[Martin Taylor 970320 11:00]
Hans Blom, 970319 to Bill Powers (970318.1115 MST))
A while ago, I thought I understood where Hans was coming from, but in this
prolonged exchange with Bill Powers about the theodolite, I begin to wonder.
Hans says:
I have some problems with your approach. Not that the computation is not
correct (it is, assuming that ...), but that it is an unlikely _model_.
Look at your formulas> x2 := x2 + v2 * dt + (t*dt*dt + dt*dt*dt/3.0)/2.0; {MY NEW X}
> v2 := v2 + (2.0*t*dt + dt*dt)/2.0; {MY NEW VELOCITY}and note that this is not the standard difference equation formulation
anymore. For one thing, you assume the controller to have an exact inner
clock that accurately delivers the time t.
What has this to do with the _controller_? If I understood Bill correctly,
he has been trying to get you to agree about the motion of the _theodolite_
when forces are applied to it. Only after you have agreed to that, is it even
reasonable to import a controller to supply those forces. It won't matter
what kind of controller you bring in, it will be the _same theodolite_ if
you are going to compare the effectiveness of different kinds of controller.
How about the two of you treating a model of the theodolite _by itself_ and
only then discussing what the controller might do? I think Bill has been
trying and trying to get Hans to do this, but always in postings that also
mention the controller. Hans has been treating the theodolite as if it
existed only insofar as the controller sees it, so the very idea of a
world containing the theodolite without any controller would seem to be
impossible to contemplate.
I don't understand Hans's world.
ยทยทยท
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I have no doubt that organisms
(we want to go there, don't we?) do have internal clocks, but those are
hardly this precise. If the clock isn't exact, these formulas introduce a
severe form of what you have called "integration error". Moreover, these
formulas -- even with an exact clock -- are of no use _in a controller_:
they are fully bound to the postulate that a=t and cannot be generalized
to something that is useful for control purposes.>My "correction term" applies to the real theodolite, and is necessary to
>represent its response to the controller's output correctly.It does not apply to any "real" theodolite, which is far more complex; it
is a (severely simplified) _model_ of a theodolite; see Martin's
comments.
You say "see Martin's comments" but you show no sign of having seen them.
If you have, how about a little message to say how you view the relationships
among (1) the real physical world that we cannot "know" precisely, (2) the
simulation world that tries to mimic the real world in important respects,
(3) the theodolite, (4) the experimenter/analyst/simulator, and (5) the
controller (without specifying how the controller might work).
Here's my take on these relationships (previously presented in other words,
in the "comments" to which you refer):
We (experimenters/analysts/simulators) have various perceptions of the real
world, and create a simulation world that would produce adequate matches to
those perceptions of the real world that are important to us in the
specific simulation.
The theodolite in the simulated world is one of those perceptions (many,
actually), and in important respects its behaviour in respect to simulation-
world forces is supposed to match the behaviour of a real world theodolite
affected by real-world forces.
The experimenter/analyst/simulator (EAS) creates the simulation world so
that its behaviour can be observed in detail not available in the real
world. Variables that might be hidden in the real world are available for
inspection in as fine or coarse detail in space and time as the EAS
chooses. If the experimenter wants to see what happens on a time
scale of nanoseconds, all that must be done is to specify that observations
on the simulation world are made every few nanoseconds. You can't do that
with real theodolites in a real world, but you can in the simulation world.
The experimenter can choose to apply forces to the simulated theodolite in
quite arbitrary ways. One of those ways is to import into the simulated world
a simulation of a controller that might be used in the real world. The
behaviour of this controller can be observed in as much detail as can that
of anything else in the simulation world. In particular, the experimenter
can observe any internal variable in the simulated controller every nanosecond,
if that is what is wanted.
The simulated controller has limitations, imposed by the need to match what
the real-world controller might be able to do. It may have limitations as
to the force it can apply to the theodolite; it may have limitations as to
the sensors available to it; it may have limitations as to the speed or
accuracy of its sensors. Or any of a host of other limitations.
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In the case in point, Hans has chosen to introduce a controller that
observes, with infinite precision, the position of the theodolite, but
it can do so only every DT seconds. Between those observations, the force
it applies to the theodolite depends on its internal nature, and not on
what the theodolite does in those intervals. Bill has taken that to mean
that the force Hans's controller applies to the theodolite is constant
between sample moments. But it need not be. Bill himself has chosen to
introduce a controller that observes the position with infinite precision
infinitely often and (sometimes) also observes velocity with infinite
precision infinitely often.
Neither is a good model of a real controller, even in the simulated world,
because of the infinite precision assumptions. Bill's controller wouldn't
change its performance measurably if its observations had limited precision,
because its mechanism accomodates random disturbance--but that has had to
be shown in other simulations; we just know it to be so from prior
experience.
What I don't understand is why Hans makes the theodolite a pure product
of the simulated controller's imagination, rather than allowing there to
exist a simulated theodolite that an external experimenter could observe
even if the simulated controller's power were shut off and the controller
removed by a simulated repair person.
Martin