category with analogue

[Martin_Taylor]

[Martin Taylor 991119 02:10]

[From Bill Powers (991118.0746 MDT)]

Anyway, you still don't understand my critique. The problem of
level-skipping is not what you describe: setting references for the angle
of the steering wheel and the location of the car in its lane at the same
time. The problem is setting a reference angle for the steering wheel
without considering that some other, higher, system, may _also already_ be
setting a reference angle for the steering wheel. That other higher system
will immediately adjust its output to restore the steering wheel to its
"proper" angle, or will adjust something else to compensate. In your verbal
description you said that the whole system would simply adjust itself to
solve the simultaneous equations just as the analog systems alone would do.
But that is incorrect, because the digital system can't be adjusted
continuously.

>>>The higher analog levels would try to counteract the effects.
>
>Yes, this always happens in a many-to-many set of connections between
>levels.

But if the systems are properly orthogonal it does _not_ happen. You can
control x+y completely independently of x-y, with no conflict at all.

Actually, I didn't talk about conflict, which I take to mean irreconcilable
settings of the reference signal values in different control units. I
talked--and I suppose you meant--mutual disturbance, and used "conflict"
loosely. On that assumption...

You seem to be saying two things: at the end of the quoted section you say
(1) If x, y, x+y, and x-y are all continuous analogue control systems, then
a change in the reference value for x+y will not disturb the perceptual
signal of x-y, no matter what the dynamics are of the environmental
feedback paths for the x and y control systems. So the x-y control unit
will therefore not alter its output to counter the non-existent
disturbance, and will not disturb the perceptual signal of the
x+y control unit. No matter what the external dynamics, each controls
without disturbing the other?

And in the previous paragraph you imply
(2) It is possible to set the reference values for x+y=7, x-y=2, and
find values for x and y that allow the values of x+y and x-y to match
their reference values, but it is NOT possible to set a reference range
(i.e. a categoric value "single digit") for "0<x+y<10", x-y=2, because
there are no values of x and y that allow this to be the case.

I find it passing strange to say it is not possible for the category
reference "true/false" to be matched by the perception derived from the
logical input function "0 < x+y < 10" at the same time as a reference
value of "2" for the perception x-y can be matched, simply because the
decision as to whether the category reference value is matched is binary
valued. But if the reference value for x+y is any exact number within
that range, you say there's no problem in getting both references
simultaneously matched.

That doesn't apply when one of the equations admits of only binary values
of its variables. The chances are extremely small that a solution will
exist.

Again, passing strange. The perceptual signal, the reference signal, and
the error signal _inside_ the category control unit are binary. The
inputs to the perceptual input function are not, and the output signal
need not be (ordinarily would not be, if the typical integrator output
function is used). How does the binary character of the internal
variables affect the simultaneous equations--quite apart from the
obvious fact that it is easier to satisfy multiple simultaneous
equations when some of them admit ranges of values rather than point
values as solutions?

Actually, as I pointed out earlier, the fact that some of the controlled
variables are categoric makes it easier to satisfy the equations together.
In fact, sometimes you can solve systems such as
   0 < x+y < 10, 2 < x-y < 20, x*y = 8,
when you couldn't solve those three equations (except by a fluke) if they
were point equalities x+y=7, x-y=2, x*y=8.

>And I must apologize again, as,I hadn't seen that you had followed
>through my moment-by moment description of the changes of signal
>values when digital and analogue units combine to set refeences for
>lower-level analogue units

I read it through but it was just a lot of arm-waving -- an analog pointing
system with an error of 30 degrees (as I remember the number) for example!

Why do you have a problem with the analogue system having an error of 30
degrees? How can this not happen when the arm has been to the right and
you now wnat it to be to the left, as in my example? Don't you believe
reference values can change?

What did you imagine that the output function would be doing?

What I said it would be doing. The output functions are perfectly
ordinary--I assumed integrators, I believe, except that I used a worst-
case situation for the category control unit and allowed its output
to be binary valued, in order to show that even this crudeness did not
create any real problem.

Martin, I'm just not going to get sucked into this.

That's been my feeling, too. I am aware that it is hard to get one's
mind around a configuration different from the one with which you have
spent so many decades, but it really isn't that complicated. I think
you are probably seeing the proposed structure in terms of a distortion
of the classical structure, whereas it is really quite easy to take it
on its own terms and see what it would do. Anyway, let's drop it for
now. I retain the opinion that none of the objections you have raised
are valid--maybe that's because I haven't understood the objections.
Anyway, in the way you have stated them they seem not to apply to the
organization I propose, or, as in the earlier part of this message,
as I interpret them, they are wrong.

