[From Bill Powers (940831.0945 MDT)]

Martin Taylor (940830.1650) --

I have a puzzlement about (a) what happens when two non-linear pull-

only conrol systems are opposed to form a virtual two-way control

system, and (b) why wrong answers are so seductive as to be agreed by

competent and well-informed people.

The first, I think we can clear up. The second is a mystery; regarding

my own agreements to errors, you see only the tip of the iceberg; I

catch most of mine before they escape into public view, but only after

waking up in the middle of the night with that "something's wrong"

feeling. Never trust nobody without checking.

Before I went off on holiday, I re-posted some maths originally posted

in Dec 1993, which seemed to show that the two opposed one-way control

systems would form a two-way virtual control system with an output law

that was the derivative of the output law of the one-way real systems.

This was a formulation of a claim Bill P. had made, and supported that

claim. Bill praised it at the time, but now has said that it was in

fact wrong.

This was in large part the result of a language mixup. Where I said

"differential" you read "derivative." It's true that the first

derivative of a square-law curve is a linear curve -- but it's the

derivative with respect to the independent variable, not with respect to

time. This, however, has no relationship to what I meant by

"differential." In a two-way system that is made of two one-way systems,

two driving signals are required, one for each side. The average value

of the two driving signals expresses the common-model driving signal,

which (in the muscle system) determines the average tension on both

sides. The difference between the two driving signals (that is, the

"differential" signal) determines the net output in one direction or the

other. One signal rises above the average value at the same time that

the other signal falls below the average value. This has nothing to do

with derivatives, either with respect to the independent variable or

with respect to time. So in the equations you cite, I wrote

F1 - F2 = (c^2 + 2cd/2 + d^2/4) - (c^2 - 2cd/2 + d^2/4) or

F1 - F2 = 2cd.

This says that the differential force produced by a differential

contraction is proportional to the common-mode contraction ...

Now we come to my agreement to a falsehood:

(Bill, in response)

Martin:

>I always keep in mind your demonstration that a pair of opposed

>square-law systems will act like a virtual two-way linear

>system.

With a balanced pair of identical one-way systems having linear

output functions, the resistance to disturbance will be zero in

the region of overlap, but it will be normal outside that region

(for larger disturbances), as Kent McClelland showed at the

meeting. The combination will be like a single output function

with a dead zone: errors will produce no net output until they

reach a threshold level, after which larger errors will produce

output as usual (from one of the systems).

...

It's interesting, isn't it, that the agonist-antagonist pairs of

muscles HAVE to have a nonlinear response if muscle tone is

nonzero and there is to be any fine control.

Those learned paragraphs of mine, and what goes between them, are

nonsense. You didn't generate the nonsense for me to agree with; I did.

I agreed with your math without actually working it out for myself, and

came to a wrong conclusion, which wasn't even what Kent McClelland had

demonstrated. This is what comes of wanting to sound knowledgeable

without actually doing the work to see if the first idea that comes into

mind is right.

Martin:

>If two opposed one-way control systems have an output function

>f(e), then the virtual two-way system that they create has an

>apparent output function that is the derivative df(e)/de.

This would appear to be true within the region of overlap for

one-way systems, but not outside it.

And in that I am agreeing to your agreement to a generalization that is

simply irrelevant -- even though it's true that the form of a linear

output function is the derivative of the form of a square-law output

function.

The output of the two systems is the SUM of the outputs:

> *

> *

> * /

> F1 * /

> * /

> * /

> * /

> * /

> * /

*| /

* | /

* | / SUM

* | /

* | /

----------------*----------------/--------------*--------------------

/ | *

/ | *

/ | *

/ *|

/ * |

/ * |

/ * |

/ * |

* |

* |

* F2 |

* |

>

The correct answer can be seen by taking my derivation for the square-

law muscles and using the first power instead of the second power:

F1 = c + d/2

F2 = c - d/2

F1 - F2 = c + d/2 - c + d/2 = d

The net force depends directly on the differential driving signal d, and

the gain is independent of the common-mode driving signal c. Outside the

region of overlap, we have half the gain.

