National Instruments NI MATRIXx Xmath - Manual (Page 65)

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 ( ) Gˆs
W ˆ ′ s
–
( )Wˆ s ( ) Gˆ s ( )Gˆ′ s
–
( )
=
PˆAˆ ′
F
Aˆ
F
Pˆ + Bˆ
F
Bˆ ′
F – =
B
W ˆ
PˆC
Gˆ
′
B
Gˆ
D
Gˆ
′
+ =
Bˆ
F
D ′ V
1
C ′
+ Pˆ DCˆ
F
B ′
W
U
1
Σ
1
+
( )′ Bˆ
F
I v
ns
T ′
–
( )D′
+ =
QˆAˆ
F
Aˆ
F
′
Qˆ + Cˆ ′
– F
Cˆ
F =
C
W ˆ
D
Gˆ
1 – C
Gˆ
B ′
W ˆ
Qˆ –
( )
=
I v
ns
T ′
–
( ) I v
ns
T
–
( ) 1 – Cˆ
F
D I v
ns
T
–
( ) [ ] 1 –
=
DCˆ
F
B ′
W
U
1
Σ
1
Bˆ
F
D ′ V
1
C ′
+
[ ]′Qˆ
–
( )
+
{ }
D
W ˆ
D ′
Gˆ
=
Fˆ W ˆ 1 – s –
( ) ( )Gˆ s ( )
Fˆ
p
Fˆ
Pˆ Σ
1
1 – U
1
′
QV
1
V
1
′
QU
1
Σ
1
1 –
= =
Qˆ V
1
′
PU
1
Σ
1
Σ
1
U
1
′
PV
1
= =
Gˆs
Gˆs