Fractional Behaviours Modelling: Analysis and Application of Several Unusual Tools (Intelligent Systems, Control and Automation: Science and Engineering, 101) 3030967484, 9783030967482

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Intelligent Systems, Control and Automation: Science and Engineering

Jocelyn Sabatier Christophe Farges Vincent Tartaglione

Fractional Behaviours Modelling Analysis and Application of Several Unusual Tools

Intelligent Systems, Control and Automation: Science and Engineering Volume 101

Series Editor Kimon P. Valavanis, Department of Electrical and Computer Engineering, University of Denver, Denver, CO, USA Advisory Editors P. Antsaklis, University of Notre Dame, Notre Dame, IN, USA P. Borne, Ecole Centrale de Lille, France R. Carelli, Universidad Nacional de San Juan, Argentina T. Fukuda, Nagoya University, Japan N. R. Gans, The University of Texas at Dallas, Richardson, TX, USA F. Harashima, University of Tokyo, Japan P. Martinet, Ecole Centrale de Nantes, France S. Monaco, University La Sapienza, Rome, Italy R. R. Negenborn, Delft University of Technology, The Netherlands António Pascoal, Institute for Systems and Robotics, Lisbon, Portugal G. Schmidt, Technical University of Munich, Germany T. M. Sobh, University of Bridgeport, CT, USA C. Tzafestas, National Technical University of Athens, Greece

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Jocelyn Sabatier · Christophe Farges · Vincent Tartaglione

Fractional Behaviours Modelling Analysis and Application of Several Unusual Tools

Jocelyn Sabatier IMS Laboratory Bordeaux University Talence, France

Christophe Farges IMS Laboratory Bordeaux University Talence, France

Vincent Tartaglione IMS Laboratory Bordeaux University Talence, France

ISSN 2213-8986 ISSN 2213-8994 (electronic) Intelligent Systems, Control and Automation: Science and Engineering ISBN 978-3-030-96748-2 ISBN 978-3-030-96749-9 (eBook) https://doi.org/10.1007/978-3-030-96749-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book deals with the modelling of fractional kinetics or dynamic behaviours. The designation “fractional behaviour” is actually a shorthand for “long memory powerlaw-type behaviours” which expresses well that the studied behaviours admit in a limited frequency or time domain band a power-law-type response. These behaviours mainly result from physical stochastic processes such as diffusion, adsorption or aggregation and can be found in numerous domains. To model these kinds of behaviours, it seems natural to think to fractional models (also known as non-integer models), models described by differential equations or partial differential equations involving fractional (or non-integer) order derivatives in place of classical derivatives. The reason is simple: a fractional model also exhibits a long memory power-law-type behaviour. But, as shown in this book, these models are doubly infinite as they can been seen as distributed parameter systems (first infinity) and as they are defined on an infinite domain (second infinity). This last infinity implies several drawbacks, recently highlighted in the literature and reminded in this book, such as difficulties to define initial conditions or to characterize internal system properties such as controllability, observability, …. Moreover fractional models are usually obtained from an input-output approach and not from an internal approach. Thus, although remaining convenient and fast-fitting tools, they should not used to draw conclusions on the underlying internal processes that produce fractional behaviours. It is for these reasons that the authors of this book have initiated research aimed at exploring several classes of models capable of capturing fractional behaviours. This research summarised in this book has shown that several classes of models other than fractional model one produce fractional behaviours. This book gives a description of these unusual modelling tools and explains how they avoid some of the weaknesses associated to fractional models. New kernels in integral operators, Volterra equations, nonlinear models and partial differential equations with spatially variable coefficients are thus successively analysed. Several modelling applications are proposed with these new tools and highlight their efficiency. Over time, the authors hope to further improve physical interpretations of these new models in real connection with the modelled phenomena.

v

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Preface

While this book contains some criticisms about the limitations of fractional models, the authors emphasize the importance of both theoretical and applied contributions they induced. This book must therefore be seen as a way of widening the field of possibilities in fractional behaviour analysis and modelling areas. Considering fractional behaviours without being limited to fractional models opens up countless avenues of research in the field of dynamic model analysis, identification, control, supervision… Talence, France

Jocelyn Sabatier Christophe Farges Vincent Tartaglione

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2 Power-Law Type Dynamic Behaviours . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definitions of Power-Law Type Long Memory Behaviours . . . . . . . 2.2.1 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System . . . . . . . . . . . . 2.2.2 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System Exhibiting a Power-Law Type (or Fractional) Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6

7 9 11

3 Fractional Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fractional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fractional Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Frequency Response of Fractional Integration and Differentiation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Fractional Models Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 How to Take into Account Initial Conditions? . . . . . . . . . . . . . . . . . . 3.6.1 Modal Decomposition of a Fractional Order System . . . . . . 3.7 Some Drawbacks Associated to Fractional Models . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 14

4 Introduction of New Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kernels η1ν (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Kernel η2ν (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Kernel η3ν (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 46 47

17 18 22 24 32 37

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Contents

4.5 Kernel η4ν (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 51 52

5 Volterra Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Pseudo State Space Description: A Particular Case of the Volterra Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Volterra-Equation-Based Model for Power-Law Type Long Memory Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 A First Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 A Second Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 A Third Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 A Fourth Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 A Fifth Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55

6 Non-linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Fractional Behaviours and Non-linear Models . . . . . . . . . . . . 6.1.2 Sand Heap Growth: An Example of Fractional Behaviour Produced by a Nonlinear System . . . . . . . . . . . . . . 6.2 Application to Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Random Sequential Adsorption (RSA) . . . . . . . . . . . . . . . . . . 6.2.3 Dynamical Modelling of RSA . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Addition of a Phenomenon of Desorption . . . . . . . . . . . . . . . . 6.2.5 Application on Experimental Data . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69

7 Partial Differential Equations with Spatially Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Prior Art on the Approximation of Fractional Order Integrators and the Resulting Electrical Networks . . . . . . . . . . . . . . . 7.3 Beyond Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Extension to Cauer Type Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Heat Equation with Spatially Variable Coefficients for Power-Law Type Long Memory Behaviour Modelling . . . . . . . . 7.6 Discussions Around Some Other Distributions for Further . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 57 58 59 60 61 62 66 68

