[Martin Taylor 970313 11:45]
Bill Powers (970312.1736 MST)]
Hans Blom, 970312 --
Let's drop the "straight-line segment" argument; it's unimportant. Let's
just skip directly to the main problem here:
Is it unimportant? I think it stems from a very fundamental difference
between the ways the two of you understand the situation being simulated.
The problem, as I see it, hinges on Hans's response to Bill's:
Perhaps you didn't _intend_ to do this, but when you use discrete
equations you are always using straight-line segments to replace
curves.This just isn't true: _nothing_ is or can be said about what happens
_between_ points; the difference equation formalism only allows us to
talk about values _in_ points.
This seems a very clear statement that Hans is dealing with a simulation
world that EXISTS only at discrete moments in time. "...only allows us
to talk about values _in_ points" could not be a sensible comment if the
simulated world (as opposed to the simulated theodolite control) had a
continuous flow of time. What the difference equation formalism allows
is relevant to the behaviour of the theodolite control system, not to
the behaviour of the theodolite, unless the theodolite inhabits a world
that exists only at the moments the control system observes it.
Hans disallows as meaningless comments such as Bill's:
You are right that I want to modify it. In the first equation, u is in fact
a constant because the controller output is constant between steps.
Therefore the velocity will increase linearly with time, and v2 := v1 +
a*dt, as you have written it. This is exact for the example we're using.
For Hans, there _is_ no "between steps" where the velocity might be constant
or changing quadratically. In the discretely jumping world, "_nothing_ is
or can be said about what happens _between_ points."
There's no point really in pursuing the discussion between the two of you
until you agree on the world in which your simulations are supposed to
operate. Either Bill should consider how a control system (PCT or MCT)
would operate in a world that does not exist between discrete moments, or
Hans should consider how a control system that observes only at discrete
moments would behave in a world that simulates the one we live in, insofar
as time in it seems to be a continuous flow.
In our world, the one we live in, we can observe between T and T+DT, no
matter how small DT, at least within time ranges of interest when talking
about a theodolite. People make instruments of ever-increasing temporal
precision, and time stuff down to femtosecond intervals.
If we are interested (as I am) in whether there is an important difference
between PCT and MCT other than the explicit and obvious structural
difference, then comparisons _have_ to be made between models working
in the same simulated world.
But during this interval, v changes from v1 to v1+a*dt. The distance
traveled is the _average_ velocity (not the final velocity) times dt. The
_average_ velocity is (v1+v2)/2, or[v1 + (v1 + a*dt)]/2,
yielding
dx/dt := v1 + a*dt/2,
and
x2 := x1 + v1*dt + 0.5*a*dt^2
This is wierd, Hans. All I am doing is going through the elementary calculus
derivation of the double integral of the initial expression.
Yes, but this calculus applies only in a continuously varying world, not
the world in which Hans's theodolite operates.
Has it been so
long since you worked with continuous systems that you have truly forgotten
all this? Perhaps now that you see the derivation, it will all come back
to you.
I'm sure Hans knows all about the calculus. He is just asserting, in many
messages, that it doesn't apply in the world of his model.
If I may take it that we agree on the theodolite equation at last, the next
step is to check out the rest of my version of the MCT program.
I'd say that "if I may take it..." is wishful thinking. But good luck,
anyway.
Martin