[From Bill Powers (930118.1600)]
Rick Marken (930118.0800) --
You spur me to add a few more entries to the "Devil's
Bibliography."
Here is Warren S. McCulloch helping to form the myth that
feedback systems go unstable when their gain exceeds unity:
McCulloch, W. S.; Finality and form; in Embodiments of Mind,
(Cambridge, MIT Press, 1965) pp. 256-275.
"When we change the magnitude of the quantity measured, a reflex
may return the system toward, but not quite to, the original
state, or it may overshoot that state. The ratio of the return to
the change that occasioned it is called the _gain_ around the
circuit. When the return is equal to the change that occasioned
it, then the gain is one.....if the gain is greater than one at
the natural frequency of the reflex, fluctuations at that
frequency begin and grow until the limitations of the structure
composing the path reduce the gain to one; then, at the level for
which the gain has become one, both the measured quantity and the
reflex activity go on fluctuating." (p. 267)
Even earlier than this confident mangling of closed-loop
properties, we have this:
McCulloch, W. S., Appendix I: Summary of the points of agreement
reached in the previous nine conferences on cybernetics.
_Cybernetics_: circular causal and feedback mechanisms in
biological and social systems. Transactions of the Tenth
Conference, April 22,23 and 24, 1953, Princeton, NJ. Josiah Macy,
Jr. Foundation, 1955. LCN 51-33199.
"The transmission of signals requires time, and gain depends on
frequency; consequently, circuits inverse for some frequencies
may be regenerative for others. All become regenerative when gain
exceeds one. Regeneration leads to extreme deviation or to
schizogenic oscillation..." (p71)
This rules out negative feedback systems with loop gains higher
than unity -- in other words, all actual control systems that
exist in organisms.
On explaining how control works:
"Wiener drew a most illuminating comparison between the
cerebellum and the control devices of gun turrets, modern
winches, and cranes. The function of the cerebellum and the
control of those machines is, in each case, to precompute the
orders necessary for servomechanisms, and to bring to rest, at a
preassigned position, a mass that has been put into motion which
otherwise, for inertial reasons, would fall short of, or
overshoot, the mark." (p. 72)
Here we have the germ of the compute-then-execute approach.
There is a certain kind of intellect which can, on hearing the
merest summary of an idea, immediately leap ahead to its most
profound implications and applications, completely unaware, or
unconcerned, that it has a superficial and mostly incorrect
understanding of the idea.
Here's another gaggle of myths, this time from W. Ross Ashby, in
_An Introduction to Cybernetics (New York: Wiley, 1966 (third
printing, copyright 1963).
"The basic formulation of s.11/4 assumed that the process of
regulation went through its successive stages in the following
order:
1. A particular disturbance threatens at D;
2. it acts on R, which transforms it to a response;
3. the two values, of D and R, act on T _simultaneously_ to
produce T's outcome;
4. the outcome is a state of E, or affects E." (p. 221)
E is an essential variable that is to be stabilized by the action
of a regulator R, acting through an environmental function T
which is fixed. Regulation is achieved when the effect of D on T
is precisely cancelled by the response of the regulator R to D,
also acting on T. It is assumed that E depends on T and T alone,
so there are no disturbances acting directly on E that can't be
sensed by the regulator.
If R and T are precisely calibrated and act with infinite
precision, then perfect regulation is possible -- but not
otherwise. Ashby tended to overlook the question of precision,
largely because in examples he tended to use small integers or
decimal fractions accurate to one decimal place to represent the
variables. As a result he greatly overestimated the capacities of
compensating systems, and therefore, by comparison, greatly
underestimated the capacities of control systems.
"_Regulation by error._ A well-known regulator that cannot react
directly to the original disturbance D is the thermostat-
controlled water bath, which is unable to say "I see someone
coming with a cold flask that is to be immersed in me -- I must
act now." On the contrary, the regulator gets no information
about the disturbance until the temperature of the water (E)
actually begins to drop. And the same limitation applies to the
other possible disturbances, such as the approach of a patch of
sunlight that will warm it, or the leaving open of a door that
will bring a draught to cool it." (p. 222).
Note the implication that a compensating regulator might exist
which, on seeing someone approach with a flask, could deduce that
it contains cold water and is about to be immersed in the bath.
Note also the unspoken assumption that merely from qualitative
knowledge about a flask of cold water, a patch of sunlight, or a
potential draught through an open door, the regulator could be
prepared to act quanititatively: to add heat to the bath that
would exactly compensate for the cooling from the water in the
flask or the evaporation due to the draught, or cooling just
sufficient to prevent any rise in the temperature of the bath.
From qualitative knowledge of the disturbance, the regulator
somehow achieves exact quantitative compensation for the
quantitative effects of the disturbance. If, of course, such a
thing were possible, the compensator would be much superior to
any form of feedback controller. But such a thing is not remotely
possible.
After doing through a series of diagrams, Ashby finally diagrams
the true error-driven control system:
" ... we have the basic form of the simple 'error-controlled
servomechanism' or 'closed-loop regulator,' with its well-known
feedback from E to R." (p. 223)
The diagram is D -> T -> E
^ |
> >
R <--
Now we get to a whole fountain of misinformation about control
systems, a series of deductions that is just close enough to
reality to be convincing, and just far enough from it to be utter
nonsense.
