Controlled variables

[Bill_Powers1]

[From Bill Powers (981016.0256 MDT)]

Bruce Abbott (981015.1840 EST)--

Rick:

That

is, they seem to have no awareness of the existence of _controlled
variables_.

Bruce A:

Come off it, Rick. You can't understand how control systems operate without
knowing that.

Would that that were true. I have never seen a text on control systems that
taught the idea that control systems control perceptual signals first, and
external variables only incidentally. In fact, the idea of controlling
inputs is not part of engineering control theory, as far as I know. In PCT,
as I hope you know, Bruce, controlled variables are defined by perceptual
input functions and have no _a priori_ definitions independent of perception.

To Rick, I must say that engineers certainly do think in terms of
controlled variables, but the variables that are controlled are not
specifically linked to the control system's internal representations of
them. They are linked to the _user's_ or _observer's_ perceptions, not the
control system's. This can work because the engineer designing the system
knows what is actually being controlled; the control system does not have
to contain any specific or explicit perception of it. If the feedback
signal is not a direct representation of the controlled quantity (and very
often it is not), filters can be added elsewhere in the system to
compensate. One common place to insert a filter is between the reference
"input" and the comparator, so the signal actually reaching the comparator
is _not_ representative of the desired state of the "output." It is
modified so that when the distorted feedback signal matches it, the
controlled variable in the environment will match the reference input prior
to the filter rather than the input to the comparator. I have always
thought this rather a strange way to design a control system, but maybe I'm
the strange one for not seeing why it's done this way. My uncharitable
guess is that the design engineer had a rather confused conception of how
control works and just started the design in the middle, compensating for
mistakes as he went.

The PCT model evolved as I tried to understand how engineers' control
systems were designed, and worked out some general principles that seemed
to apply across the board. The idea of control of input came from a book on
analog computing that pointed out the way an operational amplifier kept its
own negative input terminal at the same voltage as the positive input
terminal, through the feedback connection. The idea of controlling
_perceptual_ representations arose from seeing how sensors and signals from
sensors actually entered into the operation of real control systems, and
how malfunctions in sensors (such as unwanted thermal sensitivity) altered
the performance of control systems.

Early in the development of control theory, control systems were designed
in a simple and straightforward way, with the controlled variable being
represented by a straightforward sensor signal, and the reference signal
being an actual example of the desired value of the sensor signal. The
Honeywell strip-chart recorder was designed this way. But I think that too
much mathematics got into the act. From the mathematical point of view, it
makes no difference whether you use a distorted feedback signal and then
compensate for the distortion by altering the reference input (or some
other part of the system) in the proper way -- the way that the external
observer or designer could see was proper. There was no idea of making the
design into a model of human behavior; all that mattered was getting the
right objective variable under control.

This is, I think, why engineering control theory did not immediately become
a new model for psychologists to use. The designs were not set up for
correpondence with the physical parts of human control systems -- at least
by the time cybernetics came around. The PCT model is not just an
"adaptation" of the engineering model: it's a _rationalized_ version that
makes the functions appropriate to the human system, and shows clearly how
the parts work together to achieve control. In many engineering designs
that is not at all obvious, even to other engineers.

Best,

Bill P.

[Bruce_Gregory]

[From Bruce Gregory 9981016.0950 EDT)]

Bill Powers (981016.0256 MDT)

The PCT model is not just an
"adaptation" of the engineering model: it's a _rationalized_ version that
makes the functions appropriate to the human system, and shows clearly how
the parts work together to achieve control. In many engineering designs
that is not at all obvious, even to other engineers.

Very clear and informative. Thanks, Bill.

Bruce Gregory

[Jeff_Vancouver2]

[from Jeff Vancouver 981016.0930]

[From Rick Marken (981015.1812)] speaking to Bruce A.

1. Do you agree that controlled variables (PCT sense) are
real (observable) phenomena?

If you agree that controlled variables are observable variables
in the environment then you can be a useful informant because
you know what I know -- that there are controlled variables to
be observed. Then you can tell me what I really want to know:

2.... what are these variables called in conventional psychology?
In particular, what are these variables called in textbooks on
behavioral research (such as yours)?

Sometimes they are called behaviors (e.g., Carver & Scheier), sometimes
self-concepts (e.g., Steele).

Rick, if you want to see some psychologists who talk about control
(although they may or may not use that word, they certainly do not say
controlled variables - too easy to confused with experimental design ;))
check out some of the cites in our Goal Construct paper. In particular, I
would be interested in your reaction to Steele's work on self-affirmation.

