[From Rick Marken (930118.2000)]
Bill Powers (930118.1000) --
But whatever the degree of practice, the response of the position
control systems will always fall off as frequency increases, and
there will always be some limit beyond which increasing the
amplitude of the reference signal variations can no longer make
up for the loss of loop gain in the position control systems.
Yeah. That's what I meant to say.
This phenomenon can, therefore, be adequately modeled without
bringing in any qualitative changes, as from open-loop to closed-
loop.
And that's (sort of) what I did say. Thank you for the nice, clear
description of a closed loop trill control system.
ยทยทยท
-----
At the risk of using the net for something other than argument, I
would like to report the current status of my HyperCard based
experiments on conflict. Recall that the subject is asked to keep
a cursor at a target position on the computer screen. The
horizontal (x) and vertical (y) position of the cursor is determined
at each instant by the following equations:
x = a1Hx + b1Hy + dx
y = a2Hx + b2Hy + dy
where Hx and Hy are the time varying horizontal and vertical positions
of the handle (mouse in this case), dx and dy are time varying
disturbances and a1,b1,a2 and b2 are constants. The measure of
conflict in this tracking task is the determinant of the two equations
above -- D = a1*b2 - a2*b1. There is NO conflict when a2 = b1 = 0 so
that D = a1*b2. There is complete conflict when a1 = b2 = a2 = b1
so that D = 0. In these experiments I always kept a1 + a2 = b1 + b2 = K
(I happened to select a K = 2); this keeps the environmental "gain"
from mouse movement to cursor movement constant across conflict (D)
conditions -- and equal in both the x and y dimension.
I used a low pass filtered random disturbance -- the same disturbance at
each level of conflict -- with a cutoff frequency of about .05 Hz (3
cycles/minute). A trial lasted about 50 seconds. I ran a subject and
a control model in each conflict condition. The control model was a
pure integrator, independently controlling the x and y position of the
cursor. The model parameter (integration factor) was selected
for best fit with subject mouse movements when there was no conflict --
that is,when
x = 2Hx
y = 2Hy
The model, with the integration factor fixed, was then used to predict
the subject's actions in different conflict conditions. I measured the
deviation of model mouse movements from subject mouse movements in each
condition (as RMS deviation of model from subject values). I also measured
the subject's and the model's ability to control the cursor in each conflict
condition. Here are the results:
Level of Level of Control (RMS Model Fit
Conflict deviation of cursor from target) (RMS error)
D Subject Model
4.0 (no conflict) 0.99 1.2 .64
2.0 1.10 5.1 1.19
1.6 0.98 7.3 1.90
1.2 1.30 11.5 3.60
1.0 1.50 15.2 5.20
.8 2.20 20.1 7.40
.72 1.80 23.5 8.25
As the level of conflict increases, the subject's ability to control
the cursor get's only moderately worse -- the rms error goes from
about .99 (average of about 1 pixel from the target) to about 2
pixels average deviation. The model, however, which matches the
subject's mouse movements to within 1 pixel (.64) when there is
no conflict (correlation between subject and model is .997+), gets
precipitously worse at controlling as the conflict increases; the
subject does considerably better than the model.
What seems to be going on here (based on looking at the traces
of model and subject actions) is that the gain of the subject's control
system seems to instantly adapt to the conflict situation. The subject
is better at dealing with conflict than the simple control model; so
the simple control model does not seem to work in this situation --
or does it? There are still some possibilities; maybe the model needs
more than pure integration -- what?? I want to exhaust the possibilities
before adding a new level or something. Maybe the integration factor
depends on the size of the error to some extent?
Any suggestions regarding futher experimentation or model changes
would be greatly appreciated.
Regards
Rick