Modeling Relaxation

[Kent_McClelland2]

[From Kent McClelland (951009.1200 CDT)]

Bill Leach (951007.23:48 U.S. Eastern Time Zone)
Martin Taylor (direct)
Bill Powers

Bill Leach & Martin: Thank you for your suggestions on my question about
modeling the de-escalation of conflicts. I appreciate the plausibility of
what you say, but I can't make it fit with my models. Something must be
wrong either with my models or your intuition. Once outputs of two
conflicting have diverged, nothing reasonable that I've tried so far will
bring the OUTPUTs back to zero. (Maybe I'm too hung up with the observer's
viewpoint by looking at outputs. . . )

First, Martin's suggestion:

May I suggest a small set of experiments? Try adding a small noise disturbance
after imposing the intermediate reference, or (almost equivalently) make
the perceptions or the outputs of the two competing control systems
independently noisy. I mean add a small random number different on each
iteration.

My thinking here is that the still opposed efforts of the two control systems
are balanced, but perhaps when a disturbance makes the CEV change in
one direction it reduces the force applied by one and increases the
other, which may bring the two dynamics out of their "in-phase" relationship
and break the deadlock.

Certainly plausible suggestions, but these experiments don't in fact
produce de-escalation. Adding a disturbance, large or small, random or
fixed, is essentially irrelevant. Unless one or both of the control
systems are at maximum output, the outputs of the two control systems just
move in parallel maintaining the (conflictive) difference between them even
while they cooperate to stabilize the environmental variable (as my 1993
Durango paper on "conflictive cooperation" showed).

Now Bill Leach's comments:

... integration" factor I'm using in the formulas.) The outputs may not
ever return to zero, at least not in any reasonable length of time.

Kent, I believe you are encountering an artifact of digital computing.
A good example of the sort of problem that you might be encountering is
the "spice" simulation of a blocking oscillator circuit when exactly
symetrical componets are simulated. The circuit will not oscillate yet
in practice the real circuit will always oscillate.

I'm not familiar with the "spice" simulation. Can you give me a reference?
"Exactly symetrical components" is not my problem. The failure to
de-escalate occurs even when the gains of the two conflicting systems are
different.

Specifically, in the basic control system as used in PCT, if the error
signal value actually becomes zero then there will be no output force
generated.

I think your statement must be incorrect here. A system's error signal
value becomes zero when its output balances any disturbances (and, of
course, the output of any other conflicting/cooperating systems). What
happens next is not that the output goes to zero, but that the output
remains _unchanged_ staying at the same level whether large or small.

In the PCT model then "status quo" requires unchanged error signal
magnitude. Thus, in your spreadsheet example, when the reference was
changed so that both systems suddenly found no error between perception
and reference then the error signal value should have dropped to zero.
The actual output of both control loops should have decayed to zero at a
rate determined by the appropriate time constants. . .

Whether ouput decays toward zero or not depends on the integration factor I
use. If I use the simplest kind of control-system formula (with no
integration factor), like Tom Bourbon uses in his tracting experiments, no
decay whatever takes place. The outputs are simply stuck wherever they
happen to be when the conflict is resolved in the reference levels. If I
use "leaky integration" formulas, like those from Bill Powers's Byte
articles or Rick Marken's spreadsheet demo, very slow decay does take
place, but the time it takes for the outputs to reach approximately zero is
orders of magnitute larger than the time taken for the escalation of the
conflict. When I noted this "equilibration" effect in my 1993 presentation
at Durango, several knowledgable folks criticized my interpretation. They
said the apparent equilibration was simply an artifact of the "leaky
integration" formula I was using.

I would suggest that this explaination is an assertion that "yes,
conflicts can be fully resolved" and that one way is if the references
"match".

My demos indicate that this suggestion is also in error. A matching of
reference levels simply doesn't produce full resolution of output
differences in any finite time.

charley horse

I don't know that I believe that the cause of a "charley horse" is
necessarily a "low level organismic conflict". I always thought that a
"charley horse" was a muscle injury as a result of overstress regardless
of cause. Thus an external force applied quickly enough to a muscle that
was tensioned properly can result in the experience.

I used the muscle spasm or charley horse as an example of low-level somatic
conflict in my 1994 Sociological Perspectives paper (and in several earlier
versions of that manuscript), and nobody called me on it. Was I wrong?

Bill, I appreciate your suggestions, but my models seem to say something else.

A comment from a post by Bill Powers (951007.1500 MDT) about positive
feedback also has a bearing on my questions. He's responding to Brian
D'Agostino.

Brian:

    ... The spiral model predicts that under such circumstances,
    country A's force reductions, by reducing the threat posed to
    country B, should result in country B reducing _its_ forces and
    thus the threat it poses to country A, etc. . . .

Bill:

This is an essential insight; few people understand the other side of
positive feedback (what goes up once it starts up is just as likely to
go down once it starts down).

