Page 145 - Mathematical Techniques of Fractional Order Systems

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Chapter 5

Fractional Order

Time-Varying-Delay Systems:
A Delay-Dependent Stability

Criterion by Using Diffusive
Representation

1
1
Y. Boukal 1,2,3 , M. Zasadzinski , M. Darouach and N.E. Radhy 2
1 2
Universite ´ de Lorraine, Cosnes et Romain, France, Universite ´ Hassan II, Casablanca, Maroc,
3
Universite ´ de Valenciennes et du Hainaut-Cambre ´sis, Famars, France
5.1 INTRODUCTION

Since the physical interpretation of the fractional order derivatives given in
Podlubny (2002) has became clear to the researchers and engineers, the
modeling of physical (Battaglia et al., 2000; Vinagre, 2001; Ortigueira and
Machado, 2003; Sabatier et al., 2007; Sheng et al., 2012; Sierociuk et al.,
2013; Azar et al., 2017), biological (Magin, 2006; Freed and Diethelm,
2006), and chemical (Darling and Newman, 1997; Audounet et al., 1998;
Lederman et al., 2002) phenomena employing fractional order differentia-
tion integration and controllers have been studied by many researchers and
scientists (Bagley and Calico, 1991; Chen, 2006). Specially, the fractional
order time-delay systems can characterize a class of chaotic behaviors (Deng
et al., 2007; Lin and Lee, 2011; Yuan et al. 2013).
Generally, the time-delay phenomena included in the dynamics of frac-
tional order systems (FOS) are due to transportation of material, energy, or
information. Furthermore, the presence of time-delays, also called dead-time
or after-effect, can cause the plant instability. In fact, the problems of stabil-
ity analysis, control, and observer designs for this kind of system have
attracted the attention of many researchers and scientists. Recently, FOS
with time-varying delays or constant delays have been subject to some
research and studies (see e.g., Tarbouriech, 1997; Bonnet and Partington,
2001, 2002; Chyi and Yi-Cheng, 2006; Busłowicz, 2008; Farshad and

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