDE

In section 4.2, it was highlighted that the control of the pseudo-states x1 (t ) and x2 (t ) does not allow the true state control of the two-derivative FDE. However, we did not express the calculation of uc (t ) , allowing the transfer of the pseudo-state  x1 (t )  ( ) from an initial state, i.e. Z (0 ) =  x2 t 

 Z 1 (0 )  x1 (tc ) 0  ( ) to   =   with tc = 2Tce . Z 0  x2 (tc ) 0  2 

 u (t ) First, we calculate  1  . u 2 (t )

Using equation [4.83], we can write

Z (2) = F 2 Z (0) + F G u0 + G u1

[4.91]

Using [4.82], we obtain X (2 ) = C sys F 2 Z (0 ) + C sys F G u0 + C

sys

G u1

[4.92]

i.e.  x1 (2)  f1 (0 )  a11   a12   x (2 ) =  f (0 ) + a u 0 + a u1  2   2   21   22 

[4.93]

Thus

u   a a  u =  0  =  11 12   u1  a21 a22 

−1

 x1 (2) − f1 (0)   x (2) − f (0) 2   2

[4.94]

In this case, matrix inversion is easy. Then, with u (t ) , we compute Z 1 (t ) and Z 2 (t ) , and therefore x1 (t ) and x2 (t ) using [4.82]. Finally, we can compute the control input uc (t ) thanks to [4.90].

State Control of Fractional Differential Systems

131

4.5. Conclusion

This chapter was dedicated to the state control of fractional integrators with application to the state control of fractional order differential systems. The objective was not to propose a new control strategy but to highlight the specificities and the difficulties of FDS state control. Numerical examples were proposed to illustrate all these features, without an application objective. This methodology is an open-loop approach to state control or feed-forward state control. The corresponding practical methodology would be a closed-loop approach to feedback state control, based on the principles highlighted previously. Moreover, we demonstrated the necessary compromise between state accuracy and control “energy”, corresponding to an optimization problem with constraints. The next step would be the optimal control of the internal distributed state, but it is out of the scope of our objective in this book. However, in the next chapter, we present an application to the control of a diffusive system in order to demonstrate the interest of FDS state control.

5 Fractional Model-based Control of the Diffusive RC Line

5.1. Introduction In this chapter, we intend to demonstrate that the state control of a fractional differential system (FDS) is not only a theoretical problem; on the contrary, it can be applied to control the distributed state of a diffusive system. For this purpose, this study is based on two main principles. It was demonstrated in Chapter 4 of Volume 1 that a diffusive interface can be approximated by a fractional model. Moreover, as related in Chapter 6 of Volume 1, an infinite length RC line behaves like a fractional integrator, and there is a connection between their distributed internal variables v( x, t ) and z (ω , t ) . Hence, we intend to demonstrate that Z (ω , t ) state control of the fractional model of the RC line is equivalent to the control of the distributed RC line variable v( x, t ) .

Modeling of the diffusive interface was performed in Chapter 4 of Volume 1 using a frequency identification technique. This methodology is reserved to a theoretical analysis, making possible to justify and discriminate different models in a physical context. In practical situations, we mainly use fractional identification techniques based on time measurements [BEN 08b, MAL 08, GAB 11a, GAB 11b]. For the identification of a fractional model of the RC line, input and output data are generated by a numerical simulation. Consequently, these measurements are not disturbed by noise; therefore, we propose to use an elementary algorithm derived from the least-squares method for the identification of parameters {ai , bi } and of the fractional order n without using a complex nonlinear optimization algorithm as in Chapter 5 of Volume 1.

{ u (t ), y(t )}

Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

In the general case, the state control of a partial differential equation (PDE) is a complex mathematical problem that has given rise to a large number of publications [LIO 71, MOR 10]. More modestly, we are interested in the reset of the distributed variable v( x, t ) of the RC line after an initial excitation. Therefore, we intend to impose v( x, Tc ) = 0 ∀ x ∈ [ 0, L ]

[5.1]

for a given value Tc , with the initial condition v(x,0) ( x ∈ [ 0, L ] ), using only the interface control input. Thus, we intend to demonstrate that the reset of the RC line is equivalent imposing Z (ω , Tc ) = 0

ω ∈ [0,+∞[

[5.2]

with the initial condition Z (ω ,0) , where Z (ω , t ) is the distributed variable of the identified fractional model. 5.2. Identification of the RC line using a fractional model 5.2.1. Introduction

The diffusive interface and its approximation by a fractional model were analyzed in Chapter 4 of Volume 1, in order to justify fractional modeling in a physical framework. The different structures of differential models were validated by a frequency approach. However, as already mentioned, this identification approach is not suited in practice. Only a time approach based on the minimization of a quadratic criterion is really adapted for the treatment of input/output measurements, generated by a real experiment or a numerical simulation. Therefore, we intend to apply this methodology to the RC line. 5.2.2. An identification algorithm dedicated to fractional models 5.2.2.1. Principle

The general identification methodology, known as the model method or output error method [RIC 71, TRI 88], is based on the minimization of a quadratic criterion by nonlinear optimization [HIM 72] (see Chapter 5 of Volume 1).

Fractional Model-based Control of the Diffusive RC Line

135

Its fundamental interest is to be adapted to any type of model, linear or nonlinear. Its main drawback is related to its complex implementation. On the contrary, the least-squares technique [MEN 73, NOR 86] is very easy to implement but restricted to the case of a model linear in its parameters (LP model). With prior treatment of input/output data [KHA 15a], it is well suited to fractional models characterized by a known fractional order, such as commensurate order models (with n = 0.5 ). For non-commensurate order models, the fractional orders also have to be estimated. Several sequences of the basic algorithm with different values of the fractional order make it possible to select easily its optimal value, as in Chapter 4 of Volume 1. 5.2.2.2. One-derivative FDE model

Consider the elementary system b0 y (s ) = = H n (s ) u (s ) a0 + s n

[5.3]

corresponding to the one-derivative FDE

D n ( y(t )) + a0 y(t ) = b0u (t )

[5.4]

Consider the fractional integration with order n of the two sides of equation [5.4]:

(

)

I n D n ( y(t )) + a0 y (t ) = I n (b0u (t ))

{

[5.5]

}

Since I n D n ( y (t )) = y (t ) if the initial conditions are equal to zero, equation [5.5] is transformed into an integral equation, which can be written as

y(t ) = b0 I n (u (t )) − a0 I n ( y(t ))

[5.6]

Let us define

[

]

ϕ (t )T = I n (u (t )) − I n ( y(t ))   θ T = [b0 a0 ]

[5.7]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Then, [5.7] corresponds to the linear in the parameter form (LP) required by the least-squares method y (t ) = ϕ (t )T θ

[5.8]

5.2.2.3. Two-derivative FDE model

Consider the non-commensurate order model y (s )

u (s )

=

b0 + b1s m1 a0 + a1s m1 + s m2

= H n1 , n2 (s )

[5.9]

Let us define  m1 = n1 0 < n1 ≤ 1 with   = + m n n 1 2  2 0 < n 2 ≤ 1

[5.10]

H n1 , n2 (s ) corresponds to the two-derivative FDE D n1 + n2 ( y (t )) + a1D n1 ( y (t )) + a0 y (t ) = b0u (t ) + b1D n1 (u (t ))

[5.11]

As demonstrated previously, consider the fractional integration with the order m2 of the two sides of equation [5.11]:

{

}

{

}

I n1 + n2 D n1 + n2 ( y (t )) + a1D n1 ( y (t )) + a0 y (t ) = I n1 + n2 b0u (t ) + b1D n1 (u (t ))

[5.12]

If the initial conditions are equal to zero:

{

(

{

(

L I n1 + n 2 D n1 + n 2 ( f (t ))

)}

L I n1 + n 2 D n1 ( f (t )) =

) }= s

1 n1 + n 2

1 s

n1 + n 2

s n1 + n 2 L{ f (t )} = F (s )

s n1 F (s ) =

1 s n2

F (s )

[5.13]

[5.14]

Therefore, equation [5.12] is transformed into an integral equation

y(t ) = b0 I n1 + n2 (u (t )) + b1 I n2 (u (t )) − a 0 I n1 + n2 ( y(t )) − a1 I n2 ( y (t ))

[5.15]

Fractional Model-based Control of the Diffusive RC Line

137

Let us define

[

]

ϕ (t )T = I n1 + n2 (u (t )) I n2 (u (t )) − I n1 + n2 ( y (t )) − I n2 ( y (t ))   θ T = [b0 b1 a0 a1 ]

[5.16]

Thus, equation [5.16] corresponds to the linear in the parameter form (LP) y (t ) = ϕ (t )T θ

[5.17]

REMARK 1.– In practical terms, we proceed by successive integration of u (t ) and y(t ) : the integration with the order n2 provides I n2 (u (t )) and I n2 ( y (t )) . Then, the integration with the order n1 of the previous results provides

( (

) )

 I n1 I n2 (u (t )) = I n1 + n2 (u (t ))  n1 n2  I I ( y (t )) = I n1 + n2 ( y (t ))

5.2.2.4. Least-squares identification

Let us define: – y(t ) , which is a measurement (without noise); – yˆ (t ) , which is the prediction of y(t ) based on the knowledge of an estimation

θˆ using [5.8] or [5.17]. Then, with yk = y (kTe ) , we can express the prediction yˆ k = ϕ

T k

θˆ , where ϕ

k

(regressor vector) is a function of measurements. Consider the quadratic criterion K

J pred = 

(

k =1

yk*

2

− yˆ k

)

[5.18]

Then, the best estimation θˆ minimizing J pred [EYK 74] is  K  θ MC =   ϕ ϕ T  k k  k =1 

−1

 K    ϕ yk    k  k =1 

[5.19]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

It is necessary to distinguish yˆ pred (t ) and yˆ sim (t ) , where yˆ sim (t ) represents the simulation of the system model, based only on the knowledge of the estimation θˆ

and on the excitation u (t ) . On the contrary, yˆ pred (t ) is explained by the knowledge of u (t ) and y(t ) , and of course of θˆ . Thus

( )pred ≠ y(θˆ, t )sim

y θˆ, t

[5.20]

in the general case. In order to validate an estimation θ MC (when θ MC does not correspond to an unstable system), it is necessary to simulate the system and compute the quadratic simulation criterion K

J sim = 

k =1

(

2

yk*

− yˆ sim , k

)

[5.21]

5.2.2.5. Fractional identification with no a priori knowledge of the order

Let us consider the two cases H n (s ) =

b0 a0 + s n

and H n1 , n2 (s ) =

b0 + b1s n1

a0 + a1s n1 + s n1 + n2

with n2 = 0.5

[5.22]

where the orders n or n1 are unknown. The proposed procedure was already used in Chapter 4 of Volume 1 and applied to more complex models in [KHA 15a]. In order to determine the optimal value of the fractional order, the estimation

θ MC is repeated for increasing values of nˆ (or nˆ1 ). Therefore, we obtain the graphs J pred (nˆ ) and J sim (nˆ ) (see Figure 5.1).

Fractional Model-based Control of the Diffusive RC Line

139

Figure 5.1. Prediction and simulation quadratic criteria. For a color version of the figures in this chapter, see www.iste.co.uk/trigeassou/analysis2.zip

Of course, nˆsim ≠ nˆ pred according to the distinction between y pred and ysim . The optimal value of nˆ is the one minimizing J sim , i.e. nˆsim . Of course, this procedure can be interpreted as another nonlinear optimization algorithm. 5.2.3. Simulation of the diffusive RC line 5.2.3.1. Introduction

The objective of the numerical simulation of the RC line is to provide {uk , yk } measurements required by the identification algorithm. As yk measurements (noise free) have to exactly represent the RC line, it is necessary to minimize the possible numerical simulation errors. The dynamical behavior of the RC line obeys a diffusion equation, such as heat conduction [HOL 89]. However, simulation hypotheses for the RC line are less restrictive than those required by the simulation of heat conduction. Moreover, we can apply a negative excitation (current) to the line, whereas this excitation is not possible with heat conduction, because it would be necessary to use a heat extractor

140

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

(or cooler). This remark corresponds to a necessary requirement for the control of the RC line in the last section. Consider the line presented in Figure 5.2, which is governed by the equations (see Chapter 6 of Volume 1):  ∂ 2v ( x , t ) ∂v( x, t ) =RC   ∂x 2 ∂t   i ( x, t ) = − 1 ∂v( x, t )  R ∂x

[5.23]

Let us recall that C and R are not resistors or capacitors but capacity and resistance densities: C=

dC ( x) dx

R=

dR ( x) dx

Figure 5.2. The RC line

5.2.3.2. Time and space discretization

Consider Figure 5.3

Figure 5.3. Space discretization

[5.24]

Fractional Model-based Control of the Diffusive RC Line

141

where  xi = i Δx  tk = k Δt

[5.25]

i is the space index, whereas k is the time one.

with

L = I sp Δx Therefore,

[5.26]

∂ 2 v ( x, t ) ∂x

2

and

∂v ( x, t ) ∂t

are approximated by

 ∂ 2v( x, t ) vi +1,k − 2vi,k + vi −1, k ≈   ∂x 2 Δx 2  ∂v( x, t ) vi, k +1 − vi, k  ≈  ∂t Δt

[5.27]

Then, relation [5.23] becomes 1 vi +1, k − 2vi , k + vi −1, k ∂v( x, t ) vi , k +1 − vi , k ≈ = RC ∂t Δt Δx 2

[5.28]

Thus 2Δt  Δt  (v + vi −1, k ) for i = 0 to I sp vi , k +1 = vi , k 1 − + 2  2 i +1, k  RCΔx  RCΔx

[5.29]

Equation [5.28] represents a differential system composed of I sp + 1 first-order differential equations, while [5.29] represents a difference equation system. REMARK 2.– The heat diffusion equation, where T (x, t ) is the temperature and φ (x, t ) is the heat flux, corresponds to (see Chapter 4 of Volume 1)

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

 ∂T ( x, t ) ∂ 2T ( x, t ) =α   ∂t ∂x 2  ∂ T ( x, t )  φ ( x, t ) = − λ  ∂x

Therefore, there is an equivalence between the RC line and heat diffusion

T ( x, t ) ≡ v ( x, t ) and φ ( x, t ) ≡ i ( x, t )

[5.30]

with

α ≡

1 RC

and λ ≡

1 R

[5.31]

Thanks to this analogy, thermal resistors and thermal capacitors can be defined [HOL 89]. However, we have to be careful with this analogy. Indeed, the heat flux through a thermal resistor creates a difference in temperatures; however, it is not accompanied by a Joule effect Ri 2 as in an electrical resistor. Similarly, a thermal capacitor stores a heat charge qT ; however, this charge does 1 2 Cv as in an electrical capacitor: indeed, let us 2 recall that the heat charge qT is itself an energy!

not correspond to a stored energy

5.2.3.3. Boundary conditions

The RC line is a distributed system where current and voltage depend on the abscissa x and time t . Therefore, it is necessary to specify the boundary conditions, i.e. the values of i and v at x = 0 and x = L [FAR 82, HOL 89]. For an infinite length line, a condition exists at L = ∞ ; however, it should not be explicit (see Chapter 6 of Volume 1). On the contrary, for a finite length line, it is necessary to specify i(L ) and v(L ) . In the considered example, we assume that v(L ) = cte and in particular v(L ) = 0 , which corresponds to a short circuit of the line at x = L .

Fractional Model-based Control of the Diffusive RC Line

143

This boundary condition is known as the Dirichlet condition for heat diffusion [HOL 89]. i (L, t ) is not imposed; it depends on the excitation at x = 0 and the characteristics of the line. The point x = 0 corresponds to the excitation of the line (it is also called the diffusive interface for heat diffusion). We assume that the excitation is a current generator i0 (t ) = i (0, t ) , which is equivalent to a heat flux φ0 (t ) . Consequently, the voltage v(t ) = v(0, t ) is not imposed; it depends on i0 (t ) and the line. In terms of system theory (or automatic control), i0 (t ) is the input and v(0, t ) is the output of the RC line. 5.2.3.4. Numerical stability

The RC line is obviously a stable system. However, the stability of the difference equation system [FAR 82, HOL 89] depends on a numerical condition, linking Δt (time increment) to Δx (spatial increment). System [5.29] is stable if Δt ≤

Δx 2

[5.32]

2α

for the heat equation, which corresponds for the RC line to

Δt ≤ RC

Δx 2

[5.33]

2

As Δx =

L

[5.34]

I esp

we can also formulate this condition as Δt ≤ RC

Δt

L2

2 I esp 2

= Δtmax

[5.35]

Therefore, we can conclude that for a given value RC , with L imposed, max decreases if the spatial resolution is improved, i.e. if I sp is increased. Note

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

that if I sp is increased, Δt max decreases and global computation time is increased. Hence, a compromise is necessary. 5.2.3.5. Numerical treatment of boundary conditions

Note (see Figure 5.2) that v(0, t ) does not correspond to a capacitor CΔx voltage. Therefore, if we move the origin x = 0 on the first capacitor (Figure 5.4)

Figure 5.4. Interface at x = 0

we obtain

v(t ) = RΔx i0 (t ) + v(0, t )

[5.36]

Assume that the input is a step current i0 (t ) = I 0 H (t )

[5.37]

with the initial condition v(0,0 ) = 0 . Then, we obtain

( )

v 0+ = RΔxI0

[5.38]

Therefore, v(t ) starts with a discontinuity that does not correspond to a physical observation. In order to solve this problem, we propose to modify the spatial discretization at x = 0 and x = L .

Fractional Model-based Control of the Diffusive RC Line

145

It is necessary to impose RΔx = 0 in [5.36] in order to suppress the discontinuity, without a modification of the global repartition of resistors RΔx and capacitors CΔx . We fulfill this requirement with the diagram of Figure 5.5.

