[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:
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_
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!