IEEE Circuits and Systems Magazine - Q2 2022 - 36
Feature
The 21st Century Systems:
An Updated Vision
of Continuous-Time
Fractional Models
Manuel Duarte Ortigueira and J. Tenreiro Machado
Abstract
This paper presents the continuous-time fractional linear systems
and their main properties. Two particular classes of models
are introduced: the fractional autoregressive-moving average
type and the tempered linear system. For both classes, the computations
of the impulse response, transfer function, and frequency
response are discussed. It is shown that such systems
can have integer and fractional components. From the integer
component we deduce the stability. The fractional order component
is always stable. The initial-condition problem is analyzed
and it is verified that it depends on the structure of the system.
For a correct definition and backward compatibility with classic
systems, suitable fractional derivatives are also introduced. The
Grünwald-Letnikov and Liouville derivatives, as well as the corresponding
tempered versions, are formulated.
Digital Object Identifier 10.1109/MCAS.2022.3160905
Date of current version: 26 May 2022
©SHUTTERSTOCK.COM/TRENDSETTER
F
I. Introduction
A. On the Fractional Systems
ractional systems are becoming increasingly adopted
in the 21st century for modelling many natural
and man-made phenomena [1]-[8]. In fact, we are
presently dealing with phenomena that require a modeling
beyond traditional tools. Signals obtained when observing
many types of systems have spectra that exhibit
slopes in the amplitude Bode diagrams that are not multiples
of 20 dB per decade. This behavior can be found,
for example, in ECG, speech, music, electronic noise in
junctions, network traffic, electrochemistry, and other
36
IEEE CIRCUITS AND SYSTEMS MAGAZINE
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IEEE Circuits and Systems Magazine - Q2 2022
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