IEEE Circuits and Systems Magazine - Q3 2022 - 7

Indeed, the procedure
can be applied when
approximating continuous-time
systems
for filter implementation
and control [4],
[7]-[10], as well as in
modeling [11]-[14].
More recently, the
delta approach was
revisited soaked in
the Hilger's formulation
for a continuous/
discrete unification
[15], which is nowadays
called calculus
on time scales [16],
[17]. This formula -
tion considers a general
domain, called
time scale or more
general ly measure
chain, that can be
continuous, discrete
or mixed [16]-[19]. In
this domain, Hilger
defined two derivat
ives, named delta
and nabla, that are
the incremental ratio
or their limit to zero
when calculated at a non isolated point. With these derivatives,
we can devise the corresponding differential equations
representative of some given systems. Using a current
nomenclature we will call them nabla and delta systems in
agreement with the adopted derivatives. The nabla derivative
is causal, while the delta is anti-causal. Based on this
framework, suitable formulations for fractional discretetime
system definitions had to be sought. They are based
on discrete nabla and delta derivatives so that the resulting
systems mimic the continuous-time that emerge from
the discrete-time as a limit when the sampling rate,
rh /1=
,
increases without bound, meaning that the inter-sample
interval, h R!
+ (also called graininess) goes to zero. In
fact, it is like a return back to the origins when the discrete
were mere approximations to the CT systems [19]-[21].
However, these approaches did not propose coherent formulations
of the nabla and delta system theory, that was
developed later in a more comprehensive way [22].
The above approach has two drawbacks.
■ While in the traditional discrete-time systems the
unit circle is the reference for stability, the nabla/
delta approach uses a circle with center // ,h1+passing
at the origin [22].
■ In the traditional discrete-time signal processing
the Z transform (ZT) is the tool par excellence for
working with systems defined by difference equations.
In the case of nabla/delta systems, we need
to define new transforms [22].
As it is well known, the stability domain of causal
continuous-time system is the right half complex plane
(HCP). We can establish a one to one correspondence
between the left (right) HCP and the interior (exterior)
of the unit disk. Such relation can be expressed by a
particular case of the bilinear (or, Möbius) transformation
that is commonly named the Tustin map [23], used
for the discrete-time approximation of continuous-time
linear systems [5], [6], [24], [25]. However, no discretetime
derivative was introduced. This idea was explored
in [26]. In fact, the bilinear transformation allows us to
formulate a general discrete-time fractional calculus
that mimics the corresponding continuous-time version,
while being fully autonomous. On the other hand, it leads
to tools and concepts similar to those of continuous-time
fractional signals and systems (see the companion paper
[2]). Another important characteristic of the proposed
derivatives and systems lies in the fact that the Z transform
is the most appropriate one in this framework. As
the unit circle recovers its traditional role, the usefulness
of the Fast Fourier Transform (FFT) becomes clear
from the numerical and calculation time perspectives.
These two formulations led to discrete-time differential
equations describing input-output relations in a similar
way to what occurs with the continuous-time fractional
systems. However, there is a generalized framework that
can be used to deal with continuous-time and discretetime
systems usually called state space representation.
While the first approaches consider the system like a
" black-box " relating merely the input and output, this
one allows us to " see " what is inside the system, through
the introduction of inner variables describing the socalled
state of the system. In this case, the model is defined
by two equations that acquire the same form for
all systems: the dynamic and observation equations.
To solve the dinamic equation we need to introduce the
state transition operator verifying the semi-group properties.
We present this general formulation together with a
correct definition of state transition operator.
Manuel Duarte Ortigueira is with the Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and
Technology of NOVA University of Lisbon, Portugal (e-mail: mdo@fct.unl.pt) and J. A. Tenreiro Machado was with the Institute of Engineering, Polytechnic
of Porto, Department of Electrical Engineering,431, 4249-015 Porto, Portugal (e-mail: jtm@isep.ipp.pt).
THIRD QUARTER 2022
IEEE CIRCUITS AND SYSTEMS MAGAZINE
7

IEEE Circuits and Systems Magazine - Q3 2022

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