However, we have been down this path before. You can always find some
error in the work of the others. That error has you throw out the baby
with the bath water. The most common error is the one regarding the
confusion of controlling the results of behavior (often simply referred to
as behavior - as Bill just noted) and the perception. In other words, it
is not always clear that they are talking about qi because it sounds like
qo, but it should really be p.

I agree that this is sloppy, but sloppy is not necessarily wrong. Further,
I find I do not understand you on this point. Is CV = qi, p or is it some
other entity? I realize with this question I am going to be sent to the
back of the class, but there you have it.

I think that one very good point that you and Bill make is that
conventional psychology's methods are weak given the nature of the
phenomenon they are interested in. But there is a big difference between
saying "you do not understand" and "you are not using very good methods
given the theory you are interested in testing." It is this latter stance
that I take.

Finally, you said in a recent post that 8 years ago you said start with
phenomenon. Yet, you and particularly, Bill are very reluctant to tackle
phenomena of interest to psychologists. I think you are reluctant because
PCT quickly becomes very complex when trying to explain these phenomena
(e.g., attention switching). You are treading cautiously. This is
understandable and I do not begrudge you. However, to the outsider, that
complexity is seen as a real problem. My approach, and that of many
psychologists you all trash, is to show how, if we make some simplifying
assumptions, control theory can relatively simply help us understand some
interesting phenomenon. One of the first simplifying assumptions is to
equate qi with p. The problem with that simplification is that it makes
moot the final, crucial aspect of the TEST - blocking the sensor. Hence,
the methodological shortcoming arises. Further, and back to the sloppiness
idea, the apologists are not up front about their simplifying. It is on
this front we should speak to the apologists in psychology (i.e., you can
use these simplifying assumptions as long as you are clear you are doing
that and you understand or explain the implications of those
simplifications).

Of course, there is also no doubt that the apologists do not realize they
are making the simplifying assumptions. Otherwise, they should really be
chastised for their sloppiness. But I classify the ignorant use of
simplifying assumptions as different than complete lack of understanding.
Hence, I classify 20% of psychologists as partly understanding control. I
also sympathize with students of control theory who become exposed to it
via the simplified versions, which is the usually way as they are much more
widely read. I am one of them. I find saying "your teachers were close,
but x, y, and z" is better than "your teachers were idiots who do not
understand a thing about control."

Why do I keep getting sucked into this debate.

Sincerely,

Jeff

[Oded_Maler6]

[From Oded Maler (981016)]

(Bruce Gregory 9981016.0950 EDT)

Bill Powers (981016.0256 MDT)
>
>The PCT model is not just an
> "adaptation" of the engineering model: it's a _rationalized_ version that
> makes the functions appropriate to the human system, and shows clearly how
> the parts work together to achieve control. In many engineering designs
> that is not at all obvious, even to other engineers.

Very clear and informative. Thanks, Bill.

My opinion too!

--Oded

[R_Marken]

[From Rick Marken (981016.0945)]

Me:

... what are [controlled] variables called in conventional
psychology?

Jeff Vancouver (981016.0930) --

Sometimes they are called behaviors (e.g., Carver & Scheier),
sometimes self-concepts (e.g., Steele).

Ok. Thanks. So whenever the words "behavior" of "self-concept"
are used they are talking about controlled variables? If so,
then when they study "behavior" and "self concept" they should
be looking to see whether these variables are protected from
disturbance by the actions of the controller. Perhaps I can
see this when I read Steele's work on self-affirmation. Could
you send me the paper; it's hard for me to get to a library that
carries this stuff.

You can always find some error in the work of the others.

I'm not looking for errors; I'm just looking for evidence
that people have been testing for controlled variables.

Finally, you said in a recent post that 8 years ago you said
start with phenomenon. Yet, you and particularly, Bill are very
reluctant to tackle phenomena of interest to psychologists.
I think you are reluctant because PCT quickly becomes very
complex when trying to explain these phenomena (e.g., attention
switching).

I am not "reluctant" and I'm sure Bill isn't either. We've been
trying for years to find "phenomena of interest to psychologists"
that could be tackled from a PCT perspective. In order to
tackle a phenomenon from a PCT perspective you have to have some
data relevant to the variables that the organism might be
controlling. Bill found some relevant data in the "shock avoidance"
literature and he developed and published a model of "shock
avoidance" (it was actually a model of control of shock rate) in
1971 (LCS p.47). Given what we now know about response rates and
schedules, that model, though it worked beautifully, was probably
not exactly right. But it was a model of a phenomenon "of interest
to psychologists" and it made zero impression on the people who were
purportedly interested in this phenomenon. At least, there was
not a big rush to do further test's of Bill's model.

