[From Bill Powers (2002.12.03.1048 MST)]
Back from Thanksgiving with kids in Boulder County. My remote e-mail
access didn’t work.
Bill Williams UMKC 28 November 2002
6:00 PM CST
Not that it matters, but since committees always spend 90% of their time
on the smallest 10% of the budget …
As I understand it, a stall condition is defined as the velocity at which
laminar flow over the top of the wing begins to turn into turbulent flow:
“the boundary layer separates,” as they say. The angle of
attack (relative to the direction of travel, the relative wind, not to
the horizontal) at which this occurs depends on airspeed, lift, weight,
drag, thrust, and wing configuration. I was once co-owner of a Stinson
Station Wagon, with a high gull-wing having leading-edge slots which
could be opened with a lever. The leading-edge slots directed air
downward over the wing at high angles of attack, preserving laminar flow.
I remember flying it straight and level with the nose aimed at the sky
and the throttle wide open, at 45 MPH. Quite a sensation. Of course it
wasn’t in a stall.
I’m finding the Jagacinski and
Flach book interesting reading.
Yes, I did too, in both a positive and a negative way. Positively, they
have collected all of the main concepts of control developed over the
last 60 or 70 years, so we have something concrete against which to
compare PCT. Negatively, I think the model they have adopted from the old
engineering psychologists is not carefully thought-out even for the very
narrow tracking situation for which they use it (The “McReuer
Crossover Model”). They never realized that a subject in a tracking
experiment could create and maintain many different relationships between
cursor and target, so they thought that the error was right out there on
the display screen, rather than seeing that it had to result from
comparing a perception with an internal, not an external,
reference. In fact that point seems to have eluded practically all
control theorists in psychology.
Another interesting fact is that while Flach assures me that he has done
many simulations, even some with real analog computers, I haven’t found
any simulations in Control theory for humans. Models are
presented, but there are almost no plots of their behavior, or any
comparisons of their behavior with that of real people. There are some
Bode plots, and some examples of Fourier analysis, but these are global
representations of behavior and do not show how any variables change
through time. I don’t think it’s possible to judge how well a model fits
real behavior from a Bode plot or a Fourier analysis – the
representations are simply not detailed enough.
For those who have not seen some of the correspondence, the subject of
Bode plots comes up in Chapter 14 of the book, purporting to show that
subjects actually change their internal organization when the nature of
the external part of the loop (that we call the environmental feedback
function) is changed. The Bode plot shows the frequency response of a
control system given sine-wave variations in its reference signal, or
else sine-wave disturbances. In the simplest application, the frequency
of the sine wave is gradually raised, while a record is kept of the
amplitude and the phase of the output (as Flach defines the controlled
variable) relative to the input (the reference signal, as Flach defines
input). For human beings the data have to be obtained indirectly using
randomized inputs, but the results should be the same.
The connection between the control handle and the cursor is selected from
three choices: a direct, proportional connection, a single integration,
and a double integration. The experimental results for these conditions
are shown in Chapter 14, Figs. 14-3, 14-4, and 14-5, The conclusion is
that the human being changes internally so as to make the overall system
function look like a first-order lag in all three cases, or as we call it
in PCT, a leaky integrator. The data certainly show that this effect
occurs, no doubt about that.
I have been able to come up with a two-level model that reproduces these
effects without any changes in model parameters, a fact that calls
into doubt all conclusions about “adaptation” drawn from these
experimental findings. It is possible to set up a two-level control
system controlling velocity at the lower level and position at the higher
level, which shows the same effects as in Ch. 14 when the external part
of the loop is changed from a proportional to an integral to a double
integral response. Bode plots show the same phase behavior and the same
20-db per decade frequency rolloffs. While this does not prove that no
adaptation at all takes place, it does show that the major part of the
data can be accounted for without assuming any adaptation (that
is, any changes in the model’s parameter values from one case to
another).
At present, Flach and his people are attempting to reproduce my model by
using MatLab (as they learn it), and are having a few difficulties that
are probably temporary. When they have their simulation running, I think
we will have some interesting conversations. Flach’s remarks about a
possible link between frequency-domain and time-domain methods suggests
to me that he has somewhat limited experience with the time-domain side
(what we call simulations). I learned the frequency-domain and
time-domain stuff at the same time (possibly before Flach was born) and
decided I greatly preferred time-domain. So I probably have as much to
learn about frequency domain as Flach et. al. have to learn about time
domain. Somehow, though, I think I’m going to end up still preferring
time domain.
I don’t want to say too much now about how this exploration with Flach
will proceed. I’m about to send him a new model that shows the same
two-level control system being used to apply the brakes to a vehicle and
stop it at a specific place. Of course the kinds of control systems in
the book that would be used for this would be the sort that compute
decelerations from physical properties of the system and then apply the
braking forces open-loop. Oddly enough, if we assume that braking force
is proportional to velocity times the amount of pedal pressure applied,
the model shows the pedal pressure to be constant once the distance to
the target point has become small enough for any braking to occur. I have
now added a nonlinear braking function just to prove that the control
system can still stop at the right place, and to show that the constant
braking pressure isn’t a loophole in the model.
Best,
Bill P.
