National Instruments Xmath

Chapter 4 Frequency-Weighted Error Reduction

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

+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(

+CP)–1|

will be (approximately) small. Also outside the plant bandwidth,

|P(

+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 Equation 4-3. 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 Equation 4-4 is replaced by

–PC I PC+()PCrIPC

+()

1–

–=

–IPC+()

1–PC C

–()IPC+()

1–

≈

EMIPC+()

1–PC C

–()IPC+()

1–

∞

VV*Φ=

EMS IPC+()

1–PC C

–()IPC+()

1–V∞