# IEEE Circuits and Systems Magazine - Q2 2022 - 53

```The regularized tempered derivative is defined by
f
Df t =
m
a
,f ()
# =ft-3
()
()/xfax dx,
C()
N () ()()
-1 mm
m!
t
mG e mx a-- -1
x
-a
(115)
that generalizes the causal expression (114) to real orders
and where
N a= 6@ For stability reasons, we con.
sider
always
m ! R .0
+
B. On the Tempered LS
We introduced in (14) the notion of a tempered LS through
its TF. After having the derivatives defined in the previous
sub-section we can write the corresponding differential
equation. Let x(t) and y(t) be two functions assumed
almost everywhere continuous, with bounded variation,
and of exponential order. Therefore, they have LT with
non empty regions of convergence. We define a tempered
fractional LS with input x(t) and output y(t) as the one following
the differential equation
N
00
k
k
k
m
a
,f ()
k
where ta ,,f,,
! kN and bk = f01 M
R k
,, = 01
M
// (116)
k==
bD xt
k
c
b
k
k
,f (),
k ,, ,, ,
are real valued constant coefficients. The parameters
a k
and kb are the derivative orders that, without loss
of generality, we assume to form strictly increasing sequences
of positive real numbers. The exponential coefficients
,, ,, ,kN
m = 01 f
k
and ,, ,, ,kM are real
c = 01 f
k
numbers. The differential equation (116) is very general
in the sense that we can use forward, backward or both
types of derivatives. However, for most practical applications,
where we deal with causal systems and, therefore,
the use of the forward tempered GL or L derivatives is
more appropriate. For stability reasons, the parameters
,, ,, ,kN
m = 01 f
k
and ,, ,, ,kM must be positive.
c = 01 f
k
The TF (14) corresponding to (116) poses difficulties
for an analytic manipulation. Therefore, just consider the
commensurate case defined with
ab !a== ,.
kk kk N0
Furthermore, this case is only manageable if
,, ,, .
mc m== =
kk 0 k 123 f The TF becomes:
V. Conclusions
M
Gs
() = N
or, equivalently
()
SECOND QUARTER 2022
/
/
M
= 0
%
%
N
Gs K k=1
k=1
6
6
()
()
sp
sz
++m
m
a
a
k
k
@
@
,
(118)
k=0
k=0
as
bs
k
k
()
()
+
+
m
m
,
ka
(117)
ka
The
fractional continuous-time linear systems were
presented. Two classes were introduced, namely,
the fractional ARMA and the Tempered systems. For
both classes we showed how to handle the standard
tools, like impulse response, transfer function, and
frequency response. We considered also the stability
and the initial-condition problem. Additionally, for
backward compatibility with classic systems, suitable
fractional derivatives were introduced having
IEEE CIRCUITS AND SYSTEMS MAGAZINE
53
This expression shows clearly the presence of two factors
with different characteristics described by exponential
and fractional power functions [116].
Let us compute the impulse response merely for the
()
Cs = x e
a =+ +
a
s
sa
+
+
The inverse LT gives
()
ct = x
e x
t
that can be written as
()
ct =
CC -
() () 0#t aa 1
e
aa
x
a
-at
ut ue utd f().
-
-- ()
1
-- -cmau 1
1
x
(123)
x o
a
x
a `s
x
a j ^sah-a
a
where pk and zk 12= f are the pseudo-poles and
-zeroes and K0 is a constant. The corresponding differk
,, ,
ential equation can be written as
()
N
// (119)
k 00
ka
k
gt0
==
m00 ().
k
bD xt
k
ka
m
If we denote by () the IR of this system, corresponding
to m0 0 ,= and use the shift property of the LT,
we conclude that the IR of (119) is given by
()
gt eg t
t
=
-m0
0().
(120)
This shows that the tempering procedure leads to an
increase in the stability domain of a system.
Example IV.1. The fractional lead ()a+ compensator
used in Control to increase the phase of a system around
a chosen frequency and the lag ()acompensator,
used
to increase the static gain of a plant, are defined by the TF
[23], [114], [115]
Cs
()
= ax
+
`
x + !a
sa
sa
j ,, ,.
+
a ! R 21
(121)
M
.
a;;EE (122)
- at aa--11
-at
CC()
()
-a
f()
)
te t
a
f t
()
```

# IEEE Circuits and Systems Magazine - Q2 2022

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