Getting the Big Picture of control

<[Bill Leach 940406.19:36 EST(EDT)]

[Bill Powers (940406.0930 MDT)]

I suppose that carefully pondering this stuff for 40+ years can give one
insights at the deepest, most fundamental, and clearest levels.

Thanks. Though, I felt that I had finally picked up on what Rick was
saying to me (after how many messages?), that was wonderful.

I think that another problem is that we are often interested in
different aspect of PCT as it applies to situations.

Some of the most interesting areas of discussion are situations where
control system conflict exist. When the involved control systems are in
different packages, this can be very interesting. Musing about such
situations can be very misleading when shifting to relating "normal"
activities to PCT.

     //////////////////////////////////////////
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[From Bill Powers (940406.0930 MDT)]

I wonder how much of our discussions about the role of the
disturbance in control is biased by the way we simplify and
conventionalize examples.

Consider: in equations, we write p = o + d [omitting functions].

Suppose we choose the convention that o and d are defined as
positive when they have opposite directions of effect on the
controlled quantity and perception, with d always having a positive
effect. In that case we would change the equation to

p = d - o.

The only difference in the equations is that now the gain of the
output function must be defined as negative (-k) to make the loop
gain still come out negative ( e still = r - p).

Now it is clear that the perceptual signal represents the
_difference_ between two larger quantities. The better the control
(the higher the loop gain), the less this difference is and the less
the perceptual signal's behavior will resemble the behavior of
either of the two larger quantities. Clearly, the perceptual signal
can't "represent" the disturbance, because it represents the
_difference between_ disturbance and output.

If an integrating control system is not quite stable, given a step-
disturbance the perceptual signal might look like this:

···

*
               * *
               * *
**************** * * ***************
                             * *

That behavior of the perceptual signal obviously does not represent
a step disturbance. In general, the waveform of the perceptual
signal will not resemble the waveform of the disturbance OR the
output variable: it is always, in fact, the _difference between_
those two waveforms. This is almost too elementary an observation to
make: from seeing just the perceptual signal, it would be impossible
to guess what waveform was subtracted from what other waveform to
yield the waveform of the perceptual signal. This is the basic
reason for saying that the perceptual signal doesn't simply pass on
the effects of the disturbance.

Step-disturbances are pretty rare in nature; most disturbances are
continuous and lie within a finite bandwidth. As a result, a
perceptual signal wouldn't often have any steps in it: it would look
more like this:

+10 * *
         * *
       * *
  * * *
* * * *
                              * *
                                                *
                                                   * *
-10 * * * *
                                                                *

In this waveform there is absolutely no indication of the two
waveforms being subtracted: they could be anything, as long as this
curve represented their difference.

Another extremely important consideration gets hidden when we assume
a constant reference signal AND ASSUME IT IS CONSTANT AT ZERO.
You'll notice that I put a +10 and a -10 in the above diagram to
indicate the magnitude of the perceptual signal. This, however, is
only correct for a reference signal constant at zero. Suppose the
reference signal is constant at 20. Then we have

+30 * *
         * *
       * *
  * * *
* * * *
                              * *
                                                *
                                                   * *
+10 * * * *
                                                                 *

0 ------------------------------------------------------------------

Now the perceptual signal always is greater than zero -- with the
same disturbance! Clearly, the _average_ value of the perceptual
signal can't represent the _average_ value of either the disturbance
or the output variable. The derivative of the perceptual signal
represents the difference in derivatives of output and disturbance,
but again, not the derivative of the disturbance alone.

Now one more hidden consideration: the scale on which we are looking
at the plot. Suppose the reference signal is 100 units, and we plot
the behavior of the perceptual signal given the same disturbance,
with the same vertical plot size as above:

     * * * * * * * * * *
+100 * * * * * * *

0 ------------------------------------------------------------------

Now it is clear that the perceptual signal has an almost constant
value of 100, varying by only a small amount near 100 (with ASCII
resolution I had to exaggerate the changes to show any at all). This
obviously does not represent a disturbance with a value of 100 and
small variations above and below that amount, nor an output having
the opposite behavior. The disturbance could still have an average
value of zero -- but the perceptual signal does not.

Another consideration: scale. When we tacitly assume a constant
reference signal set to zero, we mentally magnify the changes in the
perceptual signal so we can see something happening. This gives a
false impression of how important the changes in the perceptual
signal are. We should always consider the variation in the
perception signal IN COMPARISON WITH THE TOTAL POSSIBLE RANGE, which
is also the range of useful settings of the reference signal. If, in
fact, the useful range of the perceptual signal in the above
examples is 300 units, we should show the plot with 0 at the bottom
and 300 at the top. That would produce

+300 . . . . . . . . . . . . . . .