71 76 76 78 81 87 89 94 94 97 97 98 101 107 112 113 115

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Contents

ix

Appendix A: Impulse Response of Some Transfer Functions that Exhibit Power-Law Type Long Memory Behaviours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Appendix B: Demonstration of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Introduction

After a period of very intense progression and unity in the field of fractional calculus that lasted more than 40 years, different trends seem now to set in. Here one designates by fractional calculus all the approaches involving fractional differentiation and integration operators in mathematical, applied physics and engineering domains. This situation gives rise to rather harsh publication duels (described at the end of Chap. 3) as some seek to progress along the existing path and criticise (sometimes wrongly [1]) the “weaknesses” of the others’ new ideas that have grasped the limits of fractional operators and try to offer alternatives [2–6]. The topic of this book is modelling of fractional behaviours or in other words power-law type long memory dynamic behaviours. Nowadays there is a confusion or let’s say an implicit link between fractional behaviours (of physical, biological, thermal, … systems origin) and fractional models (a tool to model them). However, there is no proof in the literature of a physical link between a fractional model and a system that exhibits a fractional behaviour. A link is established in some works but infinite spatial dimensions are considered, which question the physical inconsistency of a fractional model [7]. As shown in this book these infinite dimensions are intrinsic to a fractional model as they can be viewed after some mathematical transformations as distributed parameter systems on an infinite spatial domain. This is infinity of space dimension that is at the origin of a set of drawbacks associated to fractional models. Infinite space dimensions generate infinite memory, or equivalently infinitely slow and fast time constants which once again distances fractional models from systems modelled in terms of physical link. All these analysis and drawbacks are described in this book and justify to go further fractional models. Fractional models and fractional behaviours are two different things and even if fractional operators and fractional models remains good fitting tools to capture these behaviours other modelling tools must be found and studied to solve the mentioned drawbacks.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Sabatier et al., Fractional Behaviours Modelling, Intelligent Systems, Control and Automation: Science and Engineering 101, https://doi.org/10.1007/978-3-030-96749-9_1

1

2

1 Introduction

In order to resolve the singularity of the kernels in the definition of fractional integration and differentiation operators (one of the drawbacks) a current trend is to propose new operators with fractional behaviours and without singular kernels. These approaches are currently being discussed a lot [2, 3, 8–12] even though reformulation proposals have been made for some time [13–21]. A contribution to the definition of new kernels for operators with fractional behaviours can also be found in this book. But it does not seek to solve only the problem of singularity. It is shown how it permits to also solve several other drawbacks and how they permit to define models with fractional behaviours when introduced in a state space like description in the place of the differentiation operator. As also shown in this book, introduction of these new kernels may also have an interest in the definition of models in the form of first kind Volterra equation. A first kind Volterra equation can be viewed as a generalisation of the pseudo state space description currently used in the literature. Volterra equations was introduced by Volterra himself in the 1910s. This is 20 years before the paper by Cole and Cole [22], which is perceived by some as the first application of fractional differentiation and models. Volterra equation could thus have been used to model dispersion and adsorption phenomena in dielectrics in [22]. By defining a new class of model involving a Volterra equation, and by choosing appropriate kernels in the Volterra equation, as shown in this book various kinds of fractional power-law type long memory behaviours can be generated. But it is possible to propose tools that deviate much more from the fractional calculus field. The first one analysed in this book is in the form of a non-linear drift free affine in the control. This class of model is deduced from the analysis of a physical phenomenon that exhibits a fractional behaviour: the adsorption phenomena. Adsorption is a surface phenomenon that is exploited in many fields such as gas sensors design [23] or for liquid or gas purification [24]. Adsorption phenomenon is close to a stochastic process called Random Sequential Adsorption (RSA) process, that was deeply studied. Some of these studies proposes justifications for using nonlinear models for RSA kinetic modelling. A nonlinear model and the associated parameters tuning method are thus proposed. A discussion on the ability of the proposed model to capture the power-law kinetics, without exhibiting some of the drawbacks of fractional models, is proposed. This nonlinear model is then modified to take into account the reverse desorption process. The proposed modelling approach is applied to experimental data of CO2 capture. To conclude this book, and also to overcome the drawbacks associated to fractional diffusion equation, it is proposed in the last chapter to use diffusion equations with spatially variable coefficients. This last chapter gives several conditions (among a infinity) on variable coefficients to produce fractional behaviours with the considered class of equations. These conditions are derived from the analysis of Foster and Cauer type networks whose components (resistors and capacitors) values are distributed and linked by particular laws. The most well known law is the geometric one [25], but this chapter shows that an infinity of laws exist, some as effective as the geometric one.

1 Introduction

3

Three authors have contributed to this book. Jocelyn Sabatier is the main author contributing to all the chapters. Christophe Farges and Vincent Tartaglione have contributed to Chaps. 4, 5 and 6.

