[From Bill Powers (960715.1100 MDT)]
Hans, you're beating a dead horse.
I assume that I can interpret this as meaning that you agree with
what I said in this post.
That is too broad an interpretation. I meant only that I understand the
difference between the effects of perturbations entering various parts
of the system.
I interpret this thus: a disturbance changes slowly and smoothly,
noise rapidly and unpredictably. Let's see. I am sure that you are
familiar with the fact, that any kind of noise (colored, pink) can
be made by filtering white noise, like this
Filtering is beside the point -- we are not discussing how a particular
perturbing influence comes to have the spectral distribution that it
has. It might arise from filtering a random variable, but it might also
arise from simple interactions in the environment that do not have the
characteristics of a random variable. I'm trying to make a very simple
point and you are trying to make it complex.
I had noticed some time ago that your model didn't contain any
independent sources of perturbation that varied relatively slowly and
systematically. The random variables that were in the model were treated
simply as unpredictable variations that could not be treated moment-by-
moment, but could only be dealt with in terms of statistical measures
like means and standard deviations.
When we got to the challenge, I presented a model that had an
independent variable in it which varied slowly (relatively slowly) with
time. This is what I meant by a "disturbance." You proceeded to put this
variable into the model, and by using rapid adaptation, produced an
output that canceled the effect of this variable on the plant output x.
So this is a clear example of a disturbing variable that changes slowly
enough for the system to adjust its output to oppose it. Your model, as
well as mine, was able to react to the disturbance in real time and
prevent it from having more than a minor effect on the controlled
variable.
If I had proposed instead an independent perturbation taken directly
from a random number generator, neither your model nor mine would have
been able to oppose the effects of this disturbance on the controlled
variable. Your calculations would have included the statistical
characteristics of this disturbance as part of determining how to adjust
the parameters of the world-model, but the actual disturbance would not
have been reduced by the behavior of the model. It couldn't be, because
there would have been no way to calculate an output that would
systematically oppose the random fluctuations.
In your example of a spacecraft in orbit, one source of slow systematic
disturbances is the perturbation of the orbit due to other masses and to
zonal harmonics and mascons in the Earth itself. In a sophisticated
multi-body model, all these perturbations can be calculated. But to
control the spacecraft, it is not necessary to calculate all these
perturbations. All that is necessary is to compute the desired orbit
using 2-body dynamics, and use the reaction engines to oppose any
deviations of the measured orbit from the ideal orbit. Of course
demanding such a simple idealized orbit would turn out to be wasteful of
fuel, but my point is that such an orbit could be maintained, and it
could be maintained by a system designed exactly as your "challenge"
program was designed, one that actually computes and compensates for an
additive disturbance of low frequency. In this case these low-frequency
disturbances come from ordinary low-frequency phenomena: the slow
movement of the moon and other planets in their orbits, and the movement
of the spacecraft in the slightly aspherical field of the Earth. There
is no high-frequency noise source in the background with its effects
passing through a filter. The uncertainty in our knowledge of the orbit
is strictly due to ignoring, or being ignorant of, all the gravitational
influences that are acting. It isn't a statistical uncertainty; it is
not normally distributed.
First on terminology. If we consider the signal only, let us talk
about "noise". Noise can be white or have any coloring, depending
on the filter. And let us talk about "disturbance" when the signal
stands into a relation to something that is disturbed. In my "dead
horse" post, you could say that in case (1) the effect of the
action is disturbed because something or somebody else acts as
well; and in case (2) you could say that the perception is
disturbed. If we do this, we have meaningful definitions of both
"noise" and "disturb- ance".
This is not at all what I was trying to talk about. The place where the
disturbing influence enters is irrelevant. If, in your case (1),
somebody or something acts in a way that can be described by a random
variable, so that the effect at time t does not predict the effect at
time t + dt, then we have "noise" as I am trying to define it. If the
thing or person injects slowly-varying influences, then we have a
"disturbance" as I want to define it. We can have either one, when
someone or something is acting on the physical environment that is
involved in the control process.
