[Martin Taylor 2012.12.06.23.52]
[From Rick Marken (2012.12.06.1430)]
I suppose I should reply to Bruce A and Martin T. But because I
know that whatever I say is not going to change their minds I
would like to try to move the conversation up a level and ask why
they are so interested in information theory and it’s relationship
to control.
Now that's a good question. I can't speak for Bruce, but my interest
is that I like to have as varied a toolbox as I can. If there’s a
tool that looks as though it might sometimes be useful, I’ll keep it
in the box.
I know why I'm not interested in information theory:
it seems (and has proved) to be completely irrelevant to all of
the research I have done on control. The good ol’ PCT model has
been the basis of all my research and I find it very easy to use
the theory to understand everyday behavior. So why all the
interest in information theory? DO you use it in your research on
control? Does it explain things about control that control theory
doesn’t?
I don't understand this last question. It's like asking "does
[algebra|calculus|Laplace|Fourier transform mathematics] explain
things a bout control that control theory doesn’t?" None of those
questions would make sense to me, and neither does the question when
the mathematical operations involved are information-theoretic.
> Martin Taylor (2012.12.05.14.0)--
> RM: Oh, no. Not this again!?! I thought we had slayed the
"information
> about the disturbance in perception" dragon years ago.
Suffice it to
> say, the error signal can't convery informatoin about the
disturbance
> because the error signal depends on the difference between r
and o+d,
> not between r and d. Think about it.
> MT: Here's the argument, without the maths that so confused
you in the long-ago
> disucssion. Just think about the situation from the viewpoint
of an outside
> observer deducing the disturbance waveform from observation
of the output
> waveform. Information-theoretically, the quality of control
is specified by
> how much you don't know about the disturbance when you know
the output. If
> control is perfect, and the observer knows the environmental
feedback
> function, the output waveform is sufficient for the observer
to know the
> disturbance waveform, because the output waveform,
transformed by the
> environmental feedback function, is exactly the negative of
the disturbance
> waveform.
RM: OK. So far so good. Although you should be clear that it is
the net effect of disturbance variables on the controlled about
which you are getting information. this effect, call it d* is a
function if the all disturbances that are having an effect on the
controlled variable:
d* = g(d1,d2,...dn)
where g() is the disturbance function that related disturbance
variables, d1,d2,dn to the controlled variable.
Red Herring alert! The input quantity
sometimes called qi = d + f(qo), where f is the environmental
feedback function and d is the disturbance that you call d*.
>MT: How can this happen? The output doesn't create its
waveform spontaneously,
> nor are the output and disturbance waveforms both determined
by some common
> source of variation; yet the output waveform is influenced
(determined, if
> the reference value remains unchanged) by the waveform of the
disturbance.
> The disturbance does not influence the output by effects that
propagate
> backward through the environmental feedback function. The
only remaining
> possibility is that the output is influenced by the chain of
influences
> through the sensory system, the perceptual mechanism, the
error computation,
> through to the output function and output machinery that
produces the output
> that the outside observer can now use to determine what the
disturbance must
> be.
RM: OK. So you are saying that it works this way:
disturbance-->sensory system-->perception --> error
→ output
Three problems here. First, the disturbance variable has an effect
on a physical variable that is the environmental correlate of the
controlled perceptual variable; in B:CP it’s called the controlled
quantity… So the path above should go like this:
disturbance--> controlled quantity-->sensory
system–>perception → error → output
Second, the controlled quantity is influence by both the
disturbance and output variables simultaneously. So an arrow
should loop back from output to controlled quantity.
Finally, there is a (possibly varying) feedback function
connecting output to controlled quantity.
All true, but how does this affect what I said? Was I required to
detail every element of the control loop? I would assume that most
CSGnet readers would know all of what you call “problems”, and not
require a basic tutorial every time one wants to sketch something
about the influence flow.
Give these last two facts (or either one of them alone), it is
impossible for for the error signal to contain information about
the disturbance which would allow the output to mirror that
disturbance.
That is precisely where I showed that you were wrong! I won't repeat
the argument, because it is so trivially simple that there is hardly
any other way to say it beyond “with good control, the influence of
the output on the input quantity mirrors the influence of the
disturbance. Information about the disturbance waveform therefore
arrives at the output. The information cannot go backwards through
the environmental feedback function. The only other possibility is
that it goes through the path internal to the control system.”
Let's forget "information" in the technical sense for a moment, and
talk “correlation”. If two variables A and B are correlated, there
are several possibilities for the influences involved: (1) A
influences B, (2) B influences A, (3) something else we may call X
influences both A and B. We can substitute “have non-zero mutual
information” for “correlation” and the same thing holds. If there is
control, the mutual information between disturbance and output is
non-zero. One of the following must be true: (1) the output
influences the disturbance, (2) The disturbance influences the
output, and (3) Some unknown thing influences both output and
disturbance. In a control system we know 1 and 3 are false and from
the physical circuitry we know 2 to be true. Somehow, the
information from the disturbance appears at the output, and the only
path through which this can happen is by way of the internal
circuitry of the control system.
Be even assuming that this were possible, there is no
way for the system to know what the output should be so as to take
into account the feedback function.
Why should it? And how is this statement relevant? What control
system “knows” (I suppose you mean perceives) its output?
For example, if the feedback function multiplies the
output by 2, then if the output of the system were, based on
information about it, a perfect mirror of the disturbance, then
the output would be twice as big as it should be and there would
be no control.
True. Why mention it?
I hope you will agree that there is no information about the
feedback function that comes into the system.
Yes.
And you yourself said that one has to know the
feedback function in order to reconstruct the disturbance from
output. So clearly a control system can generate the output
appropriate to compensate for the effect of disturbances to a
controlled quantity without any information about the feedback
function that connects it’s own output to the controlled quantity.
Yes.
So if it can do that, why not go ahead and imagine
that it can also generate output that compensates for the
disturbance without any information about the effect of the
disturbance? Imagine there’s no information about the disturbance
(or feedback function). It’s easy if you try.
No it isn't. It is damned difficult, simply because I don't believe
in pure magic. I believe control systems work on normal physical
principles. — [Later] When I wrote that, I mentally substituted
“magnitude” for “effect” in your first sentence. With good control,
the disturbance has very little effect, no matter how large its
magnitude. It’s the magnitude of the disturbance that is
informationally connected to the output, not the effect of the
disturbance.
> MT: Information about the disturbance does pass through the
interior components
> of the control loop, back to the input where that information
is used to
> cancel the disturbance.
RM: So again I ask, why do you want to believe this? Or better
yet, what do I gain if I go ahead and believe it? Would it change
the questions I ask? The way I do research? Build models?
That is up to you. I can't speak for you. But it's not a question of
belief or disbelief. It’s a question of tool use. If you know how to
use a hammer, chisel, saw, and screwdriver, you can do more with
wood than if you can use only a hammer and a saw. If you don’t
believe that a chisel is useful, or don’t know what one is when you
see it, you won’t learn to use it, and your woodwork may be good,
but it will miss some effects that can easily be done with a chisel.
One thing thinking in information-theoretic terms does for me is
clarify the relationship between disturbance bandwidth, transport
lag, and limits on control quality. But there’s lots beyond that,
especially when you get into the interactions among many control
systems. Not, however, anything that interests you, which is fine by
me so long as you stop converting your personal lack of interest in
the tool into general statements of its irrelevance or
impossibility.
Martin