[From Bill Powers (960423.0950 MDT)]
Hans Blom, 960423b --
the maximum change in the first-order wants induced by the
disturbances is about 2.5% of the undisturbed value.
This is the same order of magnitude as I can change the goal
(thermostat setting) of my central heating system in Kelvin. Can I
change that goal only a little or very much? How much is much?
That's an interesting generalization from my example. By this reasoning,
we can say that no position control system can have more than a tiny,
almost vanishingly small, effect on position. For example, when I drive
my car from Durango to Boulder, a distance of 385 miles, I have actually
changed my position by only 1 part in 4.1 million, in terms of the
distance to the Sun.
A more useful estimate of the effects of control would be to compare the
range through which a disturbed variable would change in the absence of
control to the range within which it changes under the same disturbance
when control is occurring. That's the measure we use in PCT.
Finally, if a higher system were manipulating the reference signals r21
and r22, the controlled variables could be changed through their entire
possible range -- while disturbances still can change them only by about
0.25 units at most.
Obviously, when the disturbances call for more output than the systems
can produce, the values of r11 and r12 become very sensitive to small
additional changes in the disturbances.
Goals are much more influenceable when lower levels saturate.
That's the kind of answer that says "I was wrong, but I'm going to find
a way to be right anyway." You're forgetting that if you physically
overwhelm a control system, your equations no longer apply, and the
system as a whole is going to cease to function normally.
... physiological control systems often have loop gains between 2
and 5, psychological control systems may have much larger loop
gains
I hope that, having said that, you don't decide that this is a Basic
Truth, and from now on dismiss physiological control systems as having
low gains. The loop gain of a tendon-force control system that makes the
arm model behave most realistically is about 200. People have mistakenly
concluded that this loop gain is low because they have analysed this
system as a position-control system; it is actually a force (or torque)
control system. The actual position control systems, which use muscle-
length and joint-angle perceptions, have very large loop gains also:
consider how much your arm sags when you hold it out and someone puts a
1-kilogram book in your hand. The oculomotor pursuit (velocity) tracking
system has a loop gain that is probably in the thousands, considering
that the eye can follow a moving object continuosly with an error of a
couple of minutes of arc. The loop gain of a hand-eye control system
such as the one that threads a needle must also be extremely high. When
you drive a car, you maintain the position of the car in its lane within
a few feet during a trip of 20 or 100 or 1000 miles (think of where the
car would have gone without your steering control system working). Just
lately, Bruce Abbott has established that female rats maintain their
body weights on a slowly-rising curve within a few percent, despite
transient disturbances of 30 percent of body weight. Certain biochemical
control systems with allosteric enzymes in them have loop gains upward
of 50,000. The list goes on and on.
So just erase "physiological control systems have low loop gains" from
your mind. This is not true, although a few (like the iris reflex) fit
the generalization. I hope that you prefer your generalizations to fit
the majority of cases, not the minority.
Anyway, a psychologically noteworthy result would be that the goals
of a control system can be manipulated most easily when the
organism is overwhelmed.
Since you have no idea of what is going on inside an "overwhelmed"
control system, what you say is most probably wrong. Aren't you
forgetting that you CAN'T EVEN SEE the goals inside another person? How
are you going to manipulate what you can't perceive?
I'm sure that the above solutions would fit those obtainable from
his equations.
I guess so. I made a fully static analysis, you introduced leaky
integrators. The steady state results ought to be the same.
Boy, are you reluctant to admit that your argument has been destroyed!
The steady-state results ARE the same. I used leaky integrators because
that's the easiest way to stabilize a simulated control system. The
numbers I posted were the steady-state numbers. Plug them into your
equations and see - I'm not going to do it for you.
But note that the "gain" of an integrator is almost zero after a
step change of the disturbance, so right after the introduction of
the disturb- ance -- if it was introduced suddenly -- the goals
must have been influenced quite a lot.
Again, you're fishing for a way to be right even when you were wrong.
Are you telling me that you can manipulate the goals transiently by
varying disturbances, when these transient effects are determined by the
inner dynamics of the control systems? This is a fantasy, especially
considering that you can't manipulate the steady states of the goals by
more than an insignificant amount.
I can vary the thermostat setting of my home heating system only
from 5 to 30 degrees Centigrade, a range of 25 degrees on top of
about 300 Kelvin. That is less than 10% variation, yet fully
adequate for me. Yes, I know, taking Kelvins to make my point isn't
fair ;-).
Damned right it isn't fair. It's facetious. See my comments at the start
of this post.
Hans, are you going to come right out and say that your mathematical
analysis does not apply to a real system of the kind I described, or are
you just going to go on looking for a way to maintain your position in
spite of being shown that your analysis was unrealistic?
My example used what I called a "complete" control hierarchy -- all the
degrees of freedom were used up. If you want to go on claiming that
disturbances can be used to manipulate goals for this kind of system,
you're going to have to back down and add, "... over an eentsy weentsy
teensy range." That really makes your argument that the environment
_determines_ wants ridiculous. And I will come back and say, "OK, now
the loop gain is a million, or a billion -- or the system is a pure
integrator so it's effectively infinite in the steady state." If you can
stretch a point, I can outstretch you. But why bother? Your basic claim
has been disproven. If you're not going to admit that, further rational
discussion is impossible.
It's also interesting to see what happens when there are fewer control
systems than degrees of freedom in the environment. If you didn't have
an axe to grind, you might be interested in that sort of situation, too.
But if you can't accept the results of the first example, I see no
reason to go any further.
···
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In your comment to Bruce Gregory, you say
Control can be compared with hill-climbing, where, as long as the
controller controls well, it uses all its energy to remain near the
top of the hill, each time having to counter the "disturbances"
that move it away from the top.
The analogy is strained. When a control system maintains a variable near
a reference level, a disturbance that tends to increase the error does
produce more action, but a disturbance that tends to decrease the error
produces less action. If the disturbance is just enough to maintain the
controlled variable at its reference level, there is no action at all --
NO energy is required to "remain near the top of the hill."
Also, why should the controller have to use ALL its energy to maintain
the controlled variable at its reference level? What if the reference
level happens to be set, for the time being, at zero?
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Bill P.