I have now found some relevant data in the "catching baseballs"
literature and I'm going to try to publish my model of catching
to show how a control of input model can handle this phenomenon.
But even in this case I need data I don't have; and I'm not in
a position where I can do the studies that should be done to test
my model properly. So we'll see whether anyone runs with my model
if I can get it published.

Muy point is that it's not easy to find "phenomena of interest
to psychologists" that can be tackled by PCT; and the reason it's
hard to find such studies is because very few make anything
close to the correct observations -- observations that tell us
what the organism might be controlling.

Why do I keep getting sucked into this debate.

Because it's important to you. And it should be because research
on controlled variables is central to control theory.

Best

Rick

···

--
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken

[Marc_Abrams5]

From [ Marc Abrams (981016.1312) ]

[from Jeff Vancouver 981016.0930]

Great Post and I couldn't have said it better myself
althougth I have tried )

Why do I keep getting sucked into this debate

Because it matters, and I for one am thankful that it does.

Marc

[Bill_Powers1]

[From Bill Powers (981016.0952 MDT)]

Jeff Vancouver 981016.0930--

Further,
I find I do not understand you on this point. Is CV = qi, p or is it some
other entity? I realize with this question I am going to be sent to the
back of the class, but there you have it.

CV stands for "controlled variable." It is the variable that an external
observer would see being controlled. It's normally considered part of the
physical world -- that is, the world as a physicist would see it. But not
always. A person can control things that nobody else can see, like the
pleasingness of a flower arrangement.

What we hope, when we carry out the Test, is that the observer's perception
will covary exactly with the controller's perception. Both the observer and
the controller then will attribute reality to their perceptions, and can
assume that there is something in the world between them that they call the
"Controlled Variable" that corresponds to their perceptions. Each feels
that his perception has been "verified" by the other. Both will see the CV
being controlled. The more people agree that a given CV exists, and the
more of them agree on its behavior, the more confident they are that they
are talking about something in the objective, real world.

That is a mistake, for theoretical if not practical purposes. What
agreement really shows is that the people who agree have similar perceptual
input functions. The fact that they all experience the same thing shows
that (most probably) they are applying similar input functions to a common
reality. But since nobody knows how those input functions are organized,
there is no way to say what the underlying reality actually is. The best we
can hope for is consensus. We agree on a physical model, and then report
CVs's in terms of it.

The CV is therefore an imagined aspect of the common reality. It is
(theoretically) known to observer and controller only in the form of a
perceptual signal, p. The observer and controller both assume that the
experience they are calling the CV exists outside of them. PCT tells them
that it is not actually a physical variable outside them that they are
experiencing, but a perceptual signal, p, inside them. That is a
theoretical statement; it does not make p _look_ as if it's inside if it
didn't look that way before. The perceptual signal goes right on looking
like part of the external world. It looks like a CV in the external world.

In order to understand PCT -- before anything else about it can be
correctly understood -- you have to be willing to relabel all your
experiences as perceptions. No matter what it is that you're experiencing,
from a pretty sunset to the feeling of effort when you lift something heavy
to a weighty thought, it must be understood as a report from your
perceptual input functions (at some level), a report that is really a
collection of neural signals.

If you understand that, then even if you talk about CV's in the outside
world as if they were really out there, you will understand that the CV you
see depends on the perceptual function you're looking through, and it is
really a neural signal standing for something else you can't know about
directly. From what we know about the nervous system and brain, there is
really no justification for claiming that we know what's really outside.

One last thing. Just as we use a diagram of one representative control
system to stand for a whole array of them, we often use a single symbol to
stand for a whole collection of variables. For example, "the" disturbance
is shown as a single variable, but it stands for the sum of ALL influences
independent of the system's own output which, each acting through a
different set of properties of the world, contribute to the state of the
input quantity. And "the" input quantity qi stands for the set of all
physical input variables that affect the state of a perceptual signal.
"The" environmental feedback function is the set of all properties of the
world that link all the outputs qo to all the inputs qi. That's outputs and
inputs with an s.

