more about gain

[From Kent McClelland (2012.09.26.2005)]

The discussion of conflict, low gain, and dead zones in the "PCT Lament and Conflict" thread has been fascinating and informative from my point of view, and even though I haven't found the time to contribute to it myself, the exchange has made me glad that I've started following CSGnet again, after a substantial time away. It's really nice to see the current traffic on this forum zeroing in on questions directly related to the development of PCT, rather than chasing off all the time after side issues. Thank you, Martin, Rick, and Bill.

As I've worked my way through these posts, some of which have been pretty technical and not all of which I'm confident I understand, a number of questions have occurred to me, mostly connected with the idea of loop gain, which seems central to a lot of issues that interest me. Let me put them in the context of some recent posts:

[Bill Powers (in an undated post that I received on 2012.09.20.1030)]

BP: This [escalating conflict between two systems] actually depends on the loop gain of the combined control systems. This can be determined by tracing total gain around the two-system loop from any variable back to itself. Starting with one CV, we have the gain of the input, comparison, and output functions, follow by the gain of the connection from one system's output quantity to the CV of the other system, then through the other system in a similar way and back to the CV of the first system. At the same time, the feedback function of each system subtracts some amount from the magnitude of its own CV. If the input-to-output gain is high in both systems, it won't take much effect of the action on one system on the other CV to reach a threshold where the composite loop gain minus the local feedback function's gain exceeds unity, the threshold of static instability.

KM: Where can I look for some technical guidance on how to compute "composite loop gain"? I'd like to try calculating it for some of my simulations of conflict. And is the "threshold of static instability" the same thing as the "critical" value for total gain of two conflicting systems that I talked about in one of my earlier posts (Kent McClelland 2012.09.19.1029 MDT)?

In the discussion of dead zones vs. lowered gain that has been the centerpiece of the "PCT Lament and Conflict" thread, I wonder if it would help in thinking about these issues to make a sharper distinction between input gain and output gain, as Bill Powers (2012.09.28.0945) and Rick Marken (2012.09.28.1100) have started to do in their most recent posts. Is the dead zone that you've been talking about a dead zone on the input function or the output function? I could see it possibly working either way.

In the input function we might be understanding a dead zone as a threshold effect, where low levels of a particular input are not even registered as a perception (as when we need a certain level of brightness even to see a star at night), but above that threshold differences in the intensity of the input begin to be registered on a continuous scale of some kind. Or there might be some strong nonlinearities in the input function at the extremes of the scale being perceived, which would make it look like there is a dead zone at the bottom. And could low input gain make it seem like there is a dead zone, because perceptions with low gain are just not as sharp as high-gain inputs and thus might miss phenomena that to be perceived at the extremes of a scale? (Though this last seems pretty remote, I mention the possibility because stable equilibrium control with a low-gain control system leaves a bigger gap between the perception and the reference than with a high-gain control system.)

In the output function we might talk about a dead zone as a region of no output that happens because the control system is in passive-observation mode, as Bill describes it in B:CP. (Have I got the terminology right? I don't have a copy of B:CP with me at the moment to check it out.) And then a higher-level control system starts sending reference signals to the lower-level system in passive mode because the deviation from some standard at the lower level has reached a point that threatens the stability of a perception and requires adjustment at the higher level. A possible scenario: you hear a motor noise and pay no attention--don't even notice it, although it's registering out of consciousness--but then it begins to get louder and some higher-level control system begins to interpret it as an approaching vehicle, and you jump aside to save your skin.

So, would a dead zone be in input or output, both or neither?

Finally, I'm really interested in people's thoughts on perceptions with gains high enough that they approach or pass the "threshold of static instability", as Bill describes it in the post I quoted above. It seems to me that perceptions like that are interesting for a variety of reasons.

For one thing, I see the learning process as a progression in raising that threshold for a given perception. As we become more practiced and skillful in a given perception, we can control it at a higher level of gain without having it blow up in our faces, as it were.

Second, and related, it seems to me that activities that we describe as play, particularly children's play, but also many recreational pursuits for adults, involve explorations of those limits of control, seeing how much one can jack up the gain before the process goes unstable, seeing if one can operate on the knife-edge of instability. The dangerous pursuits, like rock climbing, that are also exhilarating and fun must be like that. And the sense of "flow" that comes from these activities must have to do with the concentration needed for maximal performance (like devoting all of the gain resources at one's disposal to the activity and and turning down the gain on everything else). As a sociologist, I'm particularly interested in collective activities that might operate on this level of maximal stable gain. I have a lot more I might say along this line, but I'll stop here and see if anyone thinks I'm barking up the wrong tree.

