[From Bill Powers (940831.0945 MDT)]
Martin Taylor (940830.1650) --
I have a puzzlement about (a) what happens when two non-linear pull-
only conrol systems are opposed to form a virtual two-way control
system, and (b) why wrong answers are so seductive as to be agreed by
competent and well-informed people.
The first, I think we can clear up. The second is a mystery; regarding
my own agreements to errors, you see only the tip of the iceberg; I
catch most of mine before they escape into public view, but only after
waking up in the middle of the night with that "something's wrong"
feeling. Never trust nobody without checking.
Before I went off on holiday, I re-posted some maths originally posted
in Dec 1993, which seemed to show that the two opposed one-way control
systems would form a two-way virtual control system with an output law
that was the derivative of the output law of the one-way real systems.
This was a formulation of a claim Bill P. had made, and supported that
claim. Bill praised it at the time, but now has said that it was in
fact wrong.
This was in large part the result of a language mixup. Where I said
"differential" you read "derivative." It's true that the first
derivative of a square-law curve is a linear curve -- but it's the
derivative with respect to the independent variable, not with respect to
time. This, however, has no relationship to what I meant by
"differential." In a two-way system that is made of two one-way systems,
two driving signals are required, one for each side. The average value
of the two driving signals expresses the common-model driving signal,
which (in the muscle system) determines the average tension on both
sides. The difference between the two driving signals (that is, the
"differential" signal) determines the net output in one direction or the
other. One signal rises above the average value at the same time that
the other signal falls below the average value. This has nothing to do
with derivatives, either with respect to the independent variable or
with respect to time. So in the equations you cite, I wrote
F1 - F2 = (c^2 + 2cd/2 + d^2/4) - (c^2 - 2cd/2 + d^2/4) or
F1 - F2 = 2cd.
This says that the differential force produced by a differential
contraction is proportional to the common-mode contraction ...
Now we come to my agreement to a falsehood:
(Bill, in response)
Martin:
>I always keep in mind your demonstration that a pair of opposed
>square-law systems will act like a virtual two-way linear
>system.
With a balanced pair of identical one-way systems having linear
output functions, the resistance to disturbance will be zero in
the region of overlap, but it will be normal outside that region
(for larger disturbances), as Kent McClelland showed at the
meeting. The combination will be like a single output function
with a dead zone: errors will produce no net output until they
reach a threshold level, after which larger errors will produce
output as usual (from one of the systems).
...
It's interesting, isn't it, that the agonist-antagonist pairs of
muscles HAVE to have a nonlinear response if muscle tone is
nonzero and there is to be any fine control.
Those learned paragraphs of mine, and what goes between them, are
nonsense. You didn't generate the nonsense for me to agree with; I did.
I agreed with your math without actually working it out for myself, and
came to a wrong conclusion, which wasn't even what Kent McClelland had
demonstrated. This is what comes of wanting to sound knowledgeable
without actually doing the work to see if the first idea that comes into
mind is right.
Martin:
>If two opposed one-way control systems have an output function
>f(e), then the virtual two-way system that they create has an
>apparent output function that is the derivative df(e)/de.
This would appear to be true within the region of overlap for
one-way systems, but not outside it.
And in that I am agreeing to your agreement to a generalization that is
simply irrelevant -- even though it's true that the form of a linear
output function is the derivative of the form of a square-law output
function.
The output of the two systems is the SUM of the outputs:
> *
> *
> * /
> F1 * /
> * /
> * /
> * /
> * /
> * /
*| /
* | /
* | / SUM
* | /
* | /
----------------*----------------/--------------*--------------------
/ | *
/ | *
/ | *
/ *|
/ * |
/ * |
/ * |
/ * |
* |
* |
* F2 |
* |
>
The correct answer can be seen by taking my derivation for the square-
law muscles and using the first power instead of the second power:
F1 = c + d/2
F2 = c - d/2
F1 - F2 = c + d/2 - c + d/2 = d
The net force depends directly on the differential driving signal d, and
the gain is independent of the common-mode driving signal c. Outside the
region of overlap, we have half the gain.