+200 . . . . . . . . . . . . . . .

                           (perceptual signal)
+100 ********************************************************

0 ---------------------------------------------------------------

On this scale, we can't even see the effects of the disturbance, and
the behavior of the perceptual signal gives no indication at all of
what the output or the disturbance are doing.

Even in our plots of tracking experiments, we grossly exaggerate the
unpredictable tracking errors. The eye has view of close to 180
degrees, and given the right output device allowing 180 degrees of
arm movement (and a big wrap-around screen) the handle could move a
cursor through that same range. But on our plots, the horizontal
range of movement of target and cursor is restricted to about 8
inches at a viewing distance of perhaps 20 inches: a subtended angle
of 22 degrees. A failure of prediction of the model that is 5% of
the range of movement on the screen is only 0.6% of the POSSIBLE
range of movement.

Lastly, suppose that the reference signal starts at 300 units and
declines uniformly to 0 during the plot, with the same disturbance
as in all the plots above. Now the behavior of the perceptual signal
is

+300 * . . . . . . . . . . . . . .
          *
              *
                  *
                      *
+200 . . . . . *. . . . . . . . . .
                             *
                                 *
                                     *
                                         *
+100 *
                                                 *
                                                     *
                                                         *
0 ------------------------------------------------------------*--

Not only is the effect of the disturbance (or the output) invisible
on the proper scale, but the ONLY effect on the perceptual signal
that is visible is the effect of the declining reference signal.

So by assuming a constant reference signal set to zero and mentally
magnifying the behavior of the perceptual signal enough to give a
plot that shows obvious variations, we have concealed the main kind
of information that the magnitude of the perceptual signal contains:
information about the reference signal's magnitude. When the
reference signal's magnitude is zero, the perceptual signal's
magnitude is zero. To say anything more interesting, we have to zoom
in on the perceptual signal until its tiny changes are big enough to
examine in detail. But in doing that, we are unwittingly expanding
the top of the plot by a corresponding amount, meters above the top
of the chart, and looking only at the very bottom part of it. We are
looking through a microscope and missing the life-sized picture.

Hold out one forefinger over a position on the tabletop far to your
left. In one continuous and fairly slow movement, move the finger IN
A STRAIGHT LINE until it is over a preselected point far to your
right. The total distance of travel will be perhaps three feet. All
during this movement you are correcting for disturbances due to
gravity and to the tendency of the finger to move in an arc rather
than a straight line as the arm swings at the shoulder from left to
right.

What are you looking at? To what do the position and velocity of the
fingertip correspond? Certainly not to gravity. Certainly not to the
finger's tendency to move outside or inside the straight line.
Certainly not to the tensions in muscles. What you are seeing
reflected in the perception of the finger is THE BEHAVIOR OF THE
REFERENCE SIGNAL. That is essentially the ONLY thing determining the
position of the fingertip from moment to moment. Of course there are
probably three control systems involved, and three reference
signals, but you know what I mean.

The MAIN variable represented by the perceptual signal is the
reference signal. All other effects on the perceptual signal are
unimportant in comparison. The controlled variable and hence the
perceptual signal is simply caused to vary as the reference signal
varies. This relationship is normally obvious; it becomes difficult
to see only when we arbitrarily set the reference signal to zero,
hold it constant, and switch from the Big Picture scale to a far
smaller scale.
--------------------------------------------------------------
Best,

Bill P.

<Martin Taylor 940407 19:15>

<Bill Powers (940406.0930 MDT)>

I wonder how much of our discussions about the role of the
disturbance in control is biased by the way we simplify and
conventionalize examples.

Consider: in equations, we write p = o + d [omitting functions].

Suppose we choose the convention that o and d are defined as
positive when they have opposite directions of effect on the
controlled quantity and perception, with d always having a positive
effect. In that case we would change the equation to

p = d - o.
...

I think you are absolutely right. But then I tend to believe that a lot
of what Whorf said about linguistic relativity is right, though many
linguists disagree. However, their disagreement has not, so far as I know,
been centred on the effect of our mathematical notation on how we think
of the concepts behind the notation. I've long thought that there is
a big effect there.

I'll follow your thought with another one.