References 1. Sabatier J (2020a) Fractional-order derivatives defined by continuous kernels: are they really too restrictive? Fractal Fractional 4:40 2. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl 1:73–85 3. Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci 20:763–769 4. Gao F, Yang XJ (2016) Fractional Maxwell fluid with fractional derivative without singular kernel. Therm Sci 20(suppl. 3):S873–S879 5. Losada J, Nieto JJ (2015) Properties of a new fractional derivative without singular kernel. Prog Fract Differ Appl 1:87–92 6. Sabatier J, Farges C, Tartaglione V (2020) Some alternative solutions to fractional models for modelling long memory behaviors. Mathematics 8:196 7. Dokoumetzidis A, Magin R, Macheras P (2010) A commentary on fractionalization of multicompartmental models. Pharmacokinet Pharmacodyn 37:203–207 8. Sabatier J (2020) Non-singular kernels for modelling power law type long memory behaviours and beyond. Cybern Syst 1–19 9. Sabatier J (2020d) Fractional state space description: a particular case of the Volterra equation. Fractal Fractional 4(23) 10. Zhao D, Luo M (2019) Representations of acting processes and memory effects: general fractional derivative and its application to theory of heat conduction with finite wave speeds. Appl Math Comput 346:531–544 11. Fernandez A, Özarslan MA, Baleanu D (2019) On fractional calculus with general analytic kernels. Appl Math Comput 354:248–265 12. Baleanu D, Fernandez A (2019) On fractional operators and their classifications. Mathematics 7:830 13. Lutz E (2001) Fractional Langevin equation. Phys Rev E 64. https://doi.org/10.1103/PhysRevE. 99.052125 14. Pottier N (2003) Aging properties of an anomalously diffusing particle. Phys Stat Mech Appl 317:371–382 15. Viñales AD, Despósito MA (2007) Anomalous diffusion induced by a Mittag-Leffler correlated noise. Phys Rev E 75:042102. https://doi.org/10.1103/physreve.75.042102 16. Viñales AD, Wang K-G, Despósito MA (2009) Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise. Phys Rev E 80:011101. https://doi.org/10.1103/phy sreve.80.011101 17. Sandev T, Chechkin A, Kantz H, Metzler R (2015) Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fractional Calculus and Applied Analysis 18(4):1006–1038 18. Liemert A, Sandev T, Kantz H (2017) Generalized Langevin equation with tempered memory kernel. Physica A 466:356–369 19. Sandev T (2017) Generalized Langevin equation and the Prabhakar derivative. Mathematics 5:66 20. Sandev T, Deng W, Xu P (2018) Models for characterizing the transition among anomalous diffusions with different diffusion exponents. Phys A Stat Mech Appl 51:405002 21. Sandev T, Tomovski Z, Dubbeldam JLA, Chechkin A (2018b) Phys A Stat Mech Appl 52:015101

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1 Introduction

22. Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics—I alternating current characteristics. J Chem Phys 9:341–352 23. Nikolaou I, Hallil H, Conédéra V, Deligeorgis G, Dejous C, Rebiere D (2016) Inkjet-printed graphene oxide thin layers on love wave devices for humidity and vapor detection. IEEE Sens J 16:7620 24. Bonilla-Petriciolet A, Mendoza-Castillo DI, Reynel-Ávila HE (2017) Adsorption processes for water treatment and purification. Springer, Berlin 25. Oustaloup A (1983) Systèmes asservis linéaires d’ordre fractionnaire. Ed Masson.

Chapter 2

Power-Law Type Dynamic Behaviours

2.1 Introduction The designation “fractional behaviour” used in the title of this book is actually a shorthand for “long memory power-law type behaviours” which expresses well that the studied behaviours admit, in a limited frequency or time domain a power-law type response. Moreover “power-law” seems more appropriate than “fractional”, as fractional usually refers to fractional models, that are only one of the solutions among others for modelling power-law type behaviours, and because the “power” can be other than a fractional number (a real number). The term “power-law” comes from the time series analysis field. In the analysis of time series, long memory behaviours can be characterized in terms of their autocorrelation functions [1]. The autocorrelation highlights that the coupling between values of a signal at different times decreases slowly as the time difference increases. The decay of the autocorrelation function can be power-like and so is slower than exponential decay. The concept of power-law type long memory is thus defined for signals in the time series field. The purpose of this section is to extend this concept to models that have output signals exhibiting power law type long memory behaviour. In Sect. 2.2.1, some properties of spectral density of a system output signal, and properties linking the autocorrelation functions of the input signal and the output signal are demonstrated in the general case of a Linear Time Invariant (LTI) model. In Sect. 2.2.2, these properties are particularised to systems that have output signals exhibiting power-law type long memory behaviour, allowing to propose a general definition of a power law type long memory model. In the following, the term “powerlaw type long memory” will often be reduced concisely to “power-law type”.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Sabatier et al., Fractional Behaviours Modelling, Intelligent Systems, Control and Automation: Science and Engineering 101, https://doi.org/10.1007/978-3-030-96749-9_2

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2 Power-Law Type Dynamic Behaviours

2.2 Definitions of Power-Law Type Long Memory Behaviours 2.2.1 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System Let u(t) and y(t) be respectively the input and the output of a dynamical LTI single input - single output model. Let h(t) its impulse response that verifies: y(t) = h(t) ∗ u(t). Input u(t) is assumed to be a white noise and let R y (ξ ) be the output autocorrelation defined by: ∞ y(t + ξ )y(t)dt.

R y (ξ ) =

(2.1)

−∞

Also let S y (ω) be the output power spectral density defined by: ∞ S y (ω) =

R y (ξ )e− jωξ dξ.

(2.2)

−∞

The autocorrelation function R y (ξ ) of the system output y(t) is related to the auto∞ correlation function Ru (ξ ) = −∞ u(t + ξ )u(t)dt of the system input u(t) through the relation: ∞ ∞

∞ u(t − p)h( p)dp

R y (ξ ) = −∞ −∞

u(t + ξ − q)h(q)dqdt

(2.3)

−∞

or (if permutations of integrals are permitted) ∞ ∞ R y (ξ ) =

⎛ h( p)h(q)⎝

−∞ −∞

∞

⎞ u(t − p)u(t + ξ − q)dt ⎠dqdp.

(2.4)

−∞

Using the change of variable t  = t − p, relation (2.4) becomes ∞ ∞ R y (ξ ) = −∞ −∞

or

⎡ h( p)h(q)⎣

∞

−∞

⎤

    u t u t + ξ + p − q dt ⎦dqdp

(2.5)

2.2 Definitions of Power-Law Type Long Memory Behaviours

7

∞ ∞ R y (ξ ) =

h( p)h(q)Ru (ξ + p − q)dqdp.

(2.6)

−∞ −∞

If u(t) is a white noise of variance σ , then Ru (ξ ) = σ δ(ξ ) where δ(.) is the Dirac function. Thus ∞ ∞ h( p)h(q)δ(ξ + p − q)dqdp.

