[Martin Taylor 970327 13:20]
Hans Blom, 970327c]
(Martin Taylor 970326 15:50)
>The fact that a digital computer can't make continuously varying
>signals or filter functions in no way alters the fact that the PCT
>controller being simulated observes its world all the time.Am I also allowed to say: "The fact that a digital computer can't
make continuously varying signals or filter functions in no way
alters the fact that the MCT controller BEING SIMULATED (my emphasis)
observes its world all the time"? If not, why not?
You aren't, because it brings us right back to where I unwisely jumped
into this fray a month or more ago.
I asked "what do you mean by "one-step" correction in a continuous world?"
Your answer, in effect, was that it observed only at discrete sample
moments separated by dt. All of my subsequent interventions have been
aimed at trying to get a handle at what happens in the simulation world
between moments when the controller observes the theodolite. Now you
reverse field and say that the controller observes the theodolite
continuously, and the discreteness is only in the necessary sampling
done to make it possible to use a computer to do the simulation.
That's a different controller from the one I had long thought you were
describing and trying to simulate, and it raises once again my original
question--and some others.
If in fact you are simulating an MCT controller that observes the world
continuously, you _must_ answer my original question. What is "one-step?"
There _is_ no "step" in a continuous world for the correction to be
perfect after one instance of--(pace W. Churchill and all grammarians).
If you are not talking about simulating an MCT controller that observes
the world at discrete moments and expects its correction to be exact at
the next observation moment after it observes an error, what _are_ you
talking about?
>>Martin, where does a reference level come from? From inside the
>>organism, we usually suppose. And often we find that the organism
>>is an effective control system. So not all its goals are
>>unrealistic...>That doesn't come close to answering the question of how the control
>system at time t can know/guess/model what its reference value for a
>perception will be at time t+dt in the future.Yes, that seems to be one of the central questions. I will, however,
demonstrate -- in due time; I have to go slowly -- that a PCT
controller ALSO controls for _future_ values of the perception to be
at certain reference values. That is simple logic, it seems to me: an
observed value cannot be changed anymore. Yet, more than logic seems
to be needed. And that I simply cannot understand.
Let's take this very slowly. But not as slowly as resorting to Logic (doing
that took Russell an entire volume to define the number "1", if I remember
correctly).
Point 1. Call the present moment time zero (T = 0). The world being simulated
runs in continuous time. The simulation world does not. The experimenter
can observer the behaviour of any aspect of the simulation world, but can
do so only at specified moments, which we arbitrarily say occur at uniform
intervals "dt" when we map the simulation world to the simulated world.
We can enumerate the samples taken in the simulated world, whereas we
cannot enumerate the continuously varying values taken on by the
corresponding variables as time progresses in the simulated world.
Call the sample number corresponding to time zero, zero, and give
negative numbers to samples in the past.
Point 2. If the experiment is about a control system, the experimenter has
free reign over (a) the structure of the controller, (b) the thing being
controlled, and (c) certain signal variables at positive sample numbers.
The experimenter does not have the ability to affect the observed values
at sample numbers equal to or less than zero, no matter what the variable.
The experimenter can both observe and affect the value of a variable at the
same sample number, but the observation comes before the effect of the
experimenter's intervention. In particular, an observation at sample zero
can be followed (at time T=0+epsilon in the simulated world) by an
intervention--a variable setting--at sample zero. See Points 3 and 4.
Point 3. The experimenter cannot observe the values of sample variables
for sample numbers greater than zero, but can maintain a historical record
of values of what the values were at any sample number up to and including
zero.
Point 4. The experimenter can affect the values of certain variables (Point 2c),
but only at time zero plus epsilon (by that I mean that the experimenter can
observe the value at time zero as a consequence of what happened earlier, and
at time zero can affect what may evolve in the system, the effects of which
can first be observed at sample 1.)
Point 5. A specific restriction in the case of simulating control systems
under the conditions of the Powers-Blom discussion is that the structure
and parameters of the control system and of the object being controlled
(theodolite) cannot be changed by the experimenter after the trial run has
started. There are two signals that the experimenter can affect at time
zero+epsilon (i.e., at the current sample moment, after observing the
current value). Those two signals are called "reference" and "disturbance."
Affecting those two signals is the _only_ way that the experimenter can
influence the signal values within the simulated world that includes both
the experimenter and the theodolite.
Point 6. The simulated system can be affected by the experimenter only
in respect of interventions made at samples prior to sample zero
I hope that everything is understandable up to this point. The put it in
a more colloquial way, you, the experimenter, are allowed to set up the
initial conditions any way you like, and can observe everything that happens
at discrete sample moments; but after you set the system going, you can
affect it only by injecting signals at two points, at any time in the
continuous simulated world, but only at moments notionally just after
making an observation in the discrete simulated world. And the simulated
system can't be affected by what you intend to do in the future.
>I have no qualms at all about statements that at time t one can have
>a reference value for what the perception will be at time t+dt. But
>the value of the reference is its value NOW.Yes. NOW I have to know (decide, compute, control for) where I WANT
TO go. You might call that "planning", in our case over a possibly
infinetisimally small time period dt. You are right: the reference r
for x at time t+dt must be known at time t.