You are right that words along the lines "You're wrong" "No, you're
wrong" get us nowhere. And I have a notion that maybe a variant of
Rick's 3-level demo which has a logical top level might be adequate
for looking at some of the issues. Or maybe not. But words aren't
going to change opinions--they seldom do.

At the moment I have 283 unread messages on CSGnet, and the number is
more likely to increase than to decrease. It would be interesting to
know, though, whether my impression from occasional sampling is correct--
that the "coercion" debate is covering exactly the same ground as last
year or the year before? Or has something new been said by anyone?

Martin

[Marc_Abrams3]

from [ Marc Abrams (991119.0833) ]

[Martin Taylor 991119 02:10]

. It would be interesting to
know, though, whether my impression from occasional sampling is correct--
that the "coercion" debate is covering exactly the same ground as last
year or the year before? Or has something new been said by anyone?

The "coercion", "manipulation", and "influence" debates have been going on
for 10 years. The added twist over the last 2 years is that the target has
not been a better understanding of human interaction. It has been Ed Ford's
RTP program.

Marc

[Bill_Powers1]

[From Bill Powers (991119.0525 MDT)]

Martin Taylor 991119 02:10]

You seem to be saying two things: at the end of the quoted section you say
(1) If x, y, x+y, and x-y are all continuous analogue control systems, then
a change in the reference value for x+y will not disturb the perceptual
signal of x-y, no matter what the dynamics are of the environmental
feedback paths for the x and y control systems. So the x-y control unit
will therefore not alter its output to counter the non-existent
disturbance, and will not disturb the perceptual signal of the
x+y control unit. No matter what the external dynamics, each controls
without disturbing the other?

That's reading a lot into what I said. I said that x-y can be controlled
independently of x+y. Let u = x+y, and v = x-y. You can then pick any
target values for u and v, u* and v*, and find an x,y pair that will make u
= u* at the same time v = v*. I said nothing about the means of doing this,
or whether x and y are under individual control at a lower level.

Asking about how we change u and v is another question. It is possible to
do this by using two integrating (or high-gain) output functions. The
output of the x+y or u controller affects x and y in the same direction,
while the output of the x-y or v controller affects x and y in opposite
directions (the specific directions depend on how the comparators are
hooked up and how the outputs affect x and y individually). The two
controllers will reach zero error at the same time, and also compensate for
any disturbances of x and y. They will even compensate for direct
disturbances a and b such that u = x + y + a and v = x - y + b.

So if I did say that changing u* will not disturb v, I was wrong. There
will in general be a disturbance unless the output effects on x and y are
precisely equal, but the v-control system can compensate for any inequality
and achieve a state of zero error after the transient disturbance. There is
no _steady-state_ interaction when the output functions are integrators.

And in the previous paragraph you imply
(2) It is possible to set the reference values for x+y=7, x-y=2, and
find values for x and y that allow the values of x+y and x-y to match
their reference values, but it is NOT possible to set a reference range
(i.e. a categoric value "single digit") for "0<x+y<10", x-y=2, because
there are no values of x and y that allow this to be the case.

It is possible for a higher-level system to set a reference level for 0 < u
AND u < 10. If the value of u is below or equal to zero, the reference
setting for u should be raised until u is not less than zero, and if
greater than or equal to 10, the reference setting should be lowered until
u is less than 10.

Thus the "categorical" variable (I would call it "logical") is controlled
by means of setting the reference value for u, while the reference value
for v is being set from elsewhere.

What is not feasible, I claim, is for your proposed categorical, or
digital, control system to exist _at the same level_ as the u and v
controllers. There are several reasons. First, a digital controller
controlling via a continuous variable will tend to be unstable. If it's
controlling through a high-gain output function or an integrator, it will
be unconditionally unstable (whether or not there is hysteresis). Second,
if there is a disturbance that drives the digital system you describe
beyond one of its limits by an amount in excess of the hysteresis, the
system will oscillate, again unconditionally. And third, if the same
variable is being controlled by an analog system at the same level, the
actions of the digital system will disturb the analog system and the analog
system will act against what the digital system is trying to do. The only
exception to that would occur when the required output of the analog system
just exactly matches the one value of output signal that the digital system
can produce. Then the analog system would be experiencing zero error and
would produce no output.

I find it passing strange to say it is not possible for the category
reference "true/false" to be matched by the perception derived from the
logical input function "0 < x+y < 10" at the same time as a reference
value of "2" for the perception x-y can be matched, simply because the
decision as to whether the category reference value is matched is binary
valued.

