[From Bill Powers (940523.1100 MDT)]
Standardizing conditions: a puzzle
The loop gain of a control system depends not only on the output
gain, but on the amplification/diminution factors operating in the
external part of the loop and all the way around to the comparator.
So I reasoned that if such things as mouse sensitivity, screen size,
or viewing distance changed, we would measure a different apparent
integration factor, and would have to apply a correction to obtain
the true factor characterizing the output function.
So I worked out the way of computing a correction factor, thinking
that this would improve the repeatability of our measurements of
parameters over experiments and over participants. Then, having got
the algebra worked out, I thought I should really try this out with
some real tracking data before posting my algebra to the net.
So I established 3 different conditions:
1. Normal mouse sensitivity, normal viewing distance (20 inches).
2. Normal mouse sensitivity, short viewing distance (8 inches).
3. Doubled mouse sensitivity, normal viewing distance.
The measured integration factor k under these three conditions and
identical disturbances came out as follows (a single one-minute run
for each condition):
1. k = 8.461
2. k = 8.330
3. k = 8.470
WHAT?!!!
There is less than one percent deviation of k from its mean value
over these three conditions. If we assume an approximately 2:1
change in gain in the external part of the loop, we would deduce a
2:1 change in the true value of k in the output function in order to
keep the overall loop gain from changing. In other words, the true
value of k seems to be changing in order to maintain an almost
exactly constant overall loop gain!
This considerably changes our interpretation of the simple model we
use for a tracking experiment. We are, it seems, looking at a very
good adaptive control system, one that can adapt its own output
integration factor in just a few seconds to maintain a constant loop
gain in the face of rather radical changes in the external part of
the loop.
Our model, of course, is not adaptive. It simply represents the best
equivalent non-adaptive control system. The presence of adaptation
shows up only when we change multiplication factors in the external
part of the loop -- and discover that the changes make no difference
in the apparent integration factor of the control system.
Why constant loop gain? One hypothesis might be that loop gain is
itself a controlled variable, but that would be hard to explain,
considering that the items that make up the total loop gain are not
represented by the senses. An alternative hypothesis requiring fewer
ad-hoc postulates would be that the system adjusts the total loop
gain, by adjusting k in its output function to achieve the smallest
possible mean absolute error signal.
There are probably irreducible transport lags in the real control
system, and also control is accomplished through a lower-order
control system that has to position an arm with mass. So there is an
upper limit on the output amplification that can occur; increasing
amplification above that limit will tend to produce spontaneous
oscillations, increasing the control errors. Also, of course, if the
loop gain is too low the control error will also increase because
the system will respond too sluggishly to counteract disturbances
accurately. So there must be some optimum value of loop gain, set by
basic physical characteristics of the control system. That optimum
loop gain will be independent of where in the loop the gain is
created.
When we increase the external components of loop gain, the system
would become unstable if it did not reduce the internal components
of loop gain. And that, indeed, seems to be happening. During the
first few seconds of tracking with a short viewing distance or a
doubled mouse sensitivity, there is a bit of trouble obtaining
stable control -- but control is soon recovered, and feels normal
for the rest of the run.
If the adaptation hypothesis is right, we should see a change in the
measured overall loop gain if we put an extra mass on the arm. That
would lower the frequency at which oscillation would occur, and
require a lower total loop gain to maintain stability. We then
should observe that this same loop gain is maintained when we change
viewing distance and mouse sensitivity.
I'll try to work this out when I get back from England, unless
someone else beats me to it (feel free). The added mass should be
supported against gravity and freely moving -- perhaps hanging it
from a long string would do the trick. The adaptive model will need
some simulation of the behavior of lower-level systems and the mass
of the arm. Maybe a slightly underdamped mass on a spring would
serve. If we're lucky, the main determinant of stability will be the
transport lag, which is fairly constant at around 0.15 to 0.2 sec.
···
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Best to all,
Bill P.