[From Bill Powers (960519.0330 MDT)]
Waiting around for STS-77 to launch.
Bruce Abbott (960518.1615 EST) --
If I recall correctly, Wirtshafter and Davis rejected the concept of a
set-point, but proposed that the settling point was simply the
equilibrium point between two different processes. James Lovelock, under
the influence of the same biological prejudice, made the same point
about Gaia in his Daisyworld model. If there is a shift in the settling
point, it is simply due to a change in either one of the processes. What
nobody among the many who have latched onto this clever idea realized is
that this is exactly how a control system works. The only missing part
of the discussion is an estimate of the loop gain.
Here's the general situation. You have two variables, x and y, and two
relationships between them. One relationship says that y increases when
x increases, and the other says that x decreases when y increases. A
graph of the two relationships would be like this:
White | *
># *
> # *
> # *
Y | # *
> #*
> * #
> * #
> * A # B
Black | * #
···
---------------------------#---
Cold X Hot
In Lovelock's Daisyworld, the predominant color of the daisies tends
toward white when the temperature of the planet rises (line A), and the
temperature of the planet falls as the population of daisies covering it
becomes more white -- more reflective of sunlight (line B). The
equilibrium temperature occurs where these two lines cross.
This is the usual marble-in-a-bowl sort of equilibrium process involving
two reciprocal relationships that people think of when they equate
feedback control and equilbrium. It is technically a negative feedback
situation, so the system as a whole is a rudimentary control system.
So, you ask, how would a non-rudimentary control system look? Like this:
ref
White | | * * *
># |* A
> # B |
> # *
Y | # |
> #
> * #
> > #
> > # B
Black | A *| #
-*--*--*----|--------------#---
Cold X Hot
The only difference here is that as temperature rises, there is at first
only a slow shift from black daisies toward white daisies, and at a
specific temperature there is a very rapid shift, which reaches a limit
(all white) very quickly. Most of the shift takes place in a very narrow
band of temperatures centered on the vertical line labeled "ref". That
line, of course, indicates the reference level of the control system.
The other line, B, which indicates the effect of reflectivity on the
planetary temperature, is the same as before, fixed by physical laws.
The equilibrium temperature will still be the temperature at which the
curves intersect.
In the first diagram, a movement of either line parallel to the X axis
would shift the equilibrium temperature by about the same amount. But in
the second diagram, a lateral shift of the B curve representing the
physical effect of reflectivity on temperature would have almost no
effect on temperature until it became extreme, whereas a lateral shift
in the A curve would have an effect on the temperature essentially equal
to the amount of the lateral shift. What would such lateral shifts
represent?
For the B curve, a lateral shift to the right would indicate that for
any given reflectivity, the temperature would be higher by the amount of
the shift. This might happen, for example, if the Sun got hotter. In the
first diagram, this would mean that the planet gets hotter in
proportion.
For the B curve, a lateral shift to the right would mean that the
reference temperature increases (the temperature where the slope of the
A curve is maximum). This would imply a shift in the internal response
of the daisy population, so the change from black to white occurs at a
higher temperature.
In the first diagram, the rudimentary control system, the rise in
temperature would be about equal, whichever line was shifted to the
right. In the second diagram, with a good control system, the
predominant effect on temperature is due to the shift in the A curve to
the right or left, while shifts in the B curve by the same amount to the
right or left have an effect limited to the range near the reference
level.
This is another way to express what I have called "the assymetry of
control." Qualitatively, it is true that temperature affects the color
of daisies and the color of daisies affects the temperature. But it is
not correct to say that temperature affects the color of daisies JUST AS
the color of daisies affects the temperature. The effects are far from
being similar. A change in the property of the daisies that shifts the
reference level has a far larger effect on temperature than does the
effect of a change in the Sun's brightness which causes the B-curve to
shift to the right or left. Qualitatively, we can say that the daisy
population controls the temperature and the Sun does not.
The loop gain of this control system is the negative of the product of
the slopes of the two curves at the point where they cross. In both
diagrams the slope of the environmental (B) curve is the same. But in
the second diagram, the slope of the daisy curve (A) at the crossing
point is far larger than the slope of the environmental curve. The loop
gain is high primarily because the slope of the daisy curve is high at
the reference level of temperature. And the daisies have control, while
the Sun does not, exactly because of this high slope near the reference
level.
This same graphical way of diagraming an equilibrium system applies to
all negative feedback control processes. Of course there is a continuum
of cases between that of the first diagram and that of the second. But
when I speak of control processes in organisms, I am speaking of the
cases that resemble the second diagram far more than the first.
-----------------------------------------------------------------------
STS-77 is safely in orbit.
Best,
Bill P.