IEEE Circuits and Systems Magazine - Q2 2022 - 46

The main aim in this section is to present a coherent
basis for establishing fractional operators compatible
with the corresponding classic integer order. In particular,
the formulation developed in the sequel allows currently
used tools like, IR, TF and FR, and includes the
standard derivatives obtained when the orders become
integers. To look forward FD formulations consistent
with the laws of Physics we recall the most important
results from the classic calculus. The standard definition
of derivative is
Df tf t
f ==lim
h 0
() ()
l
"
or
Df tf tl()
b ==lim
h 0
()
"
ft hf t() ()
+h
.
(49)
Substituting
- h for h interchanges the definitions,
meaning that we only have to consider h 0> In this
.
case, expression (48) uses the present and past values,
while (49) uses the present and future values. In the following,
we will distinguish the two cases by using the
subscripts f (forward-in the sense that we go from past
into future, a direct time flow) and b (backward-meaning
a reverse time flow).
It is straightforward to invert the above equations
to obtain
3
Df lim /
n=0
and
3
Df lim / " 0
-1 () ()·.
b tftnhh
h
=+
n=0
(51)
These relations motivate the following comments:
■ The different time flow shows its influence: the
causality (anti-causality) is clearly stated,
■ We have DD ft DD ft ft== and
()==
f ff f
11
()
DD ft DD ft ft
b bb b
11
Dfth a- / n!
a
f () = lim+
h " 0
Df te h a
h " 0
a
b () -=
jar
lim+
/
+3
^-ahn
n=0
n=0
-- () ().
-- () ()
We will call D ,fb
-1
antiderivative
[63].
We generalize the derivatives to the fractional case
ft nh
-
+3 ^-ahn
n!
()
ft nh
+
()
(52)
that are at present called Grünwald-Letnikov (forward
and backward) derivative, in spite of their first proposal
having been done by Liouville [26]. The symbol
^h-a n
^^ ^hh h In applications where
-= -= -+=
-
aa a
1
n
P
k
n
1
k .
ss ss s
==
which gives us two ways to solve the problem. The first
reads [26]
Df t = #3
a
f ()
x
M--a
a
1
C()
-+M
ft d
[26]. The second decomposition, ss , gives
-
aa=
-
Df tD
a () =
ff C()
M;#3
-+M
a
x
M 1--a
() ,d E
ft xx
() -xx
M () .
(56)
This is called Liouville-Caputo derivative (LC) [8],
MMs
(57)
that constitutes a derivative of the Riemann-Liouville
type, that is also called Liouville derivative [30].
In conclusion, from the IR of the differintegrator, 3 difis
the
Pochhamer representation of the raising factorial:
,
46
IEEE CIRCUITS AND SYSTEMS MAGAZINE
ferent integral formulations were obtained from where
current expressions can be derived, (55), (56), and (57).
A fair comparison of the 3 derivatives lead us to conclude
that:
SECOND QUARTER 2022
aa MM MM
--a
-1 () ()·
= "
f
tftnhh
h
(50)
where N Z0
!
+ is the greatest integer less than or equal
-
. We can write
to a , so thataa1 N1 # . However, we have two alternatives
for applying the convolution, avoiding the singularity.
Let M Z0
a#! +
ft ft h
-h
()
()
,
(48)
the variable t is not a time the constant factor e , in
-jar
the backward case, can be removed [63]. In such situations,
the derivatives can be called left and right respectively
and we can show that they verify
L [Dfts Fs
lr,
a ()]( )( ),
a
Re() ,
s
= !! $ 0
(53)
in agreement with the requirement (46). In the rest of
this section we consider merely the forward case. It
must be highlighted an important fact: equation (52) is
valid for any real (or complex) order. The relation (53)
suggests another way of expressing the FD. We only
have to remember that in (20) we obtained the impulse
response of the causal differintegrator. So, the output
for a given function, (),ft is given by the convolution
Dftftd
C() 0
a
fb, () =
1
-a #3 --a
xx x
"
1 () .
(54)
Relation (54) is an integral formulation of the FD. However,
this expression is not as handy as (52) due to the singularity
of
x a-- 1
at the origin when a 2 0 . Therefore, we
adopt it only for negative orders (anti-derivative) and, for
the positive (derivative) case we proceed with the regularization
of the integral (we will consider the causal case).
The regularized Liouville derivative is given by [91]
Dftft--= () /
1
f () = # C()
a
3 x xx x, (55)
--a
-a
N () ()
mm
-1 ft mGd
()
m!
IEEE Circuits and Systems Magazine - Q2 2022

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