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Chapter 4
Frequency-Weighted Error Reduction
4-2
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(so that
) is logical. However, a major use of weighting is in
controller reduction, which is now described.
Controller Reduction
Frequency weighted error reduction becomes particularly important in
reducing controller dimension. The LQG and
reducing controller dimension. The LQG and
design procedures lead to
controllers which have order equal to, or roughly equal to, the order of the
plant. Very often, controllers of much lower order will result in acceptable
performance, and will be desired on account of their greater simplicity.
plant. Very often, controllers of much lower order will result in acceptable
performance, and will be desired on account of their greater simplicity.
It is almost immediately evident that (unweighted) additive approximation
of a controller will not necessarily ensure closeness of the behavior of the
two closed-loop systems formed from the original and reduced order
controller together with the plant. This behavior is dependent in part on the
plant, and so one would expect that a procedure for approximating
controllers ought in some way to reflect the plant. This can be done several
ways as described in the Controller Robustness Result section. The
following result is a trivial variant of one in [Vid85] dealing with robustness
in the face of plant variations.
of a controller will not necessarily ensure closeness of the behavior of the
two closed-loop systems formed from the original and reduced order
controller together with the plant. This behavior is dependent in part on the
plant, and so one would expect that a procedure for approximating
controllers ought in some way to reflect the plant. This can be done several
ways as described in the Controller Robustness Result section. The
following result is a trivial variant of one in [Vid85] dealing with robustness
in the face of plant variations.
Controller Robustness Result
Suppose that a controller C stabilizes a plant P, and that C
r
is a (reduced
order) approximation to C with the same number of unstable poles. Then
C
C
r
stabilizes P also provided
or
An extrapolation to this thinking [AnM89] suggests that C
r
will be a good
approximation to C (from the viewpoint of some form of stability
robustness) if
robustness) if
or
VV
*
Φ
=
H
∞
C j
ω
( ) C
r
j
ω
( )
–
[
]P jω
( ) I C jω
( )P jω
( )
+
[
]
1
–
∞
1
<
I P
+
j
ω
( )C jω
( )
[
]
1
–
P j
ω
( ) C jω
( )C
r
j
ω
( )
[
]
(
)
∞
1
<
E
IS
C C
r
–
(
)P I CP
+
(
)
1
–
∞
=
E
IS
C C
r
–
(
)P I CP
+
(
)
1
–
∞
=