[Bulk] Re: least action-control efficiency

CSGnet Archive CSG2006
1

[Martin Taylor 2006.07.08.00.16]

from Tracy Harms 2006;07,07.20:30 Pacific

FRom [Marc Abrams (2006.07.07.2235)]

> > from Tracy Harms 2006;07,07.18:00 Pacific
>
> > Error must be formally
> >independent of gain.
>

   OK, not that I agree with Jim, but how do you come
to the notion that

> "error" and "gain" are in fact independent?

In a constructed control system, gain is set by the maker of the system, or is adjustable by a knob. Gain is ordinarily fixed over times of interest in teh analysis of the control function. Error depends on the disturbance and the dynamic characteristics of the control system (in which gain plays a part) and varies over time.

Sincegain is a property of the control system, error is a function of the disturbance as well as of the gain, and the disturbance has a source independent of the control system, gain and error are logically independent.

Having said that, it is true that so long as the control system is stable, for a given disturbance the error will ordinarily be lower if the gain is higher. Too high a gain could make the system go unstable, and the error could then become very large. So, if you look at it from the designer's point of view, gain can affect error.

So, Tracy is right that error is independent of gain, because the designer affects the gain whereas the disturbance that creates the error comes from some other source entirely; and Marc is right to raise the question because for a given disturbance the choice the designer made affects what the error will be.

Martin

···

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2

[From Rick Marken (2006.07.07.2210)]

Martin Taylor (2006.07.08.00.16) --

> from Tracy Harms 2006;07,07.18:00 Pacific
>
> Error must be formally independent of gain

Since gain is a property of the control system, error is a function of the disturbance as well as of the gain

Correct.

for a given disturbance the error will ordinarily be lower if the gain is higher.

You got it.

So, Tracy is right that error is independent of gain

What? You just said that error depends on gain (error will be lower the higher the gain). And now you say that error is independent of gain.

If you go to the bible (B:CP), Appendix para. 1 p. 276 you will find therein the following derivation:

(6) p = (k.e k.o r + k.d d)/(1+k.e k.o)

where k.e and k.o are equivalent to G and F, respectively, in my "Control Efficiency" post. So k.e k.o = GF = loop gain.

As the loop gain increases, equation (6) approaches

p = r

For example, if k.e k.o = 100 (and k.d = 1) then equation 6 becomes:

p = (100/101) r + (1/101) d

if the gain is increased to k.e k.o = 1000 then equation 6 becomes

p = (1000/1001) r + (1/1001) d

So as loop gain increases, p approaches r and error (the difference between r and p) approaches zero.

So error and gain are not formally independent; in fact, they are precisely inversely related.

Best

Rick

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3

From [Marc Abrams (2006.07.08.0956)]

> [Martin Taylor 2006.07.08.00.16]

> In a constructed control system...

  Do you consider "natural" control systems "constructed"? Right now, from what I have seen of physiological control processes they are
_functionally_, but not literally, the same thing.

> gain is set by the maker of the system, or is adjustable by a knob.

OK, but what is it that is being set?

  >Gain is ordinarily fixed over times of interest in teh analysis of the control function.

  OK, but is this true of natural control processes? I don't believe this may be true. I'm not saying it isn't. I'm saying I have my doubts.

  > Error depends on the disturbance and the dynamic characteristics of the control system (in which gain plays a part) and varies over >time.

OK

  >Since gain is a property of the control system, error is a function of the disturbance as well as of the gain, and the disturbance has a >source independent of the control system, gain and error are logically independent.

  Yes, given your premises there could be no other answer. But I question whether or not your premise is in fact true for natural perceptual control systems. If not, it would render your syllogism meaningless.

  > Having said that, it is true that so long as the control system is stable, for a given disturbance the error will ordinarily be lower if the >gain is higher. Too high a gain could make the system go unstable, and the error could then become very large. So, if you look at it >from the designer's point of view, gain can affect error.

  Martin, I appreciate your thoughtful (as usual) and measured reply. But you have said nothing to dissuade me from the idea that our emotions are connected in some way to both error and gain. It might be helpful if we could come up with an operational definition of exactly what "gain" would be in a physiological control process.

  >So, Tracy is right that error is independent of gain, because the designer affects the gain whereas the disturbance that creates the >error comes from some other source entirely; and Marc is right to raise the question because for a given disturbance the choice the

designer made affects what the error will be.

  I think its very much of an open question as far as natural control processes are concerned. "Gain" is set in a man made control system because it is vital for the stability of the system. Humans are not "engineered" as such and so unlike man made control processes emotions might very well be a part of our ability to help "manage" our control processes.

Regards,

Marc

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4

[From Rick Marken (2006.07.08.0740)]

Martin Taylor (2006.07.08.08.47)

Rick Marken (2006.07.07.2210)--

Martin Taylor (2006.07.08.00.16) --

So, Tracy is right that error is independent of gain

What? You just said that error depends on gain (error will be lower the higher the gain). And now you say that error is independent of gain...

So error and gain are not formally independent; in fact, they are precisely inversely related.

No they aren't, from the viewpoint of an analyst lookingat a particular control system. The gain doesn't change but the error varies all over the lot as the disturbance changes.

There is nothing about viewpoint in the equation that describes the relationship between gain and error.

From the viewpoint of a designer ... error and gain "are precisely inversely related".

So from what point of view is this not the case?

As with so much of PCT, the answer depends on the viewpoint.

I don't believe this is so in this case. I think error is inversely related to gain from every point of view.

Which is why I said that both Tracy and Marc were correct.

I think you do a disservice to education by saying that. But it's up to you.