So agreeing to a true but irrelevant fact (about the derivative of a

square-law function) led somehow to a false line of reasoning and the

conclusion that the gain for opposed linear systems had to be zero. All

I can say is that it's a good thing that you continued to be puzzled. My

own anomaly detector seemed to be down for repairs, or perhaps needs to

be.

## ···

----------------------------------------------------------------------

Rick Marken (940830.1420) --

RE: levels

You described what your beliefs and principles ARE, but you didn't say

how they relate to each other -- in terms of my levels: which are system

concepts, which are principles, which are programs, etc.. You didn't say

how elements of your statement fit a hierarchical structure ( or not).

RE: perception of statistics.

The perceptual function could compute the mean or variance of its

inputs; so the perceptual signal's value could represent variations in

the mean or variance of the inputs to the perceptual function. But this

perceptual signal is not an _estimate_ of anything (the mean or

variance it represents is not computed as a statistic); it is simply a

perceptual variable.

That's an excellent but subtle point that can stand amplification. If

the PIF actually does include a statistical calculation, such as

deriving a running mean value, the computation of the mean is part of

the PIF and defines the controlled variable as a running mean value of a

function of external variables. There is no approximation or estimate

involved. What is controlled, and perceived, is not the moment-by-moment

fluctuations in the environment, but the mean value of a function of

environmental variables. The only way for that perceptual signal to

contain any noise would be through variations in the computation of the

function or its mean -- that is, given a repeat of the same

environmental fluctuations, the reported mean value would spontaneously

vary because of glitches inside the PIF. Even then, the control system

would interpret those glitches as real changes in the controlled

perception, and would counteract them by varying the output effects on

the environment.

Whether the PIF contains noise sources or not, what such a system

controls is not a CEV computed moment-by-moment on the detailed

variations of the environment, but a mean value of a function of those

variations. The variations themselves, which are filtered out by the

computation of the mean, are uncontrolled. Only the mean is controlled.

Another control system receiving the same information and having the

necessary output equipment might compute and control the variance of a

function of the external variables. In that case, the variance would be

the controlled variable, and the mean would not be controlled by that

same system, nor would the specific fluctuations in the external

variables be controlled. ONLY the variance would be controlled.

And of course still another (and quite complex) control system might

continually perceive and control a function like U(x) - U(x|y), the

value of which we could label "the amount of information in x about y"

-- which presupposes that both x and y are available as lower-level

perceptions. This number would be perceived not as an estimate of that

information, but as the exact amount of information existing at each

moment. The output would of course have to have some effect on the

environment that could vary one or both uncertainties or the

probabilities underlying them. Again, no "estimate" is involved; the

control system will take the computed value of U(x) - U(x|y) as the

actual state of this measure of the world, and keep it matching the

reference value.

All of which reminds me of a fact that I mention from time to time, but

which seems to go unappreciated. If you look around you at the world you

are in, how much noise and uncertainty do you see? Do shapes wiggle

around and fuzz at the edges? Do intensities bobble up and down? Do

sensations sometimes seem one way and sometimes another, unpredictably?

Does "in" sometimes unpredictably change to "beside?" No, the world of

experience is almost completely smooth, stable, and predictable. The

only way to make it seem otherwise is to invent special situations where

there is truly uncertainty, not just in perceptions of the world but in

the world itself. You have to use masking noise, low light levels or

extremely brief visual images, faint touches, or apparatus that

randomizes the effects of actions. You have to get completely outside

the normal range of perceptions, and introduce uncertainties where none

naturally exist.

The dynamic range of perception is such that nearly all perceptions are

normally free of noticeable uncertainty. You can see this just by paying

attention to the world you experience -- which, according to PCT,

consists of the perceptual signals themselves.

-----------------------------------------------------------------------

Best to all,

Bill P.