Kent

[From Rick Marken (2012.09.30.1050)]

Kent McClelland (2012.09.26.2005)--

KM: In the discussion of dead zones vs. lowered gain that has been the centerpiece of the "PCT Lament and Conflict" thread, I wonder if it would help in thinking about these issues to make a sharper distinction between input gain and output gain, as Bill Powers (2012.09.28.0945) and Rick Marken (2012.09.28.1100) have started to do in their most recent posts. Is the dead zone that you've been talking about a dead zone on the input function or the output function? I could see it possibly working either way.

RM: I didn't mean to be distinguishing input from output gain. But
there can be gain on both the input and output side. Actually, I think
there are three places where gain can occur in a control loop: input
(call it g.i), output (g.o) and feedback (g.f).

Here's how I think it could work (in terms of algebraic rather than
differential equations):

p = g.i * q.i
o = g.o * (r-p)
q.i = g.f * (o + d)

I think the only gain that could be part of a higher level control
loop, such as one controlling for keeping a conflict from
accelerating, is g.o (output gain).

KM: In the input function we might be understanding a dead zone as a threshold effect, where low levels of a particular input are not even registered as a perception (as when we need a certain level of brightness even to see a star at night), but above that threshold differences in the intensity of the input begin to be registered on a continuous scale of some kind. Or there might be some strong nonlinearities in the input function at the extremes of the scale being perceived, which would make it look like there is a dead zone at the bottom. And could low input gain make it seem like there is a dead zone, because perceptions with low gain are just not as sharp as high-gain inputs and thus might miss phenomena that to be perceived at the extremes of a scale? (Though this last seems pretty remote, I mention the possibility because stable equilibrium control with a low-gain control system leaves a bigger gap between the perception and the reference than with a high-gain control system.)

RM: I'm still very tentative about this dead zone being of much use.
My simulations suggest that the dead zone does nothing but make
control worse. This might be good for keeping conflict from
accelerating but the same thing is accomplished by reducing gain. So
until I see some evidence, in terms of improved fit to data with,
rather than without, a dead zone, I will prefer output gain reduction
as the explanation of why conflicts usually don't explode into full
scale violence. And, as I said before, gain reduction is more
consistent with my subjective experience of what happens when I keep a
conflict from exploding; the error (the feeling that something is
wrong) is still there but I just lighten up my response to that error.
As you say, error remains high in low gain systems and that's what I
experience when I "lower the gain" when in a conflict; the error is
still there (which is unpleasant) but I put up with it for the sake of
keeping the conflict (relatively) in check.

KM: So, would a dead zone be in input or output, both or neither?

RM: I put it right after the comparator, as per Martin's request. So I
guess it's on the output side. I don't see how it could go on the
input side since the reference defines what an error is so if you
block out some part of the input (p) it might not be the part that is
currently the part that would result in an error.

KM: Finally, I'm really interested in people's thoughts on perceptions with gains high enough that they approach or pass the "threshold of static instability", as Bill describes it in the post I quoted above. It seems to me that perceptions like that are interesting for a variety of reasons.

RM: A control loop goes unstable when it's gain (that's loop gain; the
product of all gains around the loop: g.i*g.o*g.f) is too high
relative to it's slowing properties (and transport lag probably
affects it also). So anything that increases loop gain is likely to
lead to instability unless the system is able to compensate for this
increased gain with an increase in slowing. Conflict increases loop
gain by increasing g.f (I think; Bill will correct me if I'm wrong) so
conflict can lead to instability unless increases in loop gain are
compensated for by decreases in components of loop gain (such as a
decrease in g.o, which would offset the increase in g.f) or by
increasing the slowing factor in the loop (I think Bogarting a joint
can accomplish this;-)

KM: For one thing, I see the learning process as a progression in raising that threshold for a given perception. As we become more practiced and skillful in a given perception, we can control it at a higher level of gain without having it blow up in our faces, as it were.

RM:I don't think there is a threshold component to this on the input
side but I agree that part of what you learn to do when you learn to
control is control a perception with higher and higher gain without
going unstable.

