IEEE Circuits and Systems Magazine - Q2 2022 - 49

a state space formulation, using an expansion of the solution
in fractional Taylor series. However, a general approach
based on a reinterpretation of the role of IC was
proposed. To understand it, let us consider the simple
example of a one-pole lowpass filter
Df() () (),
taft gt+=
where a is any real. Assume that ()
f tt0
-
:
gt 0= for tt0
2 and
that we want to compute the output, f(t). This is equivalent
to say that we observe the system by means of a
unit step window ()
Df() () () () .
tt taft tt 000
ff
() ()
@
-+ -=
Without loosing generality, we set t 00 = . As
Df tt Df tt ft0ff d=-6 () () () (), we obtain
Df () () () () () ()
tt ft af tt
Now, note that we can write
Df () () () () () ()
tt ft af tt
00 (69)
@
6 ffd-+ =
@ 00 (68)
that has the well known solution () ()
6 ff f-+ =
that gives a new interpretation of the IC: it is the amplitude
of the subtracting step to make the function continuous.
In terms of the LT, we can write
sf tt ft af t00() () () ()
LL()-+ =66@@
ff
or
sF sf aF s00() ()
-+ =
()
with ()
Fs = L6ft tf() () .@ This means that we made the
sF ssFs f 0
"
L ft ts sFsf sF ssffmlf =- -= -- l
derivative, we have
() ()
66@@
This procedure can be replicated for higher order de()
() ()
f 00 2
() 00
rivatives leading to
Df ()·()()().
N
6
@
L tt sFsf s0NN k
k
f =- /
=
-
1
^^ -hh
(70)
Nk 1
that coincides with the classic formula obtained with
the ULT. The way into the fractional case is similar to
the described above with the substitutions
DD , "
a
a $ 0 . From (69), we have
Df () () () () () ()
a ff f-+ =
6 tt ft af tt
00 (71)
@
SECOND QUARTER 2022
with
substitution () () ()- . For the second order
() ().
that can be inverted by the methods introduced in section
III. Expression (75) recovers the classic formula
when the derivative orders become positive integers.
G. Other Generalizations
1) Variable Order Derivatives and Systems
In the previous sections, we assumed that the orders
of derivatives were constant. However, most definitions
and tools keep their validity when the orders become
variable, provided that suitable derivative definitions
are used. Several definitions with variable orders are
known [100]-[104], but most are incompatible with our
system framework. Suitable definitions were introduced
in [23], [105] that recover the constant order definitions
IEEE CIRCUITS AND SYSTEMS MAGAZINE
49
Ys =
f ()
k
N
//1aa aa aa^^ --hh 1
mN mm Mm
=
-
1
ys
()
00
N
-- - xs
k
=
-
1
()
(75)
/ asak
k= 0
k
M
ft te t
ff=
-
and
sf tt ft af t00() ()
a -+ =66@@() ()
LL()
ff
that gives
sf tt fsa
()
a @@ (72)66()af tLL1
() ()f -+ .
s
Therefore, the substitution sF ss Fs fs0
aa -() ()
() "
is used. The repetition of the process gives
()
L66@@() ()
N
L tt sFsf s0 ()NN m
m
6
aaf =- / ^h
@
=
-
1
Nm
ft ts sF sf sf s
sF ss ff s00
=- -
= aa aa
22 11
00
() ()
()
21 1
f
aa aa aa
-()
() --that
can be generalized to
Df ()·()()().
a- 1
aa 1-- (73)
For the noncommensurate case, defined by a secm
m 01 f=
Nat
(). NN mN m
L tt sFsf s0
m
6
ccf =- cc c
=
@
/
1
^h
--1
(74)
quence of increasing orders, ,, ,, the same
procedure led us to obtain the IC theorem of the LT [65]
Df ()·()()(),
that can be used in (47) to obtain the LT of the free response.
We have
N
// ()km Nmaa aa 1
-k
==
-
as
Ys ys
bs Xs xs
k
() -
k
N
k
^
h
=- 0// ()aa aa 1km Nm
-==
-
M
k
k
()
00
1
^
h
with
() ,Fs 0= and the LT of the free response, (),Ysf
given by
is
M
00
1
IEEE Circuits and Systems Magazine - Q2 2022

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