IEEE Circuits and Systems Magazine - Q2 2022 - 48

Remark III.7. Frequently, we find papers that, assuming
the Riemann-Liouville or Caputo derivatives, use the
results we just presented. Nonetheless, this is not consistent
with relations (60) to (62), since they are invalid for
such derivatives.
F. The Initial-Condition Problem
The initial condition (IC) problem is one of the most
discussed topics in linear systems. The IC are a set of
values that determine the output of a system when the
input is null, i.e. the free response.
Since the thirties in the last century, this problem
has been solved with the ULT. However, this approach
uses values taken at
t 0= +
, instead at t 0= - since the
,
IC depend on the past, not on the future. This led to
a modification of the ULT [93] that has been used in
the study of integer order LS. Nonetheless, we shall
avoid this approach and recall that the designation
IC refers only to inputs and outputs before the reference
instant.
Concerning the fractional systems the first approach
was based in the RL derivative and the corresponding
IC it induces [89], [94]. This procedure has been superseded
by the one coming from the C derivative, since
it uses initial conditions based on integer order derivatives
[89]. These solutions were called into question differently
by Lorenzo and Hartley [95] and Ortigueira [96]
suggesting that the IC would be dependent on the structure
of the system in question and not on the derivative
used. This means that the RL or C derivatives have
" their " own IC, not necessarily those posed by a system.
Later Trigeassou et al. [97] and Sabatier et al. [42] proposed
a new approach based on the " infinite state approach "
following the so-called diffusive representation
[98]. However, these approaches are not attractive in
Engineering, because they do not show backwards compatibility
with classic results.
The idea of having the IC as dependent on the structure
of the system was recalled by Ortigueira and Coito in [99].
In particular, it was shown how we can choose suitable IC in
-20
-40
-60
-80
-100
-120
10-3
10-2
10-1
100
101
Frequency (Hz)
102
103
104
105
-0.5
-1
-1.5
10-3
10-2
10-1
a = 0.2
100
a = 0.4
02kk ,,
.,
12 5g
101
Frequency (Hz)
a = 0.6
Figure 2. Bode diagrams of the RC circuit for a== (from above).
48
IEEE CIRCUITS AND SYSTEMS MAGAZINE
SECOND QUARTER 2022
a = 0.8
a = 1
102
103
104
105
Phase (Radian)
Amplitude (dB)

IEEE Circuits and Systems Magazine - Q2 2022

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