So agreeing to a true but irrelevant fact (about the derivative of a
square-law function) led somehow to a false line of reasoning and the
conclusion that the gain for opposed linear systems had to be zero. All
I can say is that it's a good thing that you continued to be puzzled. My
own anomaly detector seemed to be down for repairs, or perhaps needs to
be.
ยทยทยท
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Rick Marken (940830.1420) --
RE: levels
You described what your beliefs and principles ARE, but you didn't say
how they relate to each other -- in terms of my levels: which are system
concepts, which are principles, which are programs, etc.. You didn't say
how elements of your statement fit a hierarchical structure ( or not).
RE: perception of statistics.
The perceptual function could compute the mean or variance of its
inputs; so the perceptual signal's value could represent variations in
the mean or variance of the inputs to the perceptual function. But this
perceptual signal is not an _estimate_ of anything (the mean or
variance it represents is not computed as a statistic); it is simply a
perceptual variable.
That's an excellent but subtle point that can stand amplification. If
the PIF actually does include a statistical calculation, such as
deriving a running mean value, the computation of the mean is part of
the PIF and defines the controlled variable as a running mean value of a
function of external variables. There is no approximation or estimate
involved. What is controlled, and perceived, is not the moment-by-moment
fluctuations in the environment, but the mean value of a function of
environmental variables. The only way for that perceptual signal to
contain any noise would be through variations in the computation of the
function or its mean -- that is, given a repeat of the same
environmental fluctuations, the reported mean value would spontaneously
vary because of glitches inside the PIF. Even then, the control system
would interpret those glitches as real changes in the controlled
perception, and would counteract them by varying the output effects on
the environment.
Whether the PIF contains noise sources or not, what such a system
controls is not a CEV computed moment-by-moment on the detailed
variations of the environment, but a mean value of a function of those
variations. The variations themselves, which are filtered out by the
computation of the mean, are uncontrolled. Only the mean is controlled.
Another control system receiving the same information and having the
necessary output equipment might compute and control the variance of a
function of the external variables. In that case, the variance would be
the controlled variable, and the mean would not be controlled by that
same system, nor would the specific fluctuations in the external
variables be controlled. ONLY the variance would be controlled.
And of course still another (and quite complex) control system might
continually perceive and control a function like U(x) - U(x|y), the
value of which we could label "the amount of information in x about y"
-- which presupposes that both x and y are available as lower-level
perceptions. This number would be perceived not as an estimate of that
information, but as the exact amount of information existing at each
moment. The output would of course have to have some effect on the
environment that could vary one or both uncertainties or the
probabilities underlying them. Again, no "estimate" is involved; the
control system will take the computed value of U(x) - U(x|y) as the
actual state of this measure of the world, and keep it matching the
reference value.
All of which reminds me of a fact that I mention from time to time, but
which seems to go unappreciated. If you look around you at the world you
are in, how much noise and uncertainty do you see? Do shapes wiggle
around and fuzz at the edges? Do intensities bobble up and down? Do
sensations sometimes seem one way and sometimes another, unpredictably?
Does "in" sometimes unpredictably change to "beside?" No, the world of
experience is almost completely smooth, stable, and predictable. The
only way to make it seem otherwise is to invent special situations where
there is truly uncertainty, not just in perceptions of the world but in
the world itself. You have to use masking noise, low light levels or
extremely brief visual images, faint touches, or apparatus that
randomizes the effects of actions. You have to get completely outside
the normal range of perceptions, and introduce uncertainties where none
naturally exist.
The dynamic range of perception is such that nearly all perceptions are
normally free of noticeable uncertainty. You can see this just by paying
attention to the world you experience -- which, according to PCT,
consists of the perceptual signals themselves.
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Best to all,
Bill P.