[From Bill Powers (950529.1230 MDT)]
Hans Blom (950529) --
What is new in my demo is, in comparison to the "standard"
PCT-controllers:
1) an internal model of the regularities of "the world out
there" can be built, so that for instance a
square wave pattern reference level can be
accurately tracked with essentially no error and
zero response delay;
If you're speaking of the model you presented, I don't think this
is true. The optimum fit of world-model to external system would
result in control having exactly the same accuracy and the same
delay as direct control without the world-model. I think you had
better check this claim against your model run with a faster
time-scale.
3) this internal model may become so trustworthy that it is
more relied on than the perceptions; this, as
well as 2) above, was my counter-example where
the generalization "control of per- ception"
breaks down and must be replaced by "control of (by?)
internal representations".
You're making an unwarranted assumption here, that the world-
model is more trustworthy than the perceived state of the actual
system. I believe that this assumption, whether in this
particular model or in real life, is mistaken more often then
correct.
When you proposed the demonstration refuting control of
perception, you used a disturbance consisting of high-frequency
random noise. It is probably often true that perceptions
containing a large high-frequency noise component are more useful
when that component is removed by a low-pass filter. However, the
controlling system (without the aid of the omniscient engineer)
has no way of knowing how much of that noise in y is being
generated by the perceptual function itself, and how much by
actual variations in the controlled variable, xt. If any of the
"noise" in y is actually a result of high-frequency variations in
xt, then removing the noise from y will leave those high-
frequency variations uncorrected. Thus there is no guarantee that
removing the "noise" from the signal y will provide a more
reliable perception of xt; doing so might result in an incorrect
perception of xt, and less reliable control of it.
In my post to which you haven't replied yet, I discussed the
effect of adding a disturbance to y which had a spectrum lying
within the bandwidth of the normal variations in xt. The control
process (whether real-time or modeled) will necessarily treat
these variations in y as if they were real variations in xt.
There is no way to distinguish the variations in y due to the
disturbance from variations due to xt. The control process,
either model-based or real-time, will correct for the low-
frequency variations in y by causing xt to vary in the opposite
direction. Thus there is no way that x in the model can actually
represent the behavior of xt in the real world. The disturbance
of y is actually causing the real xt to vary. The result is that
y, but not xt, is following the changes in xopt.
Nothing of this is new, of course. 1) Tracking a regular
("predictable") pattern with approximately zero delay has
frequently been encountered in human operator response speed
studies.
You will find a demonstration of this effect in the 1960 paper by
myself, Clark, and MacFarland.
This can be be explained, as I said, using a hierarchical model.
The first level is an ordinary control system controlling the
cursor used for tracking. The second level perceives the
synchronization of the cursor and target movements (the principle
of the phase-locked loop will serve as the perceptual function),
and the output function consists of a variable-frequency
variable-amplitude oscillator which varies the reference signal
for the cursor-position control system. This will achieve
approximately zero average delay, without needing the kind of
world-model in your program.
If the target moves in a sine-wave, there is an even simpler way
of achieving zero delay, even with a single-level system. Simply
add some first derivative to the perceptual signal representing
target-minus-cursor. This can actually result in the cursor
movements occurring slightly _ahead_ of the target movements -- a
negative delay!
We would use that model, of course, only if the real human
behavior also showed a negative delay.
···
----------------------------------------
I believe the net result is that hidden inside this adaptive
model with
a Kalman filter is an ordinary control system that works through
the
external world in the usual way once adaptation is complete.
That depends upon what you consider an "ordinary" control
system. In the demo, control works as follows. When
adaptation is complete, a=at, b=bt, c=ct, and pax=0. The
"world" acts according to
xt [k+1] = ct + at * xt [k] + bt * u [k] + system noise
The model offers the prediction
x [k+1] = ct + at * x [k] + bt * u [k]
You are leaving out the steps in the Kalman filter that I
mentioned. The value of x is always being affected by x - y, even
after convergence is complete and the coefficients are no longer
changing. It is through this path that "ordinary" control is
being achieved. If you forced all the adjustable constants in the
world-model to be the same as those in the real system, and
constant, you would still get control through this "parasitic"
path. So the current value of external disturbances is always
being passed through the Kalman filter to the value of x. This
explains how it is that your world-model can reflect the state of
an arbitrary external disturbance, as it does.
In your solution of the system equations, you leave out this
direct path through the Kalman filter, so your analysis is
incorrect.
You say:
The system noise, being zero on average, is disregarded.
It is only through disregarding the system noise that you can
come to the conclusion that the model-based system is accurately
representing the real system.
But this is valid only for the high-frequency components of the
system noise. If you were to do a frequency analysis of the
behavior of the model, you would find that the low-frequency
components are being reduced by the loop gain of the system, and
xt is actually being made to vary at these frequencies in a way
that x does not vary. The only information about this noise that
is used in the Kalman filter is the variance and mean amplitude
of the disturbance; this can't account for the systematic
resistance to low-frequency variations.
