[Martin Taylor 960216 11:00]
Bill Powers (960216.0500 MST)
I like your answers to the 5 questions--or more properly to the problem
that underlay them. Probably I like them because they are the same as my
answers, though stated with more clarity. There do remain a couple of
niggle-points (apart from obvious misunderstandings).
A misunderstanding first, one whose genesis is mysterious:
You
proposed that a variable V was seen to change and then to return to its
former value. I said that without knowledge of the disturbing variables
on which V depends, it is impossible to do the Test. Rick said that if
the Test _is_ applied, the problem can be solved. These statements are
not contradictory. In order to do the Test, you would have to FIND other
variables that influence V, and manipulate them.
Since such other variables were introduced right at the beginning of this
problem specification, and various new ones were introduced from time to
time, it's easy to see how "the Test cannot be done in this situation" and
"the Test is easily applied" could be taken as contradictory statements.
No matter. I now know what you mean, and what Rick means, or so I think,
and we have all got back to the situation as originally proposed, to which
you and Rick have now given answers. Good.
The problem here is contained in the phrase "push the particle away."
This phrase conflates the application of a disturbing variable (push on
the particle) and a _consequence_ of that application (the particle
moves away).
True. An unfortunate consequence of using English instead of math notation.
What your way of putting it really says is "apply a
sufficient push to the particle that it moves to a new position."
You omitted the words "to me" after "really says." As the words left my
fingers, they didn't say that. They said "apply a force to the particle
that would be expected to move it to a new position." But anyway,
unless the particle were hard-glued to some mobile spike, any push would
in fact move it to a new position (at any rate it would if the particle's
position were controlled), so the difference of meaning is of little
practical importance.
To be precise while using your own terms, we would say that you apply a
known amount of push to the particle, predicting the amount of movement
of the particle away from its apparent equilibrium position (or mean
dynamic behavior) that would be expected if there is no control. If the
particle moves exactly as predicted, there is no control of position, so
the test is FAILED.
This raises one aspect of the problem, to which I alluded some days ago:
that you have to know the physics of the situation to know just how much
the particle should move. That is should move with a certain push is not
at issue. That it should move 10.3 cm is unknown. However, one could
expect it to move 10 cm rather than 1mm or 10 m, and if (as is presumed
in the original specification of the situation) it moves a lot less, then
the test is not failed.
From here on, we are more or less in sync.
It is not always possible to complete the test. We may simply have
insufficient knowledge of the environment to identify the source of the
opposing push, or to find any paths by which the state of the controlled
variable can be known to the supposed control system. We may be unable
to predict the effect of a "push." In that case we simply have to
deliver the Scottish Verdict: not proven. We have to try to learn more
about how the environment works.
Right. That's what I thought. And as applied to the Brownian motion example,
it asserts that until the chemico-mechanical transduction path is discovered,
one cannot know whether the particle is in an equilibrium or is actively
controlling.
ยทยทยท
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You have raised another question, which is why the disturbance must be
sustained rather than transient. The answer is that the opposition to
the disturbance may require time to develop.
This is no answer. It is an answer to an assertion not made, that the
only test that should be done is to apply an impulsive disturbance. If
you want to determine the time course of the development of opposition
to the disturbance, an impulsive disturbance is useful, but not as the
only kind of disturbance in the test.
If you apply the
disturbance and instantly remove it, you will never see the opposition
occurring while the disturbance is still present. The behavior you see
will not be equal and opposite to the disturbance. It will begin to
appear and then disappear again before it is fully developed, so you
will never get a simple picture of the relationship between disturbance
and action.
On the contrary, you get one _simple_ picture of that relationship. Whether
that simple picture holds under more realistic conditions cannot be
determined except by using it to predict more realistic behaviour. What
you get from the appearance and disappearance of the output is related to
the inverse Laplace transform of the loop gain function, on the assumption
that the system is linear. You can do the inversion and multiply by the
Laplace transform of any disturbing influence waveform, and see if the
resulting output waveform (and error waveform and all the others) agree
with what actually happens with a slow or sustained disturbance.
