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Fractional Order Time-Varying-Delay Systems Chapter | 5 151
5.6 NUMERICAL EXAMPLE
This section describes simulation results to show the performances of the
proposed stabilization method. The simulation and implementation of frac-
tional derivative and integrator present many difficulties. There are some
methods in the literature that are used to approximate the fractional order
integrals in the frequency domain (see Oustaloup, 1991; Matsuda and Fuji,
1993; Krishna and Reddy, 2008). In this work, the simulations are based on
the CRONE Toolbox, developed by the CRONE research group (for more
details see Oustaloup, 1991; Malti et al., 2011 and references therein).
5.6.1 Approximation of the Fractional Order Derivatives and
Consistent Initialization
One of the difficulties that have been seen in applications and simulations of
fractional order derivative and integrator terms are the complications in practi-
cable implementation of this kind of operators. This explains the use of approxi-
mations of fractional order derivative and integrator terms by using integer
order derivatives and integrals only (Oustaloup, 1991; Vinagre et al., 2000).
Among the several existing approximations in the literature, the most
used is the Crone approximation which is the one considered in this paper
(see Oustaloup, 1995; Malti et al., 2012; Lanusse et al., 2013 and the refer-
ences therein). This approximation makes use of a recursive distribution of
poles and zeros within a frequency range ½w l ; w h . A more precise approxi-
mation of the fractional differentiator in the frequency band ½w l ; w h is
obtained by increasing N. Based on this approximation, the original mathe-
matical concepts developed were integrated in the CRONE toolbox, devel-
oped since the 1990s by the CRONE team—a Matlab toolbox dedicated to
fractional calculus and its applications in signal processing and automatic
control.
For the fractional order differential equations, the initialization of this
kind of system is different from the integer order. These systems are known
for the fact that they have effectively infinite memory. In fact, the initial
conditions must be replaced by initial condition functions, or history func-
tions. For an extensive overview, refer to Sabatier et al. (2010) and the refer-
ences therein.
The initialization method used in this work to create a consistent system
initial behavior is given as:
for t ,2 t 0 the system is at rest;
for 2t 0 , t , t 1 , an input is applied to the system
for 2t 1 , t , 0, an input equal to 0 is applied to the system, that reach
time t 5 0 with a nonnull initialization.
where 2t 0 ,2 t 1 , 0.
