Diversity and Non-integer Differentiation for System Dynamics

Table of Contents

Acknowledgments

Preface

Introduction

Chapter 1 From Diversity to Unexpected Dynamic Performances

1.1. Introduction

1.2. An issue raising a technological bottle-neck

1.3. An aim liable to answer to the issue

1.4. A strategy idea liable to reach the aim

1.5. On the strategy itself

1.6. From physics to mathematics

1.7. From the unusual to the unexpected

1.8. On the nature of diversity

1.9. From the porous dyke to the CRONE suspension

1.10. Conclusion

1.11. Bibliography

Chapter 2 Damping Robustness

2.1. Introduction

2.2. From ladder network to a non-integer derivative as a water-dyke interface model

2.3. From a non-integer derivative to a non-integer differential equation as a model governing water relaxation

2.4. Relaxation expression

2.5. From a non-integer differential equation to relaxation damping robustness

2.6. Validation by an experimental simulation in analog electronics

2.7. Bibliography

Chapter 3 Non-Integer Differentiation, its Memory and its Synthesis

3.1. Introduction

3.2. From integer differentiation to non-integer differentiation

3.3. From repeated integer integration to non-integer differentiation through non-integer integration

3.4. Non-integer differentiation in sinusoidal steady state

3.5. On memory associated with non-integer differentiation

3.6. On the synthesis of non-integer differentiation

3.7. Bibliography

Chapter 4 On the CRONE Suspension

4.1. Introduction

4.2. From the porous dyke to the hydropneumatic version of the CRONE suspension

4.3. Metallic version of the CRONE suspension

4.4. Bibliography

Chapter 5 On the CRONE Control

5.1. Introduction

5.2. From the porous dyke to the CRONE control of first and second generations

5.3. Second generation CRONE control and uncertainty domains

5.4. Generalization of the vertical template through the third generation CRONE control

5.5. Bibliography

Chapter 6 Recursivity and Non-Integer Differentiation

6.1. Introduction

6.2. Indefinite recursive parallel arrangement of series RC cells

6.3. Recursive arborescent network as a lung respiratory model

6.4. Unified study of recursive parallel arrangements of RL, RC and RLC cells

6.5. A common presentation of results turning on eight RC and RL cell recursive arrangements

6.6. On unit gain frequency in non-integer differentiation or integration

6.7. On stored energy in non-integer differentiation or integration

6.8. Bibliography

Appendix 1 Damping of a Usual Automotive Suspension

A1.1. The usual automotive suspension

A1.2. Damping that the damping ratio measures

A1.3. On the importance of the transmittance denominator

A1.4. From denominator zeros to dynamics damping and rapidity

Appendix 2 Relaxation of Water on a Porous Dyke

A2.1. Context and study assumptions

A2.2. Flow-pressure differential equation

A2.3. Model of the water-dyke interface

A2.4. Non-integer order differential equation as a model governing the water relaxation

A2.5. Bibliography

Appendix 3 Systems with Explicit and Implicit Generalized Derivative

A3.1. Explicit and implicit generalized differentiation

A3.2. System with explicit generalized derivative

A3.3. System with implicit generalized derivative

A3.4. Change between systems with implicit and explicit generalized derivative

A3.5. Impulse response of a system with explicit generalized derivative

A3.6. Impulse response of a system with implicit generalized derivative

A3.7. Impulse response of a frequency bounded generalized differentiator

A3.8. On the singularities of a system with explicit generalized derivative

A3.9. Bibliography

Appendix 4 Generalized Differential Equation and Generalized Characteristic Equation

A4.1. Generalized differential equation time solving

A4.2. Generalized characteristic equation solving

A4.3. Bibliography

Appendix 5 CRONE Control Response with Initial Conditions

A5.1. Laplace transform of the generalized derivative of a time function in the presence of initial conditions

A5.2. Response to any input of the second generation CRONE control for non-zero initial conditions

A5.3. Bibliography

Appendix 6 Fractality and Non-integer Differentiation

A6.1. On interdependence between fractality and non-integer differentiation

A6.2. A metal-vacuum capacitor as a study support to roughness/non-integer differentiation change

A6.3. Roughness of recursive nature

A6.4. Roughness of fractal nature

A6.5. Bibliography

Index

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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ISBN 978-1-84821-475-0

Acknowledgments

If my acknowledgments are naturally intended for the CRONE team, I would particularly like to thank François Levron for his contributions to the mathematical aspects and Stéphane Victor for his major contribution both to the translation and to the achievement of the book.

Preface

Non-integer differentiation does not escape to the slogan different operator, different properties and performances. This is indeed the concise formula that is likely to explain the why of this operator, especially as most of its properties and performances favorably distinguish not only the operator itself but also the models that use it.

It is true that we have established non-integer models that overcome the mass-damping dilemma in mechanics and the stability-precision dilemma in automatic control; the technological achievements associated with these models have been made possible thanks to an adequate synthesis of the non-integer differentiation operator.

