Page 103 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 91

GðsÞ 5
b
s 2:7 1 2s 1:8 1 s 0:9 1 2
5 :
s 6:3 1 4:9s 5:4 1 11:05s 4:5 1 14:07s 3:6 1 10:53s 2:7 1 4:55s 1:8 1 1:05s 0:9 1 0:1
ð3:37Þ
By the change of variable w 5 s 1=10 , we obtain
GðwÞ 5
18
27
9
w 1 2w 1 w 1 2
5
63
18
36
45
54
9
27
w 1 4:9w 1 11:05w 1 14:07w 1 10:53w 1 4:55w 1 1:05w 1 0:1
ð3:38Þ
whose poles are outside the instability sector. Since all powers are multiples
of 9, in this case too, the reduction procedure can profitably be applied to an
9
integer order rational function of smaller degree in the variable z 5 w ,
namely:
2
3
B z 1 2z 1 z 1 2
GðzÞ 5
6
4
5
7
2
3
z 1 4:9z 1 11:05z 1 14:07z 1 10:53z 1 4:55z 1 1:05z 1 0:1
ð3:39Þ
whose 3rd order optimal Hankel-norm approximation (Glover, 1984) turns
out to be
2
B 0:5592z 2 0:4066z 1 0:4178
G r;HN ðzÞ 5 ð3:40Þ
2
3
z 1 0:5841z 1 0:1885z 1 0:0204
and in the s-domain with z 5 s 9=10
0:5592s 1:8 2 0:4066s 0:9 1 0:4178
G r;HN ðsÞ 5 : ð3:41Þ
b
s 2:7 1 0:5841s 1:8 1 0:1885s 0:9 1 0:0204
Let us apply now the procedure based on the retention to the asymptotic
response to the input
1
UðsÞ 5 : ð3:42Þ
b
s 0:9
To this purpose, the original forced response to input (3.42) is decom-
posed into the sum of a system-dependent component and an input-
9
dependent component, which in the domain of z 5 w 5 ðs 1=10 9
Þ turn out to
be, respectively,
5
4
3
6
2
B 2 20z 2 98z 2 221z 2 281:4z 2 209:6z 2 89z 2 20
Y Σ ðzÞ 5 ;
2
6
5
4
3
7
z 1 4:9z 1 11:05z 1 14:07z 1 10:53z 1 4:55z 1 1:05z 1 0:1
ð3:43Þ
B 20
Y U ðzÞ 5 : ð3:44Þ
z