In the attached paper, section 4 - Representations - describes a contemporary mathematical view of models. I don’t know how well it will be received by a PCT audience.
ESF-FAMuller-June2009.pdf (215 KB)
[From Rick Marken (960822.1050)]
Hans Blom (960822) --
A model does not need to capture the form of the function if its precise
form is not important for control.
But the MCT model has to come up with an _approximate_ form; a linear model
is an approximate form. The PCT model doesn't try to come up with even an
approximation to the form of the feedback function; it doesn't need to.
The PCT model linearizes (assumes a linear environment), just like
the MCT model.
Completely and utterly false. The PCT model does NOT assume a linear
environment. The PCT model (and humans) can control with almost
any environmental feedback function in effect, even one that is
non-monotonic (see Powers, Psych Review, 1978).
Me:
A world model is a representation of the _form (including sign and
coefficient values) of the environmental feedback function.
Hans:
A _representation_, yes, in the sense that it must provide a good
fit. Not in the sense that it must capture the "true" environmental
function.
Who ever said the model had to be the "true" form? That's why they
call it a _model_. PCT uses _no model_ (approximate representation) of
the environmental feedback function and it controls in the same way as
humans do. There is no evidence that human control involves "world
models"; none, zilch, nada. Your belief that world models are an
important aspect of human control is simply religious. Halleluja.
Hans Blom (960822b) --
Why do you compare HPCT with neural nets? That is kind of premature, I
think, as long as HPCT misses a mechanism to adjust weights.
The reorganization process, demonstrated in Bill's artificial cerebellum
model, is the PCT approach to adjusting weights.
Hans Blom (960822) --
Model-based control "explains" phenomena that have to do with the
availability of an internal world model (and its accuracy)...
[A lot of useless anecdotal "data" snipped]
Hans, model-based control has not "explained" a single piece of data
that has not been explained by PCT. What we need is an experiment
the produces data that can be handled by a model-based controller and
not by a PCT controller. You were on the verge of describing such an
experiment but then backed away. Why?
Bruce Abbott (960822.1040 EST) --
How the system is set up to respond to disturbance IS its model of the
environment
Oy, vey. Another definition of "model". Look, can't we just agree
that the MCT and PCT models are different? One difference is the
existence of an equation (call it a "world model" or "Jim" or whatever)
in MCT that does _not_ exist in PCT. There is no analog of "Jim" in PCT.
So let's not pretend that the control system must "know" nothing about the
environment and the effects of its own actions on it.
There is no model of the environment (in the sense of an equation that is
designed to approximate the feedback function) in the PCT. In this sense,
the PCT model knows nothing about the environment. The PCT model knows
_only_ the effects of its own actions (it controls a perception of these
effects) and it is able to adjust its parameters (via reorganization) so
that it is able to control the effects of its actions in the context of
whatever (unknown) environment it happens to be in.
As it is quite clear that people and animals are able to do the same
thing under similar circumstances, it would make more sense to examine
the MCT system carefully as one example of how this can be done rather
than to treat it as an "alternative" to PCT that must be knocked down.
That animals "do the same thing under similar circumstances" is not the
kind of data that would motivate me to consider a new control architecture.
The problem with MCT is that it does not explain _anything_ that is not
more simply and clearly explained by PCT. I have nothing against MCT
except that it's being proposed as an alternative or superset or
supplement to PCT for nothing other than religious reasons. I feel like
I'm supposed to say "I believe in MCT; I really do believe. Halleluja".
I only say such things under threat of inquisitorial torture;-)
By the way, Rick, what are Bill's "wonderful models of this
[reorganization] process" of which you speak? What is their principle
of operation?
At the meeting Bill showed me his stick figure arm model that points
at a target (not the "Little Man"; just an arm). The arm starts
"flailing about" like a baby; it slowly tunes itself up so that it's
pointing smartly and accurately. I think Bill calls this the "Artificial
Cerebellum". It works by tuning the transfer functions of the various
control systems. I believe it works in the context of a model of the
real physical environment (loads, accelerations, etc). So the feedback
functions are complex and non-linear. The PCT model adjusts control
parameters in order to produce good control. The resulting parameters
are not a model of the environment; they are just numbers that produce
good control. It is definitely a cool demo because the initial pointing
movements really look "baby-like"; after a few seconds of "learning" the
points looks quite skillful.