It is indeed the idea to synthesize non-integer differentiation (in a medium frequency range) through a recursive distribution of passive components, of transitional frequencies or of zeros and poles, which is at the origin of the non-integer differentiation operator real time use and, therefore, of both analogical and numerical applications that arise from it. As for the corresponding dates, the synthesis as we have led it has been developed by stages and thus proposed and experimented in the 70s for half-integer orders, in the 80s for real non-integer orders and in the 90s for complex non-integer orders.

The first technological applications of this operator (henceforth usable in real time) and notably the first application in 1975 of a half-integer order controller to the frequency control of a continuous dye laser, have widely contributed to take the non-integer differentiation out of the mathematician’s drawers and to give rise to new developments likely to enrich the theoretical corpus of circuits and systems.

In this way France has been the first country to experience a interest in non-integer differentiation, this renewal having been well relayed thanks to the dynamism of the foreign scientific communities, at the European level as well as at the international level.

In this context, the French institutions have encouraged research in this field through the acknowledgement of major scientific advances and the support of initiatives or actions aiming to favor the synergies between the different themes and between the academic and industrial components, the university–industry partnership having indeed been nationally rewarded by the Association Française pour la Cybernétique Economique et Technique (AFCET) ’95 Trophy distinguishing the CRONE suspension as the best technological innovation. Concerning the acknowledgements, let us cite the selection of the CRONE control as a striking fact of the Centre National de la Recherche Scientifique (CNRS) in 1997 and as Flagship Innovation of Alstom in 2000 (Hanover and Baden Baden International Fairs, 2000), a silver medal of the CNRS in 1997 and the Grand Prix Lazare Carnot 2011 of the Science Academy (founded by the Ministry of Defense). Concerning the supports, let us cite the actions financially supported by the CNRS and the Ministry of Research: the edition of La commande CRONE (Hermès, 1991) with the exceptional help of the ministry; the International Summer School Fractal and hyperbolic geometries, fractional and fractal derivatives in engineering, applied physics and economics (Bordeaux, 1994); the national project of the CNRS, Non-integer differentiation in vibratory insulation (1997–1999); the colloquium Fractional Differential systems (Paris, 1998); the launching in 1999 of the thematic action of the Ministry of Research Systems with non-integer derivatives; the launching in 2004 of the International Federation of Automatic Control (IFAC) Workshop Fractional Differentiation and its Applications through the first Workshop FDA ’04 (Bordeaux, 2004) with S. Samko as chairman of the International Programme Committee; the magisterial lecture From diversity to unexpected dynamic performances initiated by the French Science Academy (Bordeaux, 5 January 2012).

But this academic support also found a guarantee in the industrial support brought by a strong partnership with major companies such as the Peugeot Société Anonyme (PSA) Peugeot-Citroën, Bosch (Stuttgart) and Alstom, such a partnership indeed led to a high number of patents and technological transfers that have widely proved the industrial interest of non-integer approaches.

Alongside the shared efforts to inscribe these approaches in the realist frame of the university–industry relations, our efforts have never stopped being shared with those of the international scientific community to develop the best relations and collaborations within this community. Without aiming for exhaustiveness, let us cite the involvement of European countries involved in the diffusion, promotion and animation within the community: the research group Fracalmo, which originates from Fractional calculus modeling, started with a round table discussion in 1996 during the 2nd International Conference Transform methods and special functions held in Bulgaria; the journal Fractional Calculus and Applied Analysis (FCAA) started in 1998 with V. Kiryakova as managing editor; the survey on the Recent history of fractional calculus (Communications in Nonlinear Science and Numerical Simulation, 2010) at the initiative of J.T. Machado who desired to make an inventory of the major documents and events in the area of fractional calculus that had been produced or organized since 1974; the symposium Fractional Signals and Systems (FSS) launched by M. Ortigueira in 2009 in Lisbon, then held in 2011 at Coimbra (Portugal) and in 2013 at Ghent (Belgium). Both with in and out side of Europe, let us also recall the various events of the Workshop FDA after its launching at Bordeaux in 2004 (under the aegis of IFAC): Porto (Portugal) in 2006; Ankara (Turkey) in 2008; Badajoz (Spain) in 2010; Nanjing (China) in 2012; Grenoble (France) in 2013 and Catania (Italy) in 2014 (under the aegis of IEEE).

As the founder of the CRONE team, which counts today about 10 permanent researchers, I recognize this team to have always escorted me in the federative actions, launched within the national or international scientific community, to energize and harmonize research on both theoretical and applicative aspects of non-integer differentiation.

Introduction

If beyond integer differentiation many people indistinctly talk about fractional, non-integer or generalized differentiation, it seems through database analysis that fractional qualifying is used more in titles, whereas non-integer differentiation is used more in texts, with generalized differentiation appearing to be less used. So, why does non-integer qualifying benefit from our preference in this book?

