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222 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
11.5 Example
11.5.1 Model-order reduction from 4 to 1
The system H(s) to be reduced is a system of order 4 and is described in state space
form with the matrices A, B, and C:
⎛
⎞ ⎛ ⎞
− 34
15
1 0 0
2
−
A =
77
36
0 1 0
⎜
⎝ − 247
270
0 0 1
⎟
⎠ ,B= 4
⎜
⎝ −9
⎟
⎠ ,C= ( 1 0 0 0 ) . (11.75)
− 5
36
0 0 0
−2
This yields the fourth-order transfer function H(s) =C(sI − A) −1 B:
−2 − 9s +4s 2 +2s 3
H(s) =
5
36 + 247
270 s + 77
36 s2 + 34
15 s3 + s . (11.76)
4
The poles of the system H(s) are δ 1 = − 2 3 + 1 2 i, δ 2 = − 2 3 − 1 2 i, δ 3 = − 3 5 and δ 4 = − 1 3 .
Because the reduction is from order 4 to 1, it is easily possible to compute the optimal
approximation G(s) of order one, by using other numerical methods, see [89]. The
globally optimal H 2 -approximation of order one obtained in that way, is:
G(s) = −3.1894
(11.77)
0.174884 + s
which has one pole located at −0.174884.
This system G(s) is the approximation one also wants to compute with the approach
described in this chapter using the Stetter-Möller matrix method and the
Kronecker canonical form techniques for two-parameter pencils.
The system of quadratic equations (11.1) corresponding to the given transfer function
H(s) of order four, is given by:
⎛
1+δ 1 ρ 1
⎞
+δ1 2 ρ 2
e(δ 1
x 2 ⎛ ⎞ ⎛ ⎞
) 1
x 1 0
1+δ 2 ρ 1 +δ2 2 ρ 2
e(δ 2
x 2 ) 2
x 2
0
1+δ
⎜ 3 ρ 1 +δ3 2 − M(δ 1 ,δ 2 ,δ 3 ,δ 4 )
=
(11.78)
ρ 2
⎝ e(δ 3
x 2 ) 3
⎟
⎜
⎠
⎝
x 3
⎟ ⎜
⎠ ⎝
0 ⎟
⎠
0
1+δ 4 ρ 1 +δ 2 4 ρ 2
e(δ 4 )
x 2 4
with the matrix M(δ 1 ,δ 2 ,δ 3 ,δ 4 ) as before. The constraints ã N−1 and ã N−2 , as defined
in (11.2), take the form:
⎧
⎨
⎩
ã N−1 =(− 6480
2977 + 7380
2977 i)x 1 − ( 6480
2977 + 7380
2977 i)x 2 + 3375
229 x 3 − 135
13 x 4 =0
ã N−2 =( 6678
2977 − 15048
2977 i)x 1 +( 6678
2977 + 15048
2977 i)x 2 − 5625
229 x 3 + 261
13 x 4 =0.
x 4
(11.79)
In order to find the solutions of the quadratic system of equations (11.78), which
also satisfy both the linear constraints (11.79), the matrices AãN−1 (ρ 1 ,ρ 2 ) T and