Figure 5.5. Improved numerical interface at x = 0

We obtain i0 (t ) = ic (t ) + i(t ) for x = 0

[5.39]

Therefore dv(0, t ) dt

=

2 CΔx

i0 (t ) +

2 RCΔx 2

(v(1, t ) − v(0, t ))

[5.40]

Thus v(0, k + 1) =

2Δt 2Δt  2 Δt  i0 (k ) + 1 − v(0, k ) + v(1, k ) 2  CΔx Δ Δx 2 RC x RC  

[5.41]

Equation [5.41] replaces the first difference equation of system [5.29]. This modification has no impact at i = I sp since v(L, t ) = 0 . 5.2.3.6. Some general considerations

We intend to model the RC line at x = 0 using a fractional transfer function such as v(0, s ) i (0, s )

= H n (s ) or

v(0, s ) i (0, s )

= H n1 ,n2 (s )

[5.42]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Based on some elementary physical considerations, we can specify some characteristics of these models. Assume a step excitation i0 (t ) = I 0 H (t ) and that the line is initially at rest (i.e. v(x,0) = 0 ∀ x ). Therefore, at the first instants of the transitory response, only

the first cells are influenced by I 0 H (t ) , independently of line length. This means that the beginning of the graph v(0, t ) is the same ∀ L . In other terms, this means

that the line behaves like a fractional integrator

R

1

C s

0.5

for t → 0 .

For t → 0 (i.e. s → ∞ ), H n (s ) or H n1 ,n2 (s ) behave like an integrator. Thus lim H n (s ) =

s →∞

b0 s

n

≡

R

1

C s

0.5

[5.43]

Therefore, for n = 0.5 , we obtain b0 ≡

R C

.

Similarly, for n2 = 0.5 , we obtain lim H n1 , n2 (s ) =

s →∞

thus b1 ≡

R C

b1 s

n2

≡

R

1

C s

0.5

[5.44]

.

On the contrary, as t → ∞ , all the capacitors C Δx are charged and the current I 0 passes through the resistor chain. Therefore I est

v(0, ∞ ) = I 0  RΔx = RI 0 L i =1

Thus, we can define the static gain of the two fractional models as

[5.45]

Fractional Model-based Control of the Diffusive RC Line

b0 a0

= RL

147

[5.46]

We can also characterize the voltage repartition inside the line for t → ∞ . As v(0, ∞ ) = I 0 RL and v(L, ∞ ) = 0 , the internal voltage v(i ) = v(x ) varies linearly x  from v(0, ∞ ) to 0 ; therefore, v(x ) = RI 0 L1 −  . L  5.2.3.7. Experimental results

The RC line is simulated with three values of L : 50 m, 100 m and 150 m. The simulation parameters are R = 0.1Ω / m C = 0.1F / m Δx = 0.5m

[5.47]

Thus L = 50m 100  I sp = 200 for L = 100m 300 L = 150m 

[5.48]

Consequently, Δtmax remains the same: Δt ≤

RCΔx 2 2

= Δt max

[5.49]

and

Δt max = 1.25E −03s

[5.50]

Our choice for time discretization is Δt = 1E −03 s which ensures numerical stability.

The graphs of the three step responses are plotted in Figure 5.6; they correspond to i0 (t ) = I 0 H (t ) with I 0 = 1A .

148

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 5.6. Step responses of the RC line

Of course, the graphs are similar to those presented in Chapter 4 of Volume 1. The internal voltage distribution for the three lengths at t = 60 s is plotted in Figure 5.7.

Figure 5.7. Internal voltage distributions

Fractional Model-based Control of the Diffusive RC Line

149

For L = 50 , v(0, t ) reaches its asymptotic value RI 0 L = 5 volts at t = 60 s (see Figure 5.6); reciprocally, v(i,60) varies linearly from 5volts to 0 (see Figure 5.7). The asymptotic value is not reached for L = 100 ( 10 volts ) nor for L = 150 ( 15 volts ). In Figure 5.6, we can verify that the three graphs coincide for t < 10 s , confirming that the RC line behaves like a fractional integrator: R 1 C s

0.5

=

1 s

0.5

∀ L for t → 0

[5.51]

These step responses lead to two essential conclusions: b0

– in order to obtain a good estimation of the static gain

a0

include long time constant periods in the excitation u (t ) ;

, it is necessary to

– on the contrary, if the input includes many short time periods, we obtain a good estimation of high frequency modes, i.e. of the integrator behavior. If the identification data file uses Te = Δt ( Te : sampling period) with a limited number of data points, it will favor the estimation of high frequency modes. In order to avoid a very large number of data points, it is necessary to use undersampling, i.e. Te = NΔt . Consequently, the estimation of low frequency modes will be favored. 5.2.4. Experimental identification 5.2.4.1. Methodology

The RC line is identified for the three previous lengths. For each value of L , we use two data files

{ u, y}

with Te = 10−03 s and

Te = 5E −03s . Figures 5.8 and 5.9 show these two data files for L = 100 .

150

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

−03 Figure 5.8. Data file: Te = 10 s

−03 Figure 5.9. Data file: Te = 5E s

Fractional Model-based Control of the Diffusive RC Line

151

Each data file is composed of two rectangular type signals, positive and negative, in order to favor the estimation of high frequency modes. These signals are followed by a relaxation period ( i0 (t ) = 0 ) in order to favor the estimation of low frequency modes, obviously better with Te = 5E −03s . For each file, the models

H n (s )

and

H n1 , n2 (s ) are identified. The

fractional orders n and n1 are estimated using the previous procedure (see section 5.2.2.5). The graphs of J sim and J pred for the estimation of the fractional order with

Te = 5E −03s are plotted in Figure 5.1. 5.2.4.2. Experimental results

The identification of models H n (s ) and H n1 , n2 (s ) is performed with the following parameters:

ωb = 10−03 rd / s ωh = 10 +03 rd / s J = 30

[5.52]

All the identification results for H n (s ) are presented in Table 5.1 for

Te = 10−03 s , and in Table 5.2 for Te = 5E −03s . For H n1 , n2 (s ) , the results are presented in Table 5.3 for Te = 10−03 s , and in Table 5.4 for Te = 5E −03s . We do not present the results for the commensurate order model H n1 , n2 (s ) ( n1 = n2 = 0.5 ) because the corresponding models are generally unstable. Similarly, the identification of the non-commensurate order model is not possible with Te = 10−03 s and L = 150 .

152

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

L

50

100

150

n

0.65

0.51

0.5

b0

10.57

1.004

1.003

a0

0.139

0.0111 0.0021

Jsim

850

11.2

0.64

b0/a0 7.6

90.5

480

−03 Table 5.1. Identification of H n ( s ) for Te = 10 s

L

50

100

150

n

0.86

0.69

0.57

b0

0.890

0.802

0.913

a0

0.171

0.061

0.028

Jsim

1450

4400

2100

13.1

32.6

b0/a0 5.2

−03 Table 5.2. Identification of H n ( s ) for Te = 5E s

L

50

100

150

n1

0.83

0.99

---

n2

0.5

0.5

---

b0

0.241

0.0059

---

b1

1.093

1.01

---

a0

0.051

0.0011

---

a1

0.282

0.0128

---

Jsim

4.97

0.71

---

b0/a0

4.73

5.36

---

−03 Table 5.3. Identification of H n1 , n2 ( s ) for Te = 10 s

Fractional Model-based Control of the Diffusive RC Line

L

50

100

150

n1

0.84

0.83

0.94

n2

0.5

0.5

0.5

b0

0.385

0.097

0.0084

b1

1.178

1.129

1.031

a0

0.077

0.010

8.6E-04

a1

0.477

0.185

0.027

Jsim

3.47

26.9

8.31

9.7

9.8

b0/a0 5.0

153

−03 Table 5.4. Identification of H n1 , n2 ( s ) for Te = 5E s

5.2.4.3. Comments

The values of J sim obviously prove that the model H n1 , n2 (s ) provides a better

approximation of the RC line than the model H n (s ) , regardless of the value of Te . Nevertheless, the model H n (s ) is a fair approximation for Te = 10−03 s . Theoretically, this model would be inappropriate to approximate the RC line (see Chapter 4 of Volume 1). In fact, since with Te = 10−03 s , the estimation of high frequency modes is favored; thus, H n (s ) represents a high frequency approximation of the fractional integrator behavior. b0

is computed for each identified model: it represents the theoretical a0 static gain, i.e. 5 for L = 50 , 10 for L = 100 and 15 for L = 150 . The value

For Te = 10−03 s , all the values of

b0 a0

are far from their theoretical value: this is

the confirmation that the models H n (s ) and H n1 , n2 (s ) provide only a good approximation of high frequency modes. On the contrary, for Te = 5E −03s ,

b0

is a a0 better approximation of the static gain, mainly for L = 50 , i.e. the two corresponding models are a fair approximation of the long time behavior. Moreover,

154

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

as J sim has a low value for H n1 , n2 (s ) , this means that for Te = 5E −03s , this model performs a good compromise between the low and high frequency modes of the line, mainly for L = 50 and L = 100 . According to the remarks in section 5.2.3.6, we can verify that: – b0 is close to

R = 1 for H n (s ) with Te = 10−03 s , L = 100 and L = 150 . C

On the contrary, b0 ≠ 1 with Te = 5E −03s regardless of the value of L ; – b1 is close to

R

= 1 for H n1 , n2 (s ) with Te = 10−03 s , L = 50 and L = 100 .

C This confirms that H n1 , n2 (s ) gives a good compromise between the low and high

frequencies, rather for the low frequencies with L = 50 and L = 100 , and rather for the high frequencies with L = 150 . 5.3. Reset of the RC line 5.3.1. Introduction

The control of the internal state of the RC line, i.e. the control of the distribution of the v(x, t ) voltage x ∈ [0, L ] , is an ambitious objective in the general case [MOR 10]. Therefore, we limit our demonstration to the reset problem, i.e. we want to impose v(x, t ) = 0 for t ≥ TC ∀ x ∈ [0, L ]

[5.53]

verifying the distributed initial condition v(x,0) ∀ x ∈ [0, L ]

[5.54]

The reset of the RC line can be performed by natural relaxation, i.e. by the discharge of all the capacitors in the resistive components. However, this natural relaxation is not interesting because it requires a very long time response.

Fractional Model-based Control of the Diffusive RC Line

155

Therefore, the objective is to accelerate this process and to control it using the input at x = 0 in the time interval [0, TC ] . 5.3.2. Natural relaxation

In order to illustrate the natural relaxation process, we perform a numerical simulation. Before performing the simulation, it is necessary to create an initial distributed state v(x,0) , which will also be the reference for further simulations. Therefore, we apply an input i0 (t ) to the line ( L = 100 ) for 0 < t < tmax . At instant tmax , we obtain the internal distribution v(x, tmax ) which will be the initial condition v(x,0) . Figure 5.10 shows the graphs of i0 (t ) and v(0, t ) at the input of the RC line, whereas Figure 5.11 presents the distribution v(x, tmax ) , i.e. the initial distribution.

Figure 5.10. Initialization procedure

156

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 5.11. Natural relaxation of the RC line

With this initial condition v(x,0) , we perform the natural relaxation of the RC line based on i (0, t ) = 0 . Figure 5.11 shows the evolution of v(x, t ) at increasing instants. We verify that at t = 100 s , the distribution v(x,100) is far from the objective [5.53]. Thus, these graphs demonstrate that the natural relaxation is unable to satisfy an obvious speed requirement. 5.3.3. Principle of the reset technique

The reset technique is based on two principles. First, the RC line is modeled, at x = 0 , by the two transfer functions v(0, s ) i (0, s )

= H n (s ) and

v(0, s ) i (0, s )

= H n1 , n2 (s )

[5.55]

Fractional Model-based Control of the Diffusive RC Line

157

On the one hand, it has been demonstrated in section 5.2 that the diffusive interface can be approximated by a fractional model with a desired accuracy. However, this modeling at x = 0 does not provide (a priori) information on the distribution of v(x, t ) inside the RC line. On the other hand, the reset technique is inspired by the modeling of the infinite length RC line by a fractional integrator

R 1 s s 0.5

(see Chapter 6 of Volume 1).

It has been demonstrated that the real part Re(χ , t ) of the space Fourier transform of v(x, t ) verifies the same type of distributed differential equation as the internal variable z (ω , t ) of the fractional integrator ∂z (ω , t ) = −ω z (ω , t ) + u (t ) ω ∈ [0,+∞[ ∂t

[5.56]

and ∂Re(χ , t ) ∂t

=−

χ2 RC

Re(χ , t ) +

1 C

i(0, t )

χ ∈ ]− ∞,+∞[

[5.57]

The analogy between these differential equations suggests that

ω≡

χ2 RC

and z (ω , t ) ≡ Re(χ , t )

[5.58]

with TF (v(x, t )) =

+∞

 v ( x, t ) e

− jχ x

dx = Re(χ , t ) + j Im(χ , t )

[5.59]

−∞

This analogy reveals an equivalence (for the infinite length RC line) between Re(χ , t ) and z (ω , t ) , i.e. a connection between v(x, t ) and z (ω , t ) . In particular, if v(x, t ) = 0 ∀ x , Re(χ , t ) = 0 ∀ χ , implying that z (ω , t ) = 0 ∀ ω , which corresponds to the reset objective. This second principle allows us to formulate the following conjecture: the control of the distributed state of the fractional integrator makes it possible to indirectly control the distributed internal state of the RC line.

158

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Let us recall that the previous equivalence has been proved only for an infinite length line. Thus, we assume that there is also a connection between the internal state Z (ω , t ) of the fractional models H n (s ) and H n1 , n2 (s ) , and the internal

distribution v(x, t ) of a finite length RC line. 5.3.4. Proposed reset procedure

The reset of the RC line requires the control of the distributed state Z (ω , t ) . Therefore, it is necessary to have the knowledge of Z (ω ,0 ) corresponding to v(x,0) ∀ x ∈ [0, L ] . For this purpose, during the initialization procedure, we apply the same input to the RC line and to its models H n (s ) or H n1 , n2 (s ) : therefore, this simulation

provides v(x, tmax ) and Z (ω , tmax ) .

With Z (ω , tmax ) , which becomes the initial distributed state Z (ω ,0 ) of the fractional model for the reset procedure, we can compute the sequence u (t ) that has  1 1  or to the double integrator  n , n  s s 1 s 2  (see Chapter 4). Finally, thanks to this sequence u (t ) , we can compute the excitation

to be applied to the fractional integrator

1

n

u FDE (t ) (i.e. i0 (t ) ) that has to be applied to the model H n (s ) (or H n1 , n2 (s ) ) and consequently to the RC line (see Chapter 4).

Thus, we obtain the response v(0, t ) of the RC line (initialized by v(x,0) = v(x, tmax ) ) and the final internal distribution v(x, TC ) . In order to objectively appreciate the efficiency of this methodology, the direct comparison of graphs is not sufficient. Therefore, we have to define some more objective criteria: – on the distributed state Z (ω , t ) crit _ z =

(

J

J

j =0

j =0

)

 z 2j (Te ) or crit _ Z =  z12, j (Te ) + z22, j (Te )

[5.60]

Fractional Model-based Control of the Diffusive RC Line

159

– on the distribution v(x, TC ) I sp

crit _ v =  vi2 (i, TC )

[5.61]

i =0

– on the control energy TC

crit _ u =  u 2 (t )Tce

[5.62]

t =0

Moreover, the control input is characterized by its maximum and minimum values u max u min . 5.3.5. Experimental results

The objective is to compare the different resets performed for the three lengths L = { 50,100,150 } using the models H n (s ) and H n1 , n2 (s ) (corresponding to

Te = 10−3 s and Te = 5 10 −3 s ). All the control sequences are computed with Δt = 10 −3 s and kC = 300 . Therefore, Tce = kC Δt = 0.3s . The fractional models are simulated with the parameters ωb = 10−3 rd / s ,

ωh = 103 rd / s and J = 30 . Moreover, the control sequences are computed with α MC = 106 (for H n (s ) ) and

α MC = 108 (for H n1 ,n2 (s ) ) (see Chapter 4). We present some experimental graphs obtained with L = 100 . For H n (s ) , Figures 5.12 and 5.13 present the graphs of z (ω , t ) and v(x, t ) (initial and final values).

160

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 5.12. Initial and final distribution of the distributed state

Figure 5.13. Initial and final distribution of the space variable

Fractional Model-based Control of the Diffusive RC Line

161

Figure 5.14 presents the corresponding excitation u FDE (t ) , the system output ymod (t ) and the line output v(0, t ) = yline (t ) .

Figure 5.14. Excitation and response of the RC line

For

H n1 , n2 (s ) ,

we

only

present

in

Figure

5.15

the

graphs

of

Z (ω , t )T = [z1 (ω , t ) z2 (ω , t )] and in Figure 5.16 the input and output of the RC line

Figure 5.15. Initial and final distribution of the distributed states

162

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 5.16. Excitation and response of the RC line

These graphs demonstrate that it is necessary to use the criteria [5.60, 5.61 and 5.62] in order to objectively appreciate the efficiency of the reset on Z (ω , TC ) and on v( x, TC ) . In Tables 5.5–5.8, we present the different criteria obtained with the models H n (s ) and H n1 , n2 (s ) . L

50

crit_z

0.068

crit_u

5.6E+

100

150

0.13 03

1.3E

0.14 +04

1.4E+04

umax/umin 46/-62

60/-100

65/-103

crit_v

0.093

0.088

0.039

−3 Table 5.5. H n ( s ) with Te = 10 s

Fractional Model-based Control of the Diffusive RC Line

15 0

L

50

100

crit_Z

1.9E-04

4.4E-04 ----

crit_u

10.1E+03 3.6E+04 ----

umax/umin 30/-55

47/-93 ----

-05

1.6E-04 ----

crit_v

3.7E

163

−3 Table 5.6. H n1 , n2 ( s ) with Te = 10 s

L

50

100

150

crit_z

0.045

0.097

0.12

crit_u

3.1E+03

8.9E+03

1.2E+04

umax/umin

35/-36

55/-80

65/-95

crit_v

0.023

0.031

0.065

−03 Table 5.7. H n ( s ) with Te = 5E s

L

50

100

150

crit_z

8.5E-05

2.6E-04

4.4E-04

crit_u

5.5E+03

1.8E+04

3.4E+04

umax/umin

26/-44

38/-70

47/-92

crit_v

2.6E-05

6.9E-05

1.3E-04

−03 Table 5.8. H n1 , n2 ( s ) with Te = 5E s

Finally, we test the influence of the control input u FDE (t ) for L = 100 m with

H n1 , n2 (s ) and Te = 5E −03s for different values of kC . The corresponding results are presented in Table 5.9.

