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Chapter 3
Multiplicative Error Reduction
3-18
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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)
Fˆ
p
s
( )
Fˆ s
( )
v
ns
1
–
F s
( ) Fˆ s
( )
–
[
]
Gˆ
W
ˆ
Gˆ s
( )
G s
( ) W′ s
–
( ) F s
( ) Fˆ s
( )
–
[
]
–
=
W
ˆ s
( )
I v
ns
T
′
–
(
) I v
ns
T
–
(
)
1
–
=
W s
( )
F s
( ) Fˆ s
( )
–
[
]G′
–
s
–
( )
+
{
}
Gˆ s
( )
D I v
ns
T
–
(
)
(
) DCˆ
F
B
W
′
U
Σ
1
+
[
] sI Aˆ
F
–
(
)
1
–
Bˆ
F
=
(
)
W
ˆ s
( )
I v
ns
T
′
–
(
)D′
I v
ns
T
′
–
(
) I v
ns
T
–
(
)
1
–
+
=
Cˆ
F
sI Aˆ
F
–
(
)
1
–
Bˆ
F
D
′ V
1
′
C
′
+
[
]
Gˆ W
ˆ
Fˆ
Gˆ s
( )
^
Gˆ s
( ) Gˆ s
( ) Wˆ′
–
s
–
( ) Fˆ
p
s
( ) Fˆ s
( )
–
[
]
+
=
^
^
Gˆ s
( )
G
1
–
G Gˆ
–
(
)
∞
v
ns
=