Set Points & Settling points; equilibrium and control

[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.

[From Bruce Abbott (950519.1920 EST)]

Bill Powers (960519.0330 MDT) --

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.

If they did make such a proposal it wasn't in this article, which dealt only
with the question of whether the concept of an internal, neural set point
was necessary. Their alternative model, as I noted before, is itself a
control system model (assuming that the product of their output gain and
feedback gain are greater than unity), but it substitutes a set point
determined by an external stimulus for an internally-specified one.

But I think you are right that researchers at this time (probably including
Wirtshafter and Davis) failed to distinguish between control systems and
mere equilibrium systems. And for some reason they believed that what they
termed a "set point control system" must contain a specific, _neural_ device
that provided a neural reference signal, a neural perceptual signal, a
specific, neural device that functioned as the comparator, and a neural
error signal capable of driving the output so as to correct errors both
positive and negative. When they did not find such devices and signals
within the physiological matrix supporting control of body weight, or
caloric intake, or what have you, they believed they had encountered strong
evidence against a control-systems model.

An anthology edited by Toates and Halliday (1980), which was based on papers
delivered at a conference entitled "Analysis of Motivational Processes" held
in September 1979 at The Open University, is remarkable for the concensus it
reveals had emerged by that time as to the failings of control theory in its
application to physiological and behavioral systems. Nearly every author
offered some argument against it. Perhaps the most remarkable in this
context was the contribution of D. A. Booth, co-developer of several
interesting control-system models of feeding and satiety (Mark 3 of the
Booth and Toates model is presented for illustration in Booth's chapter).
On page 92 there appears a section entitled "Regulation -- a Pernicious
Concept," in which Booth launches a strong attack. Bill, I suggest you take
your anti-heartburn medicine and perhaps something really strong to keep
your blood pressure down before reading on.

In the middle of this section Booth says the following:

  Now in physiology we know of many chemical arrangements, examples of
  fluid and rigid mechanics, and neuromuscular-environmental loops (reflex
  mechanisms) which demonstrably or conceivably maintain equilibrium by
  simple negative feedback control processes. However, physiologists have
  agreed [on] a distinction between such control and something else called
  regulation. Brobeck (1960) applied it in motivational physiology. This
  distinction was read ouf of engineering control theory, not empirical
  physiology. It defined regulation as the particular and more complex
  form of control in which both the vigour of activity and its normal
  corrective direction are determined by a measurement of the deviation
  of a monitored variable (such as blood glucose concentration) from a
  predetermined reference value or set-point.

  However, the performance of a dynamic equilibrium or a simple
  (unreferenced) negative feedback system can always be described as the
  operation of one of these set-point systems. So there is the risk that
  phenomena generated by feedback control become classified as examples of
  regulation so defined. I suggest that indeed just that has happened.
  Invocation of the unnecessarily complex set-point type of formal
  description of the performance has been all too easy. There is the
  seductive convenience of comparison between thermostats which everyone
  knows about and homeostats which we hope to learn about. In control
  theory, the value of the set-point is not identical to the value of the
  controlled variable when the system has steadied, but it can be made
  similar; thus it is easy to skip the distinction and read the value
  acheived by homeostasis into the setting of the homeostat. Most serious
  of all, the set-point formalism can be interpreted as a physical
  mechanism and then what may be equilibration between body, environment,
  and brain is attributed to a mechanistic marvel within the brain. Some
  have claimed to resist this temptation and to intend their invocation of
  regulatory set-points merely as description of the system's performance
  characteristics. However, as equilibrium or unreferenced feedback
  control on the one hand and regulation on the other hand are inter-
  translabable formalisms for describing performance, the only point in
  invoking set-point regulation would appear to be the implication that a
  physical monitor-comparator mechanism exists which explains the observed
  stabilising achievements. However, such inference is fallacious. The
  postulate is arbitrary unless and until independent evidence for a
  physical comparator mechanism is obtained by anatomical observation and
  physiological or biochemical measurement at the cellular level in the
  relevant location(s).

  Thus, the result of this definition of regulation (even though not its
  logical requirement) is that stabilities and their defense that can be
  generated by equilibrium processes which are abundantly evidenced are
  being "explained" by physical mechanisms of much greater complexity
  than need to be invoked and are of a type for which there remains no
  direct physiological evidence in the whole of biology (not even for
  thermoregulation, or for metabolic biochemistry or genetic expression).
  The genes, with or without environmental aid, would have to specify a
  cellular arrangement to produce an extremely precise and unvarying
  signal of a particular strength throughout the animal's life: this would
  be the set-point. A monitor has to produce a graded signal of comparable
  accuracy and a comparator has to subtract the signals and emit another
  graded signal which is converted into physiological or behavioral
  activity. Mechanical and electronic engineers find such material systems
  easy to construct but living matter does not have the properties that
  they exploit.

In the section following this one (Stabilities without Set-Point
Mechanisms), Booth goes on to describe several marble-in-the-bowl examples
of equilibrium systems. I will quote briefly from this section:

  The notions of active rather than passive control or regulation are
  equally metaphorical, expressing little more than the user's intuition
  that we have to invoke something like a human engineer's thermostat to
  explain the observations. Yet the languages of chemistry and control
  engineering are intertranslatable. For this reason I have proposed a
  ban on use of the word "regulation" in behavioral physiology (Booth,
  1976).

This from a man who co-developed several interesting computer simulations
of feeding and weight-control, simulations that incorporate control systems.
Not only does Booth fail to distinguish between control and equilibrium
systems (he says that they are intertranslatable), he also believes that the
functional units and connections of a control system diagram must find
direct physical realization in the form of high-precision set-point signals,
comparators, and other control-system components. If a reference is
implicit then evidently we are not dealing with a control system. If there
is no physical neural or cellular comparator, it is not a control system.
Not to Booth, anyway.

Control systems are, of course, equilibrium systems, but as you point out,
equilibrium systems of a special character. The key distinction is not
between the presence or absence of an internal, separately represented
physical reference signal, but betweeen a system that passively readjusts
its "stabilized" value to changes in value of one or more of its variables,
and one that actively opposes these changes by calling on external (to
itself) energy sources in proportion to smaller changes termed "error." In
other words, such a system has a gain greater than 1.0, usually much
greater. In an equilibrium system, the error itself (change in the
stabilized feedback-stabilized variable) must do all the work.

Reference:

  Booth, D. A. (1980). Conditioned reactions in motivation. In Toates,
      F. M., & Halliday, T. R. (Eds.), _Analysis of Motivational
      Processes_, Pp. 77-102. New York: Academic Press.

Regards,

Bruce