[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.