The problem is that the system you're talking about here is _my_ kind of
hierarchy, in which the digital system operates by changing the reference
signal of a system at the highest analog level. What you say is completely
true in that case. However, if you add an (x+y) or u digital control system
_in parallel_ with the existing (x+y) or u analog control system, the
result will be very different.

To say that the two systems operate in parallel is to say that they receive
the same information from lower systems (x and y), and that they act on the
same lower variables. Imagine, then, that the reference signal for the
analog system varies smoothly from 9.5 to 10.5. The digital system, at the
same time, will experience a perceptual signal that goes from TRUE to FALSE
when u becomes greater than or equal to 10. If the reference level of the
digital system is TRUE, the output of the digital system will
simultaneously depress the values of x and y by some fixed amount (the
amount has to be large enough to cancel the largest disturbance that the
system can resist). This, of course, will drive the analog system's value
of u well below the reference value, so the analog system will experience a
very large error. Through its integrating output function, it will start
increasing the values of x and y. But at the same time, the digital system
will experience NO error since u is again (far) less than 10. But that will
turn off the output of the digital system, so x and y will be allowed to
increase in value again, which will reverse the error in the analog system,
and with x + y once again outside the digital range ....

But if the reference value for x+y is any exact number within
that range, you say there's no problem in getting both references
simultaneously matched.

As I illustrated above, if you confine the effects of the digital levels to
the reference setting of the highest analog level, it then becomes possible
to set the reference value of u independently of the reference value for v,
without causing any conflict. Of course if there are higher analog systems
also using the u and v control systems, then the digital system is
operating in parallel with _them_, and we have the case I just discussed.
This is actually just another way to describe what I have called the
"level-skipping" problem, a term I probably should not have used, since it
seems to restrict the discussion to only one example of the underlying
problem. Actually, we would have exactly the same problem if it were an
_analog_ control system being added to work in parallel with an analog
control system, except for the oscillations. Two control systems operating
in parallel to control different variables by exactly the same means are
most likely to conflict.

Actually, as I pointed out earlier, the fact that some of the controlled
variables are categoric makes it easier to satisfy the equations together.
In fact, sometimes you can solve systems such as
  0 < x+y < 10, 2 < x-y < 20, x*y = 8,
when you couldn't solve those three equations (except by a fluke) if they
were point equalities x+y=7, x-y=2, x*y=8.

You're still considering only the positive examples. Think of what happens
if disturbances drive a variable outside the stated range. Think of what
happens if analog systems controlling the same variables on a continuous
scale are operating at the same time. Think of what happens when a variable
that was true goes false and an output integrator starts to change the
output at some rate corresponding to the amount of error. If you try to
visualize ALL the effects of your proposed arrangements, you simply HAVE to
see that (1) a verbal analysis can't reveal what will really happen, and
(2) there is a strong hint that the system will be unstable.

Why do you have a problem with the analogue system having an error of 30
degrees?

Because for a normal pointing control system, that would create the maximum
possible output from the output function. In a human position control
system, an error of about 3% of the range of angles will produce full
tetanus in the muscles. An error as large as 30 degrees would overdrive the
output function manyfold.

How can this not happen when the arm has been to the right and
you now wnat it to be to the left, as in my example? Don't you believe
reference values can change?

But you have to turn off the analog system that had wanted it to be to the
right in order to let the digital system move it to the left. Otherwise the
analog system will simply send the maximum amount of signal it can generate
to the position control system's reference input, trying to restore the
position to where it wants it. You seem to think that the analog system
will go away when you turn your attention to the digital one. If the analog
system had set the reference signal to +30 degrees, and the digital one
then added enough output to change it to -30 degrees, if you looked back at
the analog output you would find it is now not +30 degrees but +3000
degrees ( or whatever its maximum is, if less than that). Actually, that
would happen only in the first instant. A little later, the analog output
would have decreased to +90, bringing the perceived angle back to +30. Of
course the digital system could do nothing more about that; it is already
producing the one and only output, of negative 60 units, that it can produce.

I won't claim that I'm imagining all this correctly, although I do have a
lot of experience with simulations so I haven't made any _small_ mistakes.
The only way to really check out these ideas is through simulation --
setting up which is also the only practical way to show exactly what you
mean by your words.

Because of that, I don't really see much point in continuing this verbal
argument. The real answers are waiting to be found by a means we are both
acquainted with. The first one to carry out the simulation will settle the
issue (as well as defining it clearly). In the meantime, should we continue
to argue about whether the moon is made of green cheese?

Best,

Bill P.