[From Bruce Abbott (950331.1050 EST)]
I've tried a few runs of the compensatory tracking task (CTRACK1) using a
square-wave disturbance in order to have some data for a time-domain
analysis. The period and amplitude of the square wave were varied across
runs (only a few values tested thus far). Below are given some preliminary
results. The variables listed are as follows:
period square-wave period in 1/60th sec
amp half-wave amplitude of disturbance in pixels (60 = 120 rail-to-rail)
RTh rise time in 1/60th sec. Rise time is the time required for the
response to go from 10% to 90% of the distance between rails. The
conversion to seconds is given in brackets.
slew rate of cursor movement computed from rise time, pixels/sec
lag best-fit lag after disturbance change
k integration coefficient for simple one-level control model
rms root mean square error, handle vs model handle
r correlation, handle vs model handle
RTmh rise time for best-fit model
-----response---- ----------model fit----------
Run period amp RTh [ sec] slew lag k rms r RTmh
000 150 60 28.5 [0.48] 227 12 0.085 22.2 0.914 32.0
001 600 60 27.9 [0.47] 232 19 0.010 9.3 0.988 27.0
002 600 60 27.2 [0.45] 238 20 0.116 14.7 0.971 23.0
003 600 60 28.1 [0.47] 231 21 0.092 9.1 0.988 30.0
004 300 120 23.7 [0.39] 548 20 0.108 21.6 0.983 25.0
005 300 60 23.8 [0.40] 273 19 0.132 12.6 0.977 20.0
006 300 120 25.2 [0.42] 515 21 0.119 18.5 0.987 26.0
Although there are not enough runs here to get a fix on the stability of
these numbers, these results are at least suggestive of some possible
relationships (or lack thereof). The three runs at a period of 600 (10 sec)
produced fairly consistent values of around 0.46 sec for the rise time.
According to Malvino's _Electronic Principles_, 3rd Ed, the rise time of a
dc amplifier subjected to a step change can be related to the upper cutoff
frequency of the amplifier, which is the frequency of a sine wave input that
will reduce the output voltage amplitude by 3 dB. The formula is
fc = 0.35/rise time.
If it does not make sense to apply this formula here, I'm sure Bill will let
me know (I feel a bit like the sorceror's apprentice here). For a rise time
of 0.46 sec, this corresponds to an upper cutoff frequency of 0.76 sec, or
about 3/4 sec. Thus far I have not dared to try the sine wave disturbance
at 3/4 sec, but judging from what happened at 1.25 sec I'd say this estimate
is not too far off the mark.
The difference between my performance and that of the simple one-level model
is that the model always approaches the new disturbance value following a
step-change as a negative exponential, like a capacitor charging or
discharging, whereas I tend to switch faster, overshoot, and then correct
(although I can force myself to act more like the capacitor by limiting my
slew rate). Fitting the model without the perceptual delay following a step
produces model slew rates that are far slower than the actual ones, but
introducing a delay factor generally produces a model whose slew rates are
not far from those actually observed.
In an earlier post, Bill Powers (950312.1715 MST) noted the following:
On this latter subject, I have seen a number of papers by Kelso and by
some MIT types who have investigated this "constant-time" phenomenon:
moving the left hand to a target 20 inches away takes the same time as
moving the right hand to a target 5 inches away. All sorts of
conjectures were given about how the computations are done to make sure
the times come out the same. In fact, if you have a control system with
fixed dynamic characteristics, and it is reasonably linear, this
constant transition time falls out of the system properties. All you're
doing is scaling up the y-axis of the time plots. If the reference
position is abruptly changed by 20 units, the initial error is 4 times
what it is when the reference position changes by 5 units, so the entire
motion occurs 4 times as fast. Four times as fast to go four times as
far: Voila! Equal times. There is no system computing how to make the
times come out the same; what would be hard would be to make them
different.
Compare the three runs done at a period of 300 and at amplitudes of 60 and
120 (runs 004, 005, & 006). The rise times are virtually identical whether
the amplitude of the disturbance was 60 (120 pixels rail-to-rail) or 120
(240 pixels rail-to-rail): about 0.4 sec. Given this constant rise time, a
doubling of the disturbance amplitude implies a doubling of the slew rate
(pixels per second of cursor movement) required to correct for the change in
disturbance value, just as Bill indicated.
The rise times were about 0.1 sec faster at period 300 than at periods of
150 or 600; I'm not sure whether this is real or just uncontrolled random
variation (some runs were done on different days, etc.). However, I have a
feeling that the change is a real effect of period. At 600 I had plenty of
time following a step before the next one would occur, and my attention
("readiness") seemed to flag somewhat. At 150 I felt "rushed"--it seemed I
was still trying to make final corrections from the previous transition when
the next one would occur. (Pushing it may also be the reason the optimal
lag was smaller at 150 than at the longer periods.) So 300 may have been in
some sense closer to optimal. Note, however, that the effect here is small.
Regards,
Bruce