Likewise for your case (2). We can have either random perturbations of
the perception, or slow perturbations. If the perturbations are random,
there will be no systematic effect on behavior, but only a lessening of
the accuracy of control. But if the perturbations are slow enough, they
will appear to the adaptive control system as real changes in the plant
output, and will be opposed. The control system will then control its
perception just as well as before, but the plant output will show
fluctuations caused by the disturbance.
So my proposed distinction between "disturbance" and "noise" is
meaningful, and yours simply ignores the distinction I am making.
There is no way for the system to distinguish between low-
frequency disturbances of the plant and direct low-frequency
disturbances of perception.
And here you are plainly wrong, as I demonstrated in my "dead
horse" post -- which may therefore not be so dead after all...
Your example of telling the difference between random fluctuations in
the plant and in perception is correct, but only for the special case
where the plant contains a pure integrator. If the plant has a
proportional response, or contains a restoring spring, there is no way
you can tell where a random disturbance is entering just from observing
the perceptual signal.
I think you're trying too hard to reduce all possible cases to the one
you're familiar with:
Considering this process in terms of a bandwidth is possible, but
it easily leads to confusion. A statistical analysis is much
clearer.
I don't get confused when speaking in terms of bandwidth. You do. You
find a statistical analysis much clearer; I find it obscure.
If you try to make the system oppose arbitrary
disturbances, you also reduce its ability to control "blind."
No. Both can coexist.
Hans, you know that's not true. If you set up your "challenge" model to
adapt slowly, it will be able to make its output (and its imagined
input) follow changes in the reference signal when feedback is lost for
a considerable period of time, but it will not be able to oppose
disturbances of the kind I used. If you set it up to be able to oppose
the disturbances I used, it will lose the ability to make the imagined
input track a changing reference signal for more than a few dt's. There
is a tradeoff between working without feedback and being able to oppose
ongoing disturbances. You can't have both.
And when control is not so good? Or absent? Do the disturbances
become noise? I will not start to use this confusing definition. It
is far too subjective for me. How do you like my distinctions?
Could you live with them? Dead horse again?
Nonsense. There is nothing subjective about my proposed distinction
between noise and disturbance. They are defined quite objectively,
relative to the speed of response of a particular control system. It is
your proposed distinctions that miss the point.
You should have cursed Mary
for her data, not thanked her: her report shows that your claim is
totally wrong.
Which claim precisely? That you CAN thread a needle in the dark?
That is ridiculous. If you meant that you CAN thread a needle in the
dark, given some completely different means that has nothing to do with
the original way of threading it, I have to agree completely. You just
turn on your needle-threading machine, and it doesn't matter whether the
room is dark or light, or whether your eyes are open or closed.
The issue was whether a model-based control model is justified by the
data. According to your model, you learn to thread a needle by building
an internal world-model that is adapted until it reacts to command
signals just as your hands, the thread, and the needle actually react to
the same command signals. When you close your eyes, the internal model
supplies, as usual, the (imagined) picture of your hands, the needle,
and the thread, and you emit exactly the same command signals as before
-- the only difference is that the adaptation process stops when the
real-time visual input is lost. Thus we should observe that your hands,
the thread, and the needle all behave just as they do when your eyes are
open.
In fact, they behave completely differently. You can't save your model
by whining "But she still threaded the needle, didn't she?"
In the dark, you are controlling a completely
different set of inputs -- not just running the same old model
without the feedback.
Now we're getting somewhere! Because we have done it so
infrequently, we are not very proficient at threading needles in
the dark.
What does that have to do with world-model-based control of threading a
needle with your eyes open and then shut? As Bill Leach said, you're
making my point, not yours. If control of needle-threading is done using
an internal world-model with your eyes open, then by your own arguments
closing your eyes should make no difference -- the same outputs should
be generated. And they're not.
That's why I suggested to Rick to contact his local Union of Blind
Seam- stresses ;-). I bet one could have a pretty good control
system for threading needles in the dark once it becomes routine.
Yes, and then what would happen if we sprayed the seamstress's fingers
with novocaine? That would be equivalent to a sighted person being
blindfolded. Can you possibly claim that the blind seamstress would be
able to thread the needle anyway?
Your arguments are getting silly, Hans.
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Best,
Bill P.