# Systems theory and PCT

[From Rick Marken (940223.1030)]

It seems that I did have a close encounter of the "systems theory"
kind once before. I just noticed a file in my e-mail folder
labelled "systems theory". It turns out that it is a reply to a
post from my sister- in- law who wrote to me in about May, 1983
as follows:

(Fritjof Capra) and he mentions that it has roots in cyberntics, as in both
von Neuman (like systems are i/o hierarchies) and Norbert Wiener (like
systems are self-organizing and self-maintaining). Where does control
theory fit in here?

it's worth, here is my reply to her. I would appreciate hearing
what Cliff Joslyn has to say about it (I am intersted in Klir's
_Facets_, by the way Cliff, but I'd appreciate a summary if possible).

···

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Hi sis-a-la;

I think of a "system" as a collection of components. The components
are typically functions that transform input variables into output var-
iables. The variables can be scalers or vectors; in the systems I care
about, the input variables are typically scalars or vectors and the
output variables are always scalars. So, in the simplest case, a
system component is a function -- y = f(x) which is a function that
transforms an input variable (x) into the output variable (y) (by the way,
in the discussion below all variables are assumed to vary over time).

What makes a system "interesting" is how the components are
hooked together; certain hook-ups result in system behavior
that is quite different than the behavior of the components. This is
what happens in a control system. The components of a control
system are "causal" functions -- the variations in the value of the
output variable (y) depend deterministically on variations in the
value of the input (x). But the behavior of a control system is NOT
causal; it is "purposeful". The output does NOT depend on the input;
rather, the input is "controlled"; it is kept at a value specified by
the system itself and maintained at this value in the face of
disturbances.

The purposeful behavior of a control system results from the fact that
the components of the system are hooked up in a negative feedback
relationship. A negative feedback control system is most easily
demonstarted (by me -- because I am a math dunce) using linear
(rather than arbitrary) functions. Two functional components define
a negative feedback control system:

(1) o = k1 (p*-p)
(2) p = k2 o + k3 d

This is a "system" of equations; but the equations could represent
real "physical" components in real physical systems (as they do in
real control systems -- like thermostats and people). The first
equation represents an "output device" like a variable intensity
heater. o is the output variable -- the actual time variations in heat
from the heater; p is an electrical variable called the "perceptual signal"
and p* is a reference signal (the signal that -- it turns out -- specifies
the desired value of p; it is the signal that results when you "set" the
thermostat). Equation 1 says that the output of the output device is
proportional (by a factor k1) to a signal that represents the difference
between the reference and perceptual signals. Equation 1 is just an
input/output function -- input (p*-p) is converted into output
(o). The conversion factor is k1.

The second equation also represents a physical device -- in this case
a "sensor device" like the thermocouple that converts heat energy into
an electrical signal. The input to this device is the heat near the sensor;
this heat depends on the output of the heater (o) AND external dist-
urbances (d), such as outdoor air temperature, people in the room, etc.
So the input to the sensor is net heat near the sensor which is the sum
of heater and disturbance generated heat. The coefficients (k2 and k3)
represent physical factors that determine the extent to which output
and disturbance variables contribute to heat near the sensor. The sensor
converts the heat input (k2o + k3d) into an electrical signal -- the
perceptual signal, p, which can be considered a continuous measure of
the heat near the sensor. So equation 2 is also an input/output function --
input (k2o + k3d) is converted into output (p).

Now we can solve equations 1 & 2 simultaneously to learn something
about the behavior of the system as a whole. First, let's solve for the
output variable, o. The result is:

o = k1/(1+k1k2) p* - k1k3/(1+k1k2) d

This can be simplified by letting k2 and k3 (the physical constants) be
large realtive to k1 and about the same size (this is generally true
in real systems) . Then we get

o = k1p* -d

So the first thing we learn about a control system is that the output (o)
of the system does NOT depend on the input (p) -- the output depends
on the reference signal, p* and disturbances THAT ARE NOT EVEN SENSED.
If the reference signal,p*, is a constant, then variations in the output
of the system depend completely on unsensed disturbances. Note that
the disturbances, d, are mixed with the system's own outputs, o, to
determine the actual input (eq. 2). So surprising finding number one
about a control system is that its output depends PRECISELY on a
variable (d) that the system does not even sense. Simple math --
heavy result. Weiner and the cyberneticists, even with all their fancy
math, never picked up on this enormous fact. Psychologists don't WANT
to pick up on it because doing so would mean that they would realize
that they have been studying an illusion. The basis of experimental
methodology in psychology is the assumption that what people do depends
on what happens to them; that is, it is assumed that

(3) o = k1 p

A basic analysis of a control system shows that, if people are control
systems, then equation (3) does not hold. This is why control theory
(the real thing) is not real popular in scientific psychology circles.

Now let's solve for the other variable that the system can influence;
the perception. If we go through the same exercise (and make the
same assumptions) we will find:

p = p*

So, in a control system, the perceptual input variable is determined by
the reference signal, p*, NOT by distal stimuli in the environment. This is
surprising because psychologists think of perception (input) as an INDE-
PENDENT VARIABLE; something that causes behavioral outputs (see
equation 3) . But in a negative feedback control system, perception is
the DEPENDENT VARIABLE; the system always acts in order to keep its
input perceptual variable matching the reference signal SET BY THE
SYSTEM ITSELF. Of course, in artificial control systems like the thermostat,
the reference signal is actually set "from outside" -- but once that signal
is set the the system operates autonomously; that is, it does whatever
is necessary (what is necessary depends on disturbances) to keep its
input matching the reference signal. In living control systems, the
dials that set the references are not accessible from "outside"; the
reference signals for the system's inputs are "set" by other "systems"
in the organism itself. So living organisms are VERY autonomous, meaning
that they cannot be "controlled" from outside at all. They cannot be
controlled from outside because no one but the system itself has access
to the reference signals (p*) that are the ultimate determiner of what the
system "does" -- meaning, what inputs it controls and at what level they
are controlled.

So a simple analysis (to do it right you would have to include dynamics
but the algebraic results are correct under the assumption that the
system is dynamically stable) shows that a system composed of cause-
effect components arranged in a negative feedback organization is
NOT a cause-effect SYSTEM; it is a purposeful system. (By the way,
the "negative" in "negative feedback" refers to the net sign of the
multipliers to the variables that travel around the loop from input to
output; the "feedback" refers to that fact that the output of the system
is one of the influences on the variable that causes that output). The
purpose of the system is p* -- the desired input perecption. The system
achieves its purpose (continuously) in the face of undetectable and
unpredictable disturbances through the operation of the negative
feedback loop.

This was a VERY elementary introduction to perceptual control system
theory. The next step is to expand the system concept to form a
system of control systems -- such an organization makes it possible
for control systems to be the source of reference signals for other
control systems. Again, the behavior of this "system of systems" is
"more than the sum of its parts". But the behavior is still purposeful.
Once you start to look at this system of systems, however, you can
start to understand the main problem confronted by any control system
in a system of control systems -- CONFLICT.

But we can deal with conflict in our next episode -- if you're still awake.

By the way, I'm not sure how this all fits into Fritjof Capra's stuff.
My completely uninformed opinion is that he is like some of the
psychologists I know -- jumping right into very abstract, complex stuff
in a search of some deep revelations. My intuition is that depth comes
out of simplicity. If psychologists had been willing to look at the simple
facts of how people are connected to their perceptions, they would have
discovered years ago that

o = - k d and p = p*

and psychology could have become what it should be -- the study
of autonomous, purposeful system (APSs) -- like you and that cutest of
all APSs -- Lauren!

I love ya!