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 non-integer 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.

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

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