Best
Rick
[From Rick Marken (960821.2130)]
Martin Taylor (960821 16:45)
The system has to model the environmental feedback function at least in
respect of its sign.
If you want to call this a "model" go ahead. But it doesn't seem much
like a model to me. For one thing, the sign of the feedback function
can be handled by the system in many ways, by changing the sign of
the perceptual function, comparator function, output function. So if the
environment function is positive there are at least four different ways to
assign signs to these three system functions (+ + -, + - +, - + +, - - -) to
make the net sign of the feedback loop negative. Since the sign of the
feedback function can be "modeled" in so many equivalent ways, there is
really nothing in the system that constitutes a "model" of the sign of the
feedback function.
For another thing, the sign of the feedback function doesn't capture anything
about the _form_ of that function. This is what is modeled in Hans' MCT
system. Actually, Hans' model assumes the form of a feedback function
and tries to find the coefficients of that function. For example, Hans' MCT
system might assume that
i = k.1 o + k.2 o ^2
The MCT model then goes about solving for k.1 and k.2 using a Kalman
filter method (essentially a running multiple regression). In principle, an
MCT system would have to solve for the actual _form_ of the feedback
function, not just the coefficients of a form that is already given. I'm not
a mathematician but I think, in practice, this would mean solving for the
coefficients of polynomials with many terms, and where the variables must
include first and second derivatives.
Anyway, a model of the feedback function (as I understand it) is an
equation the represents the _form_ of that function -- not just the sign
of that function.
Let's say that a control system can be operating in three different
environments, each characterized by a different feedback function. In
environment 1 the feedback function (relating output, o, to input, i) is
i = 10 * o (1)
In environment 2 the feedback function is
i = o^3 (2)
And in environment 3 the feedback function is
i = log (o) (3)
Because the sign of the feedback function is the same in all three cases, the
same control system works in all three environments. The model that works
is basically o = k* (r-p) where p = f(i). As you note, the amplification of
this control system (k) must change if it is to control successfully in all
three environments. But the different values of k that let the model work in
each environment (.1, .001 and .0001, respectively, say) are hardly a model
of the different environmental feedback functions.
According to your definition of "model" the number .001 is a "model" of
feedback function described by equation (2). But even this peculiar
definition of "model" doesn't really work because it is possible for
k to equal .001 when the environmental feedback function has some form
other than (2). The value of the amplification factor tells nothing about
the form of the environmental feedback function -- not even its sign
(see first paragraph).
So the PCT model works (produces the result intended by the setting of r)
without any model of the _form_ of the environmental feedback function.
This is because the output of the PCT model is driven (in a closed loop) by
r-p, not just by r. The MCT model, on the other hand, will not work at
all unless it can get an estimate of the _form_ (and the sign) of the
environmental feedback function. This is because, in the MCT model, output
is driven by r directly -- o = f(r). So the output will produce the intended
result (assuming no disturbances) only if f() (the output function) is the
inverse of the environmental feedback function.
If Hans' MCT model is placed in the three different environments described
above, it will not work (produce the intended results) unless it can determine
the _form_ of the feedback function (equation 1, 2 or 3) that exists in each
environment. The inverse of the environmental function that is used to
generate the outputs of the MCT system is actually the inverse of the
"world model" that is computed by the MCT system. A world model is a
representation of the _form (including sign and coefficient values)
of the environmental feedback function. Such a model is at best
superfluous and at worst a hindrance to a system that controls its
perception (bases its output on r-p); however, it is an absolute
necessity in system (like Hans' MCT system) that controls its output.
So there is clearly no meaninful sense in which a PCT model computes an
implicit or explicit model of the environmental feedback function.
···
-----
Summary for those who didn't follow the quantitative argument: MCT is an
unwieldy model of purpose that works only in environments where there
are no disturbances. It is also the hottest thing going in control
engineering and robotics. Go figure!
Best
Rick
[From Bruce Abbott (960823.1400 EST)]
Bruce Gregory (960823.1145 EDT) --
To call this a model does not seem to add anything helpful to
understanding the to the process. The wall reacts to my
leaning on it by generating an equal and opposite force
(Newton's third law). I suppose that I could say that this
reaction is the wall's model of the environment, but why would
I want to? What does this terminology illuminate that otherwise
might be obscure?