164

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

kC

300

600 -04

3.3E-08

crit_z

2.4E

crit_u

1800

480

umax/umin

38/-70

7/-11.5

1.35/-1

-05

-06

2.9E-07

crit_v

6.9E

TC (sec)

18

4.5E

1200 -06

1.1E 63

11.4

72

Table 5.9. Influence of kC

5.3.6. Comments

For the same value of L , the best results (i.e. the lower values of crit _ Z and

crit _ v ) are obtained with the H n1 , n2 (s ) model.

For example, with L = 100 m : – and Te = 10−03 s , we obtain crit _ v = 1.6 E −04 for H n1 , n2 (s ) which is significantly less than crit _ v = 9.3E −02 obtained for H n (s ) ; – and Te = 5E −03s , the model H n1 , n2 (s ) improves the performances since we obtain crit _ v = 6.9 E −05 < 1.6 E −04 . On the contrary, for the same model and the same Te , it is easier to control the line with L = 50m than with L = 150 m . As an example, with H n1 , n2 (s ) and

Te = 5 E −03s , we obtain crit _ v = 2.6 E −05 for L = 50m against crit _ v = 1.3E −04 for L = 150 m . The most spectacular conclusion concerns the influence of kC on the performances ( crit _ Z and crit _ v ) and in particular on the energy of the control input, as indicated by the criteria crit _ u and umax umin . For example, with L = 100 m , H n1 , n2 (s ) and Te = 5 E −03s , the amplitude of the control input is decreased from umax u min = 38 / − 70 with kC = 300 to u max u min =1.35 / − 1 with kC = 1200 .

Fractional Model-based Control of the Diffusive RC Line

165

This means that the line can be exactly reset during TC = 72s by a control input perfectly realistic (see Figure 5.17), which is obviously more interesting in comparison with the time required by the natural relaxation.

Figure 5.17. Excitation and response of the RC line

5.3.7. Conclusion

In classical automatic control, in order to control with feedback a physical system, generally nonlinear, a controller is designed using a linear model of this system, approximated at the operating point. This model, obtained by identification or mathematical approximation, is a transfer function H (s ) (or H (z ) ) called the black-box model which has no physical correspondence with the original system. This mathematical model has to faithfully reproduce the system dynamics, in order to ensure the essential requirements of the closed-loop, such as the gain and phase margins, for robustness and stability objectives [LAN 89].

166

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The RC line has been identified by a fractional model such as H n1 , n2 (s ) , with no mathematical correspondence with the physics of the line, characterized by a PDE. This fractional model can be used in a classical way for the design of a feedback controller if the objective is to control the dynamics of the pseudo-state variables. In fact, this fractional model provides a more relevant information than the integer order model H (s ) (or H (z ) ) because it makes possible to indirectly control the internal system variable v(x, t ) of the RC line by the control of the distributed variable Z (ω , t ) of the fractional model, as demonstrated in this chapter. This is supplementary proof of the relevance of fractional modeling.

PART 2

Stability of Fractional Differential Equations and Systems

Analysis, Modeling and Stability of Fractional Order Differential Systems 2: 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 Stability of Linear FDEs Using the Nyquist Criterion

6.1. Introduction In this chapter, we present a frequency technique for the stability analysis of linear differential systems, either with integer order or fractional order derivatives; moreover, these systems can include time delays. Fundamentally, a linear ordinary differential equation (ODE) (i.e. with integer order derivatives) is stable if the roots of the characteristic polynomial are not situated in the right half complex plane (RHP). Obviously, the stability problem can be solved by calculating these roots. Historically, mathematicians and control engineers preferred techniques avoiding this calculation. Two families of methods, based on the complex variable theory [LEP 80], are available: the Routh–Hurwitz criterion [ROU 77, HUR 95] and the Nyquist criterion [NYQ 32]. Based on polynomial properties, the Routh–Hurwitz criterion indicates the number of RHP roots of the characteristic polynomial (or unstable poles of the ODE) owing to relations between the coefficients of this polynomial. From a practical point of view, this criterion is easy to use, but its proof is difficult to establish [HAN 96]. Moreover, for a stable ODE, it does not quantify its stability degree. The Nyquist criterion basically deals with closed-loop systems. Based on a curve in the complex plane representing the graph of the open-loop transfer function and on a contour including the RHP, it indicates the number of unstable poles of the closed-loop transfer function. This criterion is easy to use (especially in its

170

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

simplified version) and relatively easy to prove using the complex variable theory [LEP 80, GIL 90]. Moreover, it allows the quantification of the stability degree of the closed-loop (at the origin of robust design techniques); on the contrary, it does not apply to systems in open-loop. The stability of nonlinear ODEs is analyzed with another approach, the Lyapunov technique [LYA 07, LAS 61, KRA 63]. Fundamentally, it is connected to the state space model of the differential system: this system is stable if a particular function of the state variables, representing its “energy”, is decreasing when there is no external excitation. Although these three techniques are very classical, it is interesting to recall them in order to situate the stability of fractional differential equations (FDEs) (i.e. with fractional order derivatives) in a general framework. First of all, similarly to the integer order case, the stability of linear FDEs relies on the hypothesis that the roots of their characteristic polynomial are not situated (see Chapter 7 of Volume 1) in the RHP [OUS 95a, BON 00]. Basically, FDEs are infinite dimensional systems; thus, the number of roots of their polynomial is also infinite: stability analysis by calculation of these roots is not a simple task in the general case [DAS 13]. Moreover, the usual stability results rely on a restrictive modeling of FDEs: the basic hypothesis deals with commensurability, i.e. the fractional derivative orders have to be an integer multiple of a minimal fractional order. Thanks to this hypothesis, some stability results are available, mainly based on Matignon’s theorem [MAT 98, SAB 08a]. On the contrary, despite some interesting results [RAP 15, RAP 16], the case of non-commensurate order FDEs remains an open problem. The case of time-delayed differential systems has long been investigated with integer order derivatives [DUG 97, NIC 97, RIC 03]. Two main approaches can be considered, the first one being derived from the Routh–Hurwitz criterion, and the second one from the Lyapunov technique. Because the characteristic polynomial includes time delays, the number of its roots is infinite. Stability analysis is based on frequency techniques, which indirectly determine the number of unstable roots by a generalization of the Routh–Hurwitz criterion [NIC 97]. The Lyapunov approach uses an extended quadratic function of the state variables, according to a principle formulated by Krasovskii [KRA 63]. The case of linear FDEs with time delays remains an open problem, with some existing results, such as the works of [HOT 98, BON 00, HWA 06]. Fundamentally, our approach is based on a closed-loop principle for any kind of system governed by a linear ODE or FDE [TRI 09c, MAA 09, SAB 13].

Stability of Linear FDEs Using the Nyquist Criterion

171

This closed-loop principle [BEN 08a] is derived from the analysis of the analog simulation scheme of any differential system in the controller canonical form [KAI 80]. This simulation scheme reveals a true closed-loop system, whose stability corresponds to the stability of the considered ODE or FDE. Thus, system stability can be analyzed using the Nyquist criterion. With this methodology, it is possible to give an alternate demonstration of the Routh–Hurwitz criterion for ODEs, besides with a quantification of the stability degree. In the case of FDEs, it is possible to generalize ODEs results, however, with a severe limitation due to the complexity of analytical results. The stability of time-delayed differential systems, either based on ODE or FDE models, can also be analyzed with this approach, which appears to be very general. Moreover, it is important to note that no particular modeling is required in the definition of FDEs. After a recall of FDE definitions and simulation methodology, the principle of the stability technique is presented in section 6.2. This method is applied to ODEs in section 6.3 and to FDEs in section 6.4; it is generalized to ODEs with time delays in section 6.5, and finally to FDEs with time delays in section 6.6. 6.2. Simulation and stability of fractional differential equations 6.2.1. Simulation of an FDE Consider the general linear FDE whose transfer function is

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

[6.1]

The fractional differentiation orders

m1 < m2 <  < mN

[6.2]

are real positive numbers; they are called external or explicit orders. It is necessary to define internal or implicit derivation orders such as (see Chapter 1 of Volume 1):

n1 = m1  ni = mi − mi −1  nN = mN − mN −1

[6.3]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The classical controller canonical state space form [KAI 80] is presented in Figure 6.1 with x1 (t ) = x(t ) x2 (t ) = Dn1 ( x1 (t ) )  xi (t ) = Dni −1 ( xi −1 (t ) ) 

[6.4]

xN (t ) = DnN −1 ( xN −1 (t ) )

DnN ( xN (t ) ) = − a0 x1 (t )  −aN −1 xN (t ) + u (t ) = −r (t ) + u (t ) = ε (t )

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

[6.5]

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

6.2.2. Stability of the simulation scheme

The simulation diagram in Figure 6.1, which exhibits a direct transfer composed of integration operators I ni ( s ) and a feed-back signal r (t ) composed of the sum of the weighted states ai −1 xi (t ) , with the error signal ε (t ) = u (t ) − r (t ) , explicitly refers to a closed-loop system, which is necessarily subject to a stability condition.

Of course, the stability condition of this feedback system corresponds to the 1 stability of the differential system characterized by the transfer function , A( s) i.e. to the stability condition of the considered FDE.

Stability of Linear FDEs Using the Nyquist Criterion

173

Figure 6.1. Closed-loop simulation of an FDE with fractional integrators

These considerations are not new: analog computer users were perfectly aware of this stability problem (see Chapter 1 of Volume 1) which became implicit with digital computers. Consequently, the explicit formulation of this simulation stability problem, with an open-loop transfer function H OL ( s) , provides a method for the stability analysis of H ( s) using the Nyquist criterion [NYQ 32, GIL 90]. Let us define H OL ( s) as N

H OL ( s ) =

R( s) = ε (s)

a i =1

i −1

X i ( s)

[6.6]

ε ( s)

where X i (s) =

1 ε (s) s ni + ni+1 ++ nN

[6.7]

Some elementary calculations provide H OL ( s ) =

a

s

0 n1 ++ nN

+

a

s

i −1 ni ++ nN

+

aN −1 s nN

[6.8]

It is easy to verify that

A( s ) = s mN (1 + H OL ( s) )

[6.9]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

and thus H ( s) =

s

mN

B( s) (1 + H OL ( s ))

[6.10]

The well-known term 1 + H OL ( s ) explicitly refers to the stability of a closed-loop system, although H ( s) corresponds to an open-loop FDE (or ODE).

6.2.3. Stability analysis of FDEs using the Nyquist criterion

As the previous simulation scheme refers to a closed-loop system with an open-loop transfer function H OL ( s ) , it is possible to analyze the stability of the considered FDE (or ODE) using the Nyquist criterion [NYQ 32, GIL 90]. Strict application of the Nyquist criterion is based on the definition of a D contour in the RHP. First of all, it is important to note that H OL ( s ) has only poles in s = 0 ; the particular form of H OL ( s) excludes unstable poles. Thus, the contour has to be modified to exclude the poles in s = 0 . Then, with this modified D contour (see Figure 6. 2), the net number of rotations of H OL ( s) around the (− 1 + j 0 ) point (critical point) is strictly equal to the number of unstable poles of H (s) in the RHP.

Figure 6.2. D contour excluding the origin

Stability of Linear FDEs Using the Nyquist Criterion

175

If the FDE or ODE is stable (see Figure 6.3), this net number of rotations is equal to zero, i.e. if H OL ( s) does not encircle the critical point (see [BEN 08a] for more details with FDEs). The application of the generalized Nyquist criterion provides the number of unstable poles (see Chapter 7 of Volume 1). Nevertheless, if we are only interested in the stability of the FDE, a simplified version of the Nyquist criterion can be used. It can be formulated as: if the Nyquist diagram H OL ( jω ) , plotted from ω = 0 to ω → +∞ , is situated on the “right side” of the critical point ( −1 + j 0 ) , then the system H ( s) (or the FDE) is stable; else it is unstable. Moreover, the distance between the Nyquist diagram and the critical point provides information on the stability degree (phase and gain margins) of the FDE (ODE).

Figure 6.3. Example of the Nyquist contour for a stable ODE

6.3. Stability of ordinary differential equations 6.3.1. Introduction

The principle of the stability analysis based on the Nyquist criterion is exposed thanks to a third-order ODE example. First, it is necessary to define the general

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

open-loop transfer function and to present the technique used to draw its approximate Nyquist graph. 6.3.2. Open-loop transfer function

Let us define

H (s) =

b0 + b1 s + .... + bM s M B( s) = a0 + a1 s + .... + aN −1 s N −1 + s N A( s)

M ≤N

[6.11]

which is the transfer function of a N th -order linear ODE. It is interesting to note that we obtain this transfer function from the model of the general FDE with N derivatives [6.1, 6.3] simply by imposing

n

i

= 1

∀

i = 1 to N

[6.12]

According to Lord Kelvin’s principle [THO 76], the simulation of this ODE 1 requires N integrators I ( s ) = . s As with an FDE, we obtain a closed-loop system where the active elements are the N integrators and the an parameters. The bm parameters have no effect in this closed-loop: as it is well known, this means that the stability of H ( s) depends only on the transfer

1 . A( s)

Thanks to the principle previously presented, H OL ( s) transfer function can be expressed as H OL ( s ) =

a0 sN

+

a1 s

N −1

+... +

ai s

N −i

+...+

aN −1 s

[6.13]

It is straightforward to verify that 1 + H OL ( s ) =

A( s ) sN

[6.14]

Stability of Linear FDEs Using the Nyquist Criterion

177

and B( s) s (1 + H OL ( s))

H (s) =

[6.15]

N

6.3.3. Drawing of H OL ( jω ) graph in the complex plane

The objective is to present a technique to draw an approximate graph of H OL ( jω ) . H OL ( s) is composed of N terms such as

ai s

N −i

, i.e. of N directions

ai in ( jω ) N −i

the complex plane. Let us define ai = ρi e jθi ( jω ) N −i

[6.16]

then

ρ

i

=

ai

[6.17]

ω N −i

and

θ i = −( N − i )

π 2

+ θ

sign(ai )

[6.18]

where

θ θ

sign(ai ) = 0 sign(ai ) = −π

if if

ai > 0 ai < 0

[6.19]

The corresponding graph is a ray centered at the origin and characterized by the π polar angle θ i . Since θ i is an integer multiple of , these rays coincide with the 2 real or imaginary coordinate axes.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The graph of H OL ( jω ) is the combination of these N elementary graphs. It is important to note that two of these directions are asymptotes: – if ω → 0 , then H OL ( jω ) behaves like the higher power element in ω , i.e. a0 sN

[6.20]

which is the low frequency asymptote. – if ω → ∞ , then H OL ( jω ) behaves like the lower power element in ω , i.e. aN −1 s

[6.21]

which is the high frequency asymptote. Thus, the graph of H OL ( jω ) is constrained by these two asymptotes: this is a simple way to draw an approximate graph of H OL ( jω ) . 6.3.4. Stability of the third-order ODE 6.3.4.1. Definition

Let H ( s) =

b0 a0 + a1 s + a2 s 2 + s 3

[6.22]

Then H OL ( s ) =

a0 a1 a2 + + s3 s 2 s

[6.23]

The two asymptotes are a0 s3

for ω → 0

[6.24]

Stability of Linear FDEs Using the Nyquist Criterion

179

and a2 s

for ω → ∞

[6.25]

If a0 , a1 and a2 are positive coefficients, the corresponding approximate graph of H OL ( jω ) is presented in Figure 6.4.

Figure 6.4. Open-loop graph of a third-order ODE

6.3.4.2. Stability condition

The stability condition can be deduced from the simplified Nyquist criterion, based on the position of H OL ( jω ) graph relatively to the critical point ( −1 + 0 j ) . Let us define H ( jω ) = X (ω ) + j Y (ω )

[6.26]

with X (ω ) = −

a1

ω2

and Y (ω ) =

a0

ω3

−

a2

ω

[6.27]

The ODE is at the limit of stability if the H OL ( jω ) graph is on the critical point.

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The condition X = −1 provides

ω 2 osc = a1

[6.28]

(oscillation frequency) whereas Y = 0 provides the parameter condition for oscillation

a0 = a1a2

[6.29]

According to Figure 6.4, the system is stable if X < −1 when Y = 0

[6.30]

This leads to

a0 < a1a2

[6.31]

which is of course the stability condition provided by the Routh–Hurwitz condition [ROU 77]. Moreover, it is interesting to note that the graph of H OL ( s) in the Nichols chart provides the stability margins, i.e. information on the dominant poles of H (s) (see Figure 6.5).

Figure 6.5. Nichols chart of the third-order ODE

Stability of Linear FDEs Using the Nyquist Criterion

181

6.3.4.3. Complete stability criterion

The Routh–Hurwitz criterion indicates that H ( s) is stable if all the an parameters are strictly positive with a0 < a1a2 . Is it possible to obtain all these conditions with the proposed approach? Of course, the Nyquist criterion is able to give a satisfactory answer to this question. For example, consider the a1 parameter and the graphs in Figure 6.4. If a1 = 0 , the graph of H OL ( jω ) is situated on the imaginary axis, and H ( s) is unstable. If a1 < 0 ,

a1 and H OL ( jω ) graph are in the RHP; thus, H (s) is unstable. s2

The case of the a0 parameter is interesting. Consider a0 < 0 : it is necessary to apply the complete Nyquist procedure in this case to obtain all the pertinent information. The complete graph of H OL ( s) in the complex plane is obtained using the excluding contour in the RHP (see Figure 6.2). With this contour, we obtain the graph presented in Figure 6.6. The closed-loop curve representing H OL ( s) encircles the critical point only once: this means that H ( s) has only one unstable pole. This pole is necessary real; thus, the corresponding impulse response is divergent.

