IEEE Circuits and Systems Magazine - Q2 2022 - 44

Gs
() =
1 RC s
1
+ a
6
■
a ,
where Ca is the capacitance in Fs1 a@
and a1 1. The IR
of this system has only fractional component. It is interesting
to note that when
a 1= the situation is reversed. In
figure 1 we depict the step responses of a RC circuit with
.
RC
a = 1s
- a
B. Stability
Consider a given term of (31) corresponding to one
pseudo-pole p:
gt pe t
up p
() = 1 11 pt
1/a
/aa
+
sin()
r
-+
ar #3
a
2 cos ar
()
f()
0 22eu t
f
u
- d ·( ).
ut
We can extract some conclusions [81]:
1) The fractional part is always bounded, for
bounded. Therefore,
()
#3
0 22eu A t
u
a
upcos ar
2
a-+p
- 120 (40)
utd
t
,
This term exists whenever a ! 1 and does not
contribute to unstability.
2) As mentioned previously, the integer part only
exists if
- ra 1 arg p # ra. In this case, we have
()
■
■
three situations [81]:
(/ )
arg p
arg p
() 1 2ar - the exponential increases
without bound - unstable system;
(/ )
() 2 2ar - the exponential decreases to
zero - stable system;
Step Response
1
0.8
0.6
0.4
0.2
C. General Non Commensurate Case
The non commensurate case cannot be dealt as we discussed
above, unless we know the pseudo-poles. In this
case, we can use (31). In the general case it is possible
to obtain the IR in the form of a fractional McLaurin series
through the recursive application of the general
initial value theorem [86], [87]. This theorem relates the
asymptotic behavior of a causal signal, (),gt as t 0 "
Gg tLv =
+
to the asymptotic behavior of its LT, ()
()
v = Re s " 3.
Theorem III.1 (The initial-value theorem). Assume
a = 0.2
a = 0.4
a = 0.6
a = 0.8
a = 1
02468 10 12 14 16 18 20
t
Figure 1. Step responses of the RC circuit (example III.1) for
.,
a== (from below).
02kk ,,
44
12 5g
IEEE CIRCUITS AND SYSTEMS MAGAZINE
that g(t) is a causal signal such that in some neighborhood
of the origin is a regular distribution corresponding to an
integrable function and its LT is G(s) with region of convergence
defined by ()
a real number b 2 1Re
s 2 0. Also, assume that there is
such that li ()+mgt tb
t 0 "
finite complex value. Then
()
lim+
"
t 0
gt
t
b = lim
v " 3
vv
C
b+ 1
()
()
G
b + 1
.
(41)
SECOND QUARTER 2022
exists and is a
[( )], as
a
(39)
() =- the exponential oscillates sinusoidally
- critically stable system.
arg p
ar 2(/ )
The above considerations allow us to conclude that
the behavior of stable systems can be integer, fractional,
or mixed:
■ Classical integer order systems have impulse responses
corresponding to linear combinations of
exponentials that, in general, go to zero very fast.
They are short memory systems.
■ In fractional systems without poles, the exponential
component disappears. These are long
memory systems. All pseudo-poles have arguments
with absolute values greater than /,
ra
where a11 .
■ Mixed systems have both components. Some
pseudo-poles have arguments with absolute values
larger than /.2ra
^h
Remark III.5. The three rules above are usually known
a 2 0
and any p C! . In fact, it is a simple matter to
verify that function /( )
uu cospp222ar
aa-+ arg p^h is
as Matignon's theorem, that can be put in the following
way: TF (10) is stable iff all pseudo-poles pk
^h
verify
()
k 2ar/2 .
The above procedure for studying the stability demands
knowing the pseudo-poles. In the integer order
case, there are several criteria to evaluate the stability
of a given linear system without the knowledge of
the poles. One of the most important is the RouthHurwitz
criterion that gives information on the number
of poles on the right hand half complex plane [83].
The generalization of this criterion for the fractional
case was proposed in [84] and is very similar to the
integer order case [23]. A very interesting alternative
is the Mikhailov criterion [85] that is formulated in the
frequency domain.
re(t)
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