In a lot of discussions, yours included, formulae such as p=d-o are
written, together with o=Ge, e = r-p, and then these formulae are put
together and solved...

p = d-(G(r-p)) (1)
p(1-G) = d-Gr
p = (d-Gr)/(1-G) (2)
p = d/(1-G)-r(G/1-G)

or as G approaches minus infinity

p = r (3)

as if the two occurrences of p were the same thing. They aren't, but
they seem to be, not just because the mathematical notation leaves out
the functions that should be incorporated for each section of the loop,
but also because the notation leaves out time.

The view of the loop expressed in this simplified notation is an external
one, in which things happen at different times. Only when seen from one
point in the loop do all the effects occur in the present. A proper
notation involves the operation of convolution at each stage, showing how
the effects are distributed over time:

Not o = Ge, but o(t) = G(tg) X e(t-tg) where "X" represents the convolution
   operator. (I apologise for the bastard notation, but I want to show the
   "tg" time reversal of the convolution explicitly, and putting in integral
   signs is too clumsy even for this posting.)

e(t) = C(tr,tc ) X (r(t-tr),p(t-tc)) (Here X represents the same thing
   as convolution, but for the two signals separately. For the most part,
   in the comparator, it is probably OK to treat the effect as an instantaneous
   subtraction, in accord with the usual notation, but one should be aware
   that it might not be so).

p(t) = P(tp) X s(t-tp) where s is the set of sensory inputs to the
   perceptual function. Remember that P might include differentiators,
   shift registers, and other functions of time, depending on the level
   of the hierarchy under consideration.

In the outer world, we can assume that s is immediately derived from the
CEV, because if it is not, we can subsume any delay in P(tau). And we
can take s = d+v to be immediate and in the present at the point(s)
in the outer world where the output and the disturbance have their joint
effect, where v is the effect "now" of the output on the CEV.

v(t) = F(tf) X o(t-tf) where F is the feedback function.

So the nice pretty loop of equation (1), which leads to the solution (2)
for p, should really be something quite murky, with lots of convolution.
Assuming immediacy for the comparison function, we have:

p(t) = P X s = P X (d + F X o) = P X (d+(F X G X (r(t-tr)-p(t-ts-tf-tg-tc)))

In this more accurate form, the seduction of thinking that the p is the
same on both sides of the equation fails. One can only say that if
control is good, then the two p values won't be too different, and work
from there.

Setting (tp+tf+tg+tc)=tt, a kind of loop delay, we can write

p(t) = p(t-tt) + eps(tt)

where eps is some (unknown) difference between the perceptual signal
now and at a time tt=(tp+tf+tg+tc) in the past. And even this
is misleading, because those tm parameters are only parameters in the
several convolutions, not fixed numbers, and they don't simply add,
so eps is a function of time that has to appear in the general analysis.

It isn't easy, and it isn't as obvious as the simple notation ignoring
the temporal nature of the various parts of the loop would seduce you
into believing. However, it should be possible (I haven't done it) to
complete the analysis and wind up with a relation between p and r, with
eps and d as parameters whose effects get smaller the better the control.

So I agree wholeheartedly with Bill about the power of notation both to
mislead and to lead correctly. If one is going to use shorthand notation,
one should be aware of just where the seductions and pitfalls lie. Much
of the argument on CSG-L hinges on a failure of this awareness.

Benjamin Lee Whorf is alive and well in Durango. (Actually, he studied
Navajo and Hopi, not too very far from there).

Martin

<Martin Taylor 940407 20:10>

Bill Powers (940406.0930 MDT)

On scale:

I agree with Bill that one suitable scale for display of the effect of
control is that of the full range of the perceptual variable. That
shows how precisely control is maintained in relation to the disturbances
that evolution has developed us to be able to deal with. But there is
another useful scale that evolution has provided us with, and that is
the minimum deviation that we can perceive. It is quite reasonable to
plot the deviation of control from exactness on a scale that shows the
maximum error at full-scale, because this kind of deviation is also
a limit that evolution (and training and experience) has imposed on us.

If the observed magnitude of failure of control happened often to have
lethal results, we would not be here to observe such poor controllers.
But if we are able to control substantially better than need be to
survive and propagate our genes, we are wasting resources, which is
not good for survival, either. One of my articles of faith is that
evolution seldom makes us waste resources very much, and when it does,
the reason is usually adaptability that has enhanced survival. "Very
much" are weasel words, I know. But take it as meaning "within a factor
of five or ten" and I think one won't often be far wrong.

In other words, to understand what and why humans control, it is important
both to see how well they control as a proportion of the range of controllable
perception, and to see how poorly they control in an absolute sense, at
the limit of precision.

Martin