[From Bill Powers (940413.0900 MDT)]

Martin Taylor (940412.1800) --

If the past values of p are (as they in fact are) still

reflected in o, how can you say the function does not give

previous values of p an effect on present values of o?

I'm trying to emphasize that the workings of a control system (or

any physical system) are best expressed in terms of present-time

operations. Even in a transport lag, the values of the lagged signal

that produce effects are those existing in the mechanism right now,

with past values having disappeared forever and being incapable of

having any effects in the present.

Consider a transport lag physically implemented as an analog shift

register like a charge-coupled device or an axon. In the register at

a given time is a set of stored values v of the input:

input value ----> v5 v4 v3 v2 v1 v0 ----> output value

The input value right now is v5, the output value right now is v0.

After the next shift we have

input value ----> v6 v5 v4 v3 v2 v1 ----> output value

Now the input value is v6 and the output value is v1. The value v5

still exists, having been moved into the second slot from the left.

The value v0 no longer exists; it can have no further effects. The

value v7 does not yet exist.

This is not, by the way, a convolution; it is only a transport lag.

A convolution is implemented this way:

input value ----> v6 v5 v4 v3 v2 v1 ---> shifts

> > > > > >

weightings--> w1 w2 w3 w4 w5 w6

> > > > > >

SIMULTANEOUS SUMMATION

>

output value

Again, the only effects are present-time effects. The past never

affects the present.

In an integration, there is a cumulative summation of inputs:

Input Sum

0 (initial state)

1 1

3 4

7 11

2 13

-1 12

----> -5 7

-3 4

-4 0

... ...

At the time shown by the arrow, the value of the sum before the new

input arrives is 12, and afterward is 7. From the previous value of

12, there is no way to tell what the input was at any still earlier

time; that information is lost. The same sum of 12 could have been

produced by any number of different past values of the input. So

even though the 12 does represent effects left over from past values

of the input, it is not a unique indicator any one past value. All

that is carried forward through time is the current sum. The only

value that can have any effect on a subsequent process is the final

sum of 0, which will change simultaneously with the next input. As

it happens, in the example that final value is the same as the

initial state of the integrator; the situation as far as the

integrator is concerned is just as if none of the intervening values

of input had occurred.

This is what is meant in physics by saying that the history of a

system is of no importance in determining its future behavior. If

you know all the values of variables at a given time, and all the

derivatives, it makes no difference from then on how those variables

and their derivatives got into those states. It is only the

_present_ state of the system that determines what will happen next.

Simulations necessarily have this character, because the only values

of variables that figure into the next iteration are those that

exist, somewhere, right now. The next state of a simulated system is

completely determined by its present state, and only by its present

state.

This principle is not adhered to in some approaches. A predictor-

corrector integrator, for example, uses both past and future values

of a variable as part of computing its integral at a given time.

Analytical approaches in general are free to take information from

any place in a data record, because they treat the entire data set

as if it is all equally available at all times. This is how

statistical approaches work; they treat the entire data set as a

static entity, so that, for example, in computing conditional

probabilities the bins can be filled in any sequence. While I'm not

sufficiently adept mathematically to justify my feeling, I feel that

this can lead to spurious results, if only because averages or sums

computed over a whole data set can't possibly have been correct for

a point in that data set in the middle, where later values had not

yet occured.

## ยทยทยท

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The way you talk about control systems involves ranging freely

backward and forward through time, and seems to me undisciplined.

The idea that a control system somehow uses perceptual information

in a way conditional on its probable future effects on the

controlled quantity is precisely the kind of proposal that made

scientists justifiably reject the concept of purpose. PCT explains

purpose strictly in terms of a real physical system that operates

only with present-time information. In analyzing a control system

there is NEVER a need to speak of predicting the future effects of a

variable; there is NEVER a need to explain behavior in terms of a

past value of a variable that no longer exists.

There have been numerous abortive attempts to work control theory

into mainstream ideas about behavior. Most of those have failed

because, in one way or another, they insisted on preserving a

sequential view of behavior. This sequential view fails precisely

because it fails to show that all parts of a control loop are in

operation at the same instant. In large part, this failure comes

from the tradition of trying to understand behavior as a sequence of

events, a succession of states that proceeds in jumps from one

frozen condition to another discretely-different frozen condition.

Only from this point of view can you speak of a "change" in a

variable like the CEV as if it could occur independently of the

behavior of the other variables in the system that are always

determining the state of the CEV. It leads to thinking of "typical"

inputs as impulses or step-functions -- functions that are designed

to cause changes abruptly at a given instant, before anything else

can interfere.

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

The p that you substitute from p = o + d is identically the p

in do/dt = k*(r - p), save for whatever transport lags are

present in the input function and comparator.

It is that, but the "o" in the differential equation doesn't >plug

into the other equation. It isn't a straight

substitution.

That's true only if you think that there is no output until an

effect of the input gets to it. This is the sequential view of the

system. In fact, it is the _current_ value of output that adds to

the _current_ value of disturbance to create the _current_ input to

the perceptual function. It is the _current_ value of perceptual

signal, now being generated by that function, that enters the

comparator. The CURRENT value of error is entering the output

function at the same time that the CURRENT value of output variable

is coming out of it. ALL the values of variables everywhere in the

loop are the CURRENT values.

What the functions do is to explain how those current values are

being generated out of inputs to the functions. If the system

equations don't give the right answers when you plug in all the

current values at the same time, then the functions are wrong and

need to be changed. When you have the right functions, a

simultaneous plot of all the variables in the model will match a

simultaneous plot of all the accessible variables in the real

system.

If I think of three more ways of saying this, I don't suppose you

will agree any more readily than you will now. The whole point of

speaking this way is to avoid the trap of trying to trace individual

events around and around the loop. The whole loop is always in some

state, and that state is changing with time. Transport lags and

integral lags are irrelevant; they do not alter the fact that the

whole loop operates exactly simultaneously. At a given instant, you

may find the effect of a previous input existing half-way through

the physical process that constitutes a transport lag -- but at that

same instant the input to the lag has one value and the output from

it has another. The system is not behaving according to the value

that is halfway through the transport lag, but only according to the

value presently at its output.

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Ursula Le Guin wrote what is to my mind one of the greatest science-

fiction stories ever written: The Dispossessed. Embedded in the

story was a conflict between two major world-views, which were

called Sequency (the official view) and Simultaneity, the view of

the principal character, Shevek. I think Le Guin was reading my

mind. I see the struggle between the modeling view exemplified by my

approach to PCT and certain traditional approaches as being over

exactly this point: whether we are to understand physical systems as

having overall shapes that evolve through time in an eternal now, or

as consisting of punctuate events which cause future punctuate

events along a line in which both past and future have real

existence.

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

Best,

Bill P.