(2.7)

h( p)h(q)Ru (ξ + p − q)e− jωξ dqdpdξ.

(2.8)

R y (ξ ) = σ −∞ −∞

Using relation (2.2), ∞ ∞ ∞ S y (ω) = −∞ −∞ −∞

Using τ = ξ + p − q the previous relation becomes ∞ S y (ω) =

∞ h( p)e

jωp

dp

−∞

h(q)e

−∞

− jωq

∞ dq

Ru (τ )e− jωτ dτ

(2.9)

−∞

and thus if H ( jω) denotes the frequency response, i.e. the Laplace transform H (s) = L{h(t)} s = jω, with (and H ∗ ( jω) its conjugate) of the considered dynamical system: S y (ω) = H ( jω)H ∗ ( jω)Su (ω) = σ |H ( jω)|2 .

(2.10)

2.2.2 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System Exhibiting a Power-Law Type (or Fractional) Impulse Response Let us now consider an LTI system whose impulse response is of the form h(t) =

Kt t 1−ν

He (t) and H ( jω) =

Kω 0 < ν < 2, Kt ∈ R, Kω ∈ R. (2.11) ( jω)ν

where He (t) is the Heaviside function. According to relation (2.10), the power spectral density of the system output to a white noise of variance σ is defined by

8

2 Power-Law Type Dynamic Behaviours

S y (ω) =

σ K2ω ω2ν

(2.12)

and exhibits a power-law type behaviour in the frequency domain. According to relation (2.7) for a white noise input u(t) of variance σ , the output autocorrelation is defined by ∞ ∞ h( p)h(q)δ(ξ + p − q)dqdp

R y (ξ ) = σ

(2.13)

p=0 q=0

or as the integrated function is not equal to 0 only if ξ + p = q ∞

∞ h( p)h( p + ξ )dp = σ

R y (ξ ) = σ p=0

p=0

Kt Kt dp 1−ν p ( p + ξ )1−ν

(2.14)

and thus if (.) denotes the Euler gamma function: σ K2t 4−ν (ν) R y (ξ ) = √ π

1 2

−ν

 ξ 2ν−1 .

(2.15)

Relation (2.15) demonstrates that the output signal autocorrelation exhibits a power-law type behaviour. (Power-law type long memory system) A power-law type long memory system is an LTI system that has one of the following properties in a given time or frequency range: 1.

Its impulse response h(t) slowly decays with respect to time with a power-law (or fractional) type behaviour according to: h(t) =

2.

Kt t 1−ν

He (t) 0 < ν < 2.

For a white noise input u(t) of variance σ , its output autocorrelation function exhibits a slow power-law type (or fractional) decay: σ K2t 4−ν (ν) R y (ξ ) = √ π

3.

(2.16)

1 2

−ν

 ξ 2ν−1 .

(2.17)

For a white noise input u(t) of variance σ , its output autocorrelation function is: S y (ω) =

σ K2ω . ω2ν

(2.18)

2.3 Some Examples

9

2.3 Some Examples In the literature, a confusion or rather an implicit link exists between fractional behaviours (of physical, biological, thermal, etc., origin) and fractional models (a tool to model fractional behaviours) [2]. But fractional behaviours and fractional models are two distinct concepts. One designates a property of a physical system, the other designates a model class, among a set of model classes (as shown in this book) that capture fractional behaviours. To deliberately limit this confusion and explained in the introduction, the name “fractional behaviour” is replaced by “powerlaw behaviours” in this book, which well expresses the idea that studied behaviours exhibit, in a limited frequency or time domain, a power-law-type response. It is now evident that power-law behaviours are ubiquitous in numerous domains. Power-law type behaviours often result from stochastic physical phenomena. The most popular of these phenomena are diffusion, diffusion reaction, adsorption, absorption, aggregation, fragmentation [3, 4]. These phenomena produce kinetics in t ν on space of dimension 1 (often ν = 1/2). It is when these phenomena operate on a space of dimension d that kinetics in t ν/d take place [5–7]. When these phenomena take place within a dynamic system, they in turn induce fractional dynamic behaviours. This is illustrated by Fig. 2.1 This is for instance what happens in the case of the Love wave based gas sensors developed in [8]. These sensors are based on a phenomenon of adsorption of the analysed gas. Such a phenomenon admits a kinetics of order t −1 in dimension 1 and t −1/2 in dimension 2 as shown in Chap. 6. This adsorption modifies the mass of a vibrating element, whose dynamic behaviour in turn is fractional. To illustrate this explanation in a simple way and although these sensor modelling is more complex, let us consider the resonator of Fig. 2.2. One electrode of this resonator is fixed and the other is mobile. A deposit phenomenon causes its mass to change according to the relation

Dynamic system Geometry

Physical phenomenon of stochastic nature

Fig. 2.1 Origin of fractional or power-law type dynamic behaviours

10

2 Power-Law Type Dynamic Behaviours

Fig. 2.2 Resonator description diagram

Mobile electrode

z(t)

Fixed electrode

M(t) = M0 + mt ν ,

(2.19)

where M0 is the initial mass an m an additional coefficient. A potential difference between the two armatures causes an electrostatic force to appear, defined by Fe (t) =

  ∂ 1 C(z)V 2 (t) ∂z 2

(2.20)

where C(z) is the capacitance induced by the two electrodes separated by the distance z(t). Mechanical equations permits to describe the motion of the mobile electrode by the differential equation:   d d d M(t) z(t) + μ z(t) + kz(t) = Fe (t), dt dt dt

(2.21)

in which μ and k are respectively the damping coefficients and the stiffness of the mobile electrode springs. Equation (2.21) can be rewritten as: (M0 + mt ν )

d

d2 z(t) + μ + mνt 1−ν z(t) + kz(t) = Fe (t). 2 dt dt

(2.22)

If the kinetic of mass variation is slow with respect to the resonator oscillations period, the latter can be defined by:  T (t) = 2π

M0 + mt ν . k

(2.23)

