Misc., constant ...

[Hans Blom, 960722b]

(Bill Powers (960722.1315 MDT))

Hans Blom, 960715 --

     Bill, I received your Artificial Cerebellum program. I'd like to
     have the equations that it implements. I hate reverse engineering!

I gave you more than you gave me! But I'll try to get to it. It's pretty

I remember having given a pretty thorough discussion of what the
program did, even to the point of defining what stochastic variables
are and how to compute with them. Anyway, I haven't gotten your
program to run yet, because I'm pretty unfamiliar with 80x86
assembler, both code and program. Could you get me a full Pascal
version or ship the .obj file?

     An organism is twice presented with exactly the same perceptual
     field from exactly the same position. Between the two
     presentations, there is something like a "tracking experiment",
     just like in my satellite example. On the second presentation, the
     organism "sees" more because uncertainty has been removed, although
     it _sees_ exactly the same.

I haven't the slightest idea what you mean by that.

Then this is a perfect example of what I mean. At this moment, it
could still mean pretty much anything. You're very uncertain as to
what it signifies, if anything. It could be a more or less random
collection of words strung together.

Then I could explain to you exactly what it means, until you're fully
satisfied. Maybe you will still think that I utter complete nonsense,
but that will be for a different reason ;-).

After I've explained what I mean with this text, I present it to you
a second time. This time, your reaction will be, I predict, very
different from the first time. That's what I meant with this example.
Reduction of uncertainty. "What does it mean?" versus "Oh, that!"

A constant disturbance is simply a disturbing variable that happens, for
some period of time, to be constant.

Sure. But that was not my point. My point was whether I know it
_after the fact_ or before. When I'm going to throw dice, I cannot
predict the outcome. That's random. When I've thrown them, the
outcome isn't random anymore: it's fixed, determined, even though
the outcome still may be hidden under the poker cup.

If I know that some variable _is going to be constant_ for a while, I
can use that knowledge in (predictive) control. If I don't know that,
I cannot count on it and will have to reckon with all eventualities.
If the variable turns out _to have had_ a constant value for a while,
that wouldn't have helped in my design of the control system or its
control law. It may show up in the controller's performance as better
than normal (or possibly worse than normal) behavior, but that will
be all. The term "random" applies to my knowledge of what a FUTURE
outcome will be. Once it has happened, nothing is random anymore.
(That I may not know its past values is a different problem).

            For a control system to be able to
resist disturbances, it must be able to resist them no matter how they
are behaving -- whether they are varying, or for the time being not

That's exactly my point. If a variable just HAPPENS to be constant
for a while once in a while, it's still random -- unpredictable.

Hans says of disturbances (960715d), "If it's not random, it's
predictable." But this shows the bias that comes from thinking only in
textbook terms of a formal mathematical system.

In the design of a control system, it is important to know the
environment in which it will operate, just like an organism can only
survive in its ecological niche. The more you know about the environ-
ment, the better your control(ler) will be. If you know nothing
about the environment, the "controller" might turn out to be an

as "unexpected" or "unforeseen." No matter how complete your model of
the environment, there are always potential disturbances that have not
been explicitly represented in the model. They are neither predictable
nor unpredictable; they are simply surprises.

I would equate surprising with unpredictable. If you are able to
predict an outcome, will you still be surprised by it?

     If the signal changes slowly, it is highly predictable. If
     something is highly predictable, it can be modeled with ease: just
     discover the filter. And you need only the filter to generate the
     best possible predictions.

That's assuming that you know what is going to cause a disturbance
before it does so.

Exactly! But it is not necessary to know the _cause_, though it may
help. My blood pressure controller, for instance, has built-in
knowledge that the pressure cannot vary faster than a certain rate. I
have no idea why, and that's not important either. Knowing the cause
wouldn't make control better! Knowing _that_ the signal has a certain
bandwidth is sufficient knowledge.

              I doubt that even in engineered plants this is
possible. How do you set up your model so as to take account of the fact
that the maintenance man will accidentally put 90-weight grease in a
gear box instead of 5-weight machine oil?

You don't. We cannot model everything. How do you tell yourself that
your bank may collapse at any time, loosing all your money? We
usually don't model (and don't control for) the fully unexpected.

What I _do_ design in is a "reality (self) check", where the
controller monitors its own control quality. If it detects that it
cannot control anymore, it issues an alarm and shuts off, so that the
(hopefully better qualified) clinician can (must) take over.
Controlling the unexpected is not possible; monitoring for it is.

Having said that, I must say that this is a problem, that could be
solved by a higher hierarchical control layer. But that's a different

Fine: now how about explaining this as you would to a student? What is a
covariance matrix? How is it computed? And in particular, what are the
computations by which the values of pax, pxx, and so on are found? Your
answer tells me that you know a great deal, but does not convey any of
that knowledge to me.

And I'm not willing to explain all that. As it is, my teaching load is
large enough. I've given you a number of keywords that might point
the way. I do not intend to design a course for a one or two person
audience. I suggest you find an introduction to Kalman filtering (I
suggested one in my previous post).

     By the way, why necessarily a SCALAR signal? Do you think that the
     high level concepts of the human brain are scalars? Why not

Vectors are simply lists of scalar variables treated, by convention, as
a single entity. A vector does not imply any relationship among the
scalar components: it is ONLY a list.

Be careful! A list is technically taken to mean an _unordered_ set of
values, as in a grocery list where the ordering of the entries is not
important. That is not true for a vector, where (1, 2) is something
very different from (2, 1).

Only when a vector becomes an input to some matrix operation are any
relationships among the elements created.

No. Draw vectors (1, 2) and (2, 1) into a common frame of reference
and see the difference.

We can continue this to the output function and the environment. As we
do each operation IN EXPANDED FORM, we find that the matrix-vector
notation is only that, a convenient notation. The actual physical system
has to perform each implied operation in full detail. It can't make use
of the vector-matrix notation.

That depends on the level that you intend for the "physical system".
In some BASICs, you can write c = a * b or c = a / b, where a, b and
c are matrices. Unless you understand the BASIC compiler or inter-
preter, you have no idea of how it's done. But at an even lower
level, all that happens is that signals that represent zeroes and
ones are inputs to gates and flipflops that produce different
outputs. At a lower physical level still, electrons move.

But that's not what I meant. What I mean is that a vector might stand
for a "concept". A "concept" need not necessarily have the form of a
scalar, as in the "grandmother cell" hypothesis where the firing of a
single cell or cell assembly is thought to represent the concept that
I see my grandmother.

HPCT is exactly the same as a system presented in vector-matrix form,
but the implied operations are simply spelled out in expanded form. Even
the interaction terms are represented in the forms of the individual
functions. As far as I can see, the ONLY advantage of using the matrix-
vector notation is convenience in manipulating symbols.

Yes, about in the same way as a control circuit diagram is sometimes
more convenient to bring a meaning across than a collection of
English words. But only if you understand the syntax and semantics of
its "language"...