I'm relieved to know that you accept this.
That is, the comparator
compares r(t+dt) and x(t), and the controller's output is driven by
the difference. And that in the PCT controller as well! Surprise?
No. Incredulity:-)
The comparator compares r(tau) and x(tau) in the PCT system. Your MCT system
can do what it wants. But according to the points above, it _CANNOT_ know
what the experimenter has planned that r(tau+dt) will be.
Now here's a minor concession. The notion is implicit in the simple-minded
PCT model that r(t) will be unchanged in the near future. This comes
from the fact that there is no way that the control unit can assess what it
will be asked to do. In actual practice, there may well be regularities in
the variations of the reference signal, as there may in the disturbance
signal, that allow for some prediction over short time intervals. The real
issue is how long it takes the autocorrelation function (of either signal)
to go to zero. What this says is that there may be a possibility for
predicting (to some extent) r(t>0) as a function of the waveform of r prior
to time zero. In that case, there is a possibility of using that prediction
to affect the output of the controller. However, we usually work on the
premise that the disturbance (and by implication the reference) is
band-limited white noise. But not always, and the step function disturbance
of the current dialogue is a notable exception; its spectrum is white,
but it is not Gaussian noise.
You concern yourself with the "logical" impossibility that r(tau) can be
used to affect x(tau). Should you not equally concern yourself with the
fact that r(tau-2) cannot (now) affect x(tau-1)? The fact that I (now) want
ice cream and do not have any in no way prevents the error from affecting
my output such that if, when I arrive at the dispenser of ice-cream I still
want it, I will then both want and have it. Neither does it preclude the
possibility that I arrive at the ice-cream dispenser in a state where I
no longer want any. And if I do, I can't change the fact that I used to
want it, even as recently as to allow me at my fastest to arrive at the
dispenser no longer wanting it.
>And even if NOW I can't affect the value of the perception is have
>NOW, nevertheless I can NOW want the perception to have its
>reference value, and act to make it so, without specifying any
>particular moment when it should take on that reference value.How would you want to model that, i.e. express it in mathematics or
in computer code? I need more than words, Martin...
e(t) = r(t) - p(t)
o(t) = f(e(t), e(t-1), e(t-2)...., o(t-1), o(t-2)...).
(Remember that we are enumerating the samples, not referencing them to
time in the continuous simulated world).
>All I need say is "as soon as I can, and thereafter, unless my
>reference value changes in the interim."In my MCT theodolite controller, the "as soon as I can" translates
into "after a time increment dt", which proves to be possible (unless
the controller's output saturates).
But you say that this simulation is of a controller for which dt is
literally infinitesimal (the separation between successive values on the
real number continuum). That means that the controller's output either
saturates or is _literally_ infinite. In the practical MCT controller
being simulated, saturation is the more probable;-)
"As soon as I can" means, in the simulated controller, "As soon as this
heavy theodolite deigns to move, seeing that I am pushing as hard as I can."
What does it mean in the controller in the simulation world?
Since you insist that you are simulating a realizable theodolite control
system insofar as you can when you ignore such things as friction, and
also insisting that you can make dt in the simulation system as small as
you like, you really ought not to treat a controller whose output, when
accurately simulated, is literally infinite. You should examine the behaviour
of a controller that has a specified limit to its available output.
The simulation controller should be _based on_ applying saturation force for
as long as it takes, and opposed saturation force for as long as it takes
to stop the swinging theodolite.
Notice that the PCT controller in the continuous world does not produce
infinite output, so these constraints don't affect it. What affects both
kinds of controller is transport lag, which both have omitted in the
simulation code. With transport lag, the PCT controller goes unstable
if the gain is too high, so it won't saturate if the available output
force is sufficient. The MCT controller may or may not go unstable--I have
no notion, but I'm sure you could make a model that wouldn't--but it pays
the price by being limited at small values of dt by output saturation
(or by simulating a nonsense controller capable of unlimited output).
I think (hope!) I understand you. I am saying that the reference for
x(t+dt) should already be known at time t.
I make another concession here, in that the reference value inserted after
the observations at time tau cannot be changed until time tau+1, and will
be observed at time tau+1. But it has had its effect in the simulated world
during the time since t (corresponding to sample tau), and the simulation
should take that into account.
Yet I call it r(t+dt).
That may be confusing. The alternative would be to call it r(t), but
with the understanding that it is the reference for x(t+dt).
It is the reference that the simulated system will use _until_ tau+1, but
it is the reference for which the value exists at tau (+epsilon). It is not,
at least as I understand the PCT model, the reference for the value of x
at tau+1. It is just the reference at tau for the value of x (at _any_ time,
specifically at tau).
That
seems only a change in names. I'm used to (and the literature always
uses) my notation; I'm hardly original ;-). I hesitate to start a
sect that uses a name different from the rest of the world for no
good reasons.
Fair comment.
Or do you have something deeper in mind?
I don't think that I've said anything deep at any point in the discussion.
All I've ever said was what I would think was rational common sense, based
on the notion that observations are of now, and not of some future time.
Predictions can be of what observations may be at some future time;
actions can affect what happens at future times and not at past times.
Is that too deep?
Martin