[From Bill Powers (941231.1235 MST)]
Bruce Abbott (941230.1700 EST) --
RE: methodology
Your use of standard methods is, of course, completely correct where
they apply:
Being a quantitatively minded scientist, I carefully measure the
starting positions of the stick, rudder, and elevator. Then I
systematically vary the stick position over a large number of
positions and record the rudder and elevator positions at each
stick position. I discover that every stick position yields a
unique combination of rudder and elevator positions, and that I can
decompose the "influence" of the stick into two orthogonal axes: up
and down moves the elevator without affecting the rudder, whereas
left and right moves the rudder without affecting the elevator.
This is a nice example for PCT (as you realized), because in fact your
RC control box does not emit a signal telling the receiving system how
much torque to apply using the motor that positions the control surface,
nor does it directly control the angle of the surfaces. It varies a
reference signal for the servo in the model plane. The servo compares
the sensed position of the control surface (via a potentiometer linked
to it) with the reference signal, and amplifies the error signal to run
a motor that moves the control surface. As a result, the control surface
comes to a position where the signal from the potentiometer matches the
signal sent from the ground.
The action of the control system consists of applying a torque to a
motor shaft clockwise or counterclockwise. The consequence of this
action is to change the angle of the control surfaces. The control loop
sees to it that this consequence of the actual behavior comes to a state
matching the reference state that you transmit from the ground.
The appearance is that the joystick directly controls the elevator and
ailerons. In fact, it seems that the "behavior" of the receiving system
is simply to "respond" to a signal input by creating an angle of the
control surface. Only when we examine the model plane in detail do we
find that the actual behavior of the system is a torque applied to a
motor shaft, and that there is really a control system directly sensing
and controlling a _consequence_ of that torque.
If we looked at the corresponding system on a bigger airplane, we would
find that the potentiometer sensing the control surface angle is mounted
on the shaft that holds the control surface, rather than being packaged
with the driving motor. The reason is that the control system needs to
control the actual angle of the surface, not the angle of the motor
shaft, which can be somewhat different depending on friction, loads, and
elasticity in linkages.
The problem with the standard statistical analysis has nothing to do
with the analysis per se. It's in the assumed organization of the system
that is analyzed. What your analysis would reveal would be a high
correlation between the controlled variable of a control system and the
setting of its reference signal. But a person knowing nothing of how the
system in the aircraft works might well think that the joystick controls
a stimulus which affects something in the aircraft that causes it to
produce a response in the form of a control surface angle. That is
indeed the appearance, but the implied model
joystick --> electronics --> control surface angle
is the wrong concept of how it works. The right model is
Joystick --> electronics <-sensor<--control surface angle
> >
-----> motor -->linkage --->-
The control surface angle seen as a behavioral output in the first model
turns out to be a sensory input in the second model. The behavior in the
first model is taken to be the control surface angle, but in the second
one the behavior is the output torque of a motor. If you calculated the
correlation between the joystick and the motor torque, you would find a
much lower correlation, because the torque varies as extraneous factors
come into play such as wind loads on the control surface (or
disturbances you deliberately apply, as in your example). You could
easily create a situation, by applying various patterns of disturbances,
in which the correlation between joystick and control surface angles was
essentially perfect, and the correlation between motor torque and
control surface angles was essentially zero.
In fact, you might find that for a given control surface angle, the
motor torque could be either positive or negative by large and variable
amounts. So you would have the very strange situation in which a
consequence of behavior correlates with the "stimulus input" far better
than that actual behavior corresponds with it.
Obviously the significance of the statistical analysis depends on the
underlying model.
If you happen to have a functioning model aircraft lying around, you can
try a simple experiment. Put a zero-center ammeter in series with the
power line to the motor, and push on the associated control surface. You
have already noted that the control surface will hardly be budged by
modest disturbances (don't break it!), but the motor driving current
(and hence, the torque) will vary as you apply pushes and pulls. If you
move the joystick with the other hand while you do this, varying the
reference signal, you'll see the control surface moving in the normal
manner -- but if you're pushing and pulling at the same time, you'll see
the motor current varying in a way that hardly correlates at all with
the control surface movements, except by accident.
Actually, this might not work with a real RC servo, because the gear-
down may be so high that friction in the gear train will hold the
control surface in place against disturbances without the motor current
changing. I think Dag Forssell found an RC servo with low enough
internal friction that you could see the effect of disturbances on motor
torque, but I don't know about the ones you use.
I was going to get to this point in a different way, but since it came
up here I think this example illustrates it nicely. Behavior, as action,
_influences but does not determine_ consequences. As a result, when
disturbances tend to change the consequence, the behavior changes to
oppose the disturbance. So you have the behavior changing and the
consequence NOT changing. This, I think, is the ultimate objection to
the concept of consequences selecting behavior when we're talking about
control systems. When disturbances are taken into account, the direct
connection between behavior and its consequences disappears; the
behavior can run through its entire range while the consequence remains
in one state. We can't say that consequences are determined by behavior,
nor can we say that behavior is determined by its consequences. There is
simply no regular relationship between them in either direction when
disturbances are acting.
···
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Isn't it interesting how much I was able to learn about the system
using that outdated methodology described in my text?
You could learn about observable relationships, but the standard
statistical methods can't tell you that your underlying model is right.
I don't believe that the statistical approach would have suggested that
the control surfaces would push back against your attempts to deflect
them.
Anyway, the correlations you would get in this experiment would be in
the high nineties, the kind we're interested in in PCT. I don't know
your opinion on the subject of correlations in the eighties or lower, or
on the subject of using population measures to predict the behavior of
individuals.
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Best,
Bill P.