Figure 6.6. Open-loop Nyquist’s graph of the third-order ODE

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6.3.5. Conclusion

The application of the proposed approach to ODEs provides the same results as the Routh–Hurwitz criterion. A particular feature of this methodology is that it is mainly graphical: we exhibited, with the third-order ODE example, the importance of the H OL ( jω ) approximate graph constructed on the basis of directions in the complex plane. Thus, the proofs of stability are obviously more simple to demonstrate than with the Routh–Hurwitz technique [HAN 96]. However this simplicity does not exclude rigor if necessary: the application of the complete Nyquist procedure provides all necessary information, for instance the number of unstable poles. Moreover, the graph of H OL ( jω ) in the Nichols chart makes it possible to quantify the stability degree of the ODE, which is a supplementary information compared to the simple Routh–Hurwitz approach. 6.4. Stability analysis of FDEs 6.4.1. Introduction

The stability of FDEs with one and two derivatives is investigated. Some analytic results for systems with N fractional derivatives are given, and a general numerical approach is proposed. 6.4.2. Drawing of H OL ( jω ) graph in the complex plane

H OL ( jω ) given in [6.8] is the sum of N elementary terms such as a

( jω )

i ni +1 + ...+ nN

[6.32]

Let us define

a

( jω )

i ni +1 + ...+ nN

= ρi e jθi

[6.33]

Stability of Linear FDEs Using the Nyquist Criterion

183

Then

ρ

i

=

a

ω

i ni +1 + ...+ nN

[6.34]

and

θ i = −(ni +1 + ... + nN )

π 2

+ θ

sign(ai )

[6.35]

As described previously, the corresponding graph is a ray centered at the origin, characterized by the polar angle θ i . The difference with the ODE case is that the values of θ i are fractional multiples of

π 2

.

The graph of H OL ( jω ) is the combination of these N elementary graphs. Two directions are also asymptotes – If ω → 0 , then H OL ( jω ) behaves like a

s

0 n1 + ...+ nN

[6.36]

which is the low frequency asymptote. – If ω → ∞ , then H OL ( jω ) behaves like aN −1 s nN

[6.37]

which is the high frequency asymptote As in the ODE case, H OL ( jω ) is constrained by these two asymptotes: this is a simple way to draw an approximate graph. Consider for example the graph of Figure 6.7 where all the ai parameters are positive.

184

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 6.7. Open-loop approximate graph

6.4.3. Stability of the one-derivative FDE

Let us consider the system

H (s) =

b0 a0 + s n

[6.38]

with one fractional derivative. The stability results for this simple system are well known [OUS 95a]; therefore, it is interesting to compare them with our approach. In this case a0 sn

H OL ( s ) =

[6.39]

and

H OL ( jω ) = X (ω ) + jY (ω )

[6.40]

with

X (ω ) = − Y (ω ) = −

a0

ωn a0

ωn

cos n sin n

π 2

π 2

[6.41]

Stability of Linear FDEs Using the Nyquist Criterion

185

As N = 1 , H OL ( jω ) is composed of only one ray characterized by the polar angle

θ = −n

π 2

+ θ sign(a0 )

[6.42]

The graph of H OL ( jω ) is presented in Figure 6.8 for different values of n with a0 > 0 .

Figure 6.8. Open-loop graph of the one-derivative FDE

The simplified Nyquist criterion indicates that the system is unstable for n ≥ 2 . Obviously, this result can be established analytically. The stability limit corresponds to X = −1

Y =0

[6.43]

The condition Y = 0 implies −

 π sin  n  = 0 ω  2 a0

n

which is obtained with n = 2 .

[6.44]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The condition X = −1 provides the oscillation frequency

ω 2osc = a0

[6.45]

The system remains stable if Y < 0 when X = −1

[6.46]

Thus, −

 π sin  n  < 0 ω  2 a0

n

[6.47]

which leads to n < 2 . When a0 is negative, it is preferable to use the complete Nyquist criterion. Using the excluding contour of Figure 6.2, the corresponding graph of H OL ( s) is presented in Figure 6.9 for 0 < n < 1 .

Figure 6.9. Open-loop Nyquist’s graph of the one-derivative FDE

As H OL ( s) encircles the critical point only once, H ( s) has a real pole in the RHP, and the system is unstable by divergence (see Chapter 7 of Volume 1). Thus, we can conclude as follows.

Stability of Linear FDEs Using the Nyquist Criterion

187

If a0 > 0 : – for 0 < n < 1 , the system is stable and aperiodic; – for 1 < n < 2 , the system is stable and underdamped oscillatory; – for n = 2 , the system is unstable and has two poles on the imaginary axis; – for 2 < n < 3 , the system is unstable and has two poles in the RHP. If a0 < 0 , the system is unstable and has one real pole in the RHP. 6.4.4. Stability of the two-derivative FDE 6.4.4.1. Definitions

Let us consider H (s) =

b0 s m2 + a1 s m1 + a0

[6.48]

with n1 = m1

[6.49]

n2 = m2 − m1 In this case H OL ( s ) =

a0 a + n12 s n1 + n2 s

[6.50]

The graph of H OL ( jω ) is the combination of two directions: – a low frequency asymptote a0 ( jω )

n1 + n2

ρ0 =

a0

ω

n1 + n2

θ 0 = − (n1 + n2 )

π 2

+ θ sign(a0 )

[6.51]

– a high frequency asymptote a1 ( jω )

n2

ρ1 =

a1

ω

n2

θ1 = − n2

π 2

+ θ sign(a1 )

[6.52]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6.4.4.2. Graphical analysis of stability

Consider a0 > 0 and a1 > 0 with 0 < n1 < 1 , 0 < n2 < 1 and n1 + n2 > 1 , we obtain the graph presented in Figure 6.10.

Figure 6.10. Open-loop graph of the stable FDE

Of course, we can conclude that H ( s) is stable. With 0 < n2 < 1 and 2 < n1 + n2 < 3 , we obtain the different graphs presented in Figure 6.11.

Figure 6.11. Open-loop graph of the conditionally stable FDE

Stability of Linear FDEs Using the Nyquist Criterion

189

In this case, the system is conditionally stable. REMARK 1.– It is important to note that these results have been obtained with an elementary graphical approach. 6.4.4.3. Stability condition

Let H OL ( jω ) = X ( jω ) + jY ( jω )

[6.53]

with X (ω ) =

a0

ω

Y (ω ) = − (

cos(n1 + n2 )

n1 + n2

a0

ω

1 n + n2

π

+

a1

2 ω

sin(n1 + n2 )

π

n2

cos n2

a1

π 2

π

[6.54]

+ sin n2 ) 2 ω n2 2

The stability limit corresponds to X = −1 and Y = 0 . The condition Y = 0 provides the oscillation frequency

ω

n2 osc

a =− 0 a1

sin(n1 + n2 ) sin n2

π

π 2

[6.55]

2

n2 Since ω osc is necessarily positive, it implies that

n1 + n2 > 2 when a0 > 0 and a1 > 0

[6.56]

Finally, putting ω osc in X = −1 , we obtain the parametric condition between, a0 , a1 , n1 and n2 , i.e.

a0

ω cos(n1 + n2 ) n1 osc

π

+ a1 cos n2

π 2

n2 = − ωosc

[6.57]

2

The system (with a0 > 0 , a1 > 0 ) remains stable if Y < 0 when X = −1 or equivalently

X < −1 when Y = 0

[6.58]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Using one of these two conditions, we can analytically calculate the inequality that a0 , a1 , n1 , n2 have to verify. n2 > 0 is fulfilled with REMARK 2.– Note that when a0 > 0 and a1 < 0, the condition ωosc 0 < n1 + n2 < 2 .

REMARK 3.– With ODEs, only the ai parameters have an influence on stability. On the contrary, with FDEs, the fractional orders ni also have an influence on stability, which can be discussed graphically. As an example, consider the case a0 > 0 , a1 > 0 , 0 < n2 < 1 and n1 ≥ 2

[6.59]

When n1 = 2 , the low frequency asymptote is opposite to the high frequency one, as presented in Figure 6.12. The graph of H OL ( jω ) is situated on this common line: the system is unstable. When n1 > 2 , the graph of H OL ( jω ) is situated in the upper sector corresponding to the two asymptotes (see Figure 6.13) and the system is unstable.

Figure 6.12. Instability caused by n1 = 2

Stability of Linear FDEs Using the Nyquist Criterion

191

Figure 6.13. Instability caused by n1 > 2

This analysis can be generalized to n2 , and thus we obtain a fundamental result stated as follows: the fractional system H (s) is unstable ∀ ni ≥ 2 . 6.4.4.4. Example

Consider the case a0 > 0 and a1 < 0 : with a second-order ODE, this value of a1 leads to instability. However, with an FDE, the stability depends on the respective values of n1 and n2 . Consider 0 < n1 < 1 and 0 < n2 < 1 : as a1 < 0 , the direction

a1 reverses and the s n2

corresponding graph is presented in Figure 6.14.

Figure 6.14. Two-derivative FDE with a1 < 0

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Obviously, the stability is conditional in this case. Consider n1 = n2 = 0.5 , the oscillation frequency is provided by: 0.5 ωosc =−

a0 a1

2

[6.60]

We deduce that a0 =

a12 2

[6.61]

when H OL ( jω ) = −1 . Moreover, stability is ensured if a0 >

a12 2

[6.62]

Consider a1 = − 1 , the stability is ensured by a0 > 0.5

[6.63]

For a0 = 0.7 , we present the corresponding Nichols chart in Figure 6.15 and the step response in Figure 6.16.

Figure 6.15. Nichols chart with a0 = 0.7

Stability of Linear FDEs Using the Nyquist Criterion

193

Figure 6.16. Step response with a0 = 0.7

We can note that it is possible to quantify the damping of the step response, thanks to the stability margins in the Nichols chart. Finally, for a0 = 0.4 , we verify that the system is unstable: see the step response in Figure 6.17.

Figure 6.17. Step response with a0 = 0.4

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6.4.4.5. Conclusion

We can summarize the following stability conditions for the two-derivative FDE, obtained graphically or analytically: If a0 < 0 , H ( s) is unstable (as noted previously). If a0 > 0 and a1 > 0 – for 0 < n1 + n2 < 2 , the system is unconditionally stable; – for 0 < n1 < 2 and 0 < n2 < 2 with n1 + n2 > 2 , the system is conditionally stable; – for n1 ≥ 2 or n2 ≥ 2 , the system is unstable. If a0 > 0 and a1 < 0 – for 0 < n1 < 1 and 0 < n2 < 1 , the system is conditionally stable; – for 0 < n1 < 1 and 1 < n2 < 2 , the system is conditionally stable; – for 1 < n1 < 2 and 0 < n2 < 1 , the system is conditionally stable; – for 1 < n1 < 2 and 1 < n2 < 2 , the system is unstable.

6.4.5. Stability of the N-derivative FDE

Finally, we consider the general transfer function [6.1]. It is difficult to provide general results due to the difficulty of the analytical calculations. However, it is possible to give some general rules, as the following one obtained graphically. The graph of H OL ( jω ) is constrained by the two asymptotes already presented in Figure 6.7. Consider the case ai > 0 ∀ i and −(n1 + ... + nN )

π 2

>−π

[6.64]

then the low frequency asymptote is situated under the real axis, and the system is unconditionally stable.

Stability of Linear FDEs Using the Nyquist Criterion

195

Therefore, we can formulate some simple rules: – if a0 < 0 , the system is unstable; – if ai > 0 ∀ i and

N

 n < 2 , the system is unconditionally stable; i

i =1

– if ai > 0 ∀ i and ni ≥ 2 ∀ i , the system is unstable. However, despite the lack of analytical results, it is always possible to numerically analyze the stability with the proposed frequency approach. The drawing of H OL ( jω ) in the Nichols chart gives essential information on system stability and on its stability degree thanks to the stability margins. 6.4.6. Conclusion

We demonstrated that the proposed approach to system stability applies to noncommensurate order FDEs, generalizing the Routh–Hurwitz criterion. Based on the Nyquist criterion approach, it provides information concerning stability, number of unstable poles and degree of system stability. Another important feature of this method is its graphical interpretations, which provide very simple demonstrations. Despite the complexity of analytical formulations in the general case, this method can be numerically implemented with any linear fractional system, providing information on its stability and its degree of stability. 6.5. Stability analysis of ODEs with time delays 6.5.1. Introduction

The previous technique is applied to time-delayed systems. Therefore, it is necessary to test its ability to analyze the stability of time-delayed ODEs before considering the fractional order case. The general problem is formulated, and an example is investigated for comparison with well-known results.

196

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6.5.2. Definitions

Many definitions are related to the different systems with time delays [DUG 97, NIC 97, RIC 03]. In this chapter, we are only interested in systems described by the following transfer function: H (s) =

B( s) A( s )

[6.65]

with

A( s) = s N + (aN −1 + ad N −1 e −τ N −1s ) s N −1 +  + (a1 + ad1 e −τ1s ) s + (a0 + ad0 e −τ 0 s )

[6.66]

This system is simulated using a controller canonical form as shown in 1 Figure 6.1, where I ni ( s ) = I ( s ) = and where each weighted state ai −1 xi (t ) is s replaced by ai −1 xi (t ) + adi −1 xi (t − τ i −1 ) . Thus, we obtain a0 + ad 0 e−τ 0 s a1 + ad1e−τ1s + + sN s N −1 a + ad N −1e −τ N −1s  + N −1 s −τ i s N −1 N −1 a + ad e =  i N −ii =  H i ( s) s i =0 i =0 H OL ( s ) =

[6.67]

6.5.3. Stability analysis

As described previously, H OL ( s ) has only poles in s = 0 . Thus, because in a first step we are only interested in stability, we will apply the simplified Nyquist criterion. H OL ( jω ) is composed of N terms such as H i ( jω ) =

ai + adi e−τ i jω ( jω ) N −i

[6.68]

Stability of Linear FDEs Using the Nyquist Criterion

197

H i ( jω ) is the sum of two basic elements

1)

ai

( jω )

N −i

with modulus

ai

ω

N −i

and polar angle −( N − i )

π 2

+ θ sign ( ai ) . Its graph is

a ray centered at the origin and characterized by its polar angle. 2)

ad i

( jω )

−( N − i )

π 2

N −i

e − jωτ i

with

modulus

adi

ω N −i

and

polar

angle

− ωτ i + θ signe ( adi ) .

For ω = 0 , its graph belongs to a ray similar to the previous one; on the contrary, for ω → +∞ , its graph is a spiral centered at the origin caused by the rotation of the polar angle due to the time delay. The graph of H i ( jω ) is the composition of these two basic elements. It depends on: – the respective signs of ai and adi ; – the absolute values of ai and adi . Specifically, if ai >> ad i , then the graph is close to that of the first term, and if ai a . Thus, the graph of H OL ( jω ) is close to that of

b − jωτ e . jω

Equation [6.71] has a solution:  a  

ωτ = arccos  −  b

[6.72]

For this value of ω , the system is stable if: X (ω ) > −1 when Y (ω ) = 0 . Simple calculations give

τ < τ lim

 −a  arccos    b  = 2 b − a2

Thus, the system is conditionally stable.

[6.73]

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Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Sector 2 is characterized by b < a . Thus, the graph of H OL ( jω ) is close to that of

a . jω

The condition Y (ω ) = 0 can be fulfilled only when X (ω ) → 0 (with ω → +∞ ). Thus, it means that the Nyquist diagram of H OL ( jω ) is always situated on the “right side” of the critical point: the system is always stable, independently of the value of the time delay τ . Applying this method to each sector, we finally obtain the results expressed in Figure 6.19.

Figure 6.19. Stability regions in the {a, b} plane

Of course, this stability result is identical to the results presented in [NIC 97, RIC 03]. 6.6. Stability analysis of FDEs with time delays 6.6.1. Definitions We are interested in linear FDEs with time delays whose transfer function B( s) H (s) = is a generalization of the integer order case: A( s )

A( s) = s mN + (aN −1 + ad N −1e −τ N −1s ) s mN −1 +  + (a1 + ad1e −τ1s ) s m1 + (a0 + ad0 e −τ 0 s )

[6.74]

Stability of Linear FDEs Using the Nyquist Criterion

201

with m1 < m2 <  < mN −1 < mN

[6.75]

and where the internal derivative orders ni correspond to the definition [6.3]. Then, the open-loop transfer function H OL ( s) is expressed by N −1

H OL ( s) =  i =0

ai + adi e−τ i s N −1 =  H i ( s) s ni+1 ++ nN i =0

[6.76]

6.6.2. Stability

As in the integer order case, we will consider the position of the Nyquist diagram H OL ( jω ) with respect to the critical point. H OL ( jω ) is composed of N terms such as H i ( jω ) =

ai + adi e−τ i jω ( jω ) ni+1 ++ nN

[6.77]

H i ( jω ) is the sum of two basic elements: 1)

a

( jω )

i ni +1 ++ nN

with

modulus

a

ω

i ni +1 ++ nN

and

polar

angle

π

+ θ sign ( ai ) . Its graph is a ray centered at the origin and 2 characterized by its polar angle; −(ni +1 +  + nN )

2)

ad i

( jω )

ni +1 ++ nN

−(ni +1 +  + nN )

π 2

e − jωτ i

with

modulus

ad

ω

i ni +1 + + nN

and

polar

angle

− ωτ i + θ sign ( adi ) .

For ω = 0 , its graph belongs to a ray similar to the previous one; on the contrary, for ω → +∞ , its graph is a spiral centered at the origin.

202

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The graph of H i ( jω ) is the composition of these elementary graphs, as in the integer order case. Finally, the graph of H OL ( jω ) results from the composition of the N elementary graphs: the system is stable if the Nyquist diagram lays on the “right side” of the critical point.

6.6.3. Application to an example

In order to test this method, we will apply it to a system where the stability condition is already known, thanks to another approach. This example is the following, given by [HWA 06]. Its characteristic polynomial A(s) is A( s ) =

( s)

2

+ K a1 s e −

τs

+ K a0 e −

τs

[6.78]

This example deals with the commensurate order hypothesis: then, n1 = n2 = 0,5. Moreover, the time delay is also fractional of the form e− τ s . However, this non-usual form is not an obstacle to the use of the method (with the values a0 = a1 = 1 ). Then H OL ( s ) =

Ke − s

τs

+

Ke− τ s s 0.5

[6.79]

which can be expressed as H OL ( s) = H1 ( s ) H 2 ( s )

[6.80]

with

H1 ( s) = K e−

τs

[6.81]

and H 2 ( s) =

1 1 + s s 0.5

[6.82]

Stability of Linear FDEs Using the Nyquist Criterion

203

The graph of H 2 ( jω ) lies between two asymptotes: – a ray centered at the origin, with polar angle equal to −

ω =0; – a ray centered at the origin, with polar angle equal to −

ω → +∞ .

π 2

π 4

, corresponding to

, corresponding to

The transfer function is characterized by: H1 ( jω ) = ρ1 e jθ1 with ρ1 = K e and θ1 = −

−

2 ωτ 2

2 ωτ . 2

ρ1 decreases from K to 0 , whereas θ1 varies from 0 to −∞ , when ω varies from 0 to +∞ . The graph of H1 ( jω ) is a spiral centered at the origin, and the Nyquist diagram of H OL ( jω ) is the composition of the two previous graphs. Stability is conditional, according to the value of K . Let us determine the value of K ( K lim ) corresponding to the situation where the Nyquist diagram goes through the critical point. Therefore, H OL ( jω ) = ρ e jθ with ρ = 1 and θ = −π .

