[From Bill Powers (951027.0830 MDT) --
Richard Plourde --
Hi, I used to be one of those designers of electronic control systems
for physical devices, too. Glad to hear from you, and for the chance to
stretch some rusty muscles (without risk of having to prove I'm right).
The only problem like yours that I ever tackled was that of stabilizing
a cheap helium-neon laser so it could be used for interferometer control
of a grating-ruling engine. I had read somewhere that the difference in
the modal light frequencies could be detected in a nonlinear diode
photocell as an audio-frequency signal, and sure enough that worked. I
phase-compared the audio frequency to a reference frequency to get a
variable D.C. frequency error signal and amplified the error signal to
heat a ring of resistors around the body of the laser tube, to change
the separation of the mirrors by a tiny amount (output function and
environmental feedback function). After thermal equilibrium had been
reached, switching on the control system would vary the tube length
until a stable (non-chaotic) point was found where the error signal
changed smoothly with tube length. The system would lock on and hold the
frequency and phase difference constant for months on end, long enough
to rule the largest gratings. The system just kept varying the tube
length until such a region was found, and then automatically (without
further instruction) locked on.
Your problem sounds tougher than that, with multiple path-lengths and
lots of unpredictable mode changes. There may not be any fine
adjustments of mirror separation that would find a non-chaotic operating
region, however narrow. But the detection of audio beat frequencies in a
diode photocell might be a usable trick.
According to PCT, there are two basic problems in control: converting
the variable to be controlled into a scalar perceptual signal, and
converting an error signal into a systematic effect on the controlled
variable (the comparator is always trivial). In your case, and my laser-
stabilization case, the detection problem comes down to finding a way to
perceive a variable on which the stability of the laser depends, and
turning that variable into a controlled variable. On the output side,
the solution in my system was simple: just find a way to make ultra-fine
adjustments in mirror separation, which heating proved to do very well
(although lots of other more complicated -- and faster -- approaches
could have been used). I don't know what physical manipulations, in your
case, have an effect on the variable you want to control, and maybe you
will need something much faster than heating. But I'm sure you've
already been through all this.
···
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Chris Cherpas (951026.1026 PT) --
Me:
you can't uniquely solve a system of equations in which the number of
equations is less than the number of variables.
You:
I never thought about it that way.
Suppose you have a feedback function which expresses some measure of
reinforcement r as a function of some measure of behavior b. That gives
you two variables (b and r) and one equation (the feedback function).
Assuming that neither b nor r can be varied independently of the other
(as is true in operant behavior), you need a second equation relating b
and r to arrive at a unique solution for the system behavior. In this
case the second equation would be a model of the organism which makes
the behavior a different function of the reinforcement.
If you have multiple behaviors and multiple sources of reinforcement,
with a feedback function for each pair, you then have n equations
relating 2n variables, and to solve the system of equations uniquely,
you must have another n equations expressing the reverse relationships
via the organism: a more complex model of the organism.
When you try to get more out of a system of equations than there is in
it, you start committing mathematical blunders. Herrnstein's matching
law for two simple ratio schedules says
b1/(b1 + b2) = r1/(r1 + r2)
This informative-looking expression reduces to
b1/r1 = b2/r2
But b1/r1 is simply m1, the ratio set by the apparatus for schedule 1,
and b2/r2 = m2 for the other schedule. Thus we conclude,
m1 = m2.
Therefore Herrnstein's matching law for this case says that the two
ratios are identical. That is all it means: a tautology. If there is
matching, there is matching, and the matching is strictly of one
schedule to another. The organism's behavior is irrelevant.
For the generalized case, we can do the same simplifications of the
complex-looking expressions, and arrive at
m1 = m2 = m3 = ... mn
When you have n equations and more than n variables, there is no amount
of manipulation that will yield any unique solution. If you haven't
faced that fact in the beginning, you can be led off into wildly
complicated arithmetic, introducing more conditions like time-outs and
changeover delays which do nothing but complicate the functions beyond
intuitive understanding, and still leave the result underdetermined. In
all the mathematics I have seen in JEAB this is what is going on --
attempting to get a unique answer through both verbal and mathematical
manipulations, in a situation where it is mathematically impossible to
get a unique answer. The fact that answers are apparently found shows
only how a determined mind can delude itself by burying its mistakes in
mathematical complexities where they are hard to find. The verbal
interludes connecting fits of algebraic manipulation are a fertile
source of mistakes.
