©
National Instruments Corporation
4-1
4
Frequency-Weighted Error
Reduction
This chapter describes frequency-weighted error reduction problems. This
includes a discussion of controller reduction and fractional representations.
Introduction
Frequency-weighted error reduction means that the error is measured not,
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
ω
), all the indices above are
effectively the same, with the effective weight just |
V
(
j
ω
)|, |
W
(
j
ω
)|,
or |
W
(
j
ω
)
V
(
j
ω
)|.
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
Φ
(
j
ω
), then taking
V
(
j
ω
) as a stable spectral factor
E
0
G j
ω
( )
G
r
j
ω
( )
–
∞
=
E
1
G j
ω
( )
G
r
j
ω
( )
V j
ω
( )
–
∞
=
E
2
W j
ω
( )
G j
ω
( )
G
–
r
j
ω
( )
[
]
∞
=
E
3
W j
ω
( )
G j
ω
( )
G
–
r
j
ω
( )
[
]
V j
ω
( )
∞
=