And finally, the CV is some function of the set of input quantities that
contribute to the state of a controlled perceptual signal. What function?
The same function, if you've defined it right, as the actual perceptual
input function. If the perceptual input function takes the weighted sum of
two input quantities qi1 and qi2, so

p = a*qi1 + b*qi2,

then the CV is computed just as the perceptual signal is:

CV = u*qi1 + v*qi2.

The different weights, u and v instead of a and b, reflect the fact that
the CV is measured in physical units, while the perception is measured in
Nervous System Units, NSU.

It never seems to be quite the time to expand the basic model into all its
underlying complications (even if they aren't all that complicated). When
one is doomed by his sins to teach PCT 101 forever to a freshman class that
never graduates and never gets tired of arguing, the hoped-for day when the
next step can be taken is forever postponed.

Let's turn to a discussion of THERMO3. Does anyone have any comments or
insights to communicate? I don't care how obvious they are -- what can be
learned from this model? For example, what is the CV, and what is p? What
is qo? What is qi?

Best,

Bill P.

[Jeff_Vancouver2]

[from Jeff Vancouver 981016.1354 EST]

[From Rick Marken (981016.0945)]

... what are [controlled] variables called in conventional
psychology?

Jeff Vancouver (981016.0930) --

Sometimes they are called behaviors (e.g., Carver & Scheier),
sometimes self-concepts (e.g., Steele).

Ok. Thanks. So whenever the words "behavior" of "self-concept"
are used they are talking about controlled variables?

No, probably only about 20% of the time

If so,
then when they study "behavior" and "self concept" they should
be looking to see whether these variables are protected from
disturbance by the actions of the controller. Perhaps I can
see this when I read Steele's work on self-affirmation. Could
you send me the paper; it's hard for me to get to a library that
carries this stuff.

I will look for my copy, but look at what we say about it in the Goal
Construct paper, as we perfectly reflect the original work

You can always find some error in the work of the others.

I'm not looking for errors; I'm just looking for evidence
that people have been testing for controlled variables.

Ah, you are confounding theory with method. As I said, the methodology
does not match the theory they themselves claim to be using. (I've been
there, so perhaps I am more forgiving).

Finally, you said in a recent post that 8 years ago you said
start with phenomenon. Yet, you and particularly, Bill are very
reluctant to tackle phenomena of interest to psychologists.
I think you are reluctant because PCT quickly becomes very
complex when trying to explain these phenomena (e.g., attention
switching).

I am not "reluctant" and I'm sure Bill isn't either.

Must I dredge up the posts where Bill P. rants against talking about things
psychologist talk about?

But the rest of your post reminds me of the other reason for your reticence
- having been ignored for so long.

Muy point is that it's not easy to find "phenomena of interest
to psychologists" that can be tackled by PCT; and the reason it's
hard to find such studies is because very few make anything
close to the correct observations -- observations that tell us
what the organism might be controlling.

If you could use control theory models to help understand what could be
done to reduce violence and conflict, increase learning, etc., that would
make a big difference. And guess what? That is exactly what several on
this net are doing, and sometimes they get help from you and Bill. But
these are difficult problems that do not provide clean tests of the theory
even if we are able to construct effective interventions. What this means
to me is that part of what we are witnessing is a classic conflict between
the basic researcher and the applied researcher. The conflict is much
larger than PCT and psychologists. If we recognize that, we might try to
reduce its impact.

Interestingly, I have found that I have had no luck converting I/O
psychologists to PCT, particularly with the details. On the other hand, I
have made in-roads with some more basic researchers.

Why do I keep getting sucked into this debate.

Because it's important to you. And it should be because research
on controlled variables is central to control theory.

Yes, and the sound you hear is the sound of time blowing by.

Sincerely,

Jeff

[Jeff_Vancouver2]

[from Jeff Vancouver 981016.1430 EST]

Thanks for the post, it helps. But by all means . . .

[From Bill Powers (981016.0952 MDT)]

Let's turn to a discussion of THERMO3. Does anyone have any comments or
insights to communicate? I don't care how obvious they are -- what can be
learned from this model? For example, what is the CV, and what is p? What
is qo? What is qi?

I am going to assume the focal system is the thermostat/furnace system and
not some outside observer.

CV=inside temperature
p = inside temperature*input sensitivity (observable by graphing
"perceptual signal")
qo = error signal*Output sensitivity
qi = inside temperature

In this simple example, CV=p=qi with the exception that p has a weighting
factor (that only changes of the units, not the value).

