**Author details**

Gou Nakura\*

Address all correspondence to: gg9925\_fiesta@ybb.ne.jp

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Gou Nakura\*

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**Chapter 8**

**Provisional chapter**

**Discrete-Time Fractional-Order Systems: Modeling and**

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,

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

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

> ©2012 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.

**Discrete-Time Fractional-Order Systems:**

**Stability Issues**

Maâmar Bettayeb

**1. Introduction**

[15–30].

Saïd Guermah,

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

Saïd Guermah, Saïd Djennoune and

Saïd Djennoune and Maâmar Bettayeb

Additional information is available at the end of the chapter

recursivity, diffusion and/or relaxation phenomena. [5–14].

Additional information is available at the end of the chapter

**Modeling and Stability Issues**

**Provisional chapter**