ω is a solution of the equation  2+ ω  2 τ ω + arctan  =π 2 ω   whereas K is obtained from the condition ρ = 1 (with the previous value of

[6.83]

ω ).

For τ = 1 s , we obtain ω = 3.071 and K lim = 21,51 (of course, the same value as provided by [HWA 06]).

204

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The diagrams corresponding to K = { 5;10;15; K lim } are plotted in the Nyquist plane (see Figure 6.20).

Figure 6.20. Nyquist’s diagram for different values of K

We can verify that the stability is ensured for K < 21, 51 . Moreover, the distance between H OL ( jω ) and the critical point makes it possible to quantify the stability degree of this time-delayed FDE.

7 Fractional Energy

7.1. Introduction The application of Lyapunov’s method is based on the definition of a positive quadratic function, related to energy [LYA 07, LAS 61, KRA 63, NAS 68]. For linear integer order systems, the choice of this function V (t ) is an elementary problem. Because V (t ) should be a quadratic function of state x(t ) (or X (t ) ), the choice V (t ) = x 2 (t ) or as a generalization V (t ) = X (t ) P X (t )T with P > 0 is straightforward [KAI 80, KHA 96]. Many researchers have used the same definition of V (t ) in the fractional order case [AGU 14]. Unfortunately, x(t ) (or X (t ) ) is only a pseudo-state vector and

V (t ) = x 2 (t ) is a positive semi-definite function which cannot be used as a Lyapunov function, as will be demonstrated using an elementary example. In fact, the true state of the fractional integrator is the distributed variable z (ω , t ) ( Z (ω , t ) for a multidimensional system). Therefore, V (t ) has to be a quadratic function based on z (ω , t ) . Thus, the definition of V (t ) is no longer an elementary problem as in the integer order case. We previously a priori defined V (t ) as

V (t ) =

+∞

 μ (ω ) n

z 2 (ω , t ) d ω

[7.1]

0

Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach, First Edition. Jean-Claude Trigeassou and Nezha Maamri. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

206

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

in the monovariable case [TRI 11b]. Is it possible to validate this choice using an energetic interpretation? This is a main objective of this chapter. It will be demonstrated that equation [7.1] corresponds within a factor to the energy stored in a fractional integrator. In order to prove this definition of fractional energy, we will use an electrical analog [HAR 15a] which will also be used to define the associated dissipation of energy, which is a specific feature of fractional systems. Moreover, we will raise a fundamental question: is fractional energy a simple mathematical function or a true energy? Comparisons between the energies stored in the infinite length RC line and the fractional integrator (as well as between integer and fractional order integrators) will make it possible to provide a conclusive response to this important question. Finally, the energy stored in an elementary fractional differential system will be defined using both distributed representations, i.e. the closed-loop and open-loop forms. More precisely, it will be proved that this energy is expressed as the energy stored in the associated integrator in the closed-loop form, whereas a specific expression will be established for the open-loop form. These two equivalent expressions of fractional energy will be used in the stability analysis of commensurate order fractional systems. 7.2. Pseudo-energy stored in a fractional integrator

The objective of this section is to demonstrate that the function x 2 (t ) does not correspond to an energy in the case of the fractional integrator.

We will use the signal i(t ) of Figure 7.1 as an input of the fractional integrator

Figure 7.1. Fractional integrator input. For a color version of the figures in this chapter see www.iste.co.uk/trigeassou/analysis2.zip

1 . sn

Fractional Energy

207

This signal i(t ) is realized as the sum of delayed step signals as shown in Figure 7.2.

Figure 7.2. Realization of the input

Therefore, the signal i(t ) is defined as  I1 H (t ) for 0 ≤ t < T  i (t ) = i1 (t ) + i2 (t ) = I1 H (t ) + I 2 H (t − T ) for T ≤ t < 2T i (t ) + i (t ) + i (t ) = I H (t ) + I H (t − T ) + I H ( t − 2T ) for 2T ≤ t 2 3 1 2 3 1

This signal i(t ) (see Figure 7.3).

[7.2]

is applied to the input of the fractional integrator

Figure 7.3. Fractional integrator

Thus, the input is i(t ) , and the output is v(t ) such as v( s) 1 = with 0 < n < 1 i(s) s n

[7.3]

208

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Note that the analogy with a fractional capacitor will be justified in section 7.3. Thus, we can define v (t ) at different instants t : – for 0 ≤ t < T i ( s) =

therefore v( s) =

I1 s

I1 tn v t = I1 H (t ) ( ) and Γ(n + 1) s n +1

(t − T ) tn I1 H (t ) + I 2 H (t − T ) – for T ≤ t < 2T v(t ) = Γ(n + 1) Γ(n + 1) n

We want v (2T ) = 0 . Therefore

I1 ( 2T )

n

Γ(n + 1)

+

I 2 ( 2T − T ) Γ(n + 1)

n

=0

which implies

I 2 = −2n I1 – finally, for 2T ≤ t , we want i (t ) = 0 therefore I1 + I 2 + I 3 = 0

i.e. I 3 = I1 ( 2n − 1) Consequently: v(t ) =

I1 t n − 2n ( t − T )n + ( 2n − 1) ( t − 2T )n   Γ ( n + 1) 

[7.4]

We have determined v ( t ) considering that the fractional integrator is submitted

to the different excitations i1 ( t ) , i2 ( t ) and i3 ( t ) for t ≥ 2T . However, we can also

Fractional Energy

209

interpret v ( t ) for t ≥ 2T as the result of the excitation i(t ) of Figure 7.1, i.e. the 1 for t ≥ 2T , starting from sn = 0 with the initial state z (ω , 2T ) , because i ( t ) = 0 for t ≥ 2T .

output v ( t ) represents the free response of v ( 2T )

Of course, we have already demonstrated that the free response does not depend on the pseudo-state v ( 2T ) = 0 , but on the internal state z (ω , 2T ) , which is not

equal to zero regardless of the frequency ω .

1 2 v (t ) . 2 Therefore, we can define a pseudo-energy E ( t ) for the fractional integrator, defined

For an integer order integrator, the energy E ( t ) is expressed as E ( t ) =

similarly as 1 E ( t ) = v 2 (t ) 2

[7.5]

In Figure 7.4, we present the response v ( t ) and the pseudo-energy E ( t ) for n = 0.5 .

Figure 7.4. Pseudo-energy of the fractional integrator

210

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Of course, we note that E ( 2 T ) = 0 since v ( 2T ) = 0 . Furthermore, there is an increase of E ( t ) for t ≥ 2T . This increase of E ( t ) is not coherent with the initial value of E ( t ) at t = 2 T . Thus, we can conclude that E ( t ) does not correspond to the true energy stored in

1 . sn

We know from Chapter 6 of Volume 1 that x ( t ) does not represent the true state

of the system. Thus, it will be necessary to express the true energy E ( t ) with the distributed variable z (ω , t ) instead of x ( t ) .

REMARK 1.– We can formulate a theoretical interpretation to the physical inconsistency of pseudo-energy E ( t ) . As v(t ) =

+∞

 μ (ω ) z (ω , t ) n

dω

[7.6]

0

the function +∞  1 1 E (t ) = v 2 (t ) =   μn (ω ) z (ω , t ) d ω  2 20 

2

[7.7]

is equal to zero for any combination of the components of z (ω , t ) such that +∞

 μ (ω ) z (ω , t ) n

dω = 0 .

0

It is what occurs at t = 2T thanks to the special input i ( t ) of Figure 7.1. Because there is an infinity of possibilities to obtain v ( t ) = 0 for any value of t , this means

that the pseudo-energy E ( t ) is a positive semi-definite quadratic form; thus, it cannot represent an energy nor be used as a Lyapunov function in the fractional order case [NAS 68, KHA 96]. Consequently, it is necessary to define a new expression for the energy, based on the distributed variable z (ω , t ) .

Fractional Energy

211

7.3. Energy stored and dissipated in a fractional integrator 7.3.1. Introduction

An elementary energy can be defined as de(ω , t ) = z 2 (ω , t ) d ω

[7.8]

in the frequency band d ω , and the global energy E (t ) can be defined as the weighted integral of e (ω , t ) , i.e. E (t ) =

+∞



μ n (ω ) e (ω , t ) d ω =

0

+∞

 μ (ω ) n

z 2 (ω , t ) d ω

[7.9]

0

This is the definition of the Lyapunov function V (t ) = E (t ) which has been successfully used in several papers dealing with stability [TRI 11b, TRI 13b, TRI 13c, TRI 14]. Nevertheless, this definition can be considered as arbitrary because we can ask some questions: +∞

– Is the choice

 z (ω , t ) 2

d ω realistic?

0

– Is the weighting function μ n (ω ) necessary? – Does V (t ) possess the same physical properties as the energy stored in a capacitor, i.e. in the integer order integrator? In order to provide satisfying answers to these questions, we will interpret the fractional integrator as a frequency distributed network { R (ω ) , C (ω )} fed by a current i ( t ) and providing a voltage v ( t ) , according to Figure 7.5. 7.3.2. Electrical distributed network

The structure of the distributed network cannot be chosen arbitrarily, because it 1 has to fit perfectly to the distributed equations of integrator n . s

212

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Therefore, consider an infinity of elementary parallel

{ R (ω ) , C (ω )}

cells,

connected in series, according to the diagram in Figure 7.5 [HAR 15a, HAR 15b].

Figure 7.5. Distributed electrical network

Here, C (ω ) and R (ω ) are resistor and capacitor densities at frequency ω . In

this analogy, i ( t ) is the input and v ( t ) is the output. i ( t ) and v ( t ) have to verify the distributed model of the integrator

1 , with the internal variable z (ω , t ) : sn

 ∂z (ω , t )  ∂t = −ω z (ω , t ) + i (t )   +∞ v(t ) = μ n (ω ) z (ω , t ) d ω 0 

[7.10]

According to the network structure, v ( t ) is the sum of the elementary voltages

v (ω , t ) , therefore

v(t ) =

+∞

 v (ω , t )

dω

[7.11]

0

For the elementary cell, we can write i(s) =

v (ω , s )

Z RC (ω , s )

[7.12]

Fractional Energy

213

where Z RC (ω , s ) is the complex impedance: Z RC (ω , s ) =

R (ω )

1 + R (ω ) C (ω ) s

[7.13]

Therefore v (ω , s ) =

1 C (ω ) + s

i (s) 1

[7.14]

R (ω ) C ( ω )

The Laplace transform of equation [7.10] gives z (ω , s ) =

i (s)

[7.15]

s +ω

and v( s ) =

+∞

 μ (ω ) z (ω , s ) n

dω

[7.16]

0

According to [7.11] v( s) =

+∞

 v (ω , s )

dω

[7.17]

0

Therefore, equations [7.16] and [7.17] are identical if

μ n (ω ) z (ω , t ) = v (ω , t )

[7.18]

Then, using [7.14] and [7.15], we obtain

μ n (ω )

i (s)

s +ω

=

1 C (ω )

s+

i (s) 1

[7.19]

R (ω ) C (ω )

The equality [7.19] is satisfied if

ω=

1 R ( ω ) C (ω )

[7.20]

214

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

and

μ n (ω ) =

1

[7.21]

C (ω )

7.3.3. Stored energy

The interest of the previous analogy is to allow the calculation of the energy stored in the fractional integrator thanks to the energy stored in the elementary capacitors C (ω ) . Let e (ω , t ) be the energy density stored in the elementary cell: 1 e (ω , t ) = C (ω ) v 2 (ω , t ) 2

[7.22]

As μ n (ω ) z (ω , t ) = v (ω , t ) (see [7.18]), we can write 1 1 e (ω , t ) = C (ω ) μn2 (ω ) z 2 (ω , t ) = μn (ω ) z 2 (ω , t ) 2 2

[7.23]

The elementary energy de (ω , t ) stored in the band d ω is expressed as de (ω , t ) = e (ω , t ) d ω

[7.24]

Thus, the total energy E ( t ) stored in the integrator is E (t ) =

+∞

+∞

1  de (ω, t ) = 2  μ (ω ) n

0

z 2 (ω , t ) d ω

[7.25]

0

This expression of the energy corresponds within a factor to the a priori definition [7.1]. This electrical analogy proves that E ( t ) has to be weighted by μ n (ω ) . Note +∞

that the expression

 z (ω , t ) 2

d ω would be independent of the order n , which

0

obviously would exhibit some inconsistency!

Fractional Energy

215

7.3.4. Power dissipated in the fractional integrator

The electrical analogy also reveals that an electrical power is dissipated in the distributed resistors R (ω ) . Using equations [7.20] and [7.21], we obtain R (ω ) =

μ n (ω ) ω

[7.26]

The power density dissipated in R (ω ) is expressed as

p (ω , t ) =

v 2 (ω , t )

[7.27]

R (ω )

As v (ω , t ) = μ n (ω ) z (ω , t ) [7.18], we obtain p (ω , t ) = ω μ n (ω ) z 2 (ω , t )

[7.28]

therefore dp (ω , t ) = ω μ n (ω ) z 2 (ω , t ) d ω

[7.29]

where dp (ω , t ) is the Joule power dissipated in the frequency band d ω . Finally, the total power P ( t ) dissipated in

P(t ) =

+∞

 0

dp (ω , t ) =

+∞

ω

1 is defined as sn

μ n (ω ) z 2 (ω , t ) d ω

[7.30]

0

We note that the energy storage E ( t ) in the fractional integrator is necessarily

associated with a power loss P ( t ) : this is an essential feature of the fractional integrator.

We will demonstrate in the chapters devoted to Lyapunov stability (Chapters 8–10) that this power loss increases the damping of system dynamics, i.e. it improves the natural stability of fractional order systems.

216

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

7.3.5. Energy storage 7.3.5.1. Introduction

The infinite length

{ R, C}

line has already been studied in Chapter 6 of

Volume 1. It was demonstrated that the

{

R, C} line is equivalent to a fractional

integrator with the order n = 0.5 . Basically, we can calculate the stored energy in this line thanks to physics. Furthermore, equation [7.25] provides the value of the energy stored in the equivalent fractional integrator. The comparison of these two values will allow the validation of equation [7.25].

{

The infinite length { R, C} line is composed of an infinity of elementary cells

Rdx, Cdx} , according to the diagram in Figure 7.6.

Figure 7.6. Infinite length RC line

R and C are densities per unit length, such as: R=

dR ( x ) dx

C=

dC ( x ) dx

[7.31]

Let us recall that the distributed variables v ( x, t ) and i ( x, t ) verify the diffusion equation:  ∂ 2 v ( x, t ) ∂v ( x, t ) = RC   ∂x 2 ∂t   i ( x, t ) = − 1 ∂v ( x, t )  R ∂x

[7.32]

Fractional Energy

217

with the boundary condition at x = 0 :

 i ( 0, t ) = i(t )  v(0, t ) = v(t )

[7.33]

We have proved that the

{ R, C}

line is equivalent to the fractional integrator

(see Chapter 6 of Volume 1): v ( x, s ) =

1 s 0.5

R i ( x, s ) C

[7.34]

7.3.5.2. Energy stored in the RC line

Consider a Dirac excitation of the

{

R, C} line

i ( 0, t ) = i ( t ) = I δ ( t )

[7.35]

Therefore i ( 0, s ) = I

[7.36]

and v ( 0, s ) =

1 s 0.5

R I C

[7.37]

As (see Chapter 6 of Volume 1) v ( x, s ) =

1 s 0.5

R I e− x C

sRC

[7.38]

and using R = 1 and C = 1 as a simplification, we obtain v ( x, s ) =

1 I e− x s 0.5

s

[7.39]

218

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Therefore v ( x, t ) = L−1 {v ( x, s )} =

1

e

π t

−

x2 4t

[7.40]

For each elementary cell, the stored energy density is expressed as 1 C v 2 ( x, t ) 2

e ( x, t ) =

[7.41]

Therefore, for the total stored energy:

E (t ) =

1 2

+∞

2  C v ( x, t ) dx = 0

1 C I2 2

+∞

 0

(

 − e  

x2 4t

π t

   

)

2

2

dx

[7.42]

and finally [HAR 15a] CI 2

E (t ) =

8π t

=

I2 8π t

with C = 1

[7.43]

We can also calculate E ( t ) with equation [7.25]. It is necessary to calculate z (ω , t ) , which is the solution of the differential equation [7.10]. Therefore z ( ω , t ) = e − ω t ∗ I δ (t ) = I e − ω t

[7.44]

Then E (t ) =

1 2

+∞



μn (ω ) I 2 e −2ωt d ω =

0

+∞ I 2 sin ( nπ ) − n −2 ω t 0 ω e dω π 2

[7.45]

As (Appendix A.1, Chapter 1 of Volume 1) +∞

y 0

α − ay

e

dy =

Γ (α + 1) aα +1

[7.46]

Fractional Energy

219

we obtain E (t ) =

I 2 sin ( nπ ) Γ (1 − n ) π 2 ( 2t ) 1− n

[7.47]

Finally, with n = 0.5 E (t ) =

I2

[7.48]

8π t

As equations [7.43] and [7.48] are identical, this equality proves that the energy stored in a fractional integrator is equal to the energy stored in an infinite length { R, C} line. Obviously, this equality validates equation [7.25]. 7.3.6. Integer order and fractional order integrators 7.3.6.1. Integer order integrator

The integer order integrator is characterized by the equation 1 v ( s ) = i(s) s

[7.49]

An operational amplifier realization of this integrator corresponds to the diagram in Figure 7.7 [AUV 80, CHA 82].

Figure 7.7. Integer order integrator

220

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

In this system

ve (t )  i (t ) = −ie (t ) = − R   vs (t ) = v(t )

[7.50]

Basically, this electronic system corresponds to the charge of a capacitor C with a current source i(t ) , according to the diagram in Figure 7.8.

Figure 7.8. Equivalent electrical circuit

Therefore i (t ) = C

dv(t ) dt

[7.51]

This equation corresponds to [7.49] with C = 1 . REMARK 2.– We can note that capacitor C replaces the electrical network { R (ω ) , C (ω )} of Figure 7.5, i.e. we can propose a realization of the fractional integrator using an operational amplifier and the

{ R (ω ) , C (ω )}

according to the diagram in Figure 7.9.