Bill, I don't know how to model the positive-feedback process you're
talking about. The only way I can simulate rapid de-escalation is by
suddenly switching the reference levels for the two parties (giving
reference A to system B and vice versa instead of having them agree on a
compromise value) and then arbitrarily setting both back to the compromise
value when the outputs near zero. This reverse-positive-feedback scenario
seems so remote from anything that might happen in real life that I can't
take it seriously as a simulation. When have parties to a conflict ever
simultaneously traded positions? Perhaps the reason that in real life
disarmament rarely takes place is not because of some personality bias of
national leaders, as you suggest in your post, but because control systems
have no need to change their outputs once a compromise on reference levels
is reached with the conflicting party.

I repeat my original question: How in a control-system world does any real
relaxation of tensions take place? How can one model relaxation of
conflict?

Kent

[Bill_Leach1]

<[Bill Leach 951009.18:17 U.S. Eastern Time Zone]

[Kent McClelland (951009.1200 CDT)]

Kent, let me start by agreeing with you fully that we have to have a
working model that demonstrates my assertions before what I say is
significantly more than just conjecture.

The "random noise" suggestion of Martin's was essentially based upon the
idea that your model is in the unstable state of an astable condition.
If that assertion is true then the noise should cause the system to
return to he stable state.

I notice that you mention that you had employed random noise application
to the perceptions in the past. A very real possiblity still exists for
"digital artifacting" to be occurring but I will admit that I do not
quite see how.

Spice

Spice is a computer program designed to simulate electronic circuits both
analog and digital. Very good simulations introduce thermionic noise and
can even deal with intra/inter-component coupling and standard variations
in component characteristics. The typical Spice version running on say a
PC does however treat "identical" components as though they really are
identical.

Digital artifacting refers to several problems inherent in iterative math
and logical operations when applied to analog simulation/emulation.

Two of the most obvious problems presented by digital computers
attempting to simulate/emulate analog systems is that all changes are of
discrete magnitude and occur only in specific descrete intervals.

An additional problem is that there typically is a fixed phase
relationship between all variables or events. Even when techniques such
as multitasking and multiprocessor methods are used to ensure that at
least some relationships are not fixed in phase this problem can still
persist.

It can often be extremely difficult to detect a problem that exists
because of digital artifacting.

Now back to our regularly scheduled program!

Specifically, in the basic control system as used in PCT, if the error
signal value actually becomes zero then there will be no output force
generated.

I think your statement must be incorrect here. A system's error signal
value becomes zero when its output balances any disturbances (and, of
course, the output of any other conflicting/cooperating systems). What
happens next is not that the output goes to zero, but that the output
remains _unchanged_ staying at the same level whether large or small.

You might want to go back an look at the "blood bath" between Rick Marken
and Martin Taylor a year or so ago on this subject. I ended up "siding"
with Martin on this one and eventually Bill P. agreed with what both
Martin and I were asserting.

In the "simple" control loop, output is zero when error is zero. That is
       o = k * ( r - p )
           e = ( r - p )
then:
       o = k * e

thus:

For the simple system, no matter what the gain of the system if "e"
(error) is zero then output is zero.

Another possibility for a relatively simple control loops is:

       o = k * ( r - p ) + m

where "m" would be a "fixed bias" applied to the system. "m" also could
be the "reset" value (later). I don't think that any of the literature
in PCT deals with this sort of system.

Integrator and "leaky integrator"

The "Integrator" increases the magnitude of "m" as long as the error
signal is non-zero at some time constant normally called "reset rate".
When a controller uses "reset" then it is almost always required that the
controller also have "derivative gain". That is the output formula
becomes:

       o = k * ( r - p ) + m + ( d[ r - p ]/dt * l )

where: "l" is the gain factor for the first derivative.

There are other reasons for derivative gain (like response) but the use
of integration normally requires that "k" be smaller than for a "plain"
control system to maintain stability. In "engineered control systems"
terms this was called "reset". In an engineered control system it is
necessary for the error to have the ability to reverse sign or the value
of "m" could never decrease.

The "leaky integrator" is one option when the error signal can not
reverse sign. A constant "drain" of the magnitude of "m" (toward zero)
is established. The time constant of this drain must be greater than the
time constant for the effect of the error on "m" or "m" will just "track"
error which would defeat the whole purpose of having integration in the
controller.

If the "leaky integrator" is used by itself then the output formula would
look something like this (loosely):

       o = o + ( j * ( 1 < ( k * e - o ) < e )) - ( i * o )
        (new)

where the last term is the decay term and "i" is much less than 1 and the
middle term can not become negative. In this case "k" must rather large
if the system is to respond rapidly to disturbance. The problem with
this scheme is that should "e" become less than "o" the system will be
very slow to respond.

A third use for integration is delay. In this case the formula for "o"
is (again, loosely):

         o = o + ( j * ( k * e - o ))
          (new)

Note that in this case "o" will eventually always be equal to "e * k" (or
if you prefer [ ( r - p ) * k ]). "j" just sets the rate at which the
integration takes place (and is always less than 1). What happens here
is that "o" always changes only a fraction of the difference between "e"
and "o" which slows down the response. It is still possible (and common)
to have different "sensitivities" to the direction of the difference
beween "e" and "o" (though not with the formula written above).

So with "pure" integration but no sign reversal from the error signal,
the output will never decrease in magnitude. With leaky integration the
output will decrease but at the typically much slower rate set by the
drain time constant.

