[Martin Taylor 970421 05:50]
(Martin Taylor 970420 23:57)
Bill Powers (970419.1830 MST)
Woke up at a ridiculous hour with my mind scrambled by remembrance of a
stupidity written before I went to bed. In response to:
All technical
discussions, that is, except those that Martin Taylor introduced, where he
changed the meaning of "the disturbance" from qd (the value of the variable
qd) to Fd(qd) (the value of the function Fd(qd)). If we simply go back to
the original definition of "disturbance" as a technical term in PCT, this
entire argument should disappear.
I said:
Well, at least the nonsense of saying that "there's no information about
the disturbance (qi) in the perceptual signal" should disappear.
I hope nobody was misled by that "(qi)", which should have read, of course,
either "(the effect of the disturbance on qi)", or better "(Fd(qd))", using
Bill's notation in his diagram.
I mean, of course, "xd", which I have interpolated into the diagram
in the quote below. For symmetry, I've also inserted "xo", the effect
of the output on qi. It is qi that is converted by the perceptual input
function "Fi" into the perceptual signal. It is the waveform of xd that
can be reconstructed to the degree that control is successful, using an
invariant function applied to signal "p", provided r, Fo, and Fe don't
change. That's the most dramatic demonstration that the signal "p" conveys
information about the disturbing influence xd.
r
>
p ---->[Comp] ---- e
> >
[Fi] [Fo]
> >
qi<--xo--[Fe]----- qo
^
>
xd
>
--<--[Fd] <---- qdwhere
qi = input quantity ("controlled [environmental--MMT] variable")
qo = output quantity
qd = disturbing quantity or "disturbance" for short
Fi = input function
Fo = output function
Fe = environmental (feedback) function
Fd = disturbance function
to which I add
p = controlled variable ("perceptual signal")
xo = influence of qo on qi
xd = influence of qd on qi
Clearly, any number of combinations of Fd and qd could produce the identical
contribution to the state of qi. It is not necessary, of course, for the
control system or its designer to know just how disturbances come to affect
qi -- all that matters to the control system is the state of qi, as
represented by p.
Exactly as we had always presumed it to be. Anything else is absurd, so far
as I can see--so absurd that it is a bit insulting that the presumption
we had meant "qd" rather than "xd" in the original (and, apparently,
subsequent) discussions even existed for a moment. Rather than years of
repetition of this, the first time this misapprehension was corrected
should have been the last. I had thought it securely laid to rest when we
talked about it face to face. The fact that I had never even contemplated the
notion that "disturbance" might refer to things irrelevant to the control,
until you pointed out that this is exactly what you did mean by the term,
should not now be used as an argument for anything at all.
If we simply go back to
the original definition of "disturbance" as a technical term in PCT, this
entire argument should disappear.
As I said earlier, I hope that you mean we can now get back to the meat
of the discussion and try seriously to analyse the information flow in the
control system.
ยทยทยท
------------------
The whole "info in p about d" thing started when I said that the neat thing
about a control system is that it can correct the effects of disturbances
without knowing what is causing them.
I remember it differently, and for this I'm not going to go back to the
archives because it really doesn't matter. You may well have said this
at some point, but if you did, and if it all started then, it was because
none of the "pro-information" people conceived the possibility that
"what is causing them" might have meant "qd" rather than "xd."
My memory is that I tried to tease Rick by saying something like "you
can't understand what a control system is really doing if you don't
understand the information flows." He took it very seriously and
seems still to take it so, at least until your recent message (Bill
Powers (97045.0848 MST)):
+ ...we do not have to assume [Martin] means that the control system
+itself makes any use of the knowledge of these unsensed variables and
+functions. If there is an analyst who knows about the output, the
+environmental feedback function, the disturbing function, the disturbing
+variable, and the controlled variable, then the _analyst_ can (in principle)
+do an Informational analysis of the system, showing how a hypothetical
+quantity called Information could be traced through the various processes.
+This is, of course, aside from the fact that nobody has yet done a
+successful Informational analysis of a closed-loop system, all attempts to
+do actual numerical calculations having failed so far.
The whole set of tar-baby episodes has occurred because what you say
here was not understood. Using the diagram again, the variables that might
come into play are the inputs to the loop (r and xd), and the functions in
the loop (Fe, Fi, and Fo--Fum having got lost somewhere).
And let's not forget a variable that is almost always ignored when the
algebra of control is used rather than the transform analysis--time.
You can't deal with information without considering time, because it
is the rate of gain and decay of information that is critical rather
than the amounts involved in a discrete event. We are dealing with
continuous systems for the most part. Fe, Fi, and Fo are functions
extended in time (for example, they may incorporate elements such as
integrators or differentiators). r and xd are continuously varying
waveforms, in the general case.
---------------------------
Picking up the tar-baby once more, with full realization of how ill-advised
it is to do so (but it's OK, I'm leaving for a month in a couple of weeks)...
As an example, notate the uncertainty about x given y as U(x|y). Then
if x(t) is an exact observation at time t, and y(t) is the set of all
values observed before t, U(x(t)|y(tau)) = 0 if tau >= t; but it increases
with t-tau up to U(x), the limit imposed by prior knowledge of the overall
distribution of x. U(x) = <integral over all x of>(p(x) dx), where p(x0)
is the probability density of x at x0.
Like this:
> _______---------------------
> -----
U(x(t)|y(tau)| ---
(= U(x.y.z)) | /
> /
> /
-------------------------------------
t-tau (= z) 0
Now, if you are trying to provide a value (m) that exactly matches another
value (v) for which the uncertainty function U(v.v.z) has this generic form,
but at time t you only can use observations of v prior to time tau, then after
your best effort at compensation you cannot match v exactly. The match
will leave U(v.v.t-tau) as a residual uncertainty. It's just like the
correlations we were discussing. Your observation of v at time t is
correlated 1.0 with the actual value of v (assuming perfect observation).
After a little while, the correlation of your old observation with the
current value is less, and eventually the correlation goes to zero. At that
point, the uncertainty of v is just what it would have been if you had not
made the old observation at all.
The same happens if you use not just the observation at time t, but all your
observations up to time t. Unless you have a perfect model of the
fluctuations in v, your uncertainty about its value will increase as
time goes by since you last observed it. If the bandwidth of v is
infinite, U(vv.z) goes instantaneously to U(v). But for real physical
variables it doesn't.
In a control system with non-zero transport lag, the information rate of
the disturbance (its bandwidth, in essence) and the lag together limit
the precision of control. If p is a one-to-one function of qi (i.e. Fi
is noiseless), then if z is the transport lag, U(p(t)|xd(tau)) = U(p.xd.z) in
the diagram above.
U(p.xd.z) provides a lower bound on the residual correlation between
the perceptual signal and the disturbance, and is near zero when control is
good.
I hope this all makes sense as a verbalization of what is not yet an
"actual numerical calculation". It's an attempt to set the conceptual
frame, and to show the need for considering time. Given the time of day,
I'm afraid it may make no sense at all.
Martin