# [Bulk] Re: [Bulk] Re: least action-control efficiency

[Martin Taylor 2006.07.08.08.46]

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

You really ought to read to the end of a posting before responding to it

Martin

[Martin Taylor 2006.07.08.08.47]

[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.

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

Supplement to my previous one-liner...

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.

From the viewpoint of a designer considering different possibilities for the control system he is about to make, he can't consider the disturbance as any more than a statistical quantity with range and variance to be expected, but he can vary the gain to be precisely what he wants. To him, error and gain "are precisely inversely related".

As with so much of PCT, the answer depends on the viewpoint. Which is why I said that both Tracy and Marc were correct.

Martin

[From Rick Marken (2006.07.08.0800)]

Bill Powers (2006.07.08.0747 MDET)--

Martin Taylor 2006.07.08.08.47 --

I agree with your comment on the relation between gain and error.

So you agree that there is an inverse relationship between gain and error only from the point of view of the system designer, not from the point of view of the system itself?

As the loop gain increases, the effect of a disturbance on the controlled variable decreases and the error signal resulting from a given disturbance or a given setting of the reference signal decreases.

Right, loop gain (which is what I assumed we were talking about) is inversely related to error.

...A more complete picture requires differential equations or simulations. The results in terms of steady states, however, will agree with all the above.

I didn't see anything in this post that agrees with Martin Taylor's comment on the relation between gain and error. Martin's comment was that there is an inverse relationship between gain (I assume he means loop gain but either way it's OK, G or FG) and error from the system designer's point of view but not from the system's point of view. I didn't see this comment addressed anywhere in your post. Did I miss something?

Best

Rick

···

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[From Bill Powers (2006.07.08.0925 MDT)]

From Rick Marken (2006.07.08.0800)--

So you agree that there is an inverse relationship between gain and error only from the point of view of the system designer, not from the point of view of the system itself?

Yes, From the viewpoint of the system designer, to decrease the average amount of error the system experiences over many circumstances, you increase the loop gain. There is an inverse relationship.

However, from the system's own point of view, the error signal fluctuates as disturbances and reference signals change, without any change in loop gain at all. The momentary value of the error signal can change even when the loop gain does not change.

That was the distinction Martin was making. Your interpretation of his words misled you because you took "the error signal" to mean "the average of the absolute error signal over many instances" in both cases, while failing to see the meaning "the momentary magnitude of the error signal."

Best,

Bill P.

···

As the loop gain increases, the effect of a disturbance on the controlled variable decreases and the error signal resulting from a given disturbance or a given setting of the reference signal decreases.

Right, loop gain (which is what I assumed we were talking about) is inversely related to error.

...A more complete picture requires differential equations or simulations. The results in terms of steady states, however, will agree with all the above.

I didn't see anything in this post that agrees with Martin Taylor's comment on the relation between gain and error. Martin's comment was that there is an inverse relationship between gain (I assume he means loop gain but either way it's OK, G or FG) and error from the system designer's point of view but not from the system's point of view. I didn't see this comment addressed anywhere in your post. Did I miss something?

Best

Rick
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Home 310 474-0313
Cell 310 729-1400

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[From Rick Marken (2006.07.08.0845)]

Gotta run to my piano lesson. But I'll reply to this quickly to see what I get after the lesson;-)

Bill Powers (2006.07.08.0925 MDT)]

Rick Marken (2006.07.08.0800)--

So you agree that there is an inverse relationship between gain and error only from the point of view of the system designer, not from the point of view of the system itself?

Yes, From the viewpoint of the system designer, to decrease the average amount of error the system experiences over many circumstances, you increase the loop gain. There is an inverse relationship.

However, from the system's own point of view, the error signal fluctuates as disturbances and reference signals change, without any change in loop gain at all. The momentary value of the error signal can change even when the loop gain does not change.

But the same is true for the system designer. This is not a point of view difference (as Martin said) but a difference between looking at average error versus instantaneous error.

That was the distinction Martin was making.

He was conflating the point of view distinction with the average versus instantaneous error distinction. So the fact is that there is no difference in the relationship between gain and error from the system versus observer (designer) perspective. The difference is in whether you look at average error (in which case the inverse relationship holds) or instantaneous error (in which case it doesn't).

Best

Rick

···

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From [Marc Abrams (2006.07.08.1224)]

[From Bill Powers (2006.07.08.0747 MDET)]

Thanks Bill. Although not directly addressing my questions you did indeed help shed some light for me. I believe my hunches are worthwhile exploring.

