John F.--
Generally I agree with your conclusions about tracking; the remaining divergences simply have to be worked out. Of course verbal explanations of anything technical leave a lot of room for varied interpretations, whereas mathematics (whether expressed symbolically or as numerical solutions of the - same - differential equations) usually allows only one outcome. What you say in words can be wrong when interpreted one way, but right when interpreted in another way, so the math is really our final recourse to what is really meant.
Consider first the case in which the disturbance is introduced in the form of a moving target. In your terms
error = cursor - target
and in my terms (with signs carefully chosen),
(perceived separation) = target - cursor
error = (intended separation) - (perceived separation)
If the intended or reference separation is zero, then my terms reduce to
error = 0 - (target - cursor) or
error = cursor - target
which is mathematically identical to your case. So it makes no difference whether you consider the tracking error to be defined by the display or by a relationship-perceiving perceptual control system that sets its own internal reference condition for the relationship. Those differences are what I call "ideological," since they are shaped by underlying theories that don't change the observed circumstances or the analysis.
Now consider the case in which the disturbance is introduced at the output of the plant, with the target now stationary at the zero position. First, we have to say that the cursor position in the "pursuit" case above is considered dependent on the plant's output, so everywhere we see "cursor" above we could equally well write
k*(plant output).
Thus the final line in the two cases above could be written with no change of meaning as
error = k*(plant output) - target
This allows us to tie the pursuit and (one of the) compensatory cases together as follows:
When the target is stationary and only the cursor moves, one case of compensatory tracking can be defined as
error = cursor - target
cursor = [k*(plant output) + disturbance]
If the target position is constant at zero, then clearly the error is determined jointly by the plant output and the disturbance. There is, however, no separate indication of the magnitude of the disturbance, as implied in your post by
Again -- the defining attribute of compensatory tracking is that only "error" is
displayed -- there is NO independent display of
the plant response and the disturbance/reference.
Clearly, the subject doing the tracking never sees anything but a cursor and a target on the screen either during pursuit tracking or during this case of compensatory tracking. There is no separate display of the disturbance in the second case.
It is also equally clear that mathematically, the only difference between pursuit tracking and this case of compensatory tracking is the assumed sign of the disturbance.
error = cursor - target
target = 0
cursor = [k*(plant output) + disturbance]
therefore
error = k*(plant output) + disturbance
Compare with the same case for pursuit tracking:
error = k*(plant output) - target
The form is identical to the pursuit tracking case if we say that the disturbance is the negative of the target movements, which is of course an arbitrary choice.
Finally, let's consider the case where the compensatory tracking involves a disturbance D added to the _input_ of the plant along with the controller's output U (I prefer U to C becauise C is handy as a symbol for cursor position). In this case we have
cursor = P(U+D)
where P is the plant's transfer function.
If the transfer function is linear, this is equivalent (by the superposition theorem) to saying
cursor = P(U) + P(D),
and that means that (with the target at zero)
error = P(U) - P(D)
(I hope).
Obviously P(U) is the same as (plant output) in the equations above, and P(D) is equivalent to a different disturbance D' added to the plant output, so we have
error = (plant output) + D'
which we have already seen is mathematically identical to the equation for pursuit tracking (except for a factor k which is part of the definition of P).
The only problematic part of this series of deductions is the assumptiom that
P(U+D) = P(U) + P(D)
which I believe is true of the linear cases we have so far considered. Since the integral of a sum of variables is the sum of the integrals of the variables, I think this step is OK in terms of solutions of the underlying differential equations.
There is more to be said about these three cases, but I'll pause here to see if you agree with the proof that mathematically, pursuit tracking is equivalent to both cases of compensatory tracking. While I have used words to explain the steps, I think that the basic argument above is purely mathematical and stands by itself, so to disprove it all that is needed is to show where the mathematics was done incorrectly, Is there a mistake in the argument?
Best,
Bill P.
Post to which this is a reply reproduced below.
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Bill,
What you have done is simply turned the "compensatory" task into a pursuit case. And of
course your analysis shows exactly what I proved months ago for that
case. Now what Max points out is that since we have two separate sources of
input (error position) and (vehicle velocity) the open-loop frequency response
is no longer a valid estimate of the human transfer function -- That is, it will
be confounded by plant dynamics -- which is exactly what we demonstrated.
In the pursuit case the vehicle position and the disturbance (reference or track)
are displayed independently. So -- it is reasonable that the person could
observe these things independently (so our diagram with plant velocity
being fed back is a reasonable hypothesis for this case).
In the case of compensatory tracking -- by design -- this task only provides
the relative "error" between the target and the vehicle. Whether the disturbance
is added before or after the plant -- the person only sees the joint impact of
her control action and the disturbance. This task was specifically designed because
it is a situation where the open-loop frequency response is a valid measure of
the human transfer function. So, if there are changes in the transfer function -- those
changes can be confidently attributed to changing parameters in the controller (human).
Now, if there is a criticism of McRuer's work it might be that in order to create valid
measurements he created a very "unique" context relative to most natural control tasks
where there are multiple sources of feedback (e.g., the car driver can see her speed
relative to other cars and can see her absolute speed). Thus, natural cases are more
like the pursuit case than like the compensatory case. Now the lesson here is that
you have to be careful when making inferences about the human transfer function
from the open-loop frequency response to
a disturbance on any single loop -- because the response will be confounded by
contributions of the other loops. Thus, for example, converging operations reflecting
the multiple potential sources of information will be necessary.
I think this is a very important lesson -- because much psychological research is
designed on exactly that premise -- that you can infer the "transfer function" of the
black box from manipulations of a single stimulus dimension.
The main technical point is that you are getting the same results for the
compensatory tracking task with disturbance added after the plant as for
pursuit -- because in the way you implement it -- it is isomorphic with the
pursuit case. Thus, it is NOT compensatory tracking at all. You have simply
redrawn the pursuit case.
Again -- the defining attribute of compensatory tracking is that only "error" is
displayed -- there is NOT independent display of
the plant response and the disturbance/reference.
I don't know how to make this more clear.
John