This reaction is not the wall's (implicit) model of the environment; the
wall's stiffness and resiliancy are. Walls are designed artifacts whose
properties have been chosen to match their purpose. The design of the
average gypsum wallboard tells me that the environment in which the board is
expected to function offers no forces greater than a certain rather small
amount. There is a certain "model of the environment" within which the
board is designed to function. To that extent the design of the board tells
us something about its enviornment, at least if it is successfully serving
its purpose.
Bill Powers (960822.2000 MDT) --
That sense of modeling, however, is not one I would use myself,
especially since we also use the same term to mean formal analogy. Using
the same word to designate two such widely different meanings is not, in
my view, conducive to clear thinking.
Bruce asks how this sense of "modeling" illuminates, and Bill asserts that
it obscures. I can't speak for anyone else, but in my case I brought it up
in the context of an explicit comparison between MPC and PCT with respect to
each system's "knowledge" about the world -- the extent to which something
about the environment must be taken into account in the system's design if
it is to function properly in that environment. In that context I thought
the comparison of these two senses of the term "model" was useful, because
it highlighted both the similarities and differences between the PCT and MCT
systems concerning how they accommodate to different enviornments.
So, taking a page from Rick, I don't care if you call it an implicit model
or "Fred." It is still true that the system must be tuned to reflect the
characteristics of the environment in which it must function. MCT goes
beyond this designing/tuning by developing (via experience) an explicit
model of the composite source of variation acting on the variable to be
controlled, and acting on the values generated by this model.
Well, I hate to cast the harsh light of empiricism on all this deep
speculation about the number of "world models" that can dance on the
head of a PCT hierarchy but I set up a simple little experiment last
night to help my scholastically challenged brain understand where "world
models" fit into the PCT hierarchy.
Well, Rick, I hate to cast the harsh light of logic on your efforts, but all
you have managed to demonstrate with your computer model is that control
systems are not especially sensitive to certain permutations of the feedback
function. The specific form of this function isn't all that important so
long as the sign of the feedback does not change. Try randomly varying the
delay between action and its effect on c in a similar way, and/or the
constant relating amount of mouse movement to amount of cursor movement
(change it fairly rapidly and over a wide range) and see whether you control
well.
You seem to be trying to prove that a control system does not have to be
properly tuned before it controls well: one control system with one set of
parameter values fits all. If this is true, why do we need hill-climbing
routines to find the best values? Why worry about gain and loop delay and
all that nonsense? The environment in which the system must function is
irrelevant; the system need not take account of it in any way. Or so you
seem to be arguing.
And I am mystified why you think your "variable feedback function" test has
anything to do with testing the adequacy of the MCT system. You are
basically trying to show that a system based on developing a world model
will not work if there is no stable world model. An equally fair test
between PCT and MCT would be to allow both models to run for awhile (with
stable environment functions and regular disturbance waveforms [e.g., sine
wave]) and then remove the perceptual input from both models for a few dozen
iterations. Guess who wins that one?
Also, why are you varying the feedback function? Because it's called the
environment function? And the World Model is supposed to be a model of the
environment? Maybe I'm wrong, but I thought the World Model modeles any
regular variation in disturbance (including disturbances produced by
feedback action).
Regards,
Bruce
[From Rick Marken (960823.1320)]
Bruce Abbott (960822.1040 EST)
How the system is set up to respond to disturbance IS its model of the
environment
Me, betraying traces of ethnic pride;-)
Oy, vey. Another definition of "model".
Bruce Abbott (960823.1030 EST) --
It's not "another definition of 'model,'" it's the same one everyone else
(e.g., Leach, Taylor) has been using;
But it's not the one that the clear-thinking troika of Gregory, Marken and
Powers has been using;-)
Me:
The problem with MCT is that it does not explain _anything_ that is not
more simply and clearly explained by PCT.
Bruce:
That's a bald assertion with no supporting evidence.