Figure 7.9. Fractional order integrator

electrical network,

Fractional Energy

221

The fractional integrator is characterized by the electrical equations: ve (t )  i (t ) = − ie (t ) = − R   vs (t ) = v(t )

[7.52]

i (t ) = D n ( v (t ) )  v (t ) = I n ( i (t ) )

[7.53]

i.e.

7.3.6.2. Energy stored in the integer order integrator with a step input excitation

Let us consider a step input excitation of the integer order integrator i (t ) = I H (t )

[7.54]

Therefore, equation [7.51] provides v(t ) =

+∞

 0

i (τ ) C

dτ + v(0)

[7.55]

i.e. v(t ) =

It +v(0) C

[7.56]

Then, with v(0) = 0 , we obtain v(t ) = 1 It 1 (I t) C  = 2 C 2 C 2

E1 (t ) =

It C

and

2

[7.57]

where E1 (t ) is the energy stored for n = 1 . Then, with C = 1 E1 (t ) =

1 2 (I t) 2

[7.58]

222

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

7.3.6.3. Energy stored in the fractional order integrator with a step input excitation

Consider the step input excitation i (t ) = IH (t ) . Then, z (ω , t ) is the solution of the differential equation: ∂z (ω , t ) = −ω z (ω , t ) + IH (t ) ∂t

[7.59]

Therefore z (ω , t ) = e −ω t ∗ IH (t )

[7.60]

and z (ω , t ) =

I

(1 − e ) H (t ) −ω t

[7.61]

ω

Thus

I2 En ( t ) = 2

+∞

 μ (ω ) n

(1 − e )

−ω t 2

ω

0

2

+∞ I 2 sin ( nπ ) (1 − e dω = 0 ω n+ 2 π 2

)

−ω t 2

dω

[7.62]

This integral is calculated using the integration by parts technique. Consider J (t ) =

+∞

 0

(1 − e ) −ω t

ω n+2

2

dω

[7.63]

Let us define u ( t ) = (1 − e−ω t )

2

[7.64]

and dv =

dω ω n+2

[7.65]

Fractional Energy

223

Then

J (t ) =

+∞ +∞ −2ω t  2t  e −ω t e d ω − dω    n +1 n +1  n +1  0 ω ω 0 

[7.66]

As +∞

y

α −a y

e

dy =

Γ (α + 1)

0

[7.67]

aα +1

we obtain J (t ) =

2t n +1 Γ ( −n ) 1 − 2n  n +1

Finally, as Γ (α + 1) = α Γ (α ) and Γ (α ) Γ (1 − α ) =

[7.68]

π (see Chapter 1 sin (απ )

of Volume 1), we obtain En ( t ) = ( 2 n − 1)

I 2 t n +1 Γ ( n + 2)

Let us recall that E1 ( t ) =

[7.69]

1 22 I t , for the integer order integrator. 2

Thus, if n → 1 in equation [7.69], we obtain: lim En ( t ) = ( 21 − 1) n →1

I 2 t1+1 I 2 t1+1 = = E1 (t ) Γ ( 3) 2

[7.70]

This result means that the energy stored in the fractional integrator corresponds to the energy stored in the integer order integrator, which can be interpreted as a particular case of the fractional integrator. Moreover, equation [7.69] demonstrates that the energy stored in regardless of the finite value t , as in the integer order case.

1 is finite sn

224

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

This stored energy, or fractional energy, is characterized by the same physical features as in the integer order case. However, there is an essential difference: energy dissipation is necessarily associated with energy storage in the fractional order case, in contrast to the integer order case. 7.3.6.4. Numerical simulation of stored energy

The fractional integrator is simulated thanks to a frequency discretization (see Chapter 2 of Volume 1):  dz j (t ) = −ω j z j ( t ) + i (t ) for j = 0 to J   dt  J  v(t ) =  c j z j (t )  j =0

[7.71]

After discretization, equation [7.25] becomes E (t ) =

1 J  c j z 2j (t ) 2 j =0

[7.72]

Our objective is to verify that whether this equation can provide a correct approximation of the fractional energy. Thus, we simulate the response of the integrator with a step input excitation i (t ) = IH (t ) , where the exact value of En (t ) is known: En ( t ) = ( 2 n − 1)

I 2 t n +1 Γ ( n + 2)

[7.73]

Then, we can compare En ( t ) to the numerical approximation [7.72], performed with the simulation parameters ωb = 10−3 rd / s , ωh = 103 rd / s , J = 20 , I = 1 and n = 0.5 . Consequently, consider the graphs in Figure 7.10.

Fractional Energy

225

Figure 7.10. Theoretical and simulated energies

We note the perfect fit between the two curves. 1 with the excitation i ( t ) of sn Figure 7.1, with the same simulation parameters (see the graph in Figure 7.11). Then, we can compute the energy stored in

Figure 7.11. Fractional energy

226

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

It is now obvious that E (2T ) is not equal to zero, in contrast to pseudo-energy

E (t ) . Moreover, this initial value E ( 2T ) explains the decrease of E ( t ) since t = 2T . 7.3.7. Characterization of fractional energy and its dissipation 7.3.7.1. Introduction

Equations [7.25] and [7.30] define respectively energy storage E ( t ) and power

dissipation P ( t ) in the fractional integrator. Comparison with the integer order integrator has exhibited some physical features of fractional energy. However, we can still raise some questions on the distribution of this energy between the different modes ω and also on the relation between energy storage and its dissipation. 7.3.7.2. Modal distribution of energy storage

What happens to En ( t ) with a step input excitation when t → +∞ ? Of course, lim En ( t ) = ∞ as well as lim E1 ( t ) = ∞ . t →∞

t →∞

However, what is the distribution of stored energy between the different modes as t → +∞ ? +∞ I 2 sin ( nπ ) (1 − e Let us recall [7.62]: En ( t ) = 0 ω n + 2 π 2

)

−ω t 2

dω .

Moreover, we can note that lim (1 − e −ωt ) = 1 . t →∞

Therefore, let us define

En ,ωb ( ∞ ) =

+∞ I 2 sin ( nπ ) 1 I 2 sin ( nπ ) 1 ω d = 2 n +  π π 2 2 ( n + 1) ωbn +1 ωb ω

[7.74]

It is straightforward to verify that lim

ωb → 0

En,ωb (∞ ) = ∞

as predicted by equation [7.73].

[7.75]

Fractional Energy

227

However, we can note that En ,ωb ( ∞ ) represents the energy stored in the modes

ω ranging from ω = ωb to ω → ∞ as t → ∞ . Equation [7.74] demonstrates that this energy is finite. Moreover, if ωb is increased, En ,ωb ( ∞ ) decreases, i.e. the high frequency modes provide a negligible contribution to the stored energy. In fact, En ,ωb ( ∞ ) → ∞ as predicted by [7.73], only thanks to the mode ωb = 0 . Obviously, it is the same mechanism as in the integer order case where the only mode is ω = 0 . This is a fundamental property of the fractional integrator. 7.3.7.3. Energy dissipation

Let us recall that power loss P ( t ) is defined by equation [7.30]: P(t ) =

+∞

ω

μn (ω ) z 2 (ω , t ) d ω

[7.76]

0

Let Ed ( t ) be the energy dissipated in the fractional integrator. It is defined by t

Ed (t ) =  P (τ ) dτ

[7.77]

0

What is the value of Ed (t ) with the step input excitation? We know that z (ω , t ) =

I

ω

(1 − e ) −ω t

H ( t ) , therefore

−ω t +∞ I 2 sin ( nπ ) (1 − e ) P(t ) = 0 ω n +1 dω π 2

2

[7.78]

We can calculate this integral using the integration by parts technique. Thus, we obtain P (t ) = 2

I 2t n (1 − 2n −1 ) Γ ( n + 1)

[7.79]

228

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

It is straightforward to verify that P (t ) = 0 for n = 1 , i.e. there is no power loss in the integer order integrator. Then, using [7.77], we can calculate Ed (t ) . Thus, we obtain

Ed (t ) = 2

I 2 (1 − 2n −1 ) Γ ( n + 2)

t n +1

[7.80]

7.3.7.4. Energy balance in the fractional integrator

Let EG (t ) be the energy provided by the generator. Therefore, EG (t ) = En (t ) + Ed (t )

[7.81]

Using equations [7.73] and [7.80], we obtain EG (t ) =

I2 t n +1 Γ ( n + 2)

[7.82]

I2 2 t , i.e. all the stored energy is 2 equal to the energy provided by the generator, since the power loss is equal to zero. If n → 1 , then Ed (t ) → 0 ; therefore, EG (t ) →

On the contrary, if n → 0 En (t ) → 0 , and EG (t ) → I 2 t , i.e. if n → 0 , the fractional integrator does not store energy, and all the energy provided by the generator is dissipated. 7.3.7.5. Fractional integrator auto discharge

After a charge phase with a step input excitation ( i (t ) = IH ( t ) ) , assume that i ( t ) = 0 for t ≥ T .

Let us recall the behavior of the integer order integrator: for 0 ≤ t < T v ( t ) =

It C

[7.83]

Fractional Energy

229

Let us change the time origin at T (for a simplification objective). Then, t ' replaces t − T . At t ′ = 0 ( t = T ), we have v ( 0 ) = v (T ) since i ( t ′ ) = 0 for t ′ ≥ 0 , then C

dv ( t ) dt

= 0 , so v ( t ′ ) = v ( 0 ) = cte .

Similarly, E1 ( t ′ ) =

1 2 Cv ( 0 ) = cte . 2

It is well known that the integer order integrator stores energy from 0 to T , and holds it for t ≥ T . The behavior is completely different in the fractional order case. With the new time origin, and using t for t ' , we can write (for t ≥ T ):  ∂z (ω , t ) = −ω z (ω , t )  ∂t   +∞ v(t ) = μ n (ω ) z (ω , t ) d ω 0 

[7.84]

Thus, we obtain z (ω , t ) = z ( ω , 0 ) e −ω t

[7.85]

where z (ω , 0 ) = z (ω , T ) . and v(t ) =

+∞

 μ (ω ) z ( ω , 0 ) n

e −ω t d ω

[7.86]

0

Thus, ∀ ω , z (ω , t ) → 0 as t → ∞ with a decrease speed depending on the mode ω . For the energy, we can write En (t ) =

1 2

+∞

 μ (ω ) n

0

z 2 (ω , t ) d ω =

1 2

+∞

 μ (ω ) z (ω , 0 ) 2

n

0

e −2ω t d ω

[7.87]

230

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Since e −2ωt → 0 as t → ∞ , energy En ( t ) is decreasing. Is it possible to characterize this decrease? At t = 0 , it is obvious that 1 2

En (0) =

+∞

The derivative

 μ (ω ) n

z 2 (ω , t ) d ω = En (T )

[7.88]

0

dEn ( t ) dt

characterizes the decrease

+∞ dz (ω , t ) dEn (t ) +∞ =  μn (ω ) z (ω , t ) d ω = −  ω μn (ω ) z 2 (ω , t ) d ω [7.89] dt dt 0 0

As P(t ) =

+∞

ω

μn (ω ) z 2 (ω , t ) d ω , we obtain

0

dEn ( t ) dt

= −P (t )

This means that the energy stored previously in the integrator is spontaneously dissipated in resistors R (ω ) ; this is an auto discharge phase. t

We can also write Ed (t ) =  P (τ ) dτ = En ( 0 ) − En ( t ) , i.e. the energy stored 0

during the phase charge (when i (t ) = IH ( t ) ) is completely dissipated inside the integrator when i (t ) = 0 , i.e. during the discharge phase or auto dissipation phase. We can conclude that the fractional integrator is unable to hold the energy stored during the charge phase. This is the main difference between the integer and the fractional order integrators: surprisingly, the fractional integrator (or equivalently the fractional capacitor) is not an energy storage device.

Fractional Energy

231

7.3.8. Fractional energy invariance 7.3.8.1. State-space models of the fractional integrator

The state space model of the fractional integrator is provided by the inverse 1 Laplace transform of n (see Chapter 6 of Volume 1). In fact, this technique only s provides the impulse response of the integrator:

1 hn (t ) = L−1  n  = s 

+∞

 μ (ω ) n

e−ω t d ω for 0 < n < 1

[7.90]

0

which can be interpreted as a state-space model. Moreover, according to the system theory [KAI 80, ZAD 08], the state space model of a system with input i ( t ) and v ( t ) is not unique. Thus, we can define an infinity of state space models, corresponding to the same input/output behavior. For

1 , two models can be privileged: sn

Model no. 1  ∂z (ω , t )  ∂t = −ω z (ω , t ) + i ( t )   +∞ v(t ) = μ n (ω ) z (ω , t ) d ω 0 

[7.91]

Model no. 2  ∂η (ω , t ) = −ω η (ω , t ) + μ n (ω ) i ( t )  ∂t   +∞  v(t ) =  η (ω , t ) d ω  0

[7.92]

The difference between these two models relies on their distributed variables z (ω , t ) and η (ω , t ) , and also on the position of the weighting function: on the output or on the input.

232

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

We are interested in the definition of the energy of each model, in order to verify the invariance principle [JOO 86]. 7.3.8.2. Stored energy of the two models

The stored energy of model no. 1 has been expressed using the distributed network analogy (see sections 7.3.2 and 7.3.3). Thus we have already defined [7.25], Ez (t ) =

1 2

+∞

 μ (ω ) n

z 2 (ω , t ) d ω .

0

For model no. 2, we use the same analogy (see Figure 7.5). The elementary cell is characterized by the equation v(ω , s) =

1

1

C (ω ) s +

1

i (s)

[7.93]

dω

[7.94]

R (ω ) C (ω )

As v(t ) =

+∞

 v (ω , t )

dω =

0

+∞

 η (ω , t ) 0

we obtain v (ω , t ) = η (ω , t )

[7.95]

as

η (ω , s ) =

μ n (ω ) i (s) s +ω

[7.96]

equality [7.95] implies that: 1 C (ω ) s +

1 1

R (ω ) C (ω )

=

μ n (ω ) s +ω

[7.97]

Fractional Energy

233

Therefore, this equality is verified if

ω=

1 1 and C (ω ) = μn (ω ) R (ω ) C (ω )

[7.98]

For the elementary cell 1 2 eη (ω , t ) = C (ω ) v (ω , t ) 2

[7.99]

Thus, the total energy Eη (t ) is defined as Eη (t ) =

1 2

+∞

2  C (ω ) v (ω , t ) d ω = 0

1 2

+∞

1  μ ( ω ) η (ω , t ) 0

2

dω

[7.100]

n

7.3.8.3. Invariance of energy

First, the expressions of E z (t ) and Eη (t ) seem different. In fact, we can express η (ω , t ) as a function of z (ω , t ) . Note that z (ω , s) =

μ (ω ) 1 i ( s ) and η (ω , s ) = n i (s) . s +ω s +ω

Therefore

η (ω , t ) = μ n (ω ) z (ω , t )

[7.101]

thus Eη (t ) becomes

Eη (t ) = 1 = 2

1 2

+∞

+∞

1  μ ( ω ) ( μ (ω ) z (ω , t ) ) n

0

 μ (ω ) n

n

2

dω [7.102]

z (ω , t ) d ω 2

0

i.e.

Eη (t ) = Ez (t ) = E (t )

[7.103]

We would get the same result with any other model; hence, we have revisited a fundamental principle of energy [JOO 86].

234

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Energy is an invariant of the fractional integrator. This principle can be generalized to any fractional system, i.e. energy is independent of the modeling of this system. 7.4. Closed-loop and open-loop fractional energies 7.4.1. Introduction

Energy is a fundamental topic in system stability analysis based on the Lyapunov 1 method. In the integer order case, the energy of the elementary system is s+a 1 2 expressed as x ( t ) , corresponding to the energy stored in the associated 2 integrator. This principle can be generalized directly to the fractional elementary system 1 thanks to the distributed variable z (ω , t ) . The specificity of the fractional sn + a order case is due to the availability of two distributed models, a classical one in a closed-loop form with the variable z (ω , t ) and another one in an open-loop form with the variable ξ (ω , t ) (see Chapter 7 of Volume 1). Consequently, it will be necessary to express its energy with ξ (ω , t ) . Finally, the interest of these equivalent definitions of system energy will be illustrated to express the initialization energy of the Caputo derivative. 7.4.2. Energy of the closed-loop model

Let us recall the definition of the energy of the elementary integer order system 1 which corresponds to the ODE s+a dx(t ) + ax ( t ) = u (t ) dt This model corresponds to the closed-loop diagram in Figure 7.12.

[7.104]

Fractional Energy

235

Figure 7.12. Closed-loop ODE model

x ( t ) is the output (and the state variable) of integrator

v (t ) =

1 . Its input is v ( t ) : s

dx(t ) = −ax ( t ) + u (t ) dt

[7.105]

1 is the unique storage device of this closed-loop system. The s of the ODE corresponds to the energy stored in the integrator,

The integrator energy E1 ( t )

characterized by the state variable x ( t ) . Thus E1 ( t ) =

1 2 x (t ) 2

[7.106]

Usually, this expression is simplified, i.e. this energy is defined as x 2 ( t ) , and it

is improperly said that x 2 ( t ) is the energy of state variable x ( t ) .

In fact, it is necessary to say that E1 ( t ) is the energy stored in the integrator

1 s

associated with the modeling of the ODE. Then, consider the fractional order case and the elementary model corresponding to the FDE:

1 s +a n

236

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

D n ( x ( t ) ) + ax ( t ) = u (t )

[7.107]

This model corresponds to the closed-loop diagram in Figure 7.12, where the 1 1 integrator is replaced by n . s s Therefore, x ( t ) is the output of the fractional integrator characterized by the

distributed internal variable z (ω , t ) . The input of the integrator is v ( t ) : v ( t ) = D n ( x ( t ) ) = −ax ( t ) + u (t )

[7.108]

Consequently, the distributed model of the elementary FDE is:  ∂z (ω , t )  ∂t = −ω z (ω , t ) + u ( t ) − ax ( t ) ω ∈ [ 0; +∞[  +∞  x(t ) =  μ n (ω ) z (ω , t ) d ω  0   sin ( nπ )  μ n (ω ) = 0 < n 0 for a > 0 when system [8.2] is unstable; – Q2 (t ) < 0 for a < 0 when system [8.2] is stable. This means that

dV (t ) < 0 ∀t in the stable case. dt

dV (t ) is the dt sum of two terms with opposite signs. It is this ambiguity that constitutes the fundamental specificity of the Lyapunov technique in the fractional order case.

