**1. Introduction**

Saïd Guermah,

The concepts of non-integer derivative and integral are the foundation of the fractional calculus [1–4]. Non-integer derivative has become nowadays a precious tool, currently used in the study of the behavior of real systems in diverse fields of science and engineering.

Starting from the sixties, the research in this domain of interest has progressively put to light important concepts associated with formulations using non-integer order derivative. Indeed, non-integer order derivative revealed to be a more adequate tool for the understanding of interesting properties shown by various types of physical phenomena, that is, fractality, recursivity, diffusion and/or relaxation phenomena. [5–14].

This tool has enabled substantial advances in the accuracy of description and analysis of these phenomena, namely in the domain of electrochemistry, electromagnetism and electrical machines, thermal systems and heat conduction, transmission, acoustics, viscoelastic materials and robotics. At its turn, the community of system identification and automatic control early benefited from this tool. The corresponding system representations are of the types of infinite-dimensional or distributed-parameter systems, enriched by diverse approximating non-integer-order models, indifferently called fractional-order models. There are abundant successful examples in the literature and one can see some for example in [15–30].

An important feature of fractional-order systems (FOS) is that they exhibit hereditarily properties and long memory transients. This aspect is taken into account in modeling, namely with state-space representation, in parameter estimation, identification, and controller design. Besides, these issues can be viewed either in the scope of a continuous time, or a discrete time representation. In this Chapter, we focus on some issues of modeling and stability of LTI FOS in discrete time. The extension of the results obtained with continuous-time representations to the discrete-time case is evidently motivated by

<sup>©2012</sup> Guermah et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Guermah et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Guermah; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the general trend to use digital computer-aided simulations and implementations in control engineering and signal processing. In the particular case of FOS, the discretization methods lead to new models and enlarge the possibilities of representation and simulation of such systems.

*L <sup>a</sup> <sup>D</sup><sup>α</sup>*

*<sup>t</sup> <sup>f</sup>*(*t*) = *<sup>d</sup><sup>m</sup>*

difficulty, Caputo proposed another definition given by

*C <sup>a</sup> <sup>D</sup><sup>α</sup>*

**2.2. FOS differential equation**

equation is then expressed by [45]

following transfer function

not zero

*dt<sup>m</sup>* { <sup>1</sup>

*<sup>t</sup> <sup>f</sup>*(*t*) = <sup>1</sup>

*L <sup>a</sup> <sup>D</sup><sup>α</sup>*

whereas Caputo's fractional derivative of a constant is identically zero.

*na* ∑ *i*=1

integers of a same base order *α*. In this case, it can be expressed by

*na* ∑ *i*=1 Γ(*m* − *α*)

Γ(*m* − *α*)

Naturally, as physical systems are modeled by differential equations containing eventually fractional derivatives, it is necessary to give to these equations initial conditions that must be physically interpretable. Unfortunately, the LRL definition leads to initial conditions containing the value of the fractional derivative at the initial conditions. To overcome this

> *<sup>t</sup> a*

Note the remarkable fact that the LRL fractional derivative of a constant function *f*(*t*) = *C* is

*<sup>t</sup> <sup>C</sup>* <sup>=</sup> *Ct*−*<sup>α</sup>*

Let us now consider a SISO LTI FOS. By means of its dynamic input-output relation and using (4), we can derive its continuous-time models. In all what follows, we use Caputo's definition of a fractional derivative with initial time *t* = 0, *i.e.*, *a* = 0. The derived differential

where *ai*, *bj* ∈ **R**, *αi*, *β<sup>j</sup>* ∈ **R**+, *u*(*t*) ∈ **R** is the input of the system and *y*(*t*) ∈ **R** its output. *na* and *nb* ∈ **N** are the number of terms of each side of the differential equation. The differential equation is said with commensurate order if all the differentiation orders *α<sup>i</sup>* , *β<sup>j</sup>* are multiple

*nb* ∑ *j*=1

*nb* ∑ *j*=1 *bjD<sup>β</sup><sup>j</sup>*

*aiD<sup>α</sup><sup>i</sup> y*(*t*) =

*aiD<sup>i</sup>αy*(*t*) =

**2.3. State-space model derived from a non-integer-order transfer function**

*<sup>G</sup>*(*s*) = *<sup>Y</sup>*(*s*)

By using Laplace transform, assuming null initial conditions, from (6) we derive the

*<sup>U</sup>*(*s*) <sup>=</sup> <sup>∑</sup>*nb*

*<sup>j</sup>*=<sup>0</sup> *bjs<sup>β</sup><sup>j</sup>*

∑*na <sup>i</sup>*=<sup>0</sup> *ais<sup>α</sup><sup>i</sup>*

 *<sup>t</sup> a*

*f*(*τ*)

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

*f*(*m*)(*τ*)

(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*α*−*m*+<sup>1</sup> *<sup>d</sup>τ*} (3)

http://dx.doi.org/10.5772/51983

185

(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*α*−*m*+<sup>1</sup> *<sup>d</sup><sup>τ</sup>* (4)

<sup>Γ</sup>(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*) (5)

*u*(*t*) (6)

*bjD<sup>j</sup>αu*(*t*) (7)

(8)

More specifically, the familiar state-space representation and its well-established results in the integer-order case receive here an extension to the non-integer case. This extension induces noticeable changes: there appears a neccessity of interpretation of the initial conditions [31] and a neccesity to take into account the history (or memory) of the system from the initial instant, since we have to deal, instead of the classical state, with a so-called "'pseudo state"', that is an expanded state [32, 33]. This infinite-dimensional system, as we expose it in the following sections, has been studied, from the point of view of its structural properties, i.e., controllability, observability [34, 35] and, to some extent, of its stability [36].
