Page 99 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 87

3.6 EXAMPLES
The following examples show that the response of the simplified model to
the desired input matches closely the original response even during the
transient.

3.6.1 Example 1

Consider the fractional order system described by the transfer function
4
s 1 9s 3:2 1 31s 2:4 1 58:01s 1:6 1 60:01s 0:8 1 16:03
GðsÞ 5 ; ð3:23Þ
b
4
s 4:8 1 6s 1 48s 3:2 1 286s 2:4 1 935s 1:6 1 1580s 0:8 1 888
which has also been adopted in Tavakoli Kakhki and Haeri (2009) and
Jiang and Xiao (2015), and assume that the system is driven by the input
1
UðsÞ 5 : ð3:24Þ
b
s 0:8
Remark 5: Input (3.24) can also be written as
1
0:2
UðsÞ 5 s U ð3:25Þ
b
s
whose time-domain counterpart is
0:2
^ uðtÞ 5 D HðtÞ ð3:26Þ
which is the fractional derivative of order 0:2 of the usual step function
HðtÞ. It seems reasonable to consider the inputs with Laplace transform:
1
UðsÞ 5 ð3:27Þ
b
s α
and time-domain expression (Caponetto et al., 2010)
t α21
uðtÞ 5 ; ð3:28Þ
ΓðαÞ
as the fractional order equivalents of the canonical inputs for integer order
systems (also called singularity inputs (Tewari, 2011)). Fig. 3.4 shows the
time course of (3.28) for three values of α.
By the change of variable
1
w 5 s 5 ð3:29Þ
the response of system (3.23) to input (3.24) is given in the w-domain by the
24th order rational function