I will state flatly (opening myself for some proof to the contrary) that
if your model can maintain a non-zero output when the associated error
signal IS zero then your output function is not a proper model in the PCT
paradigm (though again the reason that the model does this might even be
a digital artifacting problem -- maybe something like the order in which
various calculations are performed).

In the basic PCT control loop some non-zero error signal value is
required for a non-zero output. If the output function gain is very high
then a very tiny error signal will result in a large output but again,
the error must be non-zero for a sustained non-zero output.

I do not know if _any_ evidence of "reset" has been observed in living
control systems (and I imagine verifying a suspected case could be quite
difficult).

... When I noted this "equilibration" effect in my 1993 presentation at
Durango, several knowledgable folks criticized my interpretation. ...

I don't know if I am enough of a mathematician for this but it seems that
with almost any value of leaky integration, if the maximum output
magnitude is again the same (so that the errors really do remain at zero)
then the decay time constant for a zero error is what applies. This
assumes however that the problem is not a digital artifact such as a
computational order problem.

Unfortunately, spread sheet programs present a special problem in trying
to debug potential logic errors or digital artifacts. It is just not
practical to plot every variable. Nor is it even possible (with any
spreadsheet program that I know of) to dump variables including
"intermediate calculation results) to a file for later analysis.

Also, I think, worthy of note is that the sort of conflict that you
describe for your simulation is rarely as simple for the living systems
case. The axis of control is normally not exactly in phase even when all
parties think that it is. Actual reference values are probably never an
exact match. Such differences are probably very significant in the
dynamics of control system conflict.

Again, I do agree that determining what is going on with this program of
yours is important, particularly if a similar experiment has not been run
using a regular programming language.

Some suggestions:

Can you try changing the references at a different rate? Have you tried
doing this? With random noise present?

Have you tried "randomizing" the application of random noise or when
changes to the existing random noise occur?

Can you set the maximum output capability differently for the two control
loops? Again, have you tried this along with changing the references at
a different rate or time and applied random noise.

Try the two references just slightly overlaping as opposed to exactly the
same value when you make a step change. In fact try overlapping them by
an extreme amount as neither of these should make any difference unless
some digital artifact is involved.

                             **** Oops ****

I see from your remarks to Bill P. that you have done this (what I am
calling overlap the references). Will have to think some more about
this.

I would suggest that this explaination is an assertion that "yes,
conflicts can be fully resolved" and that one way is if the references
"match".

My demos indicate that this suggestion is also in error. A matching of
reference levels simply doesn't produce full resolution of output
differences in any finite time.

You are also dealing with pure "one-way" control systems. This is a
condition I suggest seldom exists where conflict is even possible. I
would imagine that the system set that controls the heart muscles are
probably all "one-way" control loops (but then the physical
characteristics of the heart's construction causes the muscles to return
following a "pumping action). Possibly many of the metabolic control
loops are "one-way" and require the functioning of other systems to
"drive" the perception in the opposite direction should the perception
exceed the reference.

I used the muscle spasm or charley horse as an example of low-level
somatic conflict in my 1994 Sociological Perspectives paper (and in
several earlier versions of that manuscript), and nobody called me on
it. Was I wrong?

I certainly don't know. Is there evidence that an opposing muscle or
muscle group is the source of the strain injury?

re: your comments to Bill P.

Remember also that this thing that we are calling "escallation" and
"de-escallation" is, in a very real sense, neither.

I suggest that we don't have a reference signal for "preparing for war".
We have a group of reference signals that when met we incidentally call
"being prepared for war".

Also "disarmament" is not just a lowering of the value of the set of
references for "preparing for war". This is true both from a concept
point of view and the physical activity view.

-bill

[BOB_HINTZ1]

Hi, Kent

I'm not sure if this will come to you or go out on the net, but in
either case I would like to suggest that conflict between living control
systems is a form of relating that has a beginning and an end. A chase
is a universal instance of conflict focused on controling the distance
between us. If I want to eliminate it and you want to maintain some
preferred amount, then you must counteract the effects of any action
which I perform to acheive my reference. The conflict is potential as
soon as I observe you and start to approach you. The conflict becomes
actual when you observe my approach and begin to flee. The conflict is
observeable by a third party when specific aspects of both the activity
of the chaser and the chasee can be described as contingent upon
specific aspects of the other. The conflict is over when either loses
perceptual contact with the other, although each may continue organize
activity in terms of the potential conflict with the other should
perceptual contact be reestablished.

If conflict is a form of relating between control systems, then it
doesn't relax. It might be happening or not happening. It might be
very important or not very important. It might be beginning or nearly
over. It might be intentional or accidental from the point of view of
one of the participants or an observer and either participant might be
very relaxed, depending on their confidence in the outcome, but the
conflict itself does not relax.

I have struggled with the question of form for a couple of years and
have a draft of a paper if you would like to see it. I live in Vinton,
Iowa and would be interested in exploring more about your work. I am a
family therapist who has been out of the academic loop for several years
and am just getting re-involved.

Take Care, Bob Hintz