[From Bill Powers (930315.1900 MST)]
RE: perceptual information about disturbances.
I agree with Martin Taylor that we need to stand back and try to
find some new approach to this problem. All that's happening now
is that positions are being hardened; pretty soon they will be
fortified and that will be the end of that.
Let me try to analyze the problem without bringing in information
theory at all.
If a disturbancing variable affects a controlled variable (to
speak qualitatively for a moment), it is perfectly true that the
perceptual signal will change a little. The perceptual signal
being the route by which all "information" enters the control
system, this means that the control system "has information about
the disturbing variable." I am using the term "disturbing
variable" to be perfectly sure it is understood that I am talking
about the independent physical variable that contributes to the
state of the controlled variable, not about its effects on the
controlled variable, which must be deduced. When I use the term
information here, I mean it only in the semantic sense.
The crucial question is what kind of knowledge that information
provides. If the control system is reasonably good but not
perfect, the change in the perceptual signal will give rise to a
corresponding change in the output, which will reduce the
unopposed effect of the disturbing variable by a large amount --
say 90% -- but will not remove it altogether. The net change in
the perceptual signal, even so, is only 10% as large as the
change that the disturbing variable would induce without the
control.
Furthermore, as Martin pointed out, in real control systems there
are lags. The output that is now being subtracted from the effect
of the current amount of disturbance is the output that was
generated by the perceptual signal of one lag-time ago. In
addition, the output can change only at a certain speed, so it
will rise and fall somewhat more slowly than the value of the
disturbing variable rises and falls. Changes in the perceptual
signal represent the current rate of change of the disturbance
plus the slowed rate of change of the output generated by the
perceptual signal of one lag-time ago.
So the perceptual signal at any moment represents the current
magnitude of disturbing effects plus the (opposing) lagged and
slowed influence of the just-previous perceptual signal's own
effect on the output apparatus that is opposing the disturbance.
At a given moment, the perceptual signal, being a scalar, has
just one magnitude, or it can be specified as having a mean
magnitude with some uncertainty in both time and magnitude.
Maybe we can now make clearer what it means to say that the
perceptual signal contains information about the disturbance.
Given this perceptual signal, which contains all the information
that the control system can get about the external world, how far
could the control system go (even equipping its perceptual
function with a supercomputer and a vast intelligent program)
toward reconstructing the state of the independent disturbing
variable itself?
For an external observer able to see all signals and variables,
this is not a problem: simply subtract the current magnitude of
the output from the current magnitude of the controlled variable,
and you have the magnitude of the external disturbance, plus or
minus random noise. The external observer is not limited to the
information contained in the system's perceptual signal.
But what can the ECS itself determine about the disturbance? An
ECS does not sense the state of its own output; it senses only
the state of a controlled variable affected both by the output
and by the disturbing variable. What can the information in the
perceptual signal _alone_ represent about the disturbance?
The magnitude of the perceptual variable does not represent the
magnitude of the disturbing variable averaged over the same short
time. The disturbing variable's magnitude is many times that of
the perceptual signal, by an amount that is larger as the control
system's loop gain is larger. This would be seen immediately if
somehow the output were forced to zero, so the full effect of the
disturbing variable were passed through to the controlled
variable. Then the perceptual signal would be a true (reasonably
true) representation of the disturbing variable. With the
feedback, its magnitude represents only a small part of the
magnitude of the disturbing variable.
The waveform (time-variations) of the perceptual signal do not
resemble the waveform of the disturbing variable. They represent
only the difference between time-variations in the disturbing
variable and time-variations in the slowed lagged output. So
given only the time variations in the perceptual signal, there is
no conceivable perceptual function that could calculate backward
to isolate the variations in the disturbing variable from those
in the output.
Neither would there be a channel to carry such information: in an
ECS, the only output of the perceptual function represents the
state of the controlled variable and nothing else.
So speaking strictly in terms of the effects of variables on
signals and signals on variables, it seems clear that the
perceptual signal in a single ECS is in fact somewhat affected by
the disturbing variable, but in such a way that the state of the
disturbing variable could not be uniquely reconstructed from the
information in the perceptual signal alone.
In order to oppose most of the disturbance, the output must be
adjusted nearly to the same magnitude as the disturbing variable,
and its variations must not be so slowed or lagged that the
controlled variable is allowed to change by large amounts. If
control is to be good, the output must be a reasonably close
mirror image of the amplitude and waveform of the disturbing
variable.
Thus if we thought of the disturbing variable as responsible for
the output actions, it would be necessary for the full magnitude
of the disturbing variable and its actual variations through time
to be represented in the signal that drives the output function.
But we have just seen that the perceptual signal, the source of
the driving signal, does not represent either the full magnitude
of the disturbing variable or the actual waveform of the
disturbing variable. The only conclusion possible is that the
explanation in terms of the disturbing variable quantitatively
causing the output is incorrect.
The correct explanation does not have to take the disturbing
variable into account at all -- that is, the operation of the
control system can be fully analyzed and explained without any
disturbance present. Given this explanation, we can then predict
the action of the system when any disturbance is introduced in
any part of the system. Nothing has to be added to the
explanation; the adjustment to the presence of the disturbance is
a completely automatic outcome of the operation of the system by
the same rules that apply when there is no disturbance. The
procedure is identical to that of solving the homogeneous case of
a differential equation first, and then completing the solution
with a special case, an arbitrary driving function of known form.
I believe that I can justify and demonstrate each point made
above. Rick Marken, in _Mind Readings_, has already demonstrated
most of them. This discussion has said nothing about information
theory or what it can or can't do: we've been working strictly
with basic control theory.
Is this acceptable so far?
ยทยทยท
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Best,
Bill P.