If the dynamical system linking the resonator oscillations period T (t) to the force Fe (t) is considered, relation (2.22) highlights a fractional behaviour of order t ν/2

2.3 Some Examples

11

Fig. 2.3 Impulse response of the relation (2.23) (left) and of the considered systems (system linking the resonator oscillations period T (t) to the force Fe (t)) for For M0 = 0.01 kg, m = 0.001, μ = 0.00001 N/m/s, k = 100 N/m and ν = 0.5

for the impulse response of this system. For M0 = 0.01 kg, m = 0.001, μ = 0.00001 N/m/s, k = 100 N/m and ν = 0.5 impulse response of the relation (2.23) and of the considered systems are shown in Fig. 2.3. This is how many areas show power-law type or fractional behaviours, and we can mention in particular: • • • • • •

electrochemistry with examples in batteries and ultracapacitors [9–12] thermal systems [13] biology [14] mechanics [15] earth dynamics [16] physics [2],

and many other areas.

References 1. Oppenheim AV, Willsky AS, Hamid S (1996) Signals and systems. Pearson New International Edition 2. Tarasov V (2013) Review of some promising fractional physical models. Int J Mod Phys B 27:1330005 3. Family F, Landau DP (1984) Kinetics of aggregation and gelation. Elsevier, Amsterdam 4. Krapivsky P, Redner S, Ben-Naim E (2010) A kinetic view of statistical physics. Cambridge University Press, Cambridge 5. Le Mehaute A, Crepy G (1983), Introduction to transfer and motion in fractal media: the geometry of kinetics. Solid State Ionics Part 1 9–10:17–30 6. Family F, Vicsek T (1991) Dynamics of fractal surfaces. World Scientific 7. Sapoval B (1997) Universalités et fractales : Jeux d’enfant ou délits d’initié ? Editions Flammarion, Paris

12

2 Power-Law Type Dynamic Behaviours

8. Nikolaou I, Hallil H, Conédéra V, Deligeorgis G, Dejous C, Rebiere D (2016) Inkjet-printed graphene oxide thin layers on love wave devices for humidity and vapor detection. IEEE Sens J 16:7620 9. Rodrigues S, Munichandraiah N, Shukla AK (2000) A review of state-of- charge indication of batteries by means of A.C. impedance measurements. J Power Sources 87(1–2):12–20 10. Sabatier J, Aoun M, Oustaloup A, Gregoire G, Ragot F, Roy P (2006) Fractional system identification for lead acid battery sate charge estimation. Signal Process 86(10):2645–2657 11. Allagui A, Freeborn TJ, Elwakil AS, Fouda ME, Maundy BJ, Radwan AG, Zafar S, Abdelkareem MA (2018) Review of fractional-order electrical characterization of supercapacitors. J Power Sources 400:457–467 12. Zou C, Zhang L, Hu X, Wang Z, Wik T, Pecht M (2018) A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries, and supercapacitors. J Power Sources 390:286–296 13. Malti R, Sabatier J, Akçay H (2009) Thermal modeling and identification of an aluminum rod using fractional calculus. IFAC Proc Vol (IFAC-PapersOnline), 15(PART 1):958–963 14. Ionescu C, Lopes A, Copot D, Tenreiro Machado JA, Bates JHT (2017) The role of fractional calculus in modeling biological phenomena: a review. Commun Nonlinear Sci Numer Simul 51:141–159 15. Bonfanti A, Kaplan JL, Charras G, Kabla A (2020) Fractional viscoelastic models for power-law materials. Soft Matter 16:6002–6020 16. Zhang Y, Sun H, Stowell HH, Zayernouri M, Hansen SE (2017) A review of applications of fractional calculus in Earth system dynamics. Chaos, Solitons Fractals 102:29–46

Chapter 3

Fractional Order Models

3.1 Introduction To model power-law type behaviours, many practitioners naturally think to fractional models, models described by ordinary or partial differential equations involving noninteger order derivatives in place of classical derivatives. The reason is simple: a fractional model also exhibits a long memory power-law type behaviour. In this chapter, some definitions and properties of fractional differentiation and integration operators are first reminded. It is then explained how these operators have been used to define fractional models. Some limits and drawbacks associated to these models are finally demonstrated thus justifying the next chapters aiming at proposing alternative modelling solutions.

3.2 Fractional Integration Let f (t) be a time real function, piecewise continuous on ]t0 , +∞[ and integrable on ]t0 , t[ for all t > t0 . Cauchy’s formula makes it possible to express the integer integral of order n ∈ N∗ of f (t) by: Itn0

1 f (t) = (n − 1)!

t (t − τ )n−1 f (τ )dτ.

(3.1)

t0

Inspired by Cauchy’s formula, Riemann defined in 1847 the general expression of the order ν ∈ R∗+ integral of f (t) in the form of the following expression [1]:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Sabatier et al., Fractional Behaviours Modelling, Intelligent Systems, Control and Automation: Science and Engineering 101, https://doi.org/10.1007/978-3-030-96749-9_3

13

14

3 Fractional Order Models

Itν0

1 f (t)  (ν)

t

(t − τ )ν−1 f (τ )dτ.

(3.2)

t0

in which Itν0 is the fractional integrator of order ν and (ν) is the Gamma function defined by ∞ (ν) =

e−x x ν−1 d x.

