[Martin Taylor 960215 13:30]
Rick Marken (960214.0845) to Hans Blom
in all cases, the
particle will still try to climb the gradient, in addition to being subject
to some externally applied force/disturbance.Try it. I think you will see that continuous temporal variations in the
applied force/disturbance will be completely effective, resulting in
proportional continuous temporal variations in partical position that are
highly correlated with variations in the disturbance. The position of a
Brownian particle will clearly and unambiguously fail the Test for the
Controlled Variable.
Then so will any true control system, for which:
p = d/(1+G) + rG/(1+G)
for constant r, p varies exactly proportionately (even better than "highly
correlated") with variations in the disturbance, apart from effects of
loop delay.
The fact that you were able to pick up the particle and put it somewhere else
suggests that the particle is not controlling its position;
I take it that your children never wanted to get to some place you _really_
didn't want them to be. You never picked them up when they were crawling
or toddling around? I certainly did with mine, and when they were put down
they often tried to go right back where they came from. I do think they
were controlling their position. But, as you say, the Test proved that they
weren't. My mistake.
Why do you stress _continuous_ disturbance so much?
Because what matters in control is the nearly simultanenous compensation for
disturbance by system output. You and Martin keep taking about disturbances
that come and go, and you look at what happens to the putative controlled
variable after the disturbance has gone.
Nonsense.
A control system works continuously. How to determine its parameters depends
on which parameters you want to observe. If you are most interested in loop
delay you will want to use a disturbance that changes sharply. If you are
interested in low-frequency gain, you will use a slowly changing disturbance.
And in Hans's discussion of his example he proposed both abrupt and continuous
(e.g. magnetic, I seem to remember) disturbances. In my previous discussion
of the same example, I proposed several different ways to provide
continuously varying influences on the particle's position.
You really do prefer to argue against what you wished people had said, don't
you?
Note the apparent success of the
Santa Fe Institute -- dedicated (unintentionally, I presume) to the study of
superficial similarilties between equilibrium and living systems.
It does a lot more than that--and I have not heard you or Bill P claim that
the social or economic systems are living. Others have done so, but have
been squelched. To study the dynamics of the interactions among control
systems is a very legitimate thing to do. And I am surprised to hear you
talk of the Santa Fe Institute as being even _interested_ in equilibrium
systems. As I understand it, their interest is mainly in systems that are
far from equilibrium.
You have your
own reasons for wanting to conflate control and equilibrium (stability)
phenomena.
I have not observed any such "want." And if you refer to me, I have no such
"want." But I would like to be given a clear discussion of how an observer
can discover which phenomenon is occurring, if the mechanisms for causing
the observed effect are not physically clear or accurately described.
ยทยทยท
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Quite apart from the interest I originally saw in the problem, Hans has
mentioned another reason that the specific situation is interesting. This
aspect could be described as the "inside-outside" problem.
The original notion of Brownian motion considered a small rigid
particle with no internal structure of interest. The only characteristics
of interest were its density and size. The environment completely determined
its motion.
Hans changed the situation by giving the particle a rudimentary "inside".
The particle can "take in" either some of the environmental solution or
some surrogate for the concentration of the solute. There is an analogue
of a "perceptual variable." There is also an analogue of an output function
that must be at least something like a leaky integrator--the particle changes
its size, but not its mass, as a function of the recent average concentration
it perceived. To make this a control system lacks only a reference value for
the "perceived concentration" variable.
This is an "inside-outside" problem, because in the way Hans stated it, the
solution permeates the particle, directly causing the changes in bulk. There
is no inside or outside. And yet the exact same result would happen if we
look a little more microscopically, and see that the bulk-up occurs when a
solute molecule fits into a suitably shaped "receptor key" (as an enzyme
does). The solution is entirely "outside" when you look at the particle so
closely. This is very close to what happens when we sense something by smell.
It would not be at all far fetched to say that the particle "smells" the
solute and consequently bulks up locally. The "outside" is the solution,
the "inside" is the linkage between the "smell receptor" and the bulking
action.
Given this kind of mechanochemical set-up, it is not hard to provide the
system with a reference value, using another chemical that fits into
different receptors in the molecules and makes the particle shrink. If
that "reference-setting" chemical happened to come from "inside", how would
anyone be able to show that the particle was not a control system?
"Could control have evolved from something like this?" is, I think, one
of the questions Hans was asking.
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Maybe with enough thought you will be able to see the problem, but I don't
really mind if you don't, so long as you don't mind the rest of us examining
the possibilities it opens up.
Martin