Chapter 4
Frequency-Weighted Error Reduction
4-6
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•
Form the reduced controller by interconnecting using negative
feedback the second output of
G
r
to the input, that is, set
Nothing has been said as to how should be chosen—and the end result
of the reduction,
C
r
(
s
), depends on
. Nor has the reduction procedure
been specified.
When
C
(
s
) has been designed to combine a state estimator with a
stabilizing feedback law, it turns out that there is a natural choice for
.
As for the reduction procedure, one possibility is to use a weight based
on the spectrum of the input signals to
G
—and in case
C
(
s
) has been
determined by an LQG optimal design, this spectrum turns out to be white,
that is, independent of frequency, so that no weight (apart perhaps from
scaling) is needed. A second possibility is to use a weight based on a
stability robustness measure. These points are now discussed in more
detail.
To understand the construction of a natural fractional representation for
C
(
s
), suppose that
and let
K
R
,
K
E
be matrices such
that
A
–
BK
R
and
A
–
K
EC
are stable. The controller
generates an estimate
of the feedback control
. The controller
can be represented as a series compensator
(with compensator input
y
and output
u)
. Allowing for connection with
negative feedback, the compensator transfer function matrix is:
C
r
s
( )
n
r
d
r
e
r
+
---------------
=
d
d
r
d s
( )
P s
( )
C sI A
–
(
)
1
–
B
=
x
ˆ·
Ax
ˆ
Bu K
E
Cx
ˆ
y
–
(
)
–
+
=
u
K
R
x
ˆ
–
=
K
R
x
ˆ
–
K
R
x
–
x
ˆ·
Ax
ˆ
BK
R
x
ˆ
K
E
Cx
ˆ
K
E
y
+
–
+
=
u
K
R
x
ˆ
–
=
C s
( )
K
R
sI A BK
R
K
EC
+
+
–
(
)
1
–
K
E
=
