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

Abstract

The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and discrete transfer functions. Simple transformations are presented to allow interconversions between both systems, recovering formulae obtained with the impulse invariant method. These transformations are also used to formulate a covariance equivalence. The spectral correspondence implied by the bilinear (Tustin) transformation is used to study the equivalence between the two types of systems. The general fractional CT/DT ARMA systems are also studied by considering two DT differential fractional autoregressive-moving average (FARMA) systems based on the nabla/delta and bilinear derivatives. The interrelations CT/DT are also considered, paying special attention to the systems defined by the bilinear derivatives.

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

“…One of the most interesting methods of obtaining discrete-time systems from continuous-time templates is through the conformal transformations, so that we pass from the LT context, (s), to the one of Z transform, (z) [28]. The traditional procedures can be described by…”

“…It has a great drawback: the imaginary axis in the s plane is transformed into the Hilger circles |z − 1| = 1 and |z + 1| = 1 [25]. This fact brings some inconvenience [28].…”

“…Another alternative to obtain continuous to discrete conversion is through the bilinear (Tustin) transformation [28,113]…”

“…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.…”

“…The ARMA-Type Difference Linear Systems Definition 15. We define an ARMA-type difference linear system through the following equation [25,28] The most interesting systems are those with commensurate orders:…”