User's Manual

© National Instruments Corporation 4-1 Xmath Model Reduction Module
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(jω), all the indices above are
effectively the same, with the effective weight just |V(jω)|, |W(jω)|,
or |W(jω)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 Φ(jω), then taking V(jω) as a stable spectral factor
E
0
Gjω()G
r
jω()–
∞
=
E
1
Gjω()G
r
jω()Vjω()–
∞
=
E
2
Wjω()Gjω()G–
r
jω()[]
∞
=
E
3
Wjω()Gjω()G–
r
jω()[]Vjω()
∞
=