KM: Second, and related, it seems to me that activities that we describe as play, particularly children's play, but also many recreational pursuits for adults, involve explorations of those limits of control, seeing how much one can jack up the gain before the process goes unstable, seeing if one can operate on the knife-edge of instability.

RM: Yes, I agree.

KM: The dangerous pursuits, like rock climbing, that are also exhilarating and fun must be like that. And the sense of "flow" that comes from these activities must have to do with the concentration needed for maximal performance (like devoting all of the gain resources at one's disposal to the activity and and turning down the gain on everything else). As a sociologist, I'm particularly interested in collective activities that might operate on this level of maximal stable gain. I have a lot more I might say along this line, but I'll stop here and see if anyone thinks I'm barking up the wrong tree.

RM: I think you barking up one of the right trees. But gain is not the
whole forest. For collective activities I think an important aspect of
successful collective control is learning to control the "right"
perceptions (which would include perceptions of what other people are
doing). Coordination is rarely just a matter of pushing as hard as
possible on the same perception.

By the way, I read your paper (with Worthington) on investing behavior
"Control-Theory Simulation of Buying and Selling Behavior in a Market"
and I loved it! Great job. I hope you are trying to get it published;
it's really a great application of control theory to economic
behavior.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

Regarding: `Rick Marken (2012.09.30.1050)]

RM: I didn’t mean to be distinguishing input from output gain. But

there can be gain on both the input and output side. Actually, I think

there are three places where gain can occur in a control loop: input

(call it g.i), output (g.o) and feedback (g.f).

`

Can't gain also occur with respect to the error signal? (In a sense, affecting the error signal's importance in stimulating an output)

``

With Regards,

Richard Pfau

···

From: Richard Pfau (2012.10.02 11:45 EST)

[From Rick Marken (2012.10.02.1405)]

From: Richard Pfau (2012.10.02 11:45 EST)
Regarding: Rick Marken (2012.09.30.1050)]

>RM: I didn't mean to be distinguishing input from output gain. But
there can be gain on both the input and output side. Actually, I think
there are three places where gain can occur in a control loop: input
(call it g.i), output (g.o) and feedback (g.f).

RP: Can't gain also occur with respect to the error signal? (In a sense, affecting the error signal's importance in stimulating an output)

Yes, that's what g.o is; it's the gain that determines how much output
is produced per unit error.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[From Erling Jorgensen (2012.10.02 16.15 EST)]

Richard Pfau (2012.10.02 11:45 EST)

Can't gain also occur with respect to the error signal?? (In a sense,
affecting?the error signal's?importance in stimulating an output)

This sounds similar to the output gain (g.o), since "importance" would
presumably register as an amplifying/multiplication factor for the output.

But in a similar vein, I have thought about the emotion system of the
body as a way of modulating gain. Certain emotions such as anger or fear
seem to function as a way of recruiting additional resources & energy
from the body, to assist in enacting a desired result from a given course
of action.

A further correlation that others have noted is that different emotions
may be tied to different patterns in the _changes_ of error signals.
For instance, long sustained error is similar to depressive states.
Steeply rising error is akin to alarm & fear. Sudden reduction in error
is akin to pleasure or joy. A continuing error despite effort being
expended is akin to frustration. (I realize that "akin" is not a very
rigorously defined term here.)

This raises the question of how to think about the perception of emotions
themselves. Are they (a) the corresponding perceptions of the body going
about its business of making things happen in the world? Or are they
(b) a way of perceiving error signals themselves, as a meta-perception of
how much error is going on, so that the parameter of gain can be more
specifically targeted & adjusted?

I don't know which of these options to choose. The 'qualia' of emotions
makes me question interpretation (a). I don't seem to detect low-grade
emotion perceptions at work from every little thing that I do, even though
obviously my body is implicated in the doing. But if (b) is preferable,
we certainly don't want a back-seat driver kind of system that tries to do
what the control loop is already well suited to do by matching perceptions
to references & thus reducing error signals directly.

I do note that in our models of control -- which work very, very well as
single loops or as loops operating in parallel or as a series of a few
hierarchical layers -- we do not yet have a model of how gain is adjusted
by the working simulation itself. Perhaps there has yet to be a need
for that, because our simulations are not yet operating in a very rich
environment of competing demands, where shifting decisions through raising
& lowering gain might be called for.

