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

Abstract

This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregresive-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 analysed 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 Gr ünwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated.

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

“…The initial condition problem was discussed in the companion paper [2]. Therefore, everything argued in that paper remains valid here as long as we consider the required adap-tation.…”

“…As we discussed for the CT LS [2], the usual (eternal) exponentials, e st , t ∈ R and s ∈ C, are the eigenfunctions of such systems. As it is well known, the discrete-time exponentials, z n , n ∈ Z, are the eigenfunctions of the discrete systems described by difference equations.…”

“…obtained from the generalised Grünwald-Letnikov (GL) derivative (see [2]). The symbol (−α) n stands for the Pochhamer representation of the raising factorial:…”

“…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.…”

“…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.…”