"A fundamental property of the error-controlled regulator is that
_it cannot be perfect_ in the sense of S.11/3" (p.223)
He then goes through a "formal proof" using the Law of Requisite
Variety to conclude
"It is easily shown that with these conditions _E's variety will
be as large as D's_ -- i.e., R can achieve no regulation, no
matter how R is constructed (i.e., no matter what transformation
is used to turn E's value into an R-value)."
"If the formal proof is not required, a simpler line of reasoning
can show why this must be so. As we saw, R gets its information
through T and E. Suppose that R is regulating successfully, then
this would imply that the variety of E is reduced below that of D
-- perhaps even to zero. This very reduction makes the channel
D -> T -> E
to have a lessened capacity; _if E should be held quite constant
then the channel is quite blocked_. So the more successful R is
in keeping E constant, the more does R block the channel by which
it is receiving its necessary information. Clearly, any success
by R can at best be partial." (p. 223-224)
This argument has apparently convinced many cyberneticists and
others that the Law of Requisite Variety is more general than the
principles of control, and in fact shows that control systems are
poor second cousins to compensators when it comes to the ability
to maintain essential variables constant against disturbance.
In fact this argument shows how utterly useless the Law of
Requisite Variety is for reaching any correct conclusion about
control systems.
Having swept through this dizzying exercise in proving a
falsehood, Ashby then grudgingly allows feedback control to creep
humbly back into the picture:
"Fortunately, in many cases complete regulation is not necessary.
So far, we have rather assumed that the states of the essential
variables E were sharply divided into "normal" ... and "lethal",
so occurrance of the "undesireable" states was wholly
incompatible with regulation. It often happens, however, that the
system shows continuity, so that the states of the essential
variables lie a long a scale of undesireability. Thus a land
animal can pass through many degrees of dehydration before dying
of thirst; and a suitable reverse from half way along the scale
may justly be called "regulatory" if it saves the animal's life,
though it may not have saved the animal from discomfort. "
Note the gratuitous "half way along the scale."
"Thus the presence of continuity makes possible a regulation
that, though not perfect, is of the greatest practical
importance. Small errors are allowed to occur; then, by giving
their informfation to R, they make possible a regulation against
great errors. This is the basic theory, in terms of
communication, of the simple feedback regulator." (p. 224)
The argument then veers off into "Markovian machines" and
Markovian -- stochastic -- regulation. This is billed as the most
important and far-reaching application of the error-controlled
regulator.
Note how the argument relies on qualitative statements to reach
quantitative conclusions. It is perfectly true that if a
compensating regulator affects T equally and oppositely to the
effect of D, E will not be affected at all. But by that same
argument, to the extent that R does not have perfect information
about D (and about the nature of the connection from D to T and
from R to T), T will not be affected equally and oppositely, and
thus to the extent of the imperfection, E will not be perfectly
regulated. Furthermore, if there is any disturbance at all that
is NOT detected by R (for example, a disturbance that acts
directly on E), the effects of that disturbance will not be
compensated at all. If R does not compensate for all
nonlinearities and time-functions in the connection from D to T,
compensation will not occur. When the processes involved are
thought of as real physical processes in a real environment, the
idealized assumptions behind the compensatory regulator are
easily seen to be unrealistic -- they predict regulation that is
far, far better than any that could actually be achieved in this
way.
Note also how the qualitative concept that error-regulated
control must be imperfect is used to imply that it must be _more
imperfect than compensatory regulation_. This non sequitur has
appeared in the literature over and over ever since Ashby. In his
earlier book Ashby was still toying with true feedback control
and continuous systems; the appendix is heaped with rather
aimless mathematics that is oriented in that direction. But in
this second book, Ashby shows that he never understood how an
"error-controlled" regulator works. He didn't know that the
"imperfection" inherent in such systems can be reduced to levels
of error far smaller than the error-reductions that any real
compensating system could achieve -- smaller by orders of
magnitude, in many cases, particularly cases involving human
behavioral systems.
Ashby's entire line of reasoning about feedback control in _An
introduction to cybernetics_ is spurious. Yet Ashby has been
revered in cybernetics and associated fields for 40 years as a
deep thinker and a pioneer. His Law of Requisite Variety has
nothing at all useful to say about control systems -- and in fact
led Ashby to a completely false conclusion about them -- yet it
is still cited as a piece of fundamental thinking. Whether Ashby
originated these misconceptions or simply picked them up from
others I don't know. One thing is certain: he did not get them
from an understanding of the principles of control.
Here's a little test.
I have 200 pounds of ice cubes, and you have 50 gallons of
boiling water. Desired: a nice tub of water for a bath. I get to
throw in the ice cubes (you can see exactly how may I throw in);
you get to pour in the boiling water. As you see me disturbing
the bath with ice-cubes, you estimate how much boiling water to
pour in to arrive at a bath of the right temperature. When I have
exhausted my ice cubes, you finish the process by adding more
boiling water in the amount you think is necessary.
As an alternative, I will let you see a thermometer in the tub,
but will not let you see how many ice cubes I am throwing in. You
must base your additions of boiling water entirely on the
thermometer reading.
Whichever method of filling the tub you elect, when the process
is finished you must then step into the tub and immediately sit
down in it.
Which method would you choose: compensating for known
disturbances, or basing your action on perception of the state of
the essential variable without knowing what the disturbances are?
···
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Best,
Bill P.