IEEE Circuits and Systems Magazine - Q2 2022 - 39

```■ Let a and b two complex numbers with positive
real parts. The two-parameter Mittag-Leffler function
is defined by [54]
3
Ez =
b-1
ab,
a f t
ab, () / C ab= ()k 0
k
z
+
k
,. (4)
z C
!
I n appl icat ions , the causa l f unc t ion
tE () ()pt
L () ,(),Re s 2 0
Epta
ab, -= s
s + 1
a
■ The general binomial series reads
!
1-=
z
a
k
where a n
^h3
=
k
the
raising factorial:
^^
-= -+
=
hh%
aa ,k
n
with -=
a
0 1 .
k
n
-
1
^h If a is not a negative integer, then
!
()
k
-a k
=cm
a
k
that represents the well-known binomial coefficients.
D.
Abbreviations
The following abbreviations are used in this manuscript:
ARMA Autoregressive-Moving Average
CT
DT
Continuous-Time
Discrete-Time
FARMA Fractional Autoregressive-Moving Average
FD
FP
FT
FR
IC
IR
GL
L
LTIS
LS
LT
RL
RP
RD
RFD
TF
TFD
ULT
Fractional derivative
Feller Potential
Fourier transform
Frequency response
Initial-conditions
Impulse Response
Grünwald-Letnikov
Liouville
linear time-invariant system
Linear system
(Bilateral or two-sided) Laplace transform
MLF Mittag-Leffler function
NLT
Riesz Potential
Riesz Derivative
Riesz-Feller Derivative
Transfer function
Tempered Fractional Derivative
Unilateral (one-sided) Laplace transform
SECOND QUARTER 2022
Nabla Laplace transform
Riemann-Liouville
is used, for it has LT
ab(5)
where
Re (·) is the real part of a complex number.
^h / () k
-a k
z ,
(6)
is the Pochhamer representation for
E. Outline of the Paper
We start by recalling some important concepts regarding
the study of linear systems in Section II. We define
the IR and TF with great generality (II-A), and give
examples of several types of TF. For defining such
TF, we introduce the main basic tool for expressing
the fractional systems: the differintegrator (II-C). The
previously referred systems are presented progressively
in Section III, from the simplest commensurate
(III-A), to the non-commensurate (III-C). It is shown
that fractional systems can be decomposed into two
components: integer and fractional. This decomposition
has consequences for the stability studied in
subsection III-B. The very important problem of the
IC is also analyzed in subsection III-F. The IC depend
on the structure of the system not on the used
transform. A general formula for dealing with them is
presented. The study uses the TF for starting point,
without introducing any differential equation. A type
ARMA fractional order differential equation together
with the appropriate definitions of derivatives are
presented in III-D. The FR is also studied in III-E.
This set of topics covers the most important systems,
namely the electric circuits. However, there are other
interesting subjects that we found useful and that are
studied briefly in sub-section III-G: the variable order
derivatives and systems (III-G 1)), an introduction to
fractional order stochastic processes (III-G 2)), and
the particular case of the fractional Brownian motion
(fBm) treated in III-G 3). Most of the tools introduced
in these sections are useful for dealing with another
system generalization: the tempered fractional LS
studied in Section IV. Therefore, we introduce the
tempered fractional derivatives (IV-A) suitable for expressing
the differential equations used to describe
the tempered fractional LS (IV-B). Finally, some conclusions
are presented in Section V.
II. On the Continuous-Time Linear Time-Invariant
Systems (LTIS)
A. Impulse Response and Transfer Function
The linear systems (LS) are of primordial importance
in Science and Engineering. In many situations we deal
with nonlinear systems, but the linear are very useful
and deserve study. Traditionally, linear systems are
based on integer order differential or difference equations,
if they are continuous- or discrete-time.
Let us consider continuous-time systems for which
the input-output relation assumes the general form:
()
yt = #+3
-3
IEEE CIRCUITS AND SYSTEMS MAGAZINE
39
gt xtd
x
(, )( ),xx ! R,
(7)
```

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