National Instruments NI MATRIXx Xmath User Manual
Page 76
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
=
xˆ
·
Axˆ Bu K
E
Cxˆ y
–
(
)
–
+
=
u
K
R
xˆ
–
=
K
R
xˆ
–
K
R
x
–
xˆ
·
Axˆ BK
R
xˆ K
E
Cxˆ K
E
y
+
–
+
=
u
K
R
xˆ
–
=
C s
( )
K
R
sI A BK
R
K
EC
+
+
–
(
)
1
–
K
E
=