-----------------------------------
If you define "control" as equivalent to "control of
perceptions" or, equivalently, to resistance to disturbances
(which can of course only be resisted if they are
observable, then my demo does not "control".
Good. I'm glad we agree on that. However, when you say that
disturbances can only be resisted if they are observable, I am
unsure of your meaning. One possible meaning would be correct,
the other incorrect.
Let's simplify the system equation slightly:
xt = a*xt + b*u + d
The correct meaning would be that if d affects xt by an
observable amount, then the system can control xt against the
effects of d.
The incorrect meaning would be that in order for the system to
control xt against the effects of d, it must be able to observe d
directly.
A negative feedback control system does not require any direct
information about d; all it needs to observe is the state of xt.
Do you agree with that?
----------------------------------
It also controls when there are no perceptions or when
perceptions are corrupted by random or "arbitrary"
variations. So, given my "loose" definition of the word
"control" (which I take from the control literature), can we
agree that we indeed have found a counter-example of
"control of perceptions" that functions well under some
(albeit maybe severely restricted 
circumstances?
We have found a definition of control which does NOT fit the PCT
definition. When you say this mode "functions well" you mean that
it meets your criteria for control, which do not include
resistance to ongoing arbitrary disturbances. You under your
definition of control, your system functions well, while under
mine it does not (and cannot).
----------------------------------------------
However, before we put too much emphasis on this ability to
operate
blind we should ask whether it has an important role in
explaining >human behavior (which is, after all, the primary
purpose of PCT).
Again, it is not only the ability to operate blind that I
find interesting. It is the ability to somehow store
knowledge about the world that we can use in our control
tasks IN ADDITION TO our direct perceptions.
Yes, that is an interesting subject. But it is of interest in
studies of human behavior only if we can show that there are real
behaviors that can't be accounted for economically in any simpler
way.
-----------------------------------------------
There is a tendency in computer-science circles to design a
system that
operates in some clever way, and then look around for behaviors
that >can be interpreted (and often overinterpreted) as fitting
it. This is
the wrong way around. We must start with observations of real
human
behavior in circumstances where we can measure accurately what
is
going on, and then search for a model, clever or simple, that
will
reproduce the behavior under all reasonable variations in
circumstances that we can think of.
This tendency exists not only in computer-science circles
(which, by the way, I do not consider myself to belong to),
but in all of science. There simply is no other way.
I think you're being led astray by too much generalization. There
are different levels of observation, with lower levels preceding
higher. We can measure accurately how a handle, cursor, and
target behave in a given experiment without offering a theory of
why they behave as they do. In past discussions on the net, we
have tried out various proposed models -- theories -- to explain
the observations; PCT, of course, but also two or three versions
of S-R theory.
I have accepted by now that "observations" are always model-
driven: there is so very, very much to observe that we must
set limits to what we WILL observe, what we will describe
(in language or in formula), what we deem important and what
not.
Of course, but that doesn't mean that there are no observations
to be made. If we are doing the observations correctly, however,
we will be making them at a level of observation where theories
about those observations are highly unlikely to affect what we
observe.
And what we can observe is restricted, as you note, by what
we can accurately measure, i.e. in terms of what is already
known to us. This embeds observations within a culture: the
type of observations that we can make will always depend on
whichever instruments we have and on our pre-existing
knowledge of what is observation-worthy.
Well, yeah, I suppose so, but so what? We observe what interests
us, what we think of observing, and what we can observe. Is that
suppose to discourage us from trying to explain observations? If
we were molluscs, we would have a whole different set of
observations to deal with, but we aren't molluscs.
Science, whether we like it or not, is mostly hypothesis- or
theorem- driven: someone invents a theory about how things
relate (in terms of measurables and ought-to-be measurables)
and then must proceed to demonstrate that this particular
theory works well. Theoretical physics is, I think, much
more a driving force than observational physics.
It can get to be that way, I agree. I think that's one of the
shortcomings of theoretical physics. I think that science
progresses best when we start with experience and let the
explanations follow. When you start with explanations that seem
esthetically satisfying and then try to prove that they are
right, I think that accuracy of observation suffers.
---------------------------------
Just to make sure one important point is not lost among all the
other side-issues that come up:
I believe that your Kalman-filter program achieves resistance to
external disturbances because there is an unforseen connection
between the perceptual input xt and the comparator (where xopt
comes into play). This unforseen connection gives the true value
of xt a direct effect on the signal entering the comparator, and
thus completes a negative feedback loop that is largely
independent of the adaptation processes. Whether this is
characteristic of Kalman filters in general or only of this
implementation, I do not know.
-----------------------------------------------------------------
------
Best,
Bill P.