The 21st Century Systems: An Updated Vision of Discrete-Time Fractional Models

Abstract

Two different approaches for describing discretetime 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.

Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms

“…The fractionally differenced and integrated models have gained roots and continue being used today [10,15,82]. In engineering, namely in signal processing, a fractional generalization of the ARMA model has been proposed [25,27].…”

“…Probably the first attempt was made in [21], but the systems described are not really fractional, although they use fractional delays. In the last 20 years, many texts have been published on fractional differences and derivatives in discrete time, leading to different views of what fractional systems in discrete time are and how they are characterized [22][23][24][25][26][27][28]. The purpose of this paper is exactly to describe the mathematical basis underlying the main formulations.…”

“…In the shift-invariant cases, the exponentials are the eigenfunctions and the eigenvalues are the transfer functions [27,36,116]. In the scale-invariant system, such a role is played by the powers [40].…”

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

“…The introduction of fractional (F) ARMA systems brought new different questions and solutions for the problem, since the fractionalization of DT-ARMA systems of the type ( 6) is not immediate and leads to systems with fractional delays, but that are not "fractional" in the correct assertion of the word [42]. However, we can introduce DT-ARMA versions of differential systems [3,12] that approximate CT-ARMA systems. We leave the study of these systems for a later section.…”

“…We adopted the spectral correspondence that states an approximation of the frequency responses in the interval − π T , π T . This was achieved through two fractional discrete-time FARMA models [10][11][12]. One is based on the classic Euler s to z conversion, while the second uses the bilinear (Tustin) transformation.…”