National Instruments NI MATRIXx Xmath - Manual (Page 79)

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
C s ( ) P
ˆ s
( )
+
-
-
+
P s ( ) I
+
-
I PˆG +
( )
1 –
Pˆ CsI A K
E
C + –
1 – B [
=
I C sI A K
E
C +
( ) 1 – K
E
–
( )
–
]
I PˆG +
( )
1 –
Pˆ W =
W G G
r
–
( )
H H
r
–
( )V
V
I K
R
sI A BK
R
+ –
( ) 1 – B –
C sI 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
=