Sincerely,

Jeff

[Bruce_Gregory]

[From Bruce Gregory (981016.1656 EDT)]

Jeff Vancouver 981016.1430 EST

>

I am going to assume the focal system is the thermostat/furnace system and
not some outside observer.

CV=inside temperature
p = inside temperature*input sensitivity (observable by graphing
"perceptual signal")
qo = error signal*Output sensitivity
qi = inside temperature

In this simple example, CV=p=qi with the exception that p has a weighting
factor (that only changes of the units, not the value).

I'll go along with that, with the small exception of the delay time on the
input side.

Bruce Gregory

[Abbott_Bruce]

[From Bruce Abbott (981016.1630 EST)]

Rick Marken (981016.0945) to Jeff Vancouver:

Perhaps I can
see this when I read Steele's work on self-affirmation. Could
you send me the paper; it's hard for me to get to a library that
carries this stuff.

Isn't there an excellent library just down the street from you -- at
U.C.L.A.? I'll bet they carry it.

Just a suggestion.

Regards,

Bruce

[Bill_Powers1]

[From Bill Powers (981017.0706 MDT)]

Jeff Vancouver 981016.1430 EST--

CV=inside temperature
p = inside temperature*input sensitivity (observable by graphing
"perceptual signal")
qo = error signal*Output sensitivity
qi = inside temperature

In this simple example, CV=p=qi with the exception that p has a weighting
factor (that only changes of the units, not the value).

Excellent.

Now, everybody, what is the effect of the delay time? Try a delay of 0.02
hour, then 0.2 hour. Note that the time step is 0.01 hour, so don't use
delays of less than 0.02 hour unless you reduce the time step (Model ...
Time Bounds).

Why is some delay time a good thing for this kind of control system?

Best,

Bill P.

[Tom_BOURBON3]

From Tom Bourbon [931223.1006]

[From Rick Marken (931222.2230)]

Martin Taylor (931222 1800) --

Rick:

The main goal of PCT is to discover
WHAT variables living systems control and HOW they control
them.

Martin:

Did you notice that pull-only control systems work differently?

Rick:

Different than what? They work like control systems and control their
perceptions, don't they?

Tom, now:

I second Rick's questions, Martin. Different than what? And, don't they?

Martin:

In a
structure of pull-only systems, you can't make unique determinations
of what variables living systems control.

Your comments about this in your pull-pull post were wrong, I'm
afraid. You CAN make unique determinations of what variables are
being controlled in the structure of pull-only systems presented
in your post. In that post you derive the following equation

[here Rick quoted and elaborated on Martin's equations, then developed the
argument that by using The Test an observer can determine if a variable is
controlled by a pull-only system].

Tom, now:

Martin, to Rick's analysis I would add the following brief account of a
pursuit tracking task performed by a team of six pull-only control systems,
with the results modeled by a single PCT loop. The first implementation of
this configuration was in the lab section of the undergraduate perception
course I taught, back when I was a psychologist.

The experimental setup is one familiar (to a fault, some would say) in the
PCT literature -- a horizontal target-bar [t] moves up and down on a
computer screen, with the momentary position of the target determined by
the present value of a time-indexed random series. A single 1-DF control
handle [h] affects the vertical position of a cursor, which is also a small
horizontal bar on the screen. A second random series (a disturbance [d])
also affects the position of the cursor [ c := h + d ]. In a "typical"
experiment, one person would use the handle to keep the disturbed cursor at
a selected position [ c-t ] relative to the moving target. A single-level,
single-loop PCT model can predict the results of such an experiment and
account for >.99 of the variance.

We (PCT modelers) have always known that the single-loop model does not do
full justice to the pull-only manner in which a person's muscles move the
hand that moves the handle, but the model *of the person as a whole* serves
pretty well to predict the results. The "trick" is in identifying the
perceptual variable the person controls. In this case it is the seen
separation between cursor and target [c-t], which the person creates then
defends against the disturbing effects of the two random functions *and*
his or her own actions.

In the six-person implementation, six strings are tied to the control handle
and each person uses her of his preferred hand to grasp one string. After
no more than one or two practice runs, the results are indistinguishable
from those produced by one person. In the lab demonstration, I used this
six-person routine to lay out unambiguously the pull-only nature of behavior
-- when one person uses one hand, it is too easy to look at the *results*
(pulling AND pushing of the handle) of the person's actions (which are all
pull-only muscle contractions) and never notice the fact that muscles only
pull.

