[From Bill Powers (920616.1930)]

Bruce Nevin (920616) and Mark Olson (920614) --

" Outcomes influence or determine some other environmental variables

and determine some reference signals (presumably when intrinsic error is

low?); and some environmental variables and reference signals influence

actions while other environmental variables determine actions;

and actions Control outcomes and outcomes influence or determine

other environmental variables . . ." (Nevin)

This attempt to put the basic relationships of HPCT into simple language is

getting more and more confusing. Let's go back to the basic diagram and try

to build it up in an orderly way.

First, a simplified ECS diagram showing an Input function, a Comparator,

and an Output function. In the environment we have a controlled variable or

input quantity qi, an output quantity qo, an environment function E, a

disturbance function D, and a disturbing quantity qd. Inside the organism

we have a reference signal r and a perceptual signal p (put your own arrow

heads in).

p r

## ยทยทยท

>

-----------

> I | C | O |

-----------

> >

qi<-E<- qo

>

D <- qd

Where do we start? By assuming that control is good, so the input quantity

matches the reference signal.

-1

(1) qi = I (r)

The input quantity is maintained at a value which is the inverse of the

input function of the reference signal. So:

(a) The reference signal and input function determine the state of the

input quantity in the environment.

Statement (a) is an approximation. In fact, the reference signal determines

the state of the input quantity within some region r plus or minus epsilon,

where the size of epsilon depends on the loop gain of the control system

and the maximum disturbance that the system can resist. Using the

approximation implies assuming an ideal control systems (infinite loop

gain). In that case, epsilon is zero.

Next, we explain HOW this determination is brought about. qi remains near

the specified state because variations in the disturbing quantity,

transformed through the disturbance function D, are opposed almost exactly

by variations in the output quantity transformed by the environmental

feedback function E. Therefore:

-1

(2) E(qo) + D(qd) = I (r), or for later reference,

-1 -1

qd = E (I (r) - D(qd)

The sum of the output quantity and the disturbing quantity, each

transformed by the appropriate function in the environment, must equal the

value of input quantity determined by the reference signal. This leads to

the statement

(b) The reference signal and external disturbances jointly determine the

output quantity.

Statement (b) says that given a constant reference signal, variations in

the disturbance call forth specific variations of the output quantity or

action, in the manner of an apparent causal relationship. The determining

effect of the disturbance on the output, however, is subject to the

condition that the sum of disturbance and output effects always equal a

particular value: the value of the input quantity determined by the

reference signal. This balance point, therefore, can change if the

reference signal changes. This is why the action of the system is JOINTLY

determined by disturbances and the reference signal, and not exclusively

determined by either.

Note that in both statement (a) and statement (b), the apparent causal

relationship works in the opposite direction to the direction of physical

causality. The reference signal appears to determine the input quantity

backward through the input function. The disturbance appears to affect the

output quantity (via the input quantity) backward through the environment

function. The form of the apparent causal relationship is, in both cases,

the inverse of the function actually connecting the variables by the most

direct route. These backward relationships are a direct result of the

closed-loop organization.

There is only one dyadic deterministic relationship: the reference signal

determines the input quantity. The output quantity depends on two

variables, jointly: the disturbance and the reference signal.

Now let's put together a two-level system:

(spacing for new page)

p2 r2

> >

-----------

> I2| C | O |

/ -----------

/ | |

qi2a qi2b qo2

> > /

> > /

> p r

> > >

> -----------

> > I1| C | O |

> -----------

> > >

qi1a qi1b<- E<- qo1

> >

D2 D1 <- qd1

>

qd2

This is a case in which the higher-level system generates a perception

derived from two lower-level perceptions, one of which is controlled and

one of which is not.

Now we can see that the first-level reference signal is the output of the

second-level system. The reference setting at level 2 can be seen in

behavior only if we look at the two input quantities qi1a and qi1b through

the same kind of input function that the second-level system uses. If the

second-level system perceives the sum of qi1a and qi1b, we must observe the

sum of these input quantities in order to see what is being controlled.

Furthermore, we must see the two disturbances, qd1 and qd2, using the same

perceptual function, if we are to see the net disturbance at the second

level correctly.

Considering only the first-level system, we still have the reference signal

determining the input quantity, now qi1b. This means that the output of the

second-level system is, as far as second-level control is concerned, not

qo1 but qi1b. The input quantity of the first-level system, not the output

quantity, will appear to be the action of the second-level system. If qi1a

is disturbed, qi1b will change to oppose the effect on the second-level

perceptual signal. But it is also true that if qi1b is disturbed, qi1a will

change to oppose the disturbance of the second-level input function --

although qi1b, being under control itself, will not give way much to

disturbances.

Therefore:

(c) The total disturbance, composed of d1a and d1b, and the second-level

reference signal, jointly determine the second-level output, which

translates into qi1b in this case.

(d) The second-level reference signal and the second-level input function

determine the second-level input signal, which means

-1

I2 (qi2a,qi2b) = r2

The appearance will be that an abstract variable composed of qi1a and qi1b

will be exclusively determined by the second-level reference signal. At the

same time, the output quantity of this system will appear to be qi1b, and

it will be jointly determined by the second-level reference signal and both

disturbances.

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These are fairly complex and subtle relationships. Understanding them

requires seeing how control system vary their outputs to maintain their

inputs at preselected levels or states, at the same time automatically

resisting the effects of disturbances on those inputs.

At any level of interpretation, statements (a) and (b) will hold true --

but with many systems at each level, each level has to be considered anew.

When a single control system at one level receives reference signals from

several higher systems, there can be no simple relationship between

disturbances of a given higher-level perception and the resulting change in

the lower-level net reference signal.

This is why I don't think there is much point in trying to express the

relationships of control in the familiar language of determination,

influence, and causation -- particularly not in terms of causation. In

speaking of one simple control system, I have always used causal terms in

speaking of illusions: it SEEMS that one variable is causing another to

change, but in reality, in a control system, the pathways of causation are

quite different from what they seem, and are circular instead of lineal.

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I think the closest we can come are the two basic statements:

(a) The reference signal and input function determine the state of the

input quantity in the environment.

(b) The reference signal and external disturbances jointly determine the

output quantity.

Both of these statements describe apparent causal relationships, which are

different from those that actually exist in the control system. That is,

the "determination" takes place through a path different from the one that

appears to exist. These two statements describe appearances, but not the

actual organization of a control system. Both are deductions about how

control behavior will appear to a naive observer, based on the assumption

that we are observing an ideal control system.

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Best,

Bill P.