Regards,

Marc

···

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[From Bill Powers (2006.07.08.10:40 MDT)]

Rick Marken (2006.07.08.0845)--

But the same is true for the system designer. This is not a point of view difference (as Martin said) but a difference between looking at average error versus instantaneous error.

The control system experiences only the momentary error and acts on that basis. It knows nothing about long-term average error.

Best,

Bill P.

···

That was the distinction Martin was making.

He was conflating the point of view distinction with the average versus instantaneous error distinction. So the fact is that there is no difference in the relationship between gain and error from the system versus observer (designer) perspective. The difference is in whether you look at average error (in which case the inverse relationship holds) or instantaneous error (in which case it doesn't).

Best

Rick
---

Richard S. Marken Consulting
Home 310 474-0313
Cell 310 729-1400

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From [Marc Abrams (2006.07.08.1242)]

> [From Bill Powers (2006.07.08.10:40 MDT)]

>> Rick Marken (2006.07.08.0845)--

When that is what you are controlling for what might you expect? But I think something that might be worth considering here is exactly what are they "right" about?

I know what your model is telling you but I'm less sure of how this all translates to our physiology. NSU's might be convenient, even necessary for your modeling purposes but that does nothing to address what in fact "gain" actually is or represents physiologically.

If I were interested in researching the relationship "gain" had to error empirically what would I be measuring? Indeed, this brings up a whole swarm of important and interesting questions. I'm not sure we are in a position to say we are "right" about anything at this point.

If I take our blood glucose control process as an example, what would constitute the "gain" in this system? Chemical? Neural? Some combination?

Regards,

Marc

···

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[Martin Taylor 2006.07.09.23.29]

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

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.

It's good to get back to the real thread!

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.

I have to side with Rick on this one, thinking about how useful Newton's laws of gravity and motion are in the non-relativistic, non-quantum size and speed range in which we mostly operate.

However, re-reading the quote from Jim, I see something else that hasn't been addressed in this thread as far as I remember. Jim is looking (I think) at a system working in a real world, not a toy system of a single Elementary Control System (ECS) with a simple instantaneous connection representing the environmental feedback path. He's also looking at it from outside, but as an analyst, not as an observer. (The "Analyst's Viewpoint" allows access to internal parameters such as error, whereas the "Observer's Viewpoint" allows access only to effects the observed system has on its environment.)

If an analyst with access to the values of the disturbance but not of the system parameters sees less error than might have been anticipated, it's a reasonable inference that the loop gain is greater than had been assumed. That's the sense of Jim's way of putting it: not "more gain means less error", which would be a comment from the "Designer's Viewpoint" ("How much gain should I give it?"), but "less error means more gain" ("what IS the gain of this thing, anyway?).

If we are dealing in a real(-ish) world with complex environmental feedback paths, we can't really make that kind of assertion. "Gain" isn't a simple number (it isn't really, even in the trivial case), but a time function. The output value depends on the history of the error, not on its value at any particular moment (the trivial case usually shows the output function as an integrator, so even there this is true). Now consider what happens if we simply scale the gain function by multiplying it by some number.

There are frequency-dependent phase shifts embodied in the gain function. There are frequency-dependent phase shifts in the environmental feeback path, and maybe the feedback path even has reverberations or resonances. In those kinds of condition, even if the system doesn't oscillate, increasing the gain by a multiplicative constant can easily increase the error (ripples might take longer to die down even if they don't explode to infinity). The optimum gain, meaning the gain that gives least error on average, may be well below the gain that causes the system to blow up.

In the real world, it simply isn't true that the harder you try to keep something really stable the more stable it will be. Trying too hard is a very good way not to achieve your goal in a lot of situations -- such as playing golf!

Viewpoint does matter, but so does the proper application of the maths to the real world. Write the equations in terms of scalar numbers and you get simple results. Treat them as time functions, and it's more realistic, but much harder to be sure what's going to happen with any particular manipulation, such as (Designer's Viewpoint) increasing the gain. (I call it the Designer's Viewpoint, though I'm well aware that there may well be real-time influences on the gain of specific conrol systems; I believe I remember references to Tom Boubon having done some modelling of that kind, but I don't remember what.)

I hope that's not too muddled. I've had a rather exhausting day and my brain feels foggy. I may regret this in the morning, but I doubt I'll have time to post much tomorrow, so now or never.

Martin