IEEE Circuits and Systems Magazine - Q3 2022 - 18

for the anti-causal case. We can give to expressions (77)
and (78) a form that states their similarity with the classic
fractional LS [33], [36]. For example, for the first, let
(/ )(()/(
v hz z )) . We have
M
=- +--21 111
()
Gv
/
/
()= N
k=0
k=0
^h2/h ka
Remark III.2. It is important to note that the factors
,
k ,, ,12 f=
do not have any important role in the
computations. Therefore, they can be merged with the coefficients
ak
and b .k
The procedure to invert (79) is identical to the one
we followed in the nabla system (42). For simplicity we
assume that MN1 and all the roots,
R =0
p ,k k ,, ,12 f=
k a z
N
k
k
are simple which allows us to write
()
N
Gv = /
k=1
where the Ak
vp
A
k
a - k
,
and p ,k k ,, ,,N12 f=
pseudo-poles obtained by substituting w for sa
Fs = a
d
sp
A
-
Example III.1. Consider the simple system with TF
()
Gv = a +
v 2
1
.
(82)
of
(80)
are the residues and
in (39).
The IR results from the inversion of a combination of
partial fractions such as:
()
.
(81)
av
bv
k
k
bk
.
ak
(79)
A. A Brief Introduction
Three equivalent representations of linear invariant
systems were studied in the previous sections: the differential
equation, IR, and TF. In the time-variant case,
the differential equation remains a valid representation
and we can still introduce the notion of IR, but we cannot
define a TF. On the other hand, these representations
consider the system like a " black box " relating an
input with an output, while forgetting what happens
" inside " the system. Such limitation can be avoided by
the representation in state variables, also called internal
representation. This approach has several advantages,
namely, a simple matrix formulation and the ability to
formulate, in an identical form, a large number of different
cases, such as non-linear, time-varying, or multivariate
systems [30], [37-41]. Besides the input and output
signals, this representation introduces another one, the
state, that may be vectorial or matricial. Here, we present
such a description valid for the fractional LS, not
only DT, but also CT.
B. Standard Form of the Equations
Let (),tv
Step Response of Nabla System
0.8
0.6
0.4
0.2
020406080
Time
0.8
0.6
0.4
0.2
Step Response of Bilinear System
100 120 140
Dt tt txx v
yt tt txvW= () [, (),( )],
a () [, (),( )]
=U
where []N
a aa fa
= 12
ferentiation orders. Moreover, the derivative Da
torial order a is defined as
()
Dt Dx tD xt Dx t
a aa a
x
()
= 6 12() f12 n
n()@T
y (),t and ()tx be the input, output, and state of
a system respectively described by the following set of
equations:
(83)
(84)
T is a vector with (positive) difof
vec(85)
and
represents any of the previously defined causal
derivatives, CT and DT. The expressions (83) and (84)
are called state (or dynamic), and output (or observation)
equations, respectively. We will assume that U and W
are linear functions, so that
020406080
Time
100 120 140
α = 0.5 α = 1 α = 1.5
Figure 2. Step responses of the system (82) for a 0 . ,,. .511 5
= "
h .01=
,
(from below) for nabla (above) and bilinear (below) cases, with
.
18
IEEE CIRCUITS AND SYSTEMS MAGAZINE
Dt tt ttxA xB v=+a
()
tt tt t
() () () () (),
yC xD v=+
() () () (),
(86)
(87)
where ()tx is a N 1# vector and ()tA is an N N# matrix.
The dimensions of the other matrices are chosen in
agreement with the type of system:
THIRD QUARTER 2022
In figure 2 we represent the step responses for several
values of the order, a 05= .,k k ,, ,12 3 obtained with nab=
la
and bilinear formulations.
IV. State-Variable Representation
IEEE Circuits and Systems Magazine - Q3 2022

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2022
