IEEE Circuits and Systems Magazine - Q3 2022 - 20

x tt x$xxU=
()
(, )( ).
(108)
The state transition operator is defined by
(, )( ,)
tt (, ),001
Uxx$UU=
-
with (, ).00 I= It can be shown that this operator veriU
fies
the usual properties, namely the semi-group property.
In applications, we need to compute its inverse,
1 (, ).0 x
UWe
compute it for the nabla, bilinear and CT
cases. To compute the inverse of (105) we set
(, )
3
U-1 0 nh = / zkA h
k=0
kk ()a ka +1 n
n!
where the coefficients zk , k 01 2f=
,,
f nh
(),
(110)
, need to be determined.
The product of (105) and (110) gives the indentity
matrix. With some manipulation, we obtain then
z0 1= and
z =-k !(1/)
a / zm
m
k
=
-
nl
ml m
n ``jj
1 -- +
aa
11
nn k 12
a - n
=
,, ,f
Similarly, (106) suggests that we write
(, )( ).
U z} 1f nh
--1
0 nh = / ` j
3
kA
k=0
k h
2
The values zk , k ,, ,01 f=
k
zz -- - ,, ,,
}}n
km
m
/
=
-
with z0 1 .=
For the CT case, we proceed similarly. Letting
(, )
3
U-1
0 xz
k=0
= / kAk
we can show that z0 1= and
zz j,, ,f
=
k
km k 12
m
=
-
=-/
`
m
k
a
a
With the state transition operator defined by (103) and
(109) we can obtain the output of the system using the
standard procedure [42, 40, 30].
V. Conclusions
In this paper we presented two different approaches for
describing discrete-time fractional linear systems. The
first is based on the nabla and delta discrete-time derivatives.
We found their eigenfunctions, nabla and
delta exponentials, that we used to define discrete LT.
The second approach is based on the bilinear (Tustin)
20
IEEE CIRCUITS AND SYSTEMS MAGAZINE
J.A. Tenreiro Machado graduated with
the 'Licenciatura' degree in Electrical
Engineering at the University of Porto,
in 1980, obtaining the Ph.D. and 'Habilitation'
degrees in 1989 and 1995, respectively,
in Electrical and Computer
THIRD QUARTER 2022
1
x
ka
C()
ka +1
f x
()
(112)
==a
a
1 }}n
m
n
-- -- -km a
n
k 11
are given by:
1 () 1
k 12 f
ka+1
n
ka
(109)
transformations. For both cases, appropriate algorithms
for obtaining the impulse, step, and frequency responses
were presented. Finally, the state-variable representation
was also introduced. This one is very general in the sense
that it can be used for continuous-time systems too.
Acknowledgment
A few days before the end of the revision procedure, my
friend J. Tenreiro Machado had a sudden cardio-respiratory
arrest and died. Here I want to express my gratitude
and tribute to a great man and scientist. He was a
very friendly and helpful person, with an unusual work
capacity that allowed him to publish interesting articles
on a wide range of topics.
This work was partially funded by National Funds
through the Foundation for Science and Technology of
Portugal, under the projects UIDB/00066/2020.
Manuel Duarte Ortigueira received
the Electrical Engineering degree at
Instituto Superior Técnico, Universidade
Técnica de Lisboa, in April 1975
and the PhD and Habilitation degrees
at
the same Institution in 1984 and
(111)
1991, respectively. Nowadays he is Associate Professor
with Habilitation (retired) at the Electrical Engineering
Department of the Faculty of Sciences and Technology
of Nova University of Lisbon. He was professor at Instituto
Superior Técnico and Escola Náutica Infante
D. Henrique. He published 3 books on Digital Signal Processing,
Fractional Calculus, and Fractional Signals
and Systems, over 180 papers in journals and conferences
with revision, and has 2 registered patents. His
research activity started in 1977 at Centro de Análise e
Processamento de Sinais, continued at Instituto de Engenharia
de Sistemas e Computadores (INESC), where
he was with the Digital Signal Processing and Signal
Processing Systems groups, and since 1997, at Instituto
de Novas Tecnologias (UNINOVA), where he is with the
Signal Processing group of Center of Technology and
Systems. He is regular reviewer of several international
journals and member of the scientific committee of
several international journals and conferences. Nowadays
his main scientific interests are Fractional Signal
Processing, Digital Signal Processing and Biomedical
Signal Processing.
IEEE Circuits and Systems Magazine - Q3 2022

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