Chapter 3
Multiplicative Error Reduction
3-18
ni.com
Note
The expression
is the strictly proper part of
. The matrix
is all pass; this property is not always secured in the multivariable case
when
ophank( )
is used to find a Hankel norm approximation of
F
(
s
).
5.
The algorithm constructs
and
, which satisfy,
and,
through the state variable formulas
and:
Continue the reduction procedure, starting with
,
, and
repeating the process till
G
r
of the desired degree
nsr
is obtained.
For example, in the second iteration,
is given by:
(3-4)
Consequences of Step 5 and Justification of Step 6
A number of properties are true:
•
is of order
ns
–
r
, with:
(3-5)
F
ˆ
p
s
( )
F
ˆ
s
( )
v
ns
1
–
F s
( )
F
ˆ
s
( )
–
[
]
G
ˆ
W
ˆ
G
ˆ
s
( )
G s
( )
W
′
s
–
( )
F s
( )
F
ˆ
s
( )
–
[
]
–
=
W
ˆ
s
( )
I v
ns
T
′
–
(
)
I v
ns
T
–
(
)
1
–
=
W s
( )
F s
( )
F
ˆ
s
( )
–
[
]
G
′
–
s
–
( )
+
{
}
G
ˆ
s
( )
D I v
ns
T
–
(
)
(
)
DC
ˆ
F
B
W
′
U
Σ
1
+
[
]
sI A
ˆ
F
–
(
)
1
–
B
ˆ
F
=
(
)
W
ˆ
s
( )
I v
ns
T
′
–
(
)
D
′
I v
ns
T
′
–
(
)
I v
ns
T
–
(
)
1
–
+
=
C
ˆ
F
sI A
ˆ
F
–
(
)
1
–
B
ˆ
F
D
′
V
1
′
C
′
+
[
]
G
ˆ
W
ˆ
F
ˆ
G
ˆ
s
( )
^
G
ˆ
s
( )
G
ˆ
s
( )
W
ˆ
′
–
s
–
( )
F
ˆ
p
s
( )
F
ˆ
s
( )
–
[
]
+
=
^
^
G
ˆ
s
( )
G
1
–
G G
ˆ
–
(
)
∞
v
ns
=