(3.3)

0

Few works retrace the history of the establishment of this definition resulting from scientific correspondences between several famous mathematicians like L’Hôpital, Leibnitz, Euler, Lacroix, Fourier, Abel, Liouville, Riemann, Sonin, Laurent, Grünwald or Leitnikov. For a detailed description of these correspondences, the reader can refer to the works [1–5]. Without taking into account initial conditions, Laplace transform of a fractional integral of a causal time function is defined by (* denotes the convolution product):   ν−1   ν−1   ν t 1 t ∗ f (t) = L L{ f (t)} = ν F(s), L I0 f (t) = L (ν) (ν) s

(3.4)

in which F(s) = L{ f (t)}, s = σ + jω being the Laplace variable. This relation generalizes the well-known formula   1 L I0n f (t) = n F(s) n ∈ N∗ . s

(3.5)

3.3 Fractional Differentiation If the definition of a fractional integral is limited to the relation (3.2), the situation is very different for a fractional derivative. More than 30 as listed in [6]. Here, only the definitions most used in the literature are reminded, which will be enough to illustrate the limitations and drawbacks from which fractional models suffer (and resulting from limitation of fractional differentiation and integration operators). To define the non-integer derivative of a function, the following notations are used: ν = ν + {ν},

(3.6)

3.3 Fractional Differentiation

15

where ν ∈ R∗+ , ν and {ν} denote respectively the integer and the non-integer part of ν with 0 ≤ {ν} < 1. In the Riemann–Liouville sense, fractional derivative of an order ν ∈ R∗+ of a real function f (t) is defined as the integer derivative of a fractional integral (whose integration order real part is in [0, 1]) [1]:

RL

Dνt0 f (t) =



d dt

ν+1



I 1−ν f (t) . 0

(3.7)

Using relation (3.2), Eq. (3.7) becomes: ν R L D t0

 ν+1  t d 1 f (t) = (t − τ )ν−ν f (τ )dτ t > t0 . (ν + 1 − ν) dt

(3.8)

t0

Its definition being based on a fractional integral definition, a fractional derivative is not a local operator (as integer derivative). It turns out that the fractional derivative of f (t) at time t, involves all the past of f (t) over the interval f (t). This last property highlights the long memory property of fractional derivative. Fractional derivative of a fractional integral with the same order is defined by: RL

 Dνt0 I νt0 ( f (t)) = f (t) with v > 0,

(3.9)

this relation being not always valid for Re(ν) < 0. Fractional integrator operators meet semi-group property only under some conditions. If f (t) meets conditions (3.9) then [1]: ν1 R L D0



ν2 R L D 0 ( f (t)) =

ν1 +ν2 ( f (t)) if R L D0



ν2 < ν1 + 1 . ν1 > 0

(3.10)

and r R L D t0



ν R L D t0 ( f (t)) =

r +ν R L D t0 ( f (t)) if



r ∈ N+ . ∀ν

(3.11)

Laplace transform of the fractional derivative of a causal function is given by the relation: n−1

  

s n−1−k D k I n−ν L R L Dν0 ( f (t)) = s ν F(s) − 0 ( f (t)) t=0+ k=0

with n − 1 < ν ≤ nn ∈ N

(3.12)

16

3 Fractional Order Models

which is a generalization of the relation for integer derivative. In the Caputo sense, the fractional derivative results from the permutation of integral and derivative in Eq. (3.7). Order ν Caputo’s derivative of a causal function is thus defined by: ν C D t0

 f (t) =

I0n−ν

 dn f (t) n = min(k ∈ N/k > ν)ν > 0. dt n

(3.13)

This definition was introduced by Caputo [7] because it leads to a Laplace transform easier to compute, especially since it allows taking into account the initial conditions that are integer derivatives. Therefore, these initial conditions have a physical meaning since they relate to integer derivatives. Indeed,   k n−1

  d L C Dν0 f (t) = s F(s) − s ν−k−1 f (t) dt k t=0+ k=0

(3.14)

with n − 1 < ν ≤ n and n ∈ N. However as shown in the sequel, this supposed advantage from a mathematical point of view, is an aberration if this definition is used to describe a fractional model and when initial conditions are taken into account. In the Grünwald-Letnikov sense, the fractional derivative of a function f (t) is given by l GL

Dν0 f (t) =

k k=0 (−1)

lim h→0 l→∞

  ν f (t − kh) k hν

(3.15)

  ν (ν+1) generalized Newton’s binomial coefficient. where = k!(ν−k+1) k According to Eq. (3.15), Grünwald-Letnikov fractional derivative definition takes into account past values of function f (t − kh), k = {0 . . . ∞}, h → 0, that is all past values of function f (t). Indeed, for fractional orders, weighting coefficients do not cancel as shown in Fig. 1.2b in the case k = 8 (Fig. 3.1). This last property highlights again the long memory property of fractional derivatives.

3.4 Frequency Response of Fractional Integration …

17

Fig. 3.1 Weighting coefficients variation in the case k = 8

3.4 Frequency Response of Fractional Integration and Differentiation Operators A real fractional differentiation operator is defined as the operator that links a time function y(t) (considered as the operator output) to a time function u(t) (considered as the operator input): y(t) = τ ν Dν0 (u(t))

(3.16)

where τ denotes the differentiation time constant and where ν ∈ R denotes the fractional differentiation order. Note that ν can be greater or less than 0, the operator then being respectively a derivative or an integration. In this relation (3.16), as no initial conditions are taken into account, D indifferently denotes one of the definitions given in the previous section. For null initial conditions, according to Eqs. (3.4) and (3.14), Laplace transform applied to relation (3.16) leads to: Y (s) = (τ s)ν U (s).

(3.17)

Frequency response of this fractional differentiation operator is obtained using s = jω:

18

3 Fractional Order Models

Gain (dB)

40 20 0 -20 -40 -1 10

0

10

1

10

Frequency ωτ (rad/s)

Phase (deg)

200 100 0 -100 -200 -1 10

0

10

ν=1.5 ν=1 ν=0.5 ν=0 ν=-0.5 ν=-1 ν=-1.5 101

Frequency ωτ (rad/s)

Fig. 3.2 Bode diagram of a real fractional operator

Dν ( jω) = (τ jω)ν .

(3.18)

Gain and phase of this operator are thus: | Dν ( jω)| = (τ ω)ν and ϕ( Dν ( jω)) = ν90◦ .

(3.19)

Analysis of relations (3.19) highlights two properties of a real fractional differentiation operator: • gain diagram is a line whose slope is 20ν dB per decade; • phase diagram is an horizontal line whose ordinate is ν90◦ . To illustrate the previous comments, Fig. 3.2 shows the Bode diagrams of a real fractional differentiation operator for τ = 1 and for fractional orders within −1.5 and 1.5.