But I do wonder whether the body's system of emotions -- whether with
humans or with animals, as Darwin noted -- might be an evolutionary
elaboration for situationally-sensitive, reversible modifications of gain
in control systems working on the fly. Why else are emotions there?

All the best,
Erling

[Martin Taylor 2012.09.30.20.45] (edited 2012.11.16.17.35)

Back on the western shore of the Atlantic, cleaning up a loose end

with this posting.

[From Kent McClelland (2012.09.26.2005)]
[Bill Powers (in an undated post that I received on 2012.09.20.1030)]
... If the input-to-output gain is high in both systems, it won't take much effect of the action on one system on the other CV to reach a threshold where the composite loop gain minus the local feedback function's gain exceeds unity, the threshold of static instability.
KM: Where can I look for some technical guidance on how to compute "composite loop gain"?
Try this. Consider two control systems, called control system X and

Y because they influence the values of external complexes “X” and
“Y”. Each of them is unaware of the existence of the other, but the
side-effects of each one’s control actions disturb the value of
variable the other is controlling. The situation is suggested in the
figure:

![Re more about gain.jpg|766x406](upload://4bPulU3jHFIPmqjlp6M09gvAxVk.jpeg)

System X could be an ordinary control unit, while system Y might

represent the effects of all the other control systems in the world
that are influenced by the side effects of system X’s output or
whose side-effects disturb system X’s input. Or, both systems might
be ordinary control units.

In the figure, signal values are labelled with lower-case letters,

such as px and py for the perceptual signal values in systems X and
Y, and xy and yx for the side-effect values that influence the
environmental complexes that are perceived by the other control
system. The gains of the different loops and paths are marked by G,
as in Gx and Gxy.

The "standard" conflict situation occurs when X and Y are the same

thing in the environment.

One can think about this system as having three superimposed

feedback loops, the feedback loops of the two individual local
control systems X and Y, and a “composite” loop that goes through
both, like this:

![Re more about gain1.jpg|766x406](upload://raEcm1M3jgbEGs7lYjSaSfoRTnS.jpeg)

Assuming that the path segments have gain 1.0 except for those

marked with "G"s, and remembering that the two comparators have the
effect of reversing the sign of any changes in the perceptual
signal, the gain of this composite loop is GxGxyGy*Gyx, which we
will call “Gc” for “composite loop gain” where it is convenient to
do so in what follows. Typically the composite loop gain is
positive, which would immediately signal instability if its
magnitude were greater than 1.0, but the negative loop gains of the
individual control loops can reduce or eliminate the instability.

On CSGnet, the usual way of analyzing a control system is to start

at a point in which one is interested, and work back along the
signal path to figure out where the value at that point comes from,
continuing until one arrives back where one started, eventually
winding up with an equation that has the value of interest on one
side of the equals sign, and an expression containing only constants
and the input variables on the other. The result represents a set of
values to which the value of interest would converge for those
values of the input variables, if they were held stable for a long
time and if the system is stable.

In the present case, the value of interest is px (and py, but since

the situation is symmetric, we need only consider one of them), the
constants are the Gains Gx, Gy, Gxy, and Gyx, and the input
variables are rx, dx, ry, and dy.

Because there are four constants and four input variables, the

algebra inevitably gets messy at times, and there is a very high
probability that I will make a mistake with a minus sign or invert
an x and a y or get a bracket wrong. So please take the final result
with a grain of salt unless you have gone through and carefully
checked the algebra.

Of course, what we really want is the dynamics, the way that the

system responds to changes in the values of the input variables, and
that depends on transport delays and the time courses of the
functions. For example, the algebra below treats the output gain of
a control loop as a simple multiplier, but usually we model the
output as an integrator. Nevertheless, I think it worthwhile to go
through this algebraic exercise, if only to illustrate how the
different signals interact.

If you are interested in the stable end-points of this set-up, I

suggest you do the algebra yourself rather than relying on my
results.

To simplify the visual appearance of some of the equations, and to

reduce the likelihood of making the kinds of error I just mentioned,
I will use a shorthand for some expressions that recur. In
particular, I will use

XG = Gx/(1+Gx)

YG = Gy/(1+Gy)

Gc = Gx*Gxy*Gy*Gyx

Here's the picture again. All the path gains are 1.0 except Gx, Gy,

Gxy, and Gyx.

