[Hans Blom, 951003]

(Bruce Abbott (951002.2135 EST))

This post has some relevance for model-based control:

The X observations were the numbers 1-100; the Y observations were

synthesized by multiplying X by 2 and then adding the random error.

Thus, X and Y were related by the equation

Y = 2X

but this relationship was obscured by the "influence" of

uncontrolled error. I systematically varied the standard deviation

of the error, and used linear regression to estimate the "true"

underlying relationship between X and Y. Here are the results:

Error

Pop. SD Fitted Straight Line Sest R R-sq

10 Y = +0.64 + 1.98 X 11.16 .982 96.4%

20 Y = -0.02 + 1.98 X 18.91 .951 90.4%

40 Y = -3.86 + 2.02 X 38.67 .836 69.8%

80 Y = -4.90 + 1.88 X 73.59 .598 35.8%

160 Y = +7.00 + 1.98 X 161.50 .337 11.4%

For making predictions of Y from X, the results get progressively

worse as the amount of variation in error increases relative to the

(constant) variation in X in these experiments. However, if one's

intention is to use the data to infer the "true" relationship

between X and Y (assuming that it is linear), even the worst case

shown here provides an excellent estimate of the true line.

This is exactly the kind of modelling that a model-based controller

does, and it shows what the model part of a model-based controller is

capable of. Even with a noise SD of 160, the system's gain, which is

2, is very accurately estimated as 1.98, an error of only 1%. More-

over, part of this 1% error is _compensated_ by another error, the

offset of +7.00 which ought to be zero. Although R-square is abysmal-

ly small, the _prediction_ of Y based on the knowledge of X is quite

accurate. Accurate enough for control purposes, most likely.

When controlling, however, there is an extra complication. In

control, the goal is usually to keep Y at some prescribed value by

manipulating X. If control is successful, Y will not vary much, and

neither will X. Therefore the points (X, Y) through which the "best"

line must be fitted will be much more cloud-like than in the above

case, where you could force X to vary from 1 to 100. This leads to an

extremely interesting axiom of model-based control: control _hinders_

identification.

This leads to an interesting fact: the more successful the control

actions, the less accurate the model will be. But then, an inaccurate

model leads to bad control and thus to large fluctuations in X and Y.

Eventually a compromise will be arrived at in which the model is only

good enough to allow good enough control. A paradox? Maybe. At least

a clear demonstration that a correct model is not the ultimate goal!

Thanks, Bruce!

Greetings,

Hans