The prediction probably won't agree with what actually happens, but that
is because the real control system is unlikely to be linear. It's a useful
first approximation, nevertheless, and does give you an indication of how
important any nonlinearity might be in the actual operation of the control
system. You have to do the other dynamic tests with slowly varying
disturbances as well, but that doesn't mean that the simple impulse test
isn't a good one to do.
In matters where temporal waveforms are involved, there are two points of
extreme simplicity--zero time and zero variation, or rather, infinite
bandwidth and infinite duration. To deal with finite time and finite
rate of change is more complicated. One cannot achieve either endpoint
of extreme simplicity, but one can often go outside the effective region
in which the system can act, so as to define what one might call its
envelope of behaviour.
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Wastebasket:
you have to devise means of ruling out non-control
explanations. For example, you might move the wastebasket by different
amounts and in different directions on successive trials.
Yes, in the original description of the wastebasket "Test" some months
or years ago, this was spelled out. The point of bringing it up wasn't
to change that, it was to say that the displacement in a Test for control
of location might sometimes be discontinuous. The wastebasket is picked
from its putatively controlled location and placed elsewhere.
Referring to another posting of yours yesterday, I'm wondering where
you got the idea that my old postings on information had anything to do
with the notion that the behaviour of systems is discrete rather than
continuous. I seem to remember expending a great deal of effort in
getting you to see how information theory works equally in discrete
and continuous systems, and having thought that I had succeeded in
developing that understanding in you. It's a bit discouraging having
to go back to square one, if you have reverted to thinking that it is
possible to talk about information only in the context of discrete
systems. I hope that's not the case, and that you were referring to
something else. But I can't think what that something else might be.
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What the experimenter is always doing is saying "Oh, yeah?" and coming
up with some demonstration that the explanation is wrong, or that there
is another equally plausible explanation.
...
If all challenges fail, we have no
choice but to accept the theory -- at least until a more ingenious
experimenter finds a test that is failed.
Now _that's_ what I wanted to hear you say. It's not _quite_ my idea of
how theories are compared, but in the situation at hand, it's fine.
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A tensed muscle has LESS volume than a relaxed one. Now, you knew that,
didn't you?
No. It never occurred to me to ask, but it's interesting to know.
And what might be even more interesting is that a colleague down the
corridor is working on a touch-sensing robotic hand that is worked by
tendons pulled around pulleys by muscles that _are_ pumped up in order
to tense them. Each muscle consists of a very specially shaped air
balloon that shortens its length when blown up (the surface
is flexible but only trivially stretchable). At some point I hope to
learn more about this device, which at present is slaved to a device
like a thumb-and-finger glove in which a human hand is fitten to manipulate
the machine. There is no force feedback, and the human manipulator has
to look at the machine to see how its thumb and finger are moving. But
the intention is to remedy this lack. It's intended as a hand for a robot.
The colleague knows of my interest in PCT, but I don't know what algorithms
are currently being used in the control. From what little I've been shown,
I get a distinct flavour of inverse kinematics.
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New topic: Visual hierarchy.
There is, in the 9 Feb issue of Science p 776-777, a description of an
attempt to use the known and partly known connections among areas of the
visual system to define a hierarchy. A strict hierarchy is not possible,
but using all well defined and more poorly defined connections together ,
there are many thousands of possible hierarchies that have only 6 failed
constraints (i.e. cyclic connections). If 36 uncertain connections are
removed from the analysis, there is exactly one reciprocal connection
that remains, between two areas called FST and MSTd.
In the hierarchies that have 6 failed constraints, there are between 13
and 24 distinct levels, with most having 18 or 19 levels. In the different
hierarchies, FST showed up often on level 12, and even more often on level 17.
If the visual system itself were a control system, one would expect
every connection to be reciprocal, or at least that there would be as
many downward as upward connections. But there's only one that seems
secure, between the FST that usually shows up on level 12 or 17 in the
different structures and MSTd that shows up most often on level 11.
The article says that you can find more at URL
<http://www.psychology.ncl.ac.uk/www/hierarchy.html>, with predictions of
the results of anatomical experiments that they suggest should be done, at
<http://www.psychology.ncl.ac.uk/www/predictions.html>
I don't know what one should make of this, but it looked interesting.
Martin