[From Bruce Nevin (980226.2200)]
Bruce Abbott (980224.1400 EST)--
This has probably been dealt with and I am just behind the times, but it
has been bothering me. I have to ask why your diagram does not match your
text. Here's your diagram:
Hmmm. Change in d ---> change in i -----> change in p ---->
change in e -----> change in o (all simultaneously) and you say
that change in d has _nothing to do_ with change in o? Poppycock.
Simplifying:
delta d --> delta i --> delta p --> delta e --> delta o -->
It doesn't matter if you say "all simultaneously" if you leave the loop out
of the loop! It is the loop that enables all this to go on simultaneously!
Compare now Rick Marken (980223.1310):
delta d -->
+ delta i --> delta p -> delta e -->
delta o --> |
^ |
> >
----------------------------------------------
You said they were the same:
This is the same arrangement I gave (except for substituting "delta" for
"change in." Read my post if you don't believe it!
You must have been referring to your text preceding your diagram:
Let us assume (to simplify the argument) that r is constant. At time t, d
changes, inducing a change in i. The change in i causes p to change (via
the input function), thus altering r-p similarly. Now r-p = e, so e
changes, inducing a change in o (via the output function). The change in o
induces a change in i (via the environmental feedback function), which
exactly opposes the change in i induced by d. In this ideal control system,
all these changes are assumed to occur simultaneously around the loop. But
this leads to a paradox: the influence of d on i is opposed in the same time
instant in which it occurs. Because it is completely canceled, there can
have been _no_ change in i, _no_ change in p, _no_ change in o to oppose the
influence of d on i! It can't work!
The diagram described verbally here does indeed look like Rick's:
PIF r-p
delta d --> delta i --> delta p --> delta e --> delta o ---+
^ |
> >
+-----Environmental feedback function ------+
Why did you leave the loop out of your diagram? How could you have possibly
done so if you were thinking about control of perceptual input? Whatever
the reason, the result was to ignore the influence of delta o on delta i.
Assume change in d that does not overwhelm control. This means among other
things that change between time t and time t+1 is continuous. Your sampling
of values is discrete at two instants t and t+1 but the values in fact
change continuously over that interval. Given these two assumptions
(control and continuity), delta o is as much a contributor to delta i as
delta d is. There is no moment at which delta d is an influence without
delta o also being an influence. (Even before d changes, during a period
when d was static, o may be changing slightly, maintaining control of p.)
Assume the point of view of the control system. The influences of delta o
and delta d are indistinguishable from one another. It is only to an
outside observer that d is identifiable as a distinct value. But it is
precisely to overcome the limitations of a point of view outside the black
box that we create, run, and analyze models. To understand a control system
model, you must take a point of view inside the black box, you must take
the point of view of the control system. To understand the organism being
modelled, in order to model it, you have to take the point of view of the
control system. The measurement of d and o is useful to verify the presence
of control, they do not suffice for modelling or understanding control.
I suppose this is all water over the dam now. Rick said much the same:
Note that the effect of d is irretrievably
mixed with that of o well before it enters the comparator (to
produce delta e). I guess it would have more appropriate for me
to say that d _alone_ has nothing to do with it. What I meant was
that, from the point of view of the control system, d is not
involved. The control system deals only with the variable
p which is, at any instant, a result of unknown contributions from
both d (or many d's) and o. Changes in o are not caused by changes
in d; they are caused by changes in _both_ d and o simultaneously.
But I was and am really puzzled why you would use a diagram that omitted
the feedback through the environment, while talking about the control loop,
making it look like a linear chain of causation. That was what created
Zeno's chimaera for you.
Bruce Nevin