IEEE Circuits and Systems Magazine - Q2 2022 - 37

We forecasted that fractional systems would be the 21st Century Systems
and advanced an educated prediction about their evolution,
both in the theoretical and practical perspectives.
applications [9]-[11]. Designations like /f1 noise, long
range dependence [12], fractional Gaussian noise, and
fractional Brownian motion (fBm), are ubiquitous in the
scientific literature [13]-[15]. Other applications include
the viscoelasticity [4], the J. Curie phenomenon [16], the
Schrödinger fractional equation [17], [18], and the fractional
Maxwell equations [19], [20], [21].
In a previous paper [22] (2008), and following Prof.
Nishimoto, we forecasted that fractional systems would
be the 21st Century Systems and advanced an educated
prediction about their evolution, both in the theoretical
and practical perspectives. However, the following
years revealed an unexpected unfolding that resulted in
the appearance of formulations that were not only very
inaccurate, but also far from those required when having
in mind the solid background usual in engineering.
This fact has certainly contributed to some lack of interest
by many researchers in engineering. In fact, the confusion
prevailing in the fractional world is an obstacle to
its adoption by other applied sciences. This sequence of
two articles intends to contribute for settling the dust
that overshadows Fractional Calculus (FC), by highlighting
concepts and tools that allow a correct fit between
the traditional Signals & Systems and the FC. This backwards-compatibility
is very important because it simplifies
the adoption of FC by professionals with a more
practical-oriented experience.
Instead of starting from the notion of fractional derivative
and corresponding differential equations, we begin
by introducing the concept of transfer function (TF)
and some different forms it can assume, namely those
involving irrational functions. From the TF, we define
the corresponding differential equations with a considerable
generality. In particular, we will call continuoustime
fractional autoregressive-moving average (FARMA)
those that are defined by an equation that results from
the traditional ARMA model using a mere substitution
of integer by fractional-order derivatives. This is a first
type of " fractionalization " that consists in substituting
the s Laplace variable by
s .a
This approach corresponds
to replacing the derivatives of orders 12 3 f ,, ,
by the fractional ones with orders aa f R!aa ,, ,,
yielding the so-called commensurate systems. This type
23
,
of continuous-time fractional autoregressive-moving
average models is the most interesting and is used in
practical applications [23]. Nonetheless, we go further
and use other real orders
12 ,3
commensurate systems.
It is important to compare the transient responses of
both types of systems. While the integer order models
are characterized by having exponential responses, the
fractional ones have power functions as responses. For
this reason, it is said that the first class has short memory
while, on the contrary, the second has a long one
(also called long range). However, we have applications,
namely, in physics, where we find phenomena that are
neither of short, nor of long range. These phenomena exhibit
an intermediate behavior which requires a distinct
approach. In the modelling of such kind of systems, we
need to look for some form of embedding the two types
of responses. One solution comes from the multiplication
of the derivative kernels by an exponential that gives the
required medium range systems. Several denominations
were proposed in the literature for such systems. Similarly
to what was considered in [24] we will use the designation
tempered systems. Again, we have also commensurate
and non-commensurate tempered systems. In terms
of the TF and in a simple interpretation the tempering
consists in substituting ,,s
b b ! R by^h+mb ! Rmb
s
,,
.
B. A Short History
The FC is a generalization of the conventional calculus
that was raised in 1695 from a Leibniz' idea, during the
exchange of letters between him and J. Bernoulli [25].
The FC is not as accessible as the standard calculus, but
leads to similar concepts and tools and enjoys a wider
generality and applicability. The FC allows derivative
operations of arbitrary order and represents an advance
similar to the generalization from integer to real
or complex numbers. During almost 200 years, namely
since the works of Liouville (1832), the fractional derivative
was considered as a curious and interesting topic,
but merely an abstract mathematical concept. The main
developments of the FC were accomplished by mathematicians
without having in mind real world applications.
However, the Liouville's proposals of fractional
M. Ortigueira is with the Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology
of NOVA University of Lisbon, Portugal (e-mail: mdo@fct.unl.pt) and J. A. Tenreiro Machado was with the Institute of Engineering, Polytechnic of
Porto, Department of Electrical Engineering,431, 4249-015 Porto, Portugal (e-mail: jtm@isep.ipp.pt).
SECOND QUARTER 2022
IEEE CIRCUITS AND SYSTEMS MAGAZINE
37
aa a11 f that yield non

IEEE Circuits and Systems Magazine - Q2 2022

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