IEEE Circuits and Systems Magazine - Q3 2022 - 13

The nabla and delta derivatives allow the definition of causal and
anti-causal discrete-time systems, assuming a form that mimics
the corresponding continuous-time analogues.
in practice, since it allows approaching CT systems
closely using small values of h.
For a complex conjugate pair we obtain:
3
vnhReAph " -k=1
()
2= /
kk n
, a
11 ak
n!
()
f nh
().
(48)
The properties of the nabla exponential and the
sequence of operations that we followed to compute
the NLT of the causal transform showed that,
if the poles are outside the right Hilger circle, then
the system is stable. In parallel, the partial fraction
inversions computed previously showed that
the series defining the time functions are convergent
if
;; a
ph ,12 and ;; a
ph ,11 in the first (44) and
second (46) cases, respectively. This means that
if p is outside the Hilger circle, then the system is
stable. Moreover, the system can be stable even if
the pole is located inside the Hilger circle provided
that it is outside the circle
;; ; ;=
sp .
1
discrete-time descriptions, since they have stability
domains that are defined by the Hilger circles, instead
of the unit circle that is the reference in the theory of
classic discrete-time systems. On the other hand, they
require new LT instead of the ZT. This observation motivated
the search for an alternative for defining fractional
linear systems.
In the DT approximation of CT systems, the Tustin
a This represents
the discrete counterpart of the stability
criterion for CT systems [30]. For the integer order
systems we can study the pole distribution by a
Routh-Hurwitz like criterion [31]. It is important
to remark again that the exterior (interior) of the
right Hilger circle degenerates in the left (right)
half complex plane when
h 0 " .
F. On the Initial Conditions
The initial condition problem was debated in the companion
paper [2]. Therefore, everything argued in that
paper remains valid here as long as we consider the required
adaptation. For the commensurate case, this can
be found in [22] and reads
NN () Nk a
6 ff @ / ka
gt ts ft tg hs
()
dd
a
N
() ()@=- -
Na
6 () ()
k
N
=
-
1
() () 1
-(49)
With
this expression we can insert the initial conditions
in any system as proposed in [32] and [33].
III. The Bilinear Transformation Based
Linear Systems
The systems described in the previous section are useful
for introducing the discrete-time systems defined by
fractional discrete-differential equations. We could define
the standard tools, such us the IR and TF. However,
these systems are somehow different from the standard
THIRD QUARTER 2022
(bilinear) transformation [23] is of current use. In fact,
the definition of DT LS having as base such transformation
was proposed just recently [26]. The Tustin
transformation is a particular case of the conformal
Möbius mapping. As it is well-known, it establishes a
bijection between the left (right) half complex plane
and the interior (exterior) of the unit disk. This property
allows us to consider such transformation as the
base for defining alternative discrete-time derivatives
and the fractional systems that mimic the analogous
CT versions. Such strategy enables us to adopt the
tools and the results available in the DT domain for
the CT fractional systems introduced in the companion
paper [2]. Moreover, the proposed derivatives and
systems have the important feature of being suitable
to be implemented through the FFT with the corresponding
advantages, from the numerical and calculation
time perspectives.
A. Forward and Backward Derivatives Based
on the Bilinear Transformation
The Tustin transformation is usually expressed by [5], [6]
s = 2
h
1
1
+
-
z
z
-
-
1
1
,
(50)
where h is the sampling interval, s is the derivative operator
associated with the (continuous-time) LT and z 1the
delay operator tied with the Z transform.
Let x(nh) be a discrete-time function, we define the
order 1 forward bilinear derivative ()Dx nh of x(nh) as the
solution of the difference equation
nh Dx nh h+- =- -
h
2 6xnhx nh h @
Dx () () () () .
(51)
Similarly, we define the order 1 backward bilinear derivative
()Dx nh of x(nh) as the solution of
nh hDxnh
++ =+ -
h
2 6xnhh xnh @
Dx () () () () .
(52)
IEEE CIRCUITS AND SYSTEMS MAGAZINE
13
IEEE Circuits and Systems Magazine - Q3 2022

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