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.

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 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 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 non-commensurate 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