Chapter 4 Frequency-Weighted Error Reduction
© National Instruments Corporation 4-3 Xmath Model Reduction Module
is minimized (and of course is less than 1). Notice that these two error
measures are like those of Equation4-1 and Equation 4 -2. The fact that the
plant ought to show up in a good formulation of a controller reduction
problem is evidenced by the appearance of P in the two weights.
It is instructive to consider the shape of the weighting matrix or function
P(
Ι
+CP)–1. Consider the scalar plant case. In the pass band, |PC| is likely
to be large, and if this is achieved by having |C| large, then |P(
Ι
+CP)–1|
will be (approximately) small. Also outside the plant bandwidth,
|P(
Ι
+CP)–1| will be small. This means that it will be most likely to take its
biggest values at frequencies near the unity gain cross-over frequency. This
means that the approximation Cr is being forced to be more accurate near
this frequency than well away from it—a fact very much in accord with
classical control, where one learns the importance of good loop shaping
round this frequency.
The above measures EIS and EOS are advanced after a consideration of
stability, and the need for its preservation in approximating C by Cr. If one
takes the viewpoint that the important thing to preserve is the closed-loop
transfer function matrix, a different error measure arises. With T, Tr
denoting the closed-loop transfer function matrices,
Then, to a first order approximation in C– Cr, there holds
The natural error measure is then
(4-4)
and this error measure parallels E3 in Equation4-3. Further refinement
again is possible. It may well be that closed-loop transfer function matrices
should be better matched at some frequencies than others; if this weighting
on the error in the closed-loop transfer function matrices is determined by
the input spectrum , then one really wants (T–Tr)V to be small,
so that Equation4-4 is replaced by
TT
r
–PC I PC+()PCrIPC
r
+()
1–
–=
TT
r
–IPC+()
1–PC C
r
–()IPC+()
1–
≈
EMIPC+()
1–PC C
r
–()IPC+()
1–
∞
=
VV*Φ=
EMS IPC+()
1–PC C
r
–()IPC+()
1–V∞
=