Regards

Rick

···

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5

From [Marc Abrams (2006.07.08.1022)]

> [From Rick Marken (2006.07.07.2210)]

>> Martin Taylor (2006.07.08.00.16) --

>> > from Tracy Harms 2006;07,07.18:00 Pacific

  Folks, my question and perspective does _not_ concern theoretical possibilities or engineered control processes. It has to do with real world natural control processes and in understanding if gain is indeed "independent" of error.

  If gain is inversely related to error and it is proportional to it, than its very much up in the air and an open question as to whether emotions is some how involved in this relationship.

  I think its important to understand at what level of abstraction this analysis is conducted. This relationship might only exist as an emergent property of control from "lower" levels and not actually present there. Indeed Martin, it may not be a "property" of any single control process.

Regards,

Marc

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6

[from Tracy Harms 2006;07,08.16:15]

Rick Marken wrote (2006.07.07.2210) in reply to Martin
Taylor:

What? You just said that error depends on gain
(error will be lower the
higher the gain). And now you say that error is
independent of gain.

...

So as loop gain increases, p approaches r and error
(the difference between r and p) approaches zero.

So error and gain are not formally independent; in
fact, they are precisely inversely related.

No, they are formally independent. We cannot derive
error from gain, nor can we derive gain from error,
across all situations where the terms are meaningfully
applied.

Again, what Martin Taylor wrote [2006.07.08.00.16]:

Since gain is a property of the control system,
error is a function of the disturbance as well
as of the gain, and the disturbance has a
source independent of the control system, gain
and error are logically independent.

I think Martin has put the point better than I did
when he notes that gain is a function of the control
system, isolated from environmental considerations,
but error is not. That is, however, exactly what I
was thinking about.

Immediately following this, he added:

Having said that, it is true that so long as the
control system is stable, for a given disturbance
the error will ordinarily be lower if the gain is
higher.

I'd also tried to say this in my post to which Martin
was responding. There is no contradiction between
this recognition that there is a systematic dependency
between error and gain, and maintaining that there is
no formal dependency between them.

Perhaps the term "formal" did not get the point
across.

Tracy Harms

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7

From [Marc Abrams (2006.07.09.1341)]

[from Tracy Harms 2006;07,08.16:15]

Perhaps the term "formal" did not get the point
across.

Tracy, by "formal" do you mean "fully determined by..."?

What do you see as the differences between "systematic" and "formal"

Regards,

Marc

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8

[from Tracy Harms 2006;07,09.16:50]

From [Marc Abrams (2006.07.09.1341)]

> [from Tracy Harms 2006;07,08.16:15]

>Perhaps the term "formal" did not get the point
>across.

Tracy, by "formal" do you mean "fully determined
by..."?

Yes, I do.

What do you see as the differences between
"systematic" and "formal"

Regards,

Marc

Formal relationships are those that apply to the
abstractions by which these sorts of things are
modeled, especially things such as functions by which
if x is specified, f(x) may be derived. My sense of
formalism is that this must be true for all values
that may meaningfully apply as x.

Systematic relationships, in contrast, are looser
assertions of effect and influence. Bill's comment
strikes me as falling in this category:

As the loop gain increases, variations in the
disturbance and the reference signal both have
decreasing effects on the error signal, which
remains nearer to zero when the loop gain is
high than when it is low.

That sentence deserves to be modified with the
additional qualifier, mentioned by Martin Taylor, that
there is some threshold above which increased gain
produces instability and, thus, higher average error
than does gain that is slightly below that threshold.

To my thinking, systematic theory is that which bears
in mind the iterative, interactive dynamics of the
things under contemplation, including the way things
break down and cease to produce typical ("consistent")
results at the extremes of what is possible.

I don't doubt that Rick is aware of those thresholds,
but I can't agree with his suggestion that we work
with theory as though the "normal" range of
possibilities are the only ones worth considering.
Yes, if we preclude those situations where control
systems are overwhelmed, and those situations where
the gain of control systems exceeds its ability to
reliably moderate disturbances, then average error and
gain have the nice simple inverse relationship he said
they do. What we don't get to do is correctly assert
said relationship as a formulaic truth without the
applicable qualifiers, and it has seemed to me that
this is exactly what Rick has wanted to do.

My original post on this topic was in reply to Jim
Dundon (07.07.06.1212edt)

It may be simultaneously true that a least
something corresponds simultaneously to a more
something. In the case of PCT I suppose less
error means more gain.

Not only do I think Jim was getting caught up in an
illusion, akin to the behaviorist illusion, I think he
was trying to assert a strict (mathematical, absolute,
or formal) relationship between error and gain, which
failed to include recognition of the
finite-range-of-effectiveness which actual systems
necessarily entail. Rick seems to think that we can
assume that the interior of said range may be presumed
to be under discussion whenever the terms are used,
but this does not suffice for me. A formulaic
assertion is either true for all values, or it is
false. Saying that it is true when it is identifiably
false cannot be a thing of no importance.

Tracy Harms

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9

From [Marc Abrams (2006.07.10.1219)]

Thanks for the clarification and I agree with your assessments.

  Personally, I think the use of the concepts "formal" and "systematic" might be more clearly communicated by using the terms "dependent on...." or "influenced by..." but this of course is just my personal preference

> [from Tracy Harms 2006;07,09.16:50]

>Not only do I think Jim was getting caught up in an
>illusion, akin to the behaviorist illusion, I think he
>was trying to assert a strict (mathematical, absolute,
>or formal) relationship between error and gain, which
>failed to include recognition of the
>finite-range-of-effectiveness which actual systems
>necessarily entail. Rick seems to think that we can
>assume that the interior of said range may be presumed
>to be under discussion whenever the terms are used,
>but this does not suffice for me. A formulaic
>assertion is either true for all values, or it is
>false. Saying that it is true when it is identifiably
>false cannot be a thing of no importance.

Regards,

Marc

···

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