IEEE Circuits and Systems Magazine - Q2 2022 - 45

For proof see [86] (section 8.6, pages 243-248).
The repetitive use of this theorem allows us to express
a given TF as a sum of negative power functions
plus an error term. We are going to describe the procedure
for obtaining the IR from the TF [87]. Let G(s) be a
TF and its associated ROC ()Re s
2 0. We do the following
steps:
1) Define () ()Rs Gs
=
lim HA0
v 3 " vv =
c
with inverse LT given by ()
rt0
2) Let 0c be the real value such that
()
,
Introduce an operator Da
where A0
is finite and non null. Then, let
() ()
Rs Gs As=- c- 0
10
limrt h
t 0
+ 0 () =
"
3) Repeat the process. Let 1c be the real value such
that
lim RA " vv =
c1
v 3
where A1
11
()
,
Rs Gs As As20 1
01
=- -cc
-2
with
lim R ()
rt1
v 3 " vv = 0 and
c1
is finite and non null. Again introduce
() ()
Ar t()
t
1 = lim 1
" 0+
ing () as the inverse of ().Rs1
4) In general, let nc be the real value such that
()
lim RA " vv =
cn
nn
v 3
where An
function:
,
is finite and non null. We arrive at the
nRs
Gs
and lim R ()
v 3 " vv = 0.
cn
n
As above Ar tr 0+nn to be
t
n nn- 1(),
==
"
lim+
ca
nn
=+1f nor
Gs
()
=+c
=
k
n
-
/ As Rsk
- k
1
+
! Z0
.
ca
-
- 1
1
()
We can write:
n()
(43)
leading to the conclusion that G(s) can be expanded in a
generalized Laurent series [87]
3
Gs rs0() +-ck
k 0
=
The inverse LT of each s
SECOND QUARTER 2022
-ck
allows us to obtain g(t).
If we know the pseudo-poles, we can use the algo()
= / k () .
ak
(44)
() ()
coherent with the initial value theorem and
,
nk
k
() () / As k
-
=- c
=
1
(42)
c -1 1
(), havIt
is clear that
() 0
c -+0 1
lim R ()
v 3 " vv = 0
c0
().
1
.
where ()
and A0
=
N
aD yt
k
M
// (47)
k 00
ab=
()
k
==
kk (),
k
bD xt
that is an equation defining a system with TF
M
Gs
/
/
() = N
k= 0
k= 0
We introduced the differintegrator, verifying the
property (46). For positive order, we will call it fractional
derivative.
FD is the name assigned to several mathematical operators,
namely the Gr├╝nwald-Letnikov (GL), Liouville
(L), Riemann-Liouville, Caputo, Marchaud, Hadamard,
Riesz, and others [8], [23], [30], [31], [63]. Without going
into the discussion of a methodology for deciding
if a given operator is a FD [88], we are interested in
selecting those that are suitable for generalizing wellknown
tools of Signals, Circuits and Systems and other
Applied Sciences. In particular, we must consider
the formulations that can be useful for the introduction
of fractional linear systems and their characterization
in the perspective of compatibility with integer
order definitions, as we assumed in the previous
sub-sections.
Remark III.6. We must note that
1) The traditional and more frequently used RiemannLiouvile
[30], [31] and Caputo [31], [89] derivatives
do not verify (9);
2) The Caputo-Fabrizio and Atangana-Baleanu operators
[90] are in fact highpass filters;
3) The so-called local derivatives [90] are essentially
integer order derivatives;
Therefore, these operators are incompatible with the traditional
formulations we use in Circuits and Systems.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
45
as
bs
k
k
bk
.
ak
Fs = L ft and ! s 02 We obtain from (45)
[ ()]
Re () .
- aa= ()@
6
(differintegrator) such that
L sF sD ft
1
(),
(46)
N
as Ys
k
M
// (45)
k 00
ab=
()
k
==
kk ().
k
bs Xs
rithm described in [64]. However, it must be emphasized
that there are non-factorizable pseudo-polynomials.
D. From the TF to the Differential Equation.
Fractional Derivatives
Consider the causal case (we will omit the ROC). From
(8), using the LT, we obtain () () ()
Ys GsXs=
that leads to

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