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4-1
4
Frequency-Weighted Error
Reduction
Reduction
This chapter describes frequency-weighted error reduction problems. This
includes a discussion of controller reduction and fractional representations.
includes a discussion of controller reduction and fractional representations.
Introduction
Frequency-weighted error reduction means that the error is measured not,
as previously, by
as previously, by
but rather by
(4-1)
or
(4-2)
or
(4-3)
where W,V are certain weighting matrices. Their presence reflects a desire
that the approximation process be more accurate at certain frequencies
(where V or W have large singular values) than at others (where they
have small singular values). For scalar G(j
that the approximation process be more accurate at certain frequencies
(where V or W have large singular values) than at others (where they
have small singular values). For scalar G(j
ω), all the indices above are
effectively the same, with the effective weight just |V(j
ω)|, |W(jω)|,
or |W(j
ω)V(jω)|.
When the system G is processing signals which do not have a flat spectrum,
and is to be approximated, there is considerable logic in using a weight. If
the signal spectrum is
and is to be approximated, there is considerable logic in using a weight. If
the signal spectrum is
Φ(jω), then taking V(jω) as a stable spectral factor
E
0
G j
ω
( ) G
r
j
ω
( )
–
∞
=
E
1
G j
ω
( ) G
r
j
ω
( )V jω
( )
–
∞
=
E
2
W j
ω
( ) G jω
( ) G
–
r
j
ω
( )
[
]
∞
=
E
3
W j
ω
( ) G jω
( ) G
–
r
j
ω
( )
[
]V jω
( )
∞
=