[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
···
>
-----------
> 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.
----------------------------------------------------------------------
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.