[From Bill Powers (940408.0945 MDT)]

Dag Forssell (940307.1120)--

Your comments on filling in perceptions from imagination were very

well put, and relevant to a number of discussions. It's hard to

figure out what another person is thinking just by listening to

words. Most of what you "understand" is what you imagine that the

other person is thinking.

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Bob Clark (940407.1655) --

Well, you've covered just about every way in which time has been

mentioned. I'm a bit lost, however -- why were we going into this

question?

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Martin Taylor (940407.1915) --

RE: big picture

When you get down to writing equations in which the time-delay

associated with each function in control loop is represented

explicitly, it seems to me that we are getting pretty far from the

big picture.

I prefer to approach mathematical analysis in a cruder way. The

simple algebraic equations (which actually represent steady-state

solutions of differential equations) give a prediction of real

behavior that is within about 10 percent of the actual behavior,

when the constants are adjusted. That tells me that the quasi-

steady-state representation takes care of 90 percent of the problem

I'm interested in. This is without any temporal considerations at

all.

If you then add an integrating output function, still with no

delays, the prediction comes to within about 5 percent of the actual

behavior, so now we have accounted for 95% of what we observe.

Putting in a single time-lag raises that to perhaps 97%, and at that

point I tend to start losing interest. We could, of course, go on to

introduce the detailed delays in each function, but we know that no

matter how much more detailed we get in the analysis, we aren't

going to gain much more predictivity. Whatever effects those

detailed delays may have, they can't be very important. Probably

they're so short in comparison with the speed of operation of the

loop that the pay-back for the labor of taking them into account

would be negligible.

It's fun to write something like

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)))

but the information content in such expressions is just about zero.

I'm looking for the BIG warts, not the little bumps on the warts.

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.

I don't think I've been seduced into believing anything. I've taken

lags into consideration and have decided that they make very little

difference in the way real control systems work. The real systems

are so designed that they don't.

Yah' eh t'eh to Ben Whorf.

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RE: scale

I agree with Bill that one suitable scale for display of the

effect of control is that of the full range of the perceptual

variable. ... It is quite reasonable to plot the deviation of

control from exactness on a scale that shows the maximum error

at full-scale, because this kind of deviation is also a limit

that evolution (and training and experience) has imposed on us.

When you plot the maximum error full-scale, you can no longer judge

whether it's an important amount of error or insignificant. All

errors will look the same. For control, what matters is the amount

of error in comparison with the magnitude of the controlled variable

that the system is trying to maintain. Without that information, you

can't judge whether the effort you're putting into the problem is

going to have significance or be wasted on something trivial. If an

organism can keep a variable with a range of 100 within 1 or 2 units

of any reference signal in this range, you've understood essentially

all that matters about this behavior.

One aspect of notation that is very misleading is its complexity. A

complex expression can be more wrong than a simple one, if you've

chosen the simple one carefully. When you use the simplest workable

notation, you're letting the problem drive the mathematics. Complex

notation is often just letting mathematics determine how you see the

problem.

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Best to all,

Bill P.