On the contrary, an ambiguity arises in the unstable case because

Thus, a numerical simulation will highlight the particularities of this problem in a first step, and a theoretical interpretation will be proposed in a second step.

Lyapunov Stability of Commensurate Order Fractional Systems

251

8.2.2. Numerical simulation 1 J  c j z 2j ( t ) 2 j =0 (Chapter 7), is presented for three values of a : a = −1 ; a = 0 and a = 0.4 with n = 0.5 . The case a = 0 corresponds to the simple integrator. This simulation is performed with the following parameters:

The graph of V (t ) , i.e. of its discretized approximation V (t ) =

ωb = 10 −3 rd / s ; ωh = 103 rd / s ; J + 1 = 21 cells and Te = 10 −3 s For each value of a , we use two different initial conditions: – IC1 = z j ( 0 ) = – IC2 =

1

( j + 1)

2

1

( j + 1)

in Figure 8.1;

in Figure 8.2.

Figure 8.1. Comparison of Lyapunov functions with IC1. For a color version of the figures in this chapter see www.iste.co.uk/trigeassou/analysis2.zip

In Figure 8.1, we note that the curve V ( t ) is decreasing ∀ t for a = −1 and

a = 0 . The decrease is less important for the integrator ( a = 0 ). On the contrary, for a = 0.4 , the curve V ( t ) is first decreasing, then increasing.

252

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Figure 8.2. Comparison of Lyapunov functions with IC2

Figure 8.2 corresponds to an initial condition IC2 where the higher frequency components have a lower value in comparison with IC1 . Thus, we observe the same behavior but the previous phenomenon is less important. Finally, in Figure 8.3, we present the graph of the derivative

dV (t ) for a = 0.4 , dt

with IC1 and IC2 .

Figure 8.3. Derivative of the Lyapunov function with IC1 and IC2

Lyapunov Stability of Commensurate Order Fractional Systems

253

dV (t ) is at first negative, then positive in the dt unstable case ( a = 0.4 ) as well as for IC1 or IC2 .

Thus, we demonstrate that

The first conclusion we can deduce from these simulations is that the stability dV (t ) condition < 0 is necessary, but not sufficient. Thus, the necessary and dt sufficient stability condition for fractional system is

dV ( t ) dt

0 ∀ j , i.e. corresponding to z j ( t ) > 0 ∀ j . Then, for the lower values of t , x ( t ) > 0 and a x ( t ) < 0 . The decrease of x ( t ) begins with the higher frequency terms ω j of z j ( t ) . As

ω j >> 1 , the terms −ω j z j ( t ) are highly negative; thus,

dz j ( t )

is highly negative dt (see [8.9]), and the decrease of z j ( t ) , as well as the decrease of x ( t ) , is very fast. When the higher frequency modes have vanished, the lower frequency modes (and

254

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

very low as ω j → 0 ) which are preponderant in x ( t ) remain; however, with a lower value of

dz j ( t ) dt

, which explains the slower decrease of z j ( t ) , i.e. the slower

decrease of x ( t ) . Consequently, the Lyapunov function V ( t ) , which is the weighted sum of

z ( t ) terms, decreases very fast in a first step, then more and more slowly. 2 j

Then, consider a > 0 and an initial condition such as z j (0) > 0 ∀ j , i.e. corresponding also to z j ( t ) > 0 ∀ j . Then, for the lower values of t , x ( t ) > 0 and a x ( t ) > 0 .

The higher frequency modes z j ( t ) impose, as mentioned previously, a fast initial decrease because −ω j z j ( t ) is very negative and impose (see equation [8.9]). Thus, the function V ( t )

dz j ( t )

0 . Therefore, we observe the increase of z j ( t ) , x ( t ) equation [8.9], i.e. dt and V ( t ) . This phenomenon is cumulative and V ( t ) tends to infinity. 8.2.4. Theoretical interpretation

The paradoxical behavior of V ( t ) in the unstable case has been interpreted by an elementary physical reasoning. Hopefully, we can formulate a more rigorous interpretation based on the closed-loop and open-loop representations of system [8.2] (see Chapter 7 of Volume 1): – the closed-loop model uses a fractional integrator variable z (ω , t ) ;

1 with the distributed sn

Lyapunov Stability of Commensurate Order Fractional Systems

255

– the open-loop model does not use a fractional integrator, and its distributed variable is ξ (ω ,t ) , which is different from z (ω , t ) . These two models correspond to the same energy V ( t ) = E ( t ) according to the invariance principle (Chapter 7). The impulse response hn ( t ) of [8.2] is composed of an aperiodic multimode, always stable, and of a pole, only when a > 0 . Thus (see Chapter 7 of Volume 1) +∞  μ n, a (ω ) e −ω t d ω for a < 0    0 hn (t ) =  +∞  μ ω e −ω t d ω + c e r t for a > 0   n,a ( ) 0

[8.10]

with 1

r = an

[8.11]

The state space model of the open-loop representation is expressed as ∂ξ (ω , t )  = −ω ξ (ω , t )  ∂t   dxOL (t ) = r x (t ) for a > 0 OL  dt

[8.12]

Let ξ (ω , 0 ) and xOL ( 0 ) be the initial conditions of ξ (ω ,t ) and x ( t ) . Then +∞  μn , a (ω ) ξ (ω , 0 ) e−ω t d ω for a < 0    0 x(t ) =  +∞  μ ω ξ ω , 0 e −ω t d ω + x 0 e r t for a > 0 ( ) OL ( )   n, a ( ) 0

It is important to note that x(t ) ≠ xOL (t ) .

[8.13]

256

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Because the multimode and the pole are independent, the energy of the open-loop model is the sum of the independent energies. Thus +∞  1 μn , a (ω ) ξ 2 (ω , t ) d ω for a < 0  2 0  E (t ) = V ( t ) =  +∞  1 μ ω ξ 2 ω , t d ω + 1 x 2 t for a > 0 ( ) OL ( )  2  n,a ( ) 2  0

[8.14]

Moreover, according to the invariance principle, we can also write, using the closed-loop model

E (t ) = V ( t ) =

1 2

+∞

 μ (ω ) n

z 2 (ω , t ) d ω ∀ a

[8.15]

0

Therefore, equations [8.14] tell us that V ( t ) is the sum of two independent energies, one necessarily decreasing, depending only on the initial condition ξ (ω , 0 ) , and the other one increasing, which exists only if a > 0 . Then +∞  2 −  0 ω μn,a (ω ) ξ (ω , t ) dω for a < 0 dV ( t )  =  +∞ dt − ω μ ω ξ 2 ω , t d ω + r x 2 t for a > 0 ( ) n,a ( ) OL ( )    0

[8.16]

As the energy of the multimode can be only decreasing:  dV ( t ) dV ( t ) < 0 ∀ t  lim < 0 for a < 0  t →∞  dt dt  dV ( t )  2 lim = rxOL ( t ) > 0 for a > 0  t →∞ dt

[8.17]

Moreover, as lim x ( t ) → xOL ( t ) t →∞

[8.18]

Lyapunov Stability of Commensurate Order Fractional Systems

257

for a > 0 we obtain lim

dV ( t ) dt

t →∞

= rx 2 ( t ) > 0 for a > 0

[8.19]

The ambiguity emphasized with the closed-loop model does not exist with the open-loop model. This second model makes it possible to affirm that system [8.2] is dV ( t ) 0 for a > 0

[8.22]

Note that lim ξ (ω , t ) → 0 ∀ a t →∞

[8.23]

for the open-loop representation, which is not verified for z (ω , t ) (closed-loop representation). In fact, if a > 0 , then z (ω , t ) , which is directly related to x(t ) , increases. However, the term Q2 ( t ) = ax 2 ( t ) > 0 is preponderant compared to the term +∞

Q1 ( t ) = −  ω μn (ω ) z 2 (ω , t ) d ω < 0 as t → ∞ . 0

258

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

REMARK 2.– For a > 0 , lim Q1 ( t ) < 0 , but this limit is not equal to zero. t →∞

Note that

lim

dV ( t )

t →∞

dt

= lim Q1 ( t ) + lim ax 2 ( t ) = lim rx 2 ( t ) since lim ξ (ω , t ) → 0 t →∞

t →∞

t →∞

t →∞

Thus lim ax 2 ( t ) = lim rx 2 ( t ) t →∞

t →∞

Nevertheless, we can conclude that the sign of a x 2 (t ) allows the stability characterization of system [8.2]. This stability analysis, associating closed-loop and open-loop models, will be generalized to fractional systems with N-derivatives in the next sections. 8.3. Lyapunov stability of an N-derivative FDE 8.3.1. Introduction

The one-derivative elementary FDE has been used to exhibit the essential properties of Lyapunov stability in the fractional order case, properties which result from the distributed nature of the state variable z(ω,t). In the monovariable case, the Lyapunov function V(t) corresponds to the energy of the associated fractional integrator. In the multivariable case, how do these different energies have to be associated? In order to give an answer to this question, we recall the integer order case and propose a definition of V ( t ) based on the modal representation, which provides a direct LMI stability condition when the eigenvalues are real. 8.3.2. The integer order case

Consider the autonomous system d X (t ) dt

= AX ( t ) dim ( X ) = N

and the initial condition X ( 0 ) at t = 0 .

[8.24]

Lyapunov Stability of Commensurate Order Fractional Systems

259

Usual definition of V ( t ) is [CHE 84, KHA 96]

V (t ) =

1 T X (t ) P X (t ) 2

[8.25]

V ( t ) is a scalar, i.e. V ( t ) is a quadratic positive definite form if P is a positive

definite matrix (abbreviated notation P > 0 ) [BOY 94]. Let us recall that a symmetrical matrix M is positive definite [GAN 66, KOR 68]: – if all the eigenvalues of M are positive; T

– if any quadratic form V MV associated with M is positive definite.

M is negative definite ( M < 0 ) if its eigenvalues are negative or if V T MV is negative definite. An usual positivity test (or negativity test) is to verify if all the minors of M are positive (or negative). The linear differential system [8.24] is stable if the derivative of its Lyapunov dV ( t ) < 0 , i.e. if this energy is function (or of its generalized energy) is negative, dt decreasing. Thus dV ( t )

d X (t ) 1 d X (t ) 1 T P X (t ) + X (t ) P 2 dt 2 dt dt 1 T 1 T T = X ( t ) A P X ( t ) + X ( t ) PAX ( t ) 2 2 1 T = X ( t ) ( AT P + PA ) X ( t ) 2 1 T = X (t ) Q X (t ) 2

dV ( t ) dt

T

=

[8.26]

is a quadratic form strictly negative definite if the symmetrical matrix

Q = ( AT P + PA) is negative definite.

260

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Thus, we can conclude that the system

d X (t ) dt

= AX ( t ) is stable if the LMI

condition

( A P + PA) < 0 with P > 0 T

[8.27]

is satisfied. This is a well known result. However, we may ask a question: why the scalar product X

T

(t )

X ( t ) has to be

weighted by the matrix P > 0 ? More specifically, why cannot we use P = I which is a particular positive definite matrix? Consider the system ( N = 2 )

d  x1   0  = dt  x2   −a0

1   x1  −a2   x2 

[8.28]

which is stable if a0 and a1 are positive. With P = I , we obtain V (t ) =

x12 x22 + 2 2

[8.29]

then

( A P + PA) = A T

T

+A

1 − a0   a11  0 = = 1 − a0 −2a1   a21

a12  a22 

AT + A is a symmetrical matrix ( a12 = a21 ) .

[8.30]

Lyapunov Stability of Commensurate Order Fractional Systems

Moreover,

a11

a12

a21 a22

it

is

negative

definite

if

its

minors,

i.e.

a11

261

and

= − (1 − a0 ) , are negative. 2

The first minor a11 is equal to zero; thus, AT + A is negative semi-definite: thus, we cannot claim the stability of [8.28]. Why is the choice [8.29] not appropriate, beyond the above conclusion? x12 x2 1 and 2 represent the energy of each integrator associated with [8.28]. s 2 2 However, the sum of these two energies can represent the total energy of this system only if the variables x1 and x2 are independent (or decoupled). In other words, this means that the matrix P has to perform the decoupling of state variables. This result will be demonstrated in the general case of an N-derivative fractional differential system.

Let us note that it is an elementary problem in the integer order case and, consequently, it is not usually discussed. 8.3.3. Lyapunov function of N-derivative systems

Consider the commensurate order FDE

D n ( X ( t ) ) = AX ( t )

dim ( X ) = N 0 < n 0  dt  +∞  y t = ( )  i 0 μn,λi (ω ) ξi (ω , t ) dω + yi,OL ( t ) 

[8.59]

with 1

ri = λi n

[8.60]

For each mode λi , we can write

Vi (t ) =

1 2

+∞

1 2

+∞

 μ λ (ω ) n,

i

ξi2 (ω , t ) d ω if λi < 0

[8.61]

0

or

Vi (t ) =

 μ λ (ω ) n,

i

1 2

ξi2 (ω , t ) d ω + yi2,OL ( t ) if λi > 0

0

[8.62]

and N

V ( t ) = Vi (t )

[8.63]

i =1

Similar to the monovariable case +∞ dVi (t ) = −  ω μn ,λi (ω ) ξi2 (ω , t ) d ω if λi < 0 dt 0

[8.64]

+∞ dVi (t ) = −  ω μn ,λi (ω ) ξi2 (ω , t ) d ω + ri yi2,OL ( t ) if λi > 0 dt 0

[8.65]

or

268

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Since the multimode energy tends to zero, independently of λi , we can write dVi (t ) < 0 ∀ t if λi < 0 dt

[8.66]

and lim t →∞

dVi (t ) = ri yi2,OL ( t ) if λi > 0 dt

We can conclude that

dV (t ) N dVi (t ) = < 0 dt dt i =1

[8.67]

∀ t if all the eigenvalues are

negative. Therefore, expression [8.55] is negative ∀ t if 1 T Y (t ) 2

(A

T d

+ Ad ) Y ( t ) < 0

[8.68]

i.e. if

(A

T d

+ Ad ) = 2 Ad < 0

[8.69]

However, the use of Ad requires the calculation of eigenvalues; thus, it is easier to consider the original closed-loop model. Using lemma of Appendix A.8.1 with k = 0 , the condition [8.68] is equivalent to the condition 1 T X (t ) 2

( A P+ T

PA) X ( t ) < 0

[8.70]

which corresponds to the LMI

(A P + PA) < 0 with P > 0 T

[8.71]

We can conclude that the system

D n ( X (t )) = A X (t ) 0 < n < 1

[8.72]

Lyapunov Stability of Commensurate Order Fractional Systems

269

is stable if its eigenvalues are real and negative, which is traduced by the well-known LMI

(A P + PA) < 0 with P > 0 T

[8.73]

Obviously, we can note that this LMI condition is also the LMI condition of the integer order case. However, this condition gives no information on system [8.72] stability when the eigenvalues are complex, with or without a damped oscillating behavior. Thus, it is necessary to investigate stability with complex eigenvalues. Note that in order to get rid of this problem, specific to the fractional order case, we can impose the LMI condition ( AT P + PA) < 0 with P>0 regardless of the nature of the eigenvalues of A [YUA 13]. Consequently, we obtain a conservative stability condition independent of the fractional order, as demonstrated in the next section. 8.4. Lyapunov stability of a two-derivative commensurate order FDE 8.4.1. Introduction

In section 8.3, we demonstrated that positive real eigenvalues cause the instability of a fractional order system. On the contrary, complex eigenvalues with positive real parts do not necessarily cause instability, as demonstrated in Chapter 7 of Volume 1, considering the eigenvalues and the poles of a two-derivative FDE. This means that the quadratic 1 T form X ( t ) ( AT P + PA) X ( t ) can be positive definite (caused by positive real 2 dV (t ) remains negative, due to the term parts), whereas the derivative dt +∞

−  ω μ n (ω ) Z (ω , t )T P Z (ω , t ) dω which is always negative. 0

It is this specificity of fractional order systems that makes the use of the dV (t ) < 0 ∀ t so delicate. Thus, our objective is to formulate a new LMI condition dt stability condition for N = 2, which will make it possible to get rid of the conservatism highlighted in section 8.3.