3.5 Fractional Models Definition In order to have models capable of capturing the behaviour of systems with power-law type behaviours, a first reflex of the community was to generalize well-known tools such as differential equations or state space representations (which are the models mainly used in the literature) to the fractional case.

3.5 Fractional Models Definition

19

Fig. 3.3 Bloc diagram associated to pseudo state space representation (3.21)

A fractional differential equation involves fractional derivatives of the model input u(t) ∈ Rm and of the model output y(t) ∈ R p and is described by the relation Na

νa

ak Dt0 k (y(t)) =

k=0

Nb

νb

bk Dt0 k (u(t))

(3.20)

k=0 νa

νb

in which the coefficients ak ∈ R and bk ∈ R. Dt0 k and Dt0 k denote fractional differentiation operators of orders νak ∈ R and νbk ∈ R respectively. If the orders νak and νbk verify relations νak = k/q and νbk = k/q, q ∈ N∗ , that is if all differentiation orders are multiples of 1/q, then system (3.20) is commensurate. Using the order commensurability condition and for null initial conditions, differential equation (3.20) admits a pseudo state space description of the form: 

Dνt0 x(t) = Ax(t) + Bu(t) y(t) = C x(t) + Du(t)

(3.21)

where x(t) ∈ Rn is the pseudo state vector, ν = 1/q is the fractional order of the system and A ∈ Rnxn , B ∈ Rnxm , C ∈ R pxn and D ∈ R pxm are constant matrices. As for integer order systems, pseudo state variables can be chosen to obtain a controllability or observability canonical form (without allowing conclusions to be drawn in terms of observability). The block diagram of Fig. 3.3, associated to the pseudo state space representation (3.21), can thus be obtained. Thus, contrary to what could be expected from the writing of Eq. (3.21), the pseudo state space representation does not involve explicitly the fractional derivative operator but only fractional integration operator. Consequently, it is not necessary to precise which fractional derivation definition is used in Eq. (3.21) and results developed in this book remain valid whatever definition may be used. Furthermore, even under the assumption of null initial conditions at t = t0 , that is a system supposed at rest (u(t) = y(t) = x(t) = 0, ∀t < t0 ), it is worth noticing that x(t) is not the state of the fractional system. In order to ease the understanding of this fact, let us first consider the simple case of a fractional integrator supposed at rest at t = t0 . This integrator admits a pseudo state space representation of the form (3.21) where A = 0, B = 1, C = 1 and D = 0. Associated bloc diagram is given on Fig. 3.4.

20

3 Fractional Order Models

Fig. 3.4 Bloc diagram associated to a fractional integrator

In the case of an integer order integrator, ν = 1 and (3.21) is a state space representation of the system. Value of x(t) can be computed at time t1 > t0 , provided that the input is known between t0 and t1 : t1 x(t1 ) =

u(τ )dτ.

(3.22)

t0

Values of x(t) at times t > t 1 are given by: t x(t) =

t1 u(τ )dτ =

t0

t u(τ )dτ +

t0





x(t1 )



u(τ )dτ t > t1 .

(3.23)

t1

Thus, x(t) can be computed knowing x(t1 ) and the inputs applied between t1 and t. Knowledge of integrator output at time t thus allows to summarize all the past of the system. x(t) is the state of the system according to the following definition. Condition 1 (From [7])—The present state and the chosen control function together determine the future states of the system. More precisely, given the state x(t0 ) = x0 of the system at some time t0 and a control u(.), the evolution of the system’s state x(t) is uniquely determined for all t in a suitable time interval starting  at t0 . Let us apply the same reasoning in the case of a fractional integrator of order ν. According to definition (3.2) of fractional integration, value of x(t) at time t1 > t0 can be computed if inputs are known between t0 and t1 : 1 x(t1 ) = (ν)

t1

(t1 − τ )ν−1 u(τ )dτ = x1 = constant.

t0

Value of x at any time t > t1 writes: 1 x(t) = (ν)

t t0

(t − τ )ν−1 u(τ )dτ

(3.24)

3.5 Fractional Models Definition

1 = (ν) 

t1 (t − τ ) t0



21

ν−1

1 u(τ )dτ + (ν) 

t

(t − τ )ν−1 u(τ )dτ.

(3.25)

t1

α(t)=constant=x1

Last formula allows to exhibit two noticeable differences with the integer order case. First, the term α(t) is not a constant but depends of considered time instant t. Second, knowledge of x1 = x(t1 ) does not allow to compute α(t). Output x(t) of the fractional integrator is thus not a state of the fractional order system. Same discussion can be conducted in the more general case of a pseudo state space representation of the form (3.21) where matrices A, B, C and D are arbitrary. Integrators outputs x(t) of bloc diagram of Fig. 3.5 cannot be considered as state variables. In order to compute the term α(t) for any t, knowledge of u(t) must be known on time interval [t0 , t1 ]. Fractional systems are thus qualified of infinite dimensional systems as an infinite number of values are necessary to summarize the past of the system. Vector x(t) in (3.21) is thus not a state vector of the system. Description (3.21) is thus not strictly a state space description and this is why it is better to use the following designation “pseudo state space description”.

Fig. 3.5 Comparison of the exact response of system (3.26) with the responses obtained with Riemann–Liouville and Caputo definitions (t0 = 10 s)

22

3 Fractional Order Models

Another conclusion is that the knowledge of the system past, or all its real state (of infinite dimension) at t0 is required to predict the system future. The idea is not new and was suggested for the first time in [8] but leads to the following question: how to take into account initial conditions?

3.6 How to Take into Account Initial Conditions? In a series of papers, Lorenzo and Hartley [8–12] have demonstrated that initial conditions are not taken into account in the same way whether Rieman-Liouville or Caputo definitions are considered. The goal of this section is to go further and to demonstrate that neither Rieman-Liouville nor Caputo definitions permit to take into account initial conditions in a coherent way with the system physics. The demonstration is done on a counter-example based on the system defined by the differential equation: Dν x(t) = u(t)

0 < ν < 1.

