[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.

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STS-77 is safely in orbit.

Best,

Bill P.