[Hans Blom, 960822]
(Rick Marken (960821.2130)) to (Martin Taylor (960821 16:45))
The system has to model the environmental feedback function at
least in respect of its sign.
If you want to call this a "model" go ahead. But it doesn't seem
much like a model to me.
I agree with Martin that it is a model. And I agree with you that it
is not much of a model. After all, it is just a one bit model. Yet,
having the sign right is the prime imperative. Having the gain right
as well (within a factor of 2 or 3 or 10, depending on the control
quality that is needed) would be the next step. This calls for more
bits of information.
For another thing, the sign of the feedback function doesn't capture
anything about the _form_ of that function. This is what is modeled
in Hans' MCT system. Actually, Hans' model assumes the form of a
feedback function and tries to find the coefficients of that
function.
This is a misunderstanding. A model does not need to capture the form
of the function if its precise form is not important for control.
This can be understood best by noting that a control system usually
has an operating point or operating region which only consists of
_part of_ the full curve, and NOT from minus infinity to plus
infinity. In many cases, the first derivative of the curve suffices;
that is why linear models are so popular. Only if taking the first
derivative is not sufficient will it be necessary to explore which
higher order terms might need to be incorporated into the model as
well. Then, too, an incremental approach -- adding just one higher
order term at a time -- works well. In practice, more than three or
four terms are hardly ever necessary.
This has to do with the mathematical fact that over a limited range
any function can be fitted well by a polynomial with just a few
terms. This has also to do with the fact that a model is only an
approximation. The MCT approach does not require the _form of the
function_ to be known; it only requires a (more or less close) fit to
the function.
I'm not a mathematician but I think, in practice, this would mean
solving for the coefficients of polynomials with many terms, and
where the variables must include first and second derivatives.
Practice shows that more than just a few (3 or 4) terms are not
needed if the function is smooth (in the mathematical sense). Some
functions aren't smooth; a hysteresis, for example, is impossible to
model this way.
So the PCT model works (produces the result intended by the setting
of r) without any model of the _form_ of the environmental feedback
function.
The PCT model linearizes (assumes a linear environment), just like
the MCT model. In simulations of PCT models with a fixed gain, you
will therefore find that the response speed of the controller depends
on the slope (gain, in a certain operating point) of the
environmental function. If this makes control too sluggish in certain
regions, there are two ways (in MCT, at least) in which to solve this
problem. The first is to get a better approximation of the function
by including one or more higher order terms. The second is to keep a
linear model but to make its gain adaptive -- dependent on the
operating point. The latter solution is possible only if there _is_ a
well-defined operating "point" always, i.e. if the function is
traversed slowly.
If Hans' MCT model is placed in the three different environments
described above, it will not work (produce the intended results)
unless it can determine the _form_ of the feedback function
(equation 1, 2 or 3) that exists in each environment.
If the MCT model is linear, the system will control well if the
reference value changes slowly. In that case, the slope of the
function describes the function well, and the controller's gain can
be adapted. If the reference value changes rapidly (e.g. square wave
like) to points of the function where its slope is very different, a
linear model will not provide good control anymore. But the same is
true for a PCT model, I bet (don't; I checked ;-).
A world model is a representation of the _form (including sign and
coefficient values) of the environmental feedback function.
A _representation_, yes, in the sense that it must provide a good
fit. Not in the sense that it must capture the "true" environmental
function. In fact, different approximations may provide approximately
equally good fits. A model CANNOT KNOW reality in any absolute sense.
It can, however, get to know HOW TO INTERACT WITH reality upto any
precision required -- at least if the "laws of reality" don't change.
Greetings,
Hans