[From Bill Powers (940717.0630 MDT)]
Bill Leach (940716.1026 EDT) --
This is an issue that I have been musing over for a couple of days
and at least now think that I now know a couple of points the
"bother" me.
D --------{1}---------> E
> ^
> >
{2} {3}
> >
--------->R------>-----
For this to work at all, then one of the following (as I see it,
unspecified conditions) must exist:
{1} is a 'lossey' path with a known set of characteristics.
The regulator output {3} has no effect on {2} (or it then IS a
negative feedback control system).
You raise an important point about our diagrams that is often left
implicit, and so needs to be brought out and dusted off now and then.
You are thinking right, to see that {1} mustn't be just a rigid
connection. But first a couple of generalities.
In a symbol-and-arrow diagram where variables are at the nodes (as
above, and in the way we usually diagram the environment of a control
system in PCT), the _arrows_ are the functions connecting _variables_.
This is the "dual" of the other way, the way we use inside the control
system, where the arrows are variables and the nodes, the boxes, are the
functions.
The rationale (never formally agreed upon by anyone) is this. Inside the
nervous system or in an electronic circuit, signal-conducting paths span
the distances between active or passive elements. Whether you use
current, voltage, or impulses per second to measure signals, you would
measure the same signal anywhere along the path between its source and
its destination. So it is natural to draw signals as arrows, and label
each arrow with the name assigned to the signal. Computing functions
then become boxes with inputs and outputs, receiving signals, performing
some operation on their values, and emitting signals representing the
outcome of the operation. In some cases the functions are performed by
single neurons. It is natural, then, to associated functions with
specific places in the nervous system and signals with the routes betwen
functions.
In the environment, the situation is reversed. We generally measure
physical variables at a specific position in space, or in a restricted
region. We now have variables at different locations in space connected
by arrows labeled as physical laws (diffusion, inverse-square laws,
aerodynamic laws, etc.). It is logical, then, to represent the
connecting properties or functions as arrows, and the measured values of
variables as symbols at specific locations.
Now apply this to your diagram above. D ---> E says that the disturbing
variable D, which is measured at one point in space, has an influence on
essential variable E in a different place, via some properties of the
physical world represented by the arrow. If D is the ONLY influence on
E, then we say that D "determines" E via some intervening physical law.
If D is the brightness of a light bulb and E is the illumination of some
surface at a distance d from the light bulb, we would put a label on the
arrow such as "inverse-square law" or some such.
If there is more than one influence acting on E, we have
D -------> E
> ^
> >
-->R------
When two arrows converge, by convention linear addition of effects is
assumed unless the arrows are labeled with + and - signs, in which case
linear subtraction is assumed. Any other kind of combination of effects
becomes more awkward to diagram.
Most physical variables -- attributes of matter and energy -- can be
influenced by many other physical variables. We assume that the state of
any one physical variable is _determined_ by the sum of effects from all
other variables that can influence it, but it is not determined by any
one other physical variable. So the concept of a disturbance is really
inherent in physics. The output of a control system is a physical
variable like a force or an energy outflow, and via physical laws this
output influences -- but does not determine -- the state of the
controlled variable. In general there are disturbances that also
directly influence -- but do not determine -- the state of the
controlled variable. The state of the controlled variable is
_determined_ only by the sum of all influences on it, which includes the
control system's output and all other influences from other physical
variables, or disturbances.
You have hit upon the fundamental reason for which we define a
disturbance D as a physical variable, and make explicit, as a function,
the connection between D and its effects on a controlled variable. We
can manipulate physical variables like positions, temperatures, and so
forth. We can't directly manipulate the effect of these variables on
other variables. Martin Taylor and I have gone around and around on this
point: why can't we just define a disturbance as the _effect_ on a
variable, rather than explicitly stating that other variable with an
explicit connecting function? The answer is that we can't directly
manipulate the effect; the effect is something calculated, but not a
separate physical quantity. A force cannot be conjured up out of thin
air: it must be produced by a physical interaction of two things. In
mathematics, you can postulate anything you please, but in modeling, you
have to postulate things that have physical meaning.
