IEEE Circuits and Systems Magazine - Q2 2022 - 38

However, in which concerns fractional System Theory we cannot say that we had a
clear formulation generalizing the classic tools, namely impulse response (IR),
TF, and frequency response (FR), keeping a backward compatibility.
derivative had physical problems as their main motivation
[26]-[28]. It is important to refer that Fourier proposed
a general analysis/synthesis of the Fourier transform
(FT) contemplating the fractional case. Liouville
based his formulations on decompositions of the functions
in terms of exponentials. However, he had a limitation,
since, at that time, the Bromwich integral inverse
of the Laplace transform (LT) was unknown. Therefore,
he was not sufficiently convincing. The appearance of
Riemann's proposal [29] allowed the formulation of a
synthesis called Riemann-Liouville derivative (RL) [30],
[31] that remained almost like a " standard " until the
end of the 20th century, when it was substituted by the
(Dzherbashian-) Caputo (C) derivative, a particular case
of second Liouville's formula. Heaviside considered
some problems where fractional behavior appears and
applied his operational approach [1], [32]. Since the first
Liouville's paper, several other definitions of derivative
and integral operators were formulated, not necessarily
compatible in the sense of giving always the same
results. This state of affairs created difficulties when
trying to extend specific tools based on the traditional
integer order to the more general arbitrary order context
[30], [31].
Since the early 1990's, some scientists and engineers
have been working with those different forms,
having in mind the perspective of practical applications
[33]-[36]. It is important to refer the pioneering
works of Oustaloup's group in control and identification
[37]-[42]. The fractional electrical circuits started
being investigated [43]-[48]. In the last 10 years a number
of papers were published describing electronic realizations
of fractional systems [49]. However, in which
concerns fractional System Theory we cannot say that
we had a clear formulation generalizing the classic
tools, namely impulse response (IR), TF, and frequency
response (FR), keeping a backward compatibility. Apparently,
this problem was examined, for the first time,
in [50] and revisited in [22], proposing that FC would
be the tool for system modelling in the 21th century
systems. A more general vision of fractional systems
and their applications was presented in [51]. In the last
decade, many applications appeared and important
themes like, analysis, modelling, and synthesis, have
been considered [52]. However, a closer look reveals
that there are many integer order tools that need to be
extended to the fractional framework, while keeping a
backward compatibility [53]. Not all proposed formula38
IEEE
CIRCUITS AND SYSTEMS MAGAZINE
tions for the fractional operators are suitable for doing
this task.
In this paper, the state-of-the-art of the compatible
fractional system theory is described. We start by introducing
the continuous-time ARMA like systems with
commensurate and non commensurate orders. The traditional
tools are introduced and computed with generality.
The stability is also considered. The important
initial condition problem is discussed.
C. Remarks
We assume that
■ We work always on .R
■ We use the bilateral LT:
L ft Fs
6 ()@== () -
()
R
# ft etdst
,
(1)
where f(t) is any real or complex function defined
on R and F(s) is its transform, provided it has a
non void region of convergence (ROC)
■ The inverse LT is given by the Bromwich integral
()
1st
2rj
6 ()
aaj
j
+
3
Fs es t ! R (2)
() ,,
d
j =-1 .
ft L- Fs @ = 1 # 3
=
where a R! is inside the region of convergence of
the LT and
■ The FT is obtained from the LT through the substitution
sj~= with
~ ! R .
■ The functions and distributions have LT and/or FT.
■ Current properties of the Dirac delta distribution,
d (),t and its derivatives will be used.
■ The standard convolution is given by
() ()
ft gt fg t
R
d
) xx x=-# () () .
(3)
■ The order of the fractional derivative is assumed
to be any real number.
■ The multi-valued expressions sa
and ()sa
are
used. To obtain functions from them we will fix for
branch-cut lines the negative real half axis for the
first and the positive real half axis for the second.
For both expressions, the first Riemann surface
is selected.
■ The " floor " of a real number a is denoted as the
integer N a= 6@ verifying
NN .11# a
+
■ The Heaviside unit step and the signum function
are represented by ()tf
and sg (),tn
=-f
spectively. These functions are related through
sg () () .tt21n
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