<img alt="" src="cid:part1.00080607.05070206@mmtaylor.net" height="234" width="441">

First we consider the local X control loop, which has two sources of

disturbance, dy and yx.

px = sx = ox + dx + yx

     = Gx*rx - Gx*px + dx + yx

px*(1+Gx) = rx*Gx + dx + yv

px = rx*XG + dx*XG/Gx + yx  (remember XG = Gx/(1+Gx) )

This is fine except for that pesky yx, which is a variable we will

have to hunt down. That’s where the algebra begins to get messy. We
will find an expression for yx and then substitute it back into the
expression for px.

yx = Gyx*oy        now we track oy backwards around the Y control

loop

     = Gyx*Gy*ry - Gyx*Gy*py

     = Gyx*Gy*ry - Gyx*Gy*oy - Gyx*Gy*dy - Gyx*Gy*xy

Oops. We have come back to oy again, which is where we started in

tracking yx. So we had better simplify this to find an expression
for oy. We can multiply that expression by Gyx to find yx, which we
can then substitute back into the expression for px.

oy =Gy*ry - Gy*oy - Gy*dy - Gy*xy

oy*(1+Gy) = Gy*ry - Gy*dy - Gy*xy

oy = YG*ry - YG*dy - YG*xy

Since yx = Gyx*oy we have

yx = Gyx*YG*ry - Gyx*YG*dy - Gyx*YG*xy

Substituting back into the expression for px

px = rx*XG + dx*XG/Gx + Gyx*YG*ry - Gyx*YG*dy - Gyx*YG*xy

We still have an unwanted variable xy on the right hand side, so we

had better find an expression for it and substitute the expression
we find into the equation.

xy = Gxy*ox

ox = Gx*rx - Gx*px

This gets us back to px, so we need go no further. Substituting xy =

GxyGxrx - GxyGxpx into the expression for px, we have

px = rx*XG + dx*XG/Gx + Gyx*YG*ry - Gyx*YG*dy - Gyx*YG*Gxy*Gx*rx +

GyxYGGxyGxpx

writing Gx*Gxy*Gy*Gyx as Gc for visual simplicity,

px = rx*XG + dx*XG/Gx + Gyx*YG*ry - Gyx*YG*dy - Gc*rx + Gc*px

px*(1-Gc) = rx*(XG-Gc) + dx*XG/Gx + ry*Gyx*YG - dy*Gyx*YG

px = (rx*(XG-Gc) + dx*XG/Gx + ry*Gyx*YG - dy*Gyx*YG)/(1-Gc)

The divisor 1-Gc looks as though it could cause a lot of trouble if

Gc approaches or exceeds 1.0. Now is the time to expand the XG, YG,
and Gc shorthands to see whether it really does. Let’s do it one
term at a time, to avoid long lines of text full of lots of brackets
and divisions.

rx multiplier (XG-Gc)/(1-Gc)

XG-Gc = (Gx/(1+Gx) - Gc)/(1-Gc)

           = Gx*(1-Gc)/((1+Gx)*(1-Gc))

           = Gx/(1+Gx)

dx multiplier XG/(Gx*(1-Gc))

XG/(Gx*(1-Gc)) = 1/((1+Gx)*(1-Gc))

So long as Gc is not near 1.0, the dx multiplier approaches zero as

Gx increases, but control is not as good as it is when the composite
loop does not exist. If Gc>1.0 the multiplier is negative.

ry multiplier Gyx*YG/(1-Gc)

Gyx*YG/(1-Gc) = Gyx*Gy/((1+Gy)*(1-Gc))

                        => Gyx/(1-Gc)    as Gy increases

  or

                        => -1/(Gx*Gy*Gxy)  as both Gy and Gc

increase

Apart from the sign, the dy multiplier is the same.

None of this, if the algebra is correct, implies that the effect of

the Y system’s control will help the X system to control, and some
values of the parameters (e.g. the composite loop gain being near
1.0) lead to very unwelcome results. But it should be emphasized
that the situation analyzed is very artificial, especially as the
two individual loop gains are taken to be simple numbers rather than
as time-varying functions.

Social systems have other possible connections, in particular

connections in which the effect of the action of one control system
on another is to influence its feedback path. Such interconnections
may prove to be more important than the mutual disturbance
connection discussed here.

I hope the above helps to clarify the concept of the composite loop,

even though the algebra is not much help in modelling the waveforms
expected after variations in one or more of the input variables.

Martin