[Martin Taylor 960213 16:00]
Bill Powers (960213.0300 MST)
OK, here's a control system:
Controlled variable: temperature of a water bath.
Reference level: 80 degrees c (equivalent).
Output function: heater.
Disturbance: refrigeration unit removing heat at a variable rate.
Please explain how it will help us if we think of the control system as
a cooling device and why the CEV would be cooler than it would be under
the influence of the disturbance.
I assume you are using a conversational convention here, that "Please"
means "I don't think you can." I think I can.
Remember that the CEV is not the water bath, which gets hotter or colder
as the case may be. The CEV is the _temperature_ of the water bath, which
has its fluctuations reduced when it is under control. Reduction in
fluctuation is a measure of thermodynamic temperature (Variance = kT or
something like that...don't trust that particular formula dredged from
a faulty memory).
The temperature of the bath is increased by the heater that is the output
of the control system. The perceived temperature of the bath is controlled,
and with it the actual temperature, one hopes; so the temperature of the
temperature is what is reduced (equivalent statement: the fluctuations in
the temperature of the bath are reduced in RMS value. Second equivalent
statement: the temperature of the bath, not the bath, is what is cooled).
In fact, the temperature of the bath is one degree of freedom in the
total number of degrees of freedom that describe the motions of the
molecules in the bath (order of Avogadro's number). Both the heater and
the refrigerator act on the individual molecules, but do so in a way
that affects this average. The fluctuation of temperature of the bath
due to internal randomness in the near-equilibrium conditions of the
bath is very, very small compared to the fluctuations induced by the
heater and the refrigerator, so the fact that the sensor is measuring
temperature is almost irrelevant to the issue of whether the control
is heating or cooling the CEV. So the heating and cooling effects on
the bath represent a degree of freedom whose changes are rapidly dissipated
among the many molecules of water in the bath, but which is recoverable
by an averaging sensor. It is the one degree of freedom recovered by
the averaging sensor whose fluctuations are reduced by the control process.
Now, I think the core of the question is really not "why is the control
system a cooling device" but "how it will help us if we _think of_ the
control system as a cooling device."
If you are thinking only of a single control loop in an environment that
contains one well defined disturbance, it doesn't help. Where it helps
is when you get into the situation that Hans was apparently thinking of
in his "N-dimensional" postings. The one-dimensional control system
is acting in a high-dimensional world, and the effects of its actions
may be felt in dimensions not represented in the perceptual signal.
The perceptual function P defines some direction in N-space. The output
O also defines some function. If O is precisely colinear with P, then
all of the output energy is used to cool the CEV (reduce its fluctuations).
But O may not be precisely colinear with P, in which case there is a component
orthogonal to P. This component represents energy dissipated into the
environment, which heats the environment (in the same sense of increasing
its fluctuations).
If all of that dissipated energy is carried away, having no effect, either
then or later, on anything that contributes to the disturbance influencing
the CEV, well and good. But that happy circumstance is not always the case.
Side effects often come back to haunt you by contributing to fluctuations
of the disturbance. And no matter how good the control system may be, more
fluctuations in the disturbance mean greater mean-square error. In a high
entropy environment, some part of the side effect _will_, with probability
near unity, be found later in the disturbance.
If your thinking is not attuned to thermodynamic concepts such as entropy
and temperature, it may not help you to think of the control system as
a cooling device. As I just showed, you can get often the same result in
a cumbersome way from a first-principles argument. But thinking of the
_set_ of control systems in a very high-dimensional environment, the
ideas of thermodynamics make it easy to see what will almost certainly
happen.
What I'm trying to say, I think, is that to use thermodynamic arguments
is _not_ arguing by analogy, any more than to use results from electronic
servo systems in discussing living control systems is arguing by analogy.
Whether you find the tool useful depends on your ability to handle it,
and whether the situation makes it a good tool for your purpose.
To determine the waveform of error following a step-function disturbance,
one would not look to thermodynamics. To see why reorganization or
evolution might develop a more nearly optimum organization even in the
absence of resource competition or scarcity, one might appeal to
thermodynamics, noting the differential cooling and heating properties
of "behaviour" (that effects control) and "action" (that includes side
effects along with control). It is harder to maintain a cool interior
when the external environment is hotter, so it is a good idea for your
refrigerator (i.e. control system) to be efficient and not emit too much
heat into its own neighbourhood.
Hope this helps. Incidentally, I am trying to revise my old tutorial
posting on entropy, putting it more in a control-system context. With
luck, it will result in a posting before long. Rick will think it is
putting a theoretical cart into a place where there are no horses, but
then he doesn't see that the cart uses one of M. Carnot's magic engines.
Martin