Each person works to control the present seen value of c-t. The present
position of the handle [h] is determined by the net pull from six strings.
Everyone must pull on his or her string at all times (the equivalent of
muscle tone in one person?) The direction of movement for h is determined
by momentary accelerations in one direction or the other. The team performs
*nearly as well as* one person. (I assumed back then that the surprisingly
small difference was accounted for by the fact that there were six reference
perceptions, in six heads, rather than a single reference in one head.) One
single-level, single-loop PCT model predicts the results of the six-string
team, again, *nearly* as accurately as it predicts the results for one
person. In the model, the reference signal is for a selected value of the
perceptual signal (which is the present distance, c-t).

As Martin said, "In a structure of pull-only systems, you can't make unique
determinations of what variables living systems control." I would extend
that caveat to include structures of *any kinds of control systems," so far
as our ability to make *unique* determinations; our determinations are
*always* provisional, to some degree. But in my one-person and six-person
implementations of pursuit tracking, the cursor-target separation seems a
likely candidate for designation as a controlled variable. In either case,
the control is achieved (as is all human control?) by way of pull-only
devices.

So, Martin, it looks as though neither Rick nor I can yet see how a system
of pull-only devices differs from ideas at least implicit (and sometimes
openly stated) in PCT. However, I agree, and I am sure Rick does as well,
with what you said in your first post on the topic: the pull-only nature of
human behavior (and the behavior of all other species of pullers) warrants
more (direct, explicit) attention.

Until later,

Tom

(Season's greetings to everyone for whom they matter.)

[Human_Factors_Lab]

[From Rick Marken (931222.2230)]

Martin Taylor (931222 1800) --

Me:

The main goal of PCT is to discover
WHAT variables living systems control and HOW they control
them.

Martin:

Did you notice that pull-only control systems work differently?

Different than what? They work like control systems and control their
perceptions, don't they?

In a
structure of pull-only systems, you can't make unique determinations
of what variables living systems control.

Your comments about this in your pull-pull post were wrong, I'm
afraid. You CAN make unique determinations of what variables are
being controlled in the structure of pull-only systems presented
in your post. In that post you derive the following equation

Geq = 2G(r1-s) + 2G(s-r2) = 2G*(r1 - r2)

which is the gain of two pull only square law systems combined as an
equivalent simple ECS. You go on to say:

The reason this formulation may be useful is that the functions G may in
reality be applied in any direction, as in the case of the three rubber bands.

Good point.

One can compute the equivalent control gain for any direction at all in the
space, not just in the direction of one of the individual pulling ECSs, by
taking the function G to represent the component of the output in the
direction of interest.

OK. But you must remember that the gain is a measure of the ability to
control the INPUT variable -- which, in your derivation, was s, a scalar.
So s (and the perceptions thereof) varied in only ONE dimension. The only
"direction of interest" in terms of the effect of G is the direction in
which the output influences the controlled perceptual variable, p(s).

For example, given the three ECSs pulling in the
directions 1, 2, and 3 in the figure, an observer can determine the index of
control in the direction A-B, in a direction in which none of the ECSs are
pulling. An experimenter wishing to apply the Test to see whether there is a
controlled variable along A-B will find that there is, even though no ECS
actually controls in this direction.

     1
      \ .B
       \ . .
      . O----- 3
  . . /
A /
     2

What is important about the ECSs (in terms of the Test) is not the
direction in which they are pulling but the variable that they are
perceiving. There is a controlled variable along A-B if there is
a control system or systems perceiving the position of O along that line.
If system 1, 2 and 3 are all perceiving the position of O along A-B and
their configuration makes it possible for them to exert a net output
that opposes disurbances along A-B then position of O along axis A-B
will show up clearly as a controlled variable (using the Test).
If the alignment of systems 1,2 and 3 does not allow control along
A-B, even though all are perceiving the position of O along this
axis, the position of O along A-B will NOT show up as a controlled
variable -- simply because it is not well controlled.

Your derivation is still useful, I believe, becuase it shows how the
orientation of pull-only systems (like those shown in your figure
above) affect the gain of control when those systems are all controlling
perceptions of s (O in your diagram) along a horizontal line. Obviously,
certain pull orientations, when compared to an equivalent ECS, will allow
better control of the input (higher gain) than others. But this finding has
no particularly interesting implications for the Test, except, perhaps,
to show that it's hard to tell what a person is controlling if the
gain of the control system is (for whatever reason) low -- that is,
if control is poor.

Best

Rick