<Martin Taylor 940407 19:15>

<Bill Powers (940406.0930 MDT)>

I wonder how much of our discussions about the role of the

disturbance in control is biased by the way we simplify and

conventionalize examples.

Consider: in equations, we write p = o + d [omitting functions].

Suppose we choose the convention that o and d are defined as

positive when they have opposite directions of effect on the

controlled quantity and perception, with d always having a positive

effect. In that case we would change the equation to

p = d - o.

...

I think you are absolutely right. But then I tend to believe that a lot

of what Whorf said about linguistic relativity is right, though many

linguists disagree. However, their disagreement has not, so far as I know,

been centred on the effect of our mathematical notation on how we think

of the concepts behind the notation. I've long thought that there is

a big effect there.

I'll follow your thought with another one.

In a lot of discussions, yours included, formulae such as p=d-o are

written, together with o=Ge, e = r-p, and then these formulae are put

together and solved...

p = d-(G(r-p)) (1)

p(1-G) = d-Gr

p = (d-Gr)/(1-G) (2)

p = d/(1-G)-r(G/1-G)

or as G approaches minus infinity

p = r (3)

as if the two occurrences of p were the same thing. They aren't, but

they seem to be, not just because the mathematical notation leaves out

the functions that should be incorporated for each section of the loop,

but also because the notation leaves out time.

The view of the loop expressed in this simplified notation is an external

one, in which things happen at different times. Only when seen from one

point in the loop do all the effects occur in the present. A proper

notation involves the operation of convolution at each stage, showing how

the effects are distributed over time:

Not o = Ge, but o(t) = G(tg) X e(t-tg) where "X" represents the convolution

operator. (I apologise for the bastard notation, but I want to show the

"tg" time reversal of the convolution explicitly, and putting in integral

signs is too clumsy even for this posting.)

e(t) = C(tr,tc ) X (r(t-tr),p(t-tc)) (Here X represents the same thing

as convolution, but for the two signals separately. For the most part,

in the comparator, it is probably OK to treat the effect as an instantaneous

subtraction, in accord with the usual notation, but one should be aware

that it might not be so).

p(t) = P(tp) X s(t-tp) where s is the set of sensory inputs to the

perceptual function. Remember that P might include differentiators,

shift registers, and other functions of time, depending on the level

of the hierarchy under consideration.

In the outer world, we can assume that s is immediately derived from the

CEV, because if it is not, we can subsume any delay in P(tau). And we

can take s = d+v to be immediate and in the present at the point(s)

in the outer world where the output and the disturbance have their joint

effect, where v is the effect "now" of the output on the CEV.

v(t) = F(tf) X o(t-tf) where F is the feedback function.

So the nice pretty loop of equation (1), which leads to the solution (2)

for p, should really be something quite murky, with lots of convolution.

Assuming immediacy for the comparison function, we have:

p(t) = P X s = P X (d + F X o) = P X (d+(F X G X (r(t-tr)-p(t-ts-tf-tg-tc)))

In this more accurate form, the seduction of thinking that the p is the

same on both sides of the equation fails. One can only say that if

control is good, then the two p values won't be too different, and work

from there.

Setting (tp+tf+tg+tc)=tt, a kind of loop delay, we can write

p(t) = p(t-tt) + eps(tt)

where eps is some (unknown) difference between the perceptual signal

now and at a time tt=(tp+tf+tg+tc) in the past. And even this

is misleading, because those tm parameters are only parameters in the

several convolutions, not fixed numbers, and they don't simply add,

so eps is a function of time that has to appear in the general analysis.

It isn't easy, and it isn't as obvious as the simple notation ignoring

the temporal nature of the various parts of the loop would seduce you

into believing. However, it should be possible (I haven't done it) to

complete the analysis and wind up with a relation between p and r, with

eps and d as parameters whose effects get smaller the better the control.

So I agree wholeheartedly with Bill about the power of notation both to

mislead and to lead correctly. If one is going to use shorthand notation,

one should be aware of just where the seductions and pitfalls lie. Much

of the argument on CSG-L hinges on a failure of this awareness.

Benjamin Lee Whorf is alive and well in Durango. (Actually, he studied

Navajo and Hopi, not too very far from there).

Martin