The _anything_ I am talking about here is the data from any experiment
other than one in the corpus of PCT control experiments, the results of
which we have successfully modeled using PCT. Actually, I haven't seen
the behavior of an MCT system compared to the behavior of subjects in
any of our PCT experiments so I don't really know whether or not MCT can
explain anything at all.
PCT (even HPCT) does a lot of hand-waving (to use a currently popular term)
about switching to internally-generated perceptions (memory) but thus far
there are no working models that actually implement this suggestion...MCT
has an explicit mechanism and a working demonstration of the prinicple.
So currently it's your religious faith that HPCT _will_ be able to handle
such data against a working computer model of MCT. How about developing
the PCT equivalent so we can evaluate it?
Wouldn't it be better to first have the data that seems to require this
kind of model? It seems to me that this "controlling without perception"
thing is very likely a non-phenomenon invented by the believers in output
generation models of behavior to justify their model. I have never seen
any convincing experimental evidence that organisms can continue to
control when they can no longer perceive. When the lights go out, for
example, people don't stop perceiving; any behavior involves the control
of MANY perceptual variables. When you eliminate one or two of these
variables (by turning out the lights or severing afferent neurons) you
don't get rid of all them all. I think it's very likely that the attempt
to build a control system that controls for some time without perception
is an attempt to solve a problem that probably doesn't exist. Let's see
the data first.
Bruce Abbott (960823.1400 EST) --
the [conrol] system must be tuned to reflect the characteristics of the
environment in which it must function.
Yes. And I'd prefer to call the tuned state of the system a "tuned state"
rather than a "model" of the environment to which the system is tuned.
That's because the tuned state is not necessarily a model (analog) of
the environment.
Well, Rick, I hate to cast the harsh light of logic on your efforts, but
all you have managed to demonstrate with your computer model is that
control systems are not especially sensitive to certain permutations of
the feedback function.
That's all I aimed to show; I have very modest goals;-)
You seem to be trying to prove that a control system does not have to be
properly tuned before it controls well:
No. I was trying to show that the parameters of a control system do not
need to be an implicit model (analog) of the feedback function in order
for a control system to control. If the control parameters are constant
while the feedback function changes, then these parameters cannot be
considered a model (analog) of anything about the feedback function. The
fact that the control system can still maintain pretty good control shows
that modeling (in any sense) of the environmental feedback function is not
a necessary characteristic of control of input.
And I am mystified why you think your "variable feedback function" test
has anything to do with testing the adequacy of the MCT system.
I didn't think it would test the "adequacy" of an MCT system. It just
seemed like a situation where an MCT system might make different (and
better) predictions of the behavior of a human subject than the PCT system.
I'd have to run both models to see if this is true. If both models make
about the same predictions in both situations then this variable feedback
function test is not a good basis for comparison.
An equally fair test between PCT and MCT would be to allow both models to
run for awhile (with stable environment functions and regular disturbance
waveforms [e.g., sine wave]) and then remove the perceptual input from
both models for a few dozen iterations. Guess who wins that one?
Wouldn't the model that wins be the one that does the best job of
accounting for data from organisms controlling perceptual variables under
different conditions?
Phenomena phirst, remember;-)
Best
Rick
<[Bill Leach (960824.0044)]
From Bruce Abbott (960823.1400 EST)]
... between PCT and MCT would be to allow both models to run for
awhile (with stable environment functions and regular disturbance >waveforms [e.g., sine wave]) and then remove the perceptual input from >both models for a few dozen iterations. Guess who wins that one?
Depends upon your view point. With respect to how well the model's
"quality of control" matches the human's "quality of control", the PCT
model wins (the PCT model doesn't control worth a sh*t but then neither
does the human.
With regard to the "nature" of the observed output behaviour, the MCT
based model and the human behave similarly -- that is both show a
continuing "output pattern" (though I believe that if the experiment
could be performed where the human tactile perception of such things as
mouse position could be removed along with the visual field the human's
performance would be even more like the PCT model.
While this is conjecture at this point, I also suspect that if the PCT
model were equipped with similar tactile perceptions it to would control
very closely in the "nature" of the observed output as well as in
"quality" of control.
This may be a bit too strong but... the idea of controlling current
perception without the presence of a current perception is a myth.
···
--
bill leach
b.leach@worldnet.att.net
ars KB7LX