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Chapter 4 Frequency-Weighted Error Reduction
© National Instruments Corporation 4-9 Xmath Model Reduction Module
Figure 4-4. Redrawn; Individual Signal Paths as Vector Paths
It is possible to verify that
and accordingly the output weight can be used in an
error measure . It turns out that the calculations for frequency
weighted balanced truncation of G and subsequent construction of C
r
(s) are
exceptionally easy using this weight.
The second
fracred( ) option is the dual of this. The error measure is
where:
It is possible to argue heuristically the relevance of these error measures
from a second point of view. It turns out that:
K
R
C
sI A BK
R
+–()
1–
K
E
Cs() P
ˆ
s()
+
-
-
+
Ps()I
+
-
IP
ˆ
G+()
1–
P
ˆ
CsI A K
E
C+–
1–
B[=
ICsI AK
E
C+()
1–
K
E
–()– ]
IP
ˆ
G+()
1–
P
ˆ
W=
WG G
r
–()
HH
r
–()V
V
IK
R
sI A BK
R
+–()
1–
B–
CsI A BK
R
+–()
1–
B
=
D
L
N
L
W
1
– W
2
V
1
N
R
–
V
2
D
r
I 0
0 I
=