Page 101 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 89

B
YðzÞ 5
2
5
4
3
2 0:0181z 1 0:8917z 1 8:1335z 1 25:8372z 1 41:1316z 1 31:4882
5
3
2
4
5
6
z 1 6z 1 48z 1 286z 1 935z 1 1580z 1 888
0:0181
1 :
z
ð3:32Þ
Applying the shifted Pade ´ approximation method suggested in
Tavakoli Kakhki and Haeri (2009) with s 0 5 133, we find the following 2nd
order approximation of the system-dependent component (first addendum at
the right-hand side of (3.32)):
B2 2 0:0181z 1 1:0407
Y ðzÞ 5 : ð3:33Þ
Σ
2
z 2 2:2538z 1 40:6760
Adding the input-dependent component (second addendum at the right-
hand side of (3.32)) to (3.33) as well as an auxiliary 1st order term with the
far-off pole at z 52 100 (step (v) of Procedure 3.5.1), we find the following
3rd order approximation of the z-domain system transfer function:
2
B z 1 100:7402z 1 73:4276
G r ðzÞ 5 ð3:34Þ
3
2
z 1 97:7z 2 184:7z 1 4067:6
which, via the change of variable z 5 s 4=5 , corresponds to the
stable fractional order transfer function:
s 1:6 1 100:7402s 0:8 1 73:4276
G r ðsÞ 5 : ð3:35Þ
b
s 2:4 1 97:7s 1:6 2 184:7s 0:8 1 4067:6
Instead, by applying the shifted Pade ´ method with s 0 5 133 directly to
(3.23),asin Tavakoli Kakhki and Haeri (2009), without consideration of
the input component of the forced response, the following simplified
stable fractional order model is obtained:
s 1:6 1 5:0349s 0:8 1 0:3743
G r;TH ðsÞ 5 : ð3:36Þ
b
s 2:4 1 2:0349s 1:6 1 29:2696s 0:8 1 145:3930
The responses to input (3.24) of the reduced system (3.35) obtained
according to the method suggested in this section and of the reduced system
(3.36) are compared in Fig. 3.5 with the response to the same input of the
original system (3.23). It is apparent that the retention of the steady-state
component leads to a better approximation in the medium to long run.
Since fractional order systems are infinite-dimensional, or long-memory,
systems (Sabatier et al., 2014), it might be argued that the transfer function
of a fractional order system can be considered “simple” if it contains a small
number of parameters. From this point of view, functions (3.35) and (3.36)
in the previous subsection are indeed simpler than the original transfer