Page 18 - Mathematical Techniques of Fractional Order Systems

8 Mathematical Techniques of Fractional Order Systems

In all the models below, a GL variable order derivative of type-D was
used (Sierociuk et al., 2015a,b). This is another possible formulation with a
memory of past values of the order, and is derived from the approximations
of the GL definition in Eqs. (1.12a) and (1.12b), which are respectively an
iterative approximation and a recursive approximation. The step time is
1
hAR , the lower integration limit is c 5 0, and n 5 bt=hc. If implemented
for the entire length of the simulation, without any truncation of the series,
both formulations are equivalent (Sierociuk et al., 2015c).

n

α 1 X r α
0 D fðtÞ α ð21Þ fðt 2 rhÞ ð1:12aÞ
t
h r
r50
!
n 2 α
rec α fðtÞ X r r α
ð
t
0 D ftðÞ α 2 ð21Þ 0 D t2rh ft 2 rhÞ ð1:12bÞ
h r
r51
The GL type-D variable order derivative can be approximated according
to Eq. (1.13a). This derivative was already successfully used for modeling
the heat transfer process in a media with a time-varying structure (Sierociuk
et al., 2013).
!
n
fðtÞ X 2αðtÞ
D D αðtÞ 2 r D D αðtÞ
0 t fðtÞ ð21Þ 0 t2rh fðtÞ ð1:13aÞ
h αðtÞ r
r51
" #
n
fðtÞ X 2αðtÞ
D D αðtÞ 2 j D D αðt2jhÞ fðtÞ 2 c 1 c
2N t fðtÞ ð21Þ 2N t2jh
h αðtÞ j
j51
ð1:13bÞ
The type-D construction corresponds to an input-reductive strategy that
assumes the rejection of input differentiators, which translates in an immedi-
ate effect of order switching (Fig. 1.2). Additionally, this construction also
allows for the effect of initial conditions to be without memory of accumu-
α
lated values, as formulated in Eq. (1.13b) where D D fðtÞ 5 c 5 constant,
2N
l
for l 5 ð2N; 0Þ, and represented in the Fig. 1.3. These characteristics are
i-r Ξ{α(t)}
f (t) ser
b b b b
i-r D f (t)
α(t)
0 t
α αN–1 αN
a 1 a a a
S S S N–1 S N
1 2
FIGURE 1.2 Schematic representation of the input-reductive switching order scheme, in serial
form, from orders α 1 to α 2 . From Sierociuk, D., Malesza, W., Macias, M., 2015b. Fractional
Variable Order Derivative Simulink User Guide, https://www.mathworks.com/matlabcentral/
fileexchange/38801-fractional-variable-order-der.