IEEE Circuits and Systems Magazine - Q3 2022 - 6
Feature
The 21st Century
Systems:
An Updated
Vision of
Discrete-Time
Fractional Models
Manuel Duarte Ortigueira
and J.A. Tenreiro Machado
IMAGE LICENSED BY INGRAM PUBLISHING
Abstract
Two different approaches for describing discrete-time fractional
linear systems are presented. The first is based on the nabla
and delta discrete-time derivatives. In this case, suitable exponentials
are introduced and used to define discrete Laplace
transforms. The second approach is based on the bilinear (Tustin)
transformations. For both cases, appropriate algorithms for
obtaining the impulse, step, and frequency responses are presented.
The state-variable representation is also analysed.
I. Introduction
D
6
iscrete-Time Fractional Models must be truly regarded
as representative of the 21st century systems.
Indeed, while the continuous-time fractional models
exist since the 19th century, the discrete-time versions
have only recently been introduced. This state of affairs
happens because it is not easy to " fractionalize " discrete
integer order systems defined by difference equations. An
attempt was rehearsed in [1] where a model based on a
Digital Object Identifier 10.1109/MCAS.2022.3160908
Date of current version: 5 September 2022
IEEE CIRCUITS AND SYSTEMS MAGAZINE
fractional delay equation was introduced. It was possible
to define and compute the traditional tools like, impulse response
(IR) and transfer function (TF). However, it could
not be considered as a fractional discrete-time system, in
the sense followed in [2], since the corresponding TF is not
truly fraccional. This failure showed that we had to give a
deeper thought to the problem and, the most natural way
to do it, was to go back to the origins.
Discrete-time systems began as a mere set of numerical
tecniques to approximate and solve continuous-time
differential equations. Such procedure was based on
the incremental ratia used to aproximate the derivatives
and is currently known as Euler method [3]. This
approach gave rise to the delta systems [4], but, with a
slight modification of the perspective, originated the important
class of the discrete-time systems based on the
difference equations [5], [6]. Nonetheless, the original
Euler procedure was not completely abandoned, even
if there was no formal introduction of such systems.
Besides remaining important as an intermediate step
to obtain difference equations from continuous-time
systems, they were used under the delta system format.
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THIRD QUARTER 2022
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IEEE Circuits and Systems Magazine - Q3 2022
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