In the formulation of generalized differentiation, the differentiation order is integer or non-integer, the generalized indeed including the integer and the non-integer. But the specificity of our book is certainly on the non-integer, notably through the link of non-integer differentiation with the recursivity and fractality that diversity covers. Besides, the real or complex non-integer differentiation synthesis, which is major in our contribution, is exclusively a matter for the non-integer, hence our preference. As for the fractional itself that excludes the irrational, it cannot be preferred to the non-integer that does include the irrational and the non-integer rational.

Beyond the originality and the interest of the book’s contents that the thematic nature confers, the originality of the contents’ structuration results from the author’s will to offer readings of the book at several speeds.

To that effect, the book is structured so as to offer to the reader several reading levels corresponding to an increasing level of complexity and/or of specialization. The first three chapters have indeed a more general character whereas the next three chapters are more specialized, notably Chapters 4 and 5 that respectively deal with the CRONE suspension and the CRONE control (CRONE being the French abbreviation of Commande Robuste d’Ordre Non Entier, in English, non-integer order robust control).

Thus, the reader who only requires a consciousness-raising to the non-integer approach and to the most striking results on the subject can limit his reading to Chapter 1. The specificity of this chapter is indeed its approach, which is both very general to constitute an overview on the non-integer approach and very targeted to enable a simplified presentation, even an educational example, liable to quickly sensibilize the reader to the thematic interest and thus to give him the desire to go further in his reading.

The reader, who is concerned about the proofs and notably about a deepening of the proofs briefly led in Chapter 1, can partially or fully read the chapters and appendices cited in Chapter 1 with a link enabling then to directly find the expected supplement.

Finally, the reader eager to benefit from the subtleties acquired by the author in 40 years of sustained researche in the non-integer field, is invited to browse the whole book, including the Appendices, which constitute mini-chapter, complementary at the physics level as well as the mathematical level.

Chapter 1 largely contributes to the originality of the book. It indeed results from the magisterial conference presented in the framework of the Grand Prix Lazare Carnot 2011 of the French Academy of Sciences. On the form, given the various disciplines within this academy, our contribution is a simplified presentation at the limit of scientific popularization. On the content, through a structured approach of diversity, this chapter unquestionably offers a framework, on the one hand, to the introduction of non-integer derivative as a modeling tool, on the other hand, to the use of such a modeling form to put into light dynamic performances (and notably of damping) unsuspected in an integer approach of mechanics and automatic control. The non-integer approach indeed enables us to overcome the mass-damping dilemma in mechanics and consequently the stability-precision dilemma in automatic control. Furthermore, the metallic and hydropneumatic versions of the CRONE suspension are the subject of a unified presentation through various forms of diversity leading to non-integer differentiation. To illustrate our strategy through a study system, we have studied the relaxation of water on a porous dyke. Such as obtained, the model of the water-dyke interface and the one governing the water relaxation are non-integer models of order between 0 and 1 for the interface and of order between 1 and 2 for the relaxation, these models being valid in a medium frequency range.

Chapter 2 proves and validates the damping robustness of the water relaxation on a porous dyke, the various study stages going from the object to its performances and to their experimental verification. The first stage (section 2.2) consists of starting off with a recursive parallel arrangement of serial RC cells to obtain a non-integer differentiation as a model of the water-dyke interface. Obtaining a differentiation non-integer order results from a smoothing of the admittance Bode asymptotic diagrams. The second stage (section 2.3) consists of using the dynamics fundamental principle to change from the non-integer derivative so obtained to a non-integer differential equation as a model governing water relaxation. After the analytical determination of the relaxation, which shows damping robustness in the time domain and constitutes the third stage (section 2.4), the fourth stage (section 2.5) illustrates damping robustness in operational domain and in the frequency domain. Lastly, the fifth stage (section 2.6) consists of experimentally verifying damping robustness through an electronic circuit made of operational amplifiers. This circuit is achieved in such a way that its transmittance respects the non-integer differential equation, which governs the water relaxation on a porous dyke.

Chapter 3 deals with non-integer differentiation, its memory and its synthesis. Section 3.2 presents, in discrete time, non-integer differentiation through the extension, to the non-integer case, of the generic form of integer differentiation. Section 3.3 presents, in continuous time, non-integer differentiation from repeated integer integration via non-integer integration. Thus presented in discrete time then in continuous time, non-integer differentiation is the subject, in section 3.4, of a study of its properties in a sinusoidal steady state. In this operating state and as regards kinematic magnitudes, therefore in terms of position, speed and acceleration, the non-integer derivative of position takes into account position and speed or speed and acceleration whether the differentiation order n is between 0 and 1 or between 1 and 2. Section 3.5 is devoted to the memory phenomenon associated with non-integer differentiation. The discrete form of the non-integer derivative as presented in section 3.2, directly shows that the function to be differentiated is taken into account through its values at all the past instants. Non-integer differentiation thus introduces a memory notion such that the past attenuation or accentuation is imposed by the differentiation order. As this memory notion is not without evoking a subtle form of memory, an investigation trail is proposed by considering an