(3.26)

As in Lorenzo and Hartley work, the demonstration is based on the shift of the time origin. System (3.26) is supposed to be at rest when time t < 0. Input u(t) is also supposed to be defined by: u(t) = H (t) − H (t − t0 )

(3.27)

where H (.) denotes the Heaviside function. Time response of system (3.26) is thus defined by: x(t) =

tν (t − t0 )ν H (t) − H (t − t0 ). (ν + 1) (ν + 1)

(3.28)

The change of variable τ = t − 2t0

(3.29)

is now used so that at time τ = 0, the system is not at rest. Using Riemann–Liouville definition, Laplace transform of the fractional derivative of x(τ ) is defined by [13]: n−1

   ν ν L R L Dτ =0 (x(τ )) = s x(s) − s k R L Dν−k−1 τ =0 (x(τ )) τ =0 k=0

(3.30)

3.6 How to Take into Account Initial Conditions?

23

with n − 1 < ν < n and where x(s) denotes the Laplace transform of x(τ ) with zero initial conditions. Laplace transform applied to Eq. (3.26), leads to: ν

s x(s) −

n−1

sk

 RL

Dν−k−1 τ =0 (x(τ )) τ =0 = 0

(3.31)

k=0

(here n = 1) and the corresponding time response is defined by x(τ ) = L−1

 RL

 Dντ =0 (x(τ )) I 1−ν τ =0 (x(τ )) τ =0

(3.32)

where 



I 1−ν τ =0 (x(τ )) τ =0

   2t0 s  1 e et0 s −1 = L − ν+1 s 1−ν s ν+1 s τ =0  2t0 s   t0 s e e = L−1 − 2 = t0 . s2 s τ =0

(3.33)

The time response of system (3.26) is thus given by: x(τ ) = t0

τ ν−1 . (ν)

(3.34)

Using Caputo definition, Laplace transform of the fractional derivative of x(τ ) is defined by [13]: n−1

   L C Dντ =0 (x(τ )) = s ν x(s) − s ν−k−1 C Dkτ =0 (x(τ )) τ =0 .

(3.35)

k=0

Laplace transform applied to Eq. (3.26), leads to: s ν x(s) −

n−1

s ν−k−1

 RL

Dkτ =0 (x(τ ))

τ =0

=0

(3.36)

k=0

(here n = 1) and the corresponding time response is given by x(τ ) = L−1 where, using (3.28),



 s ν−1 [x(τ )]τ =0 sν

(3.37)

24

3 Fractional Order Models

[x(τ )]τ =0 =

(2t0 − t0 )ν (2t0 )ν − . (ν + 1) (ν + 1)

(3.38)

Time response of system (3.26) is thus given by: x(τ ) =

(2t0 )ν − t0 ν H (τ ) τ ≥ 0. (ν + 1)

(3.39)

In Fig. 3.5, system (3.26) response to the input given by relation (3.27) is compared to the response obtained using Riemann–Liouville and Caputo definitions, initial conditions being taken into account. This figure highlights that the three time responses are completely different. Using a counter-example, it has been demonstrated in this section that neither the Riemann–Liouville nor the Caputo definitions are compatible with the real behaviour of a fractional system. Using these definitions, system response u(t) does not mach system response to non-zero initial conditions. To solve this problem another representation is now introduced. This representation proposed in this section was developed in [14] for SISO systems and is extended here to MIMO systems. This representation is also known as diffusive representation [15, 16] and is directly derived here from the impulse response of a fractional order system as shown in the sequel.

3.6.1 Modal Decomposition of a Fractional Order System As for integer systems, modal decomposition of system (3.21) is given by: 

Dνt0 x J (t) = J x J (t) + B J u(t) y(t) = C J x J (t) + D J u(t)

(3.40)

where J is a Jordan matrix whose diagonal is constituted of matrix A eigenvalues denoted λl , l ∈ {1, . . . , r }, of multiplicity n l . If the system is supposed at rest and introducing inverse Laplace transform, output y(t) is then given by:   y(t) = L−1 C J (s ν I − J )−1 B J ∗ u(t) + D J u(t)

(3.41)

where matrix (s ν I − J )−1 is: (s ν I − J )−1 

−1

−1

−1  = diag s ν I − Jn 1 (λ1 ) , . . . , s ν I − Jnl (λl ) , . . . , s ν I − Jnr (λr ) (3.42)

3.6 How to Take into Account Initial Conditions?

25

where Laplace variable s is supposed to belong to a subset of C such that (s ν I − J )−1 exists and where each Jordan block is given by ⎡ ⎢ ⎢ ν

−1 ⎢ s I − Jnl (λl ) =⎢ ⎢ ⎢ ⎣

1 s ν −λl



1 s ν −λl 1 s ν −λl

2

···



1 s ν −λl

n l ⎤

⎥ ⎥ ⎥ r ⎥ n l = n. . .  1 2 ⎥ ⎥ l=1 . s ν −λ ⎦ l .. .

(3.43)

1 s ν −λl

Output y(t) is thus produced by a linear combination of n terms called modes and defined by h λl ,q (t) = L−1



1 ν s − λl

q  q ∈ {1, . . . , n l } l ∈ {1, . . . , r }.

(3.44)

Each mode given by (3.44) can be decomposed into two parts λ ,q

λl ,q l h λl ,q (t) = h ex p (t) + h di f f (t)

(3.45)

λ ,q

h pl (t) is produced by the poles of the modes (similar to the impulse response of an integer order system); λ ,q h dl (t) is the diffusive part of the mode, characterized by a long range memory. 3.6.1.1

Exponential Part of a Mode λ ,q

The exponential part h pl (t) results from the computation of the residues of the poles associated to the mode h λl ,q (t). Note that, unlike integer order systems, poles (which are solutions of the equation s ν − λl = 0 are not equal to matrix A eigenvalues. In order to extend the notion of poles to fractional order models, let us associate to each eigenvalue λl the set of integers K λl defined as:  arg(λl ) ν arg(λl ) ν