User's Manual

Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-19 Xmath Model Reduction Module
• and stand in the same relation as W(s) and G(s), that is:
–
– With , there holds
or
– With there holds
or
–
– is the stable strictly proper part of .
• The Hankel singular values of (and ) are the first as – r Hankel
singular values of F,
• has the same zeros in Re[s]>0 as G(s).
These properties mean that one is immediately positioned to repeat the
reduction procedure on , with almost all needed quantities being on
hand.
W
ˆ
s() G
ˆ
s
W
ˆ
′ s–()W
ˆ
s() G
ˆ
s()G
ˆ
′ s–()=
P
ˆ
A
ˆ
′
F
A
ˆ
F
P
ˆ
+ B
ˆ
F
B
ˆ
′
F
–=
B
W
ˆ
P
ˆ
C
G
ˆ
′
B
G
ˆ
D
G
ˆ
′
+=
B
ˆ
F
D′ V
1
C′+ P
ˆ
DC
ˆ
F
B′
W
U
1
Σ
1
+()′B
ˆ
F
Iv
ns
T′–()D′+=
Q
ˆ
A
ˆ
F
A
ˆ
F
′
Q
ˆ
+ C
ˆ
′–
F
C
ˆ
F
=
C
W
ˆ
D
G
ˆ
1–
C
G
ˆ
B′
W
ˆ
Q
ˆ
–()=
Iv
ns
T′–()Iv
ns
T–()
1–
C
ˆ
F
DI v
ns
T–()[]
1–
=
DC
ˆ
F
B′
W
U
1
Σ
1
B
ˆ
F
D′ V
1
C′+[]′Q
ˆ
–()+{}
D
W
ˆ
D′
G
ˆ
=
F
ˆ
W
ˆ
1–
s–()()G
ˆ
s()
F
ˆ
p
F
ˆ
P
ˆ
Σ
1
1–
U
1
′
QV
1
V
1
′
QU
1
Σ
1
1–
==
Q
ˆ
V
1
′
PU
1
Σ
1
Σ
1
U
1
′
PV
1
==
G
ˆ
s
G
ˆ
s