270

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Recall that the system is defined as dim( X ) = 2

D n ( X (t )) = A X (t )

 0 with A =  − a0

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

[8.74]

1  − a2 

Once again, the duality closed-loop/open-loop representations will be used to express this stability condition. 8.4.2. State space model of the open-loop representation

The state space model of the two-derivative open-loop representation is characterized by an equivalent aperiodic multimode, which is always stable and by possible poles, which is stable or unstable (see Chapter 7 of Volume 1). Consequently, the state space model is ∂ξ (ω , t ) = −ω ξ (ω , t ) ∂t

[8.75]

for the equivalent multimode and d x OL (t )

= [AOL ] x OL (t )

dt

[8.76]

for possible poles. If the eigenvalues λ1 and λ2 are real:

[AOL ] =  1 r 0

0 r2 

[8.77]

with (see Chapter 7 of Volume 1)

ri = (λi )1 n if λi > 0 and ri = 0 if λi < 0

[8.78]

and if the eigenvalues are complex (and conjugate; see Chapter 7 of Volume 1)

λi = a ± jb = ρ e ± jθ α + jβ

[AOL ] =  

0

0  α − jβ 

[8.79] [8.80]

Lyapunov Stability of Commensurate Order Fractional Systems

271

where α = r cos (ϕ )   β = r sin (ϕ )

[8.81]

with    ρ = 

1

r=ρ n

ϕ= 2

a0 =

a +b

2

θ n

+

2kπ

n b θ = a tan  a

(0 < n < 1)

[8.82]

Let ξ (ω ,0) and x OL (0 ) be the initial conditions at t = 0 . Then  ξ (ω , t ) = ξ (ω ,0) e −ω t    x (t ) = e[ AOL ]t x (0) OL  OL

[8.83]

Thus ∞

x(t ) = x1 (t ) =  μ n,ξ (ω )ξ (ω ,0) e −ω t dω + C e[ AOL ] t x OL (0)

[8.84]

0

8.4.3. State space models of the closed-loop representation

Let us recall that the distributed model of [8.74]

D n ( X (t )) = A X (t ) dim( X (t )) = 2 is expressed as  ∂ Z (ω , t ) = −ω Z (ω , t ) + A X (t )  ∂t  +∞   X (t ) =  μ n (ω ) Z (ω , t ) dω  0

[8.85]

272

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Let Z (ω ,0) be the initial condition at t = 0 . Then (see Chapter 9 of Volume 1)    ∞ Z (ω , t ) = exp t  F (ω , ξ ) dξ  Z (ω ,0)    0  +∞  X (t ) =  μ n (ω ) Z (ω , t ) dω  0 

[8.86]

Using the modal form

D n (Y (t )) = Ad Y (t )

[8.87]

The distributed state space model is  ∂W (ω , t ) = −ω W (ω , t ) + Ad Y (t )  ∂t  +∞   Y (t ) =  μ n (ω )W (ω , t ) dω  0

[8.88]

Let W (ω ,0 ) be the initial condition at t = 0 . Then (see Chapter 9 of Volume 1)    ∞ W (ω , t ) = exp t  Fd (ω , ξ ) dξ  W (ω ,0)    0  +∞  Y (t ) =  μ n (ω ) W (ω , t ) dω  0 

[8.89]

The original model and its modal equivalent are linked by the relations X (t ) = M Y (t ) and Y (t ) = M −1 X (t )

[8.90]

8.4.4. Energy and stability of the open-loop representation

The invariance principle (Chapter 7) tells us that: V (t ) = VOL (t ) = VCL (t )

[8.91]

Lyapunov Stability of Commensurate Order Fractional Systems

273

where VOL (t ) is the energy of the open-loop representation and VCL (t ) is the energy of the closed-loop one. Let us recall that ξ (ω , t ) and x OL (t ) are independent. Thus

V (t ) = VOL (t ) =

∞

1 1 μ n,ξ (ω )ξ 2 (ω , t ) dω + X TOL (t ) X OL (t )  20 2

[8.92]

and +∞

[

]

1 dV (t ) T = −  ω μ n,ξ (ω )ξ 2 (ω , t ) dω + X TOL (t ) AOL + AOL X OL (t ) 2 dt 0

[8.93]

= QOL ,1 (t ) + QOL ,2 (t ) QOL ,1 (t ) < 0

∀ t and lim QOL,1 (t ) = 0 t → +∞

[8.94]

Consider real and complex eigenvalues. 8.4.4.1. Real eigenvalues

– QOL , 2 (t ) = 0 if λ1 and λ2 are real and negative; thus, the system is stable.

r 0  1 T X OL (t )  1  X OL (t ) . If λ1 and/or λ2 are real and positive, 2  0 r2  QOL ,2 (t ) is positive definite or positive semi-definite. Thus, the system is unstable. – QOL,2 (t ) =

8.4.4.2. Complex eigenvalues

QOL,2 (t ) =

[

]

1 H H H X OL (t ) AOL + AOL X OL (t ) = α X OL (t ) X OL (t ) 2

Then – QOL , 2 (t ) < 0 if α < 0 , and the system is stable. – QOL , 2 (t ) > 0 if α > 0 , and the system is unstable.

[8.95]

274

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

As the aperiodic multimode is always stable, system stability only depends on QOL ,2 (t ) . Unfortunately, QOL ,2 (t ) requires pole calculation, which is unrealistic in practice. Therefore, it is necessary to express the stability condition in the context of the closed-loop representation. 8.4.5. Energy and stability of the closed-loop representation

Let us recall that

V (t ) =

1 2

+∞

T  μn (ω ) Z (ω, t ) P Z (ω, t ) dω

[8.96]

0

and +∞

[

]

1 dV (t ) = −  ω μ n (ω ) Z T (ω , t ) P Z (ω , t ) dω + X T (t ) AT P + PA X (t ) 2 dt 0

[8.97]

= Q1 (t ) + Q2 (t )

and for the modal model

V (t ) = +

1 2

+∞

T  μn (ω ) W (ω, t )W (ω, t ) dω

[8.98]

0

+∞

[

]

dV (t ) 1 = −  ω μ n (ω ) W T (ω , t ) W (ω , t ) dω + Y T (t ) AdT + Ad Y (t ) dt 2 0

[8.99]

= Q1 (t ) + Q 2 (t ) '

'

When λ1 and λ2 are real, then

0  2λ AdT + Ad =  1   0 2λ2  Thus, AdT + Ad < 0 if λ1 and λ2 are negative.

[8.100]

Lyapunov Stability of Commensurate Order Fractional Systems

275

We already demonstrated (see section 8.3) that the system is stable if T

A P + PA < 0 with P = PT > 0 . When λ1 and λ2 are complex

λ1, 2 = a ± jb

[8.101]

Since 2a 0  AdH + Ad =    0 2a 

[8.102]

then

[

]

1 Q '2 (t ) = Y H (t ) AdH + Ad Y (t ) = a Y H (t ) Y (t ) 2

[8.103]

Q2' (t ) is negative definite if a < 0 , which corresponds to complex eigenvalues with negative real parts. Then, if a < 0 , as Q1' (t ) < 0 and Q2' (t ) < 0 , we obtain

dV (t )

< 0 ∀ t . Thus, dt the system is stable, such as when the eigenvalues are real and negative.

Thus, as described previously, the system is stable if the LMI condition AT P + PA < 0 with P = PT > 0

[8.104]

is satisfied. Note that this condition is the same as in the integer order case; moreover, it is independent of the fractional order.

dV (t ) is the sum of a negative term ( Q '2 (t ) ) and dt of a positive one ( Q '2 (t ) ); therefore, it is not possible to conclude directly. Thus, it is necessary to define a new LMI condition able to discriminate the different possibilities corresponding to a > 0 . On the contrary, when a > 0,

276

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

8.4.6. Definition of a stability test for a > 0

The objective is to define a stability test for the closed-loop representation (with variable Z (ω , t ) ) based on the open-loop representation. Recall that dV (t ) dt

=Q

OL ,1

(t ) + Q (t )

[8.105]

OL , 2

QOL,1(t ) is the result of the equivalent multimode. Therefore +∞

Q

OL ,1

(t ) = −  ω μ n (ω )ξ 2 (ω,0) e − 2ω t dω

[8.106]

0

Thus lim Q

t →∞

OL ,1

(t ) = 0

[8.107]

even though the system is unstable. REMARK 4.– Recall that +∞

Q1' (t ) = −  ω μ n (ω )W T (ω , t )W (ω , t ) dω 0

Thus, lim Q1' (t ) = 0 only if the system is stable, i.e. if W (ω , t ) → 0 as t → ∞ . t →∞

+∞

T Consequently, Q1 (t ) = −  ω μ n (ω ) Z (ω , t ) P Z (ω , t ) dω tends to zero as t → ∞ 0

only if the system is stable. On the contrary, in the instability case, lim Q1' (t ) = ∞ and lim Q1 (t ) = ∞ . t →∞

t →∞

Lyapunov Stability of Commensurate Order Fractional Systems

277

Let us define ~ dV (t ) dt

=

dV (t ) dt

−Q

OL ,1

(t )

[8.108]

then ~ 2α dV (t ) 1 H (t )  = Q (t ) = X OL OL , 2 dt 2 0

0 X (t ) 2α  OL

[8.109]

Therefore ~ d V (t ) H = α X OL (t ) X OL (t ) dt

~ dV (t ) dt

can be used to characterize system stability as in the integer order case:

–

~ dV (t )

–

~ dV (t )

dt

dt

[8.110]

< 0 if α < 0 , and the system is stable.

> 0 if α > 0 , and the system is unstable.

Obviously, this criterion has no practical interest because it assumes that α is available, i.e. we have previously analyzed the system! However,

~ dV (t )

is a useful indicator that will be used to define a stability dt indicator for the closed-loop representation. Ideally, it would be necessary to define ~ dV (t ) the equivalent of with the closed-loop representation. However, this is not dt possible because Q1' (t ) (and Q1 (t ) ) do no tend to zero as t → ∞ if the system is unstable. A new indicator

dV (t ) dt

 dV~ (t )   is defined using  , i.e. at the limit of stability.  dt   α = 0

278

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Recall that dV (t )

=Q

dt

OL ,1

(t ) + Q (t )

[8.111]

OL , 2

and

dV (t ) = Q1' (t ) + Q 2' (t ) dt

[8.112]

Therefore, Q

As Q

OL ,1

(t ) + Q

OL , 2

H (t ) = α X OL (t ) X OL (t ) ,

we obtain Q

OL , 2

OL , 2

(t ) = Q ' (t ) + Q ' (t )

(t ) = 0

1

[8.113]

2

for α = 0 , i.e. for (see Chapter 7 of Volume 1):

π  nπ  a = a lim = ρ cos  = a0 cos(n ) = k 2  2 

[8.114]

Then

(Q

OL ,1

(t ))α = 0 = Q ' (t ) + 1 Y T (t ) [ 2 alim ] Y (t ) 1

2

= Q1 (t ) + k Y (t ) Y (t ) '

[8.115]

T

therefore  dV~ (t )  dV (t )   = − Q (t )  dt  OL ,1 α= 0 dt  α = 0 1 = Q 1' (t ) + Y T (t ) AdT + Ad Y (t ) − Q 1' (t ) − k Y T (t )Y (t ) 2

(

)

[

]

[8.116]

Let us define ~ dV (t )  dV (t )  1  =  = Y T (t ) AdT + Ad Y (t ) − k Y T (t )Y (t )  dt  dt α =0 2

[

]

[8.117]

Lyapunov Stability of Commensurate Order Fractional Systems

dV (t )

Note that

dt

is equal to

~ dV (t ) dt

279

only for α = 0 .

On the contrary, it is straightforward to test the sign of

dV (t ) dt

.

As

[A

T d

]

+ Ad = 2aI

[8.118]

we obtain

dV (t ) = Y T (t ) [a − k ] Y (t ) = Y T (t ) [a − alim ] Y (t ) dt

[8.119]

Thus: – – –

dV (t ) dt

dV (t ) dt dV (t ) dt

= 0 for a = alim : limit of stability;

< 0 for a < alim : stability; > 0 for a > alim : instability.

This means that the two-derivative system is stable if the matrix

[

]

1 T Ad + Ad − kI 2

is negative definite.

Obviously, the indicator sign as

dV (t ) dt

dV (t ) dt

is not equal to

[8.120] dV (t ) dt

; however, it has the same

when t → ∞ .

Condition [8.120] is not directly usable; it has to be transposed to the closed-loop representation characterized by Z (ω , t ) .

280

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Using the lemma of Appendix A.8.1, we obtain the LMI condition: with P = PT > 0

AT P + PA − 2 k P < 0

[8.121]

where

 nπ  k = a0 cos   2 

[8.122]

This condition depends on the order n of the system, and it does not require prior calculations, as a0 is directly available. REMARK 5.– LMI condition [8.121] has been established for complex eigenvalues. However, is it not invalidated in the case of real eigenvalues? Recall that a0 = λ1λ2 .  nπ It is possible that λ1 < 0 and λ 2 > 0 , so a0 < 0 . As k = ρ cos  2

  with 

ρ = a0 , condition [8.122] is invalidated, i.e. the system is unstable. On the contrary, the situation λ1 > 0 and λ2 > 0 gives a0 > 0 , then

AT P + PA > 0 (with P = PT > 0 ). Thus, the system is declared unstable. However, what is the response to the LMI condition [8.121]? Assume λ1 ≠ λ2 so λ1 < a0 and λ2 > a0 (or reciprocally). Then, λ2 − k > 0 and

1 2

[A

T d

]

+ Ad − kI is not negative, so the LMI condition

AT P + PA − 2kP < 0 (with P = PT > 0 ) is not satisfied.

Lyapunov Stability of Commensurate Order Fractional Systems

Finally, if λ1 = λ2 = λ , then λ =

281

a0 and

 λ1 − k = 0  λ2 − k = 0 Then,

[A 2

1

T d

]

+ Ad − kI is not negative definite, and condition [8.121] is not

satisfied. Thus, we can conclude that the LMI condition is not invalidated by the existence of real eigenvalues. 8.5. Lyapunov stability of an N-derivative FDE ( N > 2 ) 8.5.1. Introduction

For an N-derivative commensurate order FDE, stability can be tested with the LMI condition [8.73]:

AT P + PA < 0

with P = PT > 0

[8.123]

This means that real eigenvalues are negative and that the real part of complex eigenvalues is also negative. Obviously, the condition [8.123] is independent of the fractional order n . On the contrary, when eigenvalues are complex, a positive real part does not necessarily imply instability. For N = 2 , it has been demonstrated that the system remains stable if the LMI

AT P + PA − 2 k P < 0

with P = PT > 0

 nπ is satisfied, where k = ρ cos  2

[8.124]

  and ρ is the modulus of the eigenvalues. 

Is it possible to generalize this LMI condition, with the same formulation, for any value N (number of fractional derivatives)? After statement of the problem in the general case, we propose the treatment of the case N = 3 in order to illustrate a global methodology.

282

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

8.5.2. Problem statement

In order to simplify notations, assume that the system corresponds to the following transfer function: H N (s ) =

1

[8.125]

DN (s )

where

DN (s ) = a0 + a1s n +  + ai s in +  + a( N −1)s ( N −1)n + s Nn ( 0 < n < 1 )

[8.126]

H N (s ) corresponds to the differential system

D n ( X (t )) = A X (t ) dim( X (t )) = N 0 < n < 1

[8.127]

with

 0  0  A=    0 − a0 

         0 0 1   − a1 − a2  − a N −1  1 0

 

0 1

0 0

[8.128]

Assume that the roots of DN (s ) (i.e. the eigenvalues of A ) are all distinct. Then, we can express H N (s ) using a partial fraction expansion:

δ

δ

δ

δ

H N (s ) = n 1 + n 2 ++ n i ++ n N s − λ1 s − λ2 s − λi s − λN

[8.129]

Eigenvalues can be real or complex. Complex eigenvalues can be clustered under the form H 2' (s ) =

1 a0'

+

a1' s n

+ s 2n

[8.130]

Real unstable poles appear when the eigenvalues are real positive, i.e. λi > 0 .

Lyapunov Stability of Commensurate Order Fractional Systems

283

Similarly, oscillatory poles, stable or unstable, can appear for complex eigenvalues, i.e. λi = a ± jb . When all the eigenvalues are real negative, or complex with negative real parts, the system H N (s ) is stable, and it satisfies the LMI condition [8.123]. The true difficulty concerns complex eigenvalues with positive real parts, characterized by the LMI condition T T  nπ A' P ' + P ' A' − 2k ' P ' < 0 , P ' = P ' > 0 , k ' = ρ ' cos  2

  

[8.131]

Obviously, modulus ρ ' of cluster [8.130] is unknown and explicit calculation of

ρ ' is without interest. Thus, we propose an adaptation of LMI condition [8.131] to the particular case N = 3 in the next section. 8.5.3. LMI generalization for N = 3 8.5.3.1. Problem statement

Consider the three-derivative system: H 3 (s ) =

1 D3 (s )

=

1 n

a0 + a1s + a2 s 2 n + s 3n

[8.132]

which can be expressed as a differential system

D n ( X (t )) = A X (t ) dim( X (t )) = 3 0 < n < 1

[8.133]

with

 0 A =  0 − a0

1 0 − a1

0  1  − a 2 

[8.134]

284

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

The polynomial D3 (s ) can be expressed as

(

)(

D3 (s ) = a0' + a1' s n + s 2 n s n + γ

)

[8.135]

If γ < 0 , the system is unstable because the corresponding eigenvalue is positive. Then, a0 < 0 and it is easy to discriminate this situation. Consider γ > 0 (negative eigenvalue); therefore, stability depends only on the

(

)

roots of D2' (s ) = a0' + a1' s n + s 2n . Let us define

 0 A' =  ' − a0 Thus, subsystem

1  − a1'  1 D2'

(s )

[8.136]

is stable if the LMI condition

T

T

A' P ' + P ' A' − 2k ' P ' < 0 , P ' = P ' > 0 where k ' =

 nπ a0' cos  2

[8.137]

  is satisfied. 

The eigenvalues of A' are

λ1' = a + jb ; λ'2 = a − jb

[8.138]

and the eigenvalues of A are

λ1 = a + jb ; λ2 = a − jb ; λ3 = − γ

[8.139]

Let us define k = k ' and consider the modal matrix  a − jb 0 0  a + jb 0 0   1    Nd =   0 a + jb 0  +  0 a − jb 0   − k I 2  0 − γ   0 0 − γ     0

[8.140]

Lyapunov Stability of Commensurate Order Fractional Systems

285

Therefore

a − k N d =  0  0

0 a−k 0

 0  − γ − k  0

[8.141]

This matrix is negative definite if a−k< 0

[8.142]

−γ − k < 0

[8.143]

and

As γ > 0 and k > 0 , then condition [8.137] is satisfied if  nπ a − a0' cos  2

  < 0 

[8.144]

i.e. if the LMI condition (using lemma of Appendix A.8.1) AT P + PA − 2kP < 0 where k =

with P = PT > 0

[8.145]

 nπ  a0' cos  is satisfied.  2 

Unfortunately, a0' is unknown: thus, the LMI [8.145] can be used if it is not necessary to calculate explicitly a0' . 8.5.3.2. Proposed algorithm

From equation [8.135], we obtain the relations

a0 = a0' γ

[8.146]

a1 = a0' + a1' γ

[8.147]

a2 = a1' + γ

[8.148]

286

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Let aˆ1' be an arbitrary value of a1' and [ γˆ , aˆ0' ] be the corresponding values of

γ and a0' satisfying [8.146] and [8.147]. According to these relations, we obtain a0 γˆ

[8.149]

a a1 = 0 + aˆ1' γˆ γˆ

[8.150]

a 0 − γˆ a1 + aˆ1' γˆ 2 = 0

[8.151]

aˆ0' =

i.e.

Then, for an arbitrary value aˆ1' , γˆ is the solution of the polynomial [8.151] with Δ = a12 − 4 a0 aˆ1'

[8.152]

and

γˆ =

a1 ± Δ

[8.153]

2a1'

γˆ can be only real, i.e. Δ has to be positive Δ > 0 implies aˆ1'