The mistakes are hardly worth looking for, because there is a
fundamental principle we can use as a shortcut. You must have at least n
_independent_ equations to solve for the values of n variables.
To understand the combination of organism and feedback function, you
MUST, ABSOLUTELY MUST, either dissect the organism and determine its
input-output function, or _propose_ such a function -- a model of the
organism -- and test its adequacy experimentally. For every feedback
function in a system, you must have another function that describes the
behaving system. Somehow you must supply the missing equations. What
this means for the experimental analysis of behavior is that it is
literally impossible to understand the organism strictly on the basis of
observable variables and the environmental connections between them.
When you try to do that, you necessarily end up one or more equations
short of the number you need to determine the system. One way or
another, you must propose a model of the organism, and this says that
pure empiricism (without mapping or guessing at the insides of an
organism) can't succeed in explaining behavior.
I leave it to you to consider the implications of all this for the
theoretical content of JEAB and other similar journals.
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Bruce Abbott (951026.1605 EST) --
This apparent gaff came up in the context my
discussing the descriptions engineers gave (and still give) of
the standard ECU, usually illustrated by something like the
following diagram:
>
disturbance
error V
reference --->[comparator] ----->[output function] --X---> output
^ |
input | feedback |
-----------------------------------------
This is not the standard engineering diagram, but the standard
engineering PSYCHOLOGY diagram. Missing is a function between the output
and the input, which is usually present in engineering diagrams as some
of the system stabilization is done there.
Even in a voltage regulator, the feedback voltage is equal to the output
voltage only for low output voltages. If you want to stabilize a 10,000
volt power supply, you don't feed back 10,000 volts. You divide the
output down by using a (precision) resistive voltage divider in the
feedback line, and set the reference voltage correspondingly lower. For
example, the feedback function would divide the output by 1000.0, and
the reference voltage would be 10.000 volts.
Your second diagram is correct:
ref sig.---->[comparator]-- error sig.-------+
^ |
percept. sig.| |
[input function] [output function]
^ |
input | | output
+-----------X<------------+
^
>
disturbance
The problem is that this system does nothing to resist the effect
of any disturbances on the output, and thus cannot account for the
fact that organisms like rats, pigeons, and people vary their
actions so as to achieve a constant end.
Since the output is what is usually called the "behavior" or "action" of
the system, you have one more question to answer before passing PCT 101:
If the input is a reinforcer, and the behavior of the system comes to
maintain that reinforcer at a specific level, what is the relationship
between the reinforcer and the behavior (1) without any disturbance
acting, and (2) with a randomly varying disturbance acting?
This question counts for 50% of your grade.
-------------------------------------------
--- (951026.1650) --
In fact, this is what I have been driving at for about the last two
weeks. I have been arguing several points, but a main one is that
such established phenomena are worth investigating as areas in
which PCT could be applied, tested, and developed. This strategy
has the advantage of making contact with established research
traditions and theories, with the potential for placing PCT
squarely within mainstream psychology. (Am I an optimist or what?)
No argument there. What I have been arguing is that because of assuming
a wrong model in the beginning, many researchers end up describing and
investigating phenomena that don't exist. But I encourage your strategy,
particularly in connection with Bowlby where there seems to be
acceptance of his work, but not of his theory. All you have to do, it
seems, is to add the last span to the bridge, which I admit is a lot
easier than starting from scratch.
I, too, regret that Bolwby is not here to enjoy what we are doing, as I
think he would have done.
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Chuck Tucker (951025) --
Thanks very much, Chuck, for posting that wonderful paper by Dennis
Delprato. I had seen parts of it before, but never the whole thing
(unless I've lost some more critical neurons). It is a beautiful
systematic introduction of PCT using a subject of great interest as the
focal point. This one should go on our Web page.
And Dennis, congratulations. Whatever flak you may get from readers, you
will get none from me.
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Bill Leach (951026.20:19 U.S. Eastern Time Zone) --
Thanks for the education about Ben Franklin. I should have realized that
he would hardly have flown a kite on a metal string if he hadn't
suspected that electricity was involved. Or do I have that wrong, too?
I also appreciate your other insightful posts.
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i.kurtzer (951025.1830) --
There are, in fact, ways of determining loop gain without getting inside
the control system box, defining it strictly in terms of observable
variables. But I support your basic idea, which is to keep observation
and theory as separate as possible. If you have good observations, you
can try out many different theories on them, but if a particular theory
is woven into the observations, you're biasing the data against other
interpretations.
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That is much too long a day at the keyboard!
Best,
Bill P.