National Instruments NI MATRIXx Xmath User Manual
Page 65
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-19
•
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