···
---------------------------
Now, your specific problem:
The diagram as described and coupled with my own understanding of
the process for which control is being attempted should be
represented a little differently:
D ->--------f1(D)---------> E
> ^
> >
f2(D) f1(R)
> >
> >
--------->R------>-----
You can now see that "R" is wrongly labeled according to the above
conventions: it's shown as a variable when it should be a function. So
we really should draw this as
f1(D)
D ------------------------> E
> ^
> R(D) |
------------->-------------
Now it's obvious that we have two paths from D to E, one through an
environmental function f1, and the other through a "regulating" function
R. Effects on E via the path through R() are supposed to compensate for
effects on E via the path through f1().
--------------------------------------------
Now you bring up a very interesting point:
Again, I think that we are trying to talk about a "non-negative
feedback control system) that supposedly works. If it actually
works then the following diagram holds true:
>------<----f1(E)-----<-----|
> >
> ^
V |
D ->--------f1(D)---------> E
> ^
V |
f2(D) f1(R)
> >
> >
--------->R------>-----
Thus, even if R is not sensing the state of E (the thing that it is
supposedly controlling), D is and therefore f2(D) IS a function of
E.
What you are expressing here is the fact that normal physical
relationships are bidirectional: if D affects E, there is at the same
time an effect of E on D. If D is exerting a force on E, there is
necessarily a reaction force acting on D.
As I started to say above, mathematical modeling often falls into error
or at least confusion at points like this. If you're modeling a physical
system, you can't just say "let there be a disturbance D that affects E"
unless you have in mind some physical way of establish D so there will
be an effect on E. You can't arbitrarily set D unless you can show that
the reflected effect from E can't change it.
If you don't pay attention to HOW D is established, then what you're
really doing is setting up initial conditions that hold only in the
first instant. We move one end of a spring (the position relative to E
being D) to establish a force on E by stretching the spring. But then
what? If we let go of that end of the spring, BOTH ends will move,
because E is pulling on D just as much as D is pulling on E. If we hold
the position D constant, we will have to anchor it to a very large mass,
or else set up a control system so that the reflected force from E can't
move the position D. That, however, will make the applied force depend
on movements of E, so the initial force we apply is still valid only for
the first instant.
If we want to apply a "force-disturbance" to E, then we have to arrange
for the applied force to be independent of the position of E. That can
be done with a different kind of control system, or by using a uniform
gravitational field, or hanging a weight over a pulley (except that the
mass of the weight would make the force vary with accelerations). But
this is a very special case, requiring special conditions or elaborate
equipment. By saying "Let there be a disturbing force..." we have
unwittingly introduced many complications into the model and restricted
the validity of our calculations to a few special cases.
If we want D to be a true independent variable, then we must set it up
so that E can't affect it. This is normally assumed, often incorrectly,
in modeling a physical situation. When we set up a disturbance in a
computer model, we can easily make D independent of any reflected
effects, so it is a true independent variable. But for modeling in
general, the reciprocal relationship MUST be taken into account if the
model is to be correct.
All this has a resemblance to what you said about digital design. It's
relatively easy to model the logical dependencies you first think of.
But when you run the model, you discover that you haven't handled all
the cases.
-----------------------------------
Finally, the feedback effects from E to D. You have made a brilliant
discovery here. In fact, if R affects E, it is true that given a
bidirectional relationship between D and E, there is a feedback path.
R() is then a control system, even though its designers didn't intend
for it to be one.
The only question left is, how good a control system and what would the
system control? If we gave R a reference input, we could let its output
gain be very high, and then we could have a good control system. But
controlling what? Control systems control their sensory inputs, so this
system would be controlling not E, but D. It would act on E in order to
make D match its reference signal. This would not lead to control of E.
If you adjust the properties of R() so that E is maintained constant,
then there will be no control system and E will be protected against
variations in D. The reflected effect of E on D will be reduced to the
extent that the forward effect through R() exactly cancels the direct
effect through F1().
So the compensatory "control" system is still viable. It is a
mathematical curiosity, because matching the two balanced effects on E
with any precision would in general put impossible requirements on R().
All the examples analyzed by Ashby involve exact arithmetic
relationships, often using only small whole numbers, and require a
complete absence of disturbances other than the one being compensated.
Toss in an integral or two, some nonlinearities, some drifts in sensor
and actuator properties, and a few unsensed disturbances, and the whole
idea goes down the tube.
Nice provocative post.
---------------------------------------------------------------------
Best,
Bill P.