[From Bill Powers (920526.1500)]
If anyone has trouble getting a Sunday flight out of Durango (Aug. 2),
contact Mary immediately. If we know how may people would like to leave on
Sunday, maybe we can get an airline to lay on a larger plane.
Chris Love (920523) --
The random reorganizing method is certain less efficient than just going
the right way and keeping on in that direction. However, the relationship
between a change in error and a change in gain (or whatever the criterion
is) isn't necessarily single-valued -- raising gain may lower error up to a
point, and then make it increase again.
The point of using the random method is to have a single simple principle
applicable to all kinds of optimizations, with the least amount of "smarts"
possible. Intelligent methods of reorganization can easily be devised for
specific applications, but in modeling behavior we then have to ask how the
intelligent method got organized, where it gets the information it uses,
and what perceptual and computational equipment it needs (and where that
came from). A realistic reorganizing system has to be able to work before
there is any organization at all, so it can't rely on higher functions that
will appear only later (as a RESULT of reorganization), and it can't have
any knowledge of what "makes sense" in the current environment. It's too
easy for an adult human being to think of optimization methods that use
logic, computation, and information about the objective situation. It's
harder to think of a method when all those advanced capabilities are ruled
out.
Why not write yourself a little program to play with random control? All
you need to do is set a dot moving at constant speed on the screen, and
arrange it so every time you tap the space bar the angle of travel changes
at random. Put a little circle up as a target, then steer the dot to the
circle by tapping the space bar! You'll be amazed at how easy it is.
Once you have this working, you can replace the display with a single dot
that moves along a one-dimensional line toward a target, and position the
visible dot according to the radial distance between the invisible dot and
the invisible target in the now-invisible two-dimensional display. So
invisibly, there will be a dot moving at constant speed and variable
direction toward a target located in two dimensions, but what you will see
shows only the radius along a single line. You will still be able to get
the dot to the target in two dimensions, even though all you're seeing is a
one-dimensional display of the radial distance. This little demonstration
is worth pondering at length.
Thanks for the clarification on "putting your feet up." Now I remember.
--- (920526) --
Judging from the way you laid out that control-system diagram, you must
have got it from some standard text on control engineering. If so, that "H"
would probably refer to some sort of INTERNAL feedback connection, which we
don't need (as you suspected). Or else it's meant to represent the
perceptual function. In any case there isn't any indication of the
controlled variable in the system! The "O" is the control engineer's
concept of output: it has some sort of effect in the external world useful
to somebody else. This form of the diagram is very limiting because it
doesn't allow seeing the effects of action on the controlled variable along
with effects of independent disturbances. Also the role of the perceptual
function gets completely lost. How the heck does the environment have a
DIRECT effect on the comparator (X), without going through a perceptual
function? I think the diagram is just too confused to use.
Here's a rearrangement that might make things clearer:
.
comparator ==========. output ============
R ===>X ===> Error signal ====> G .==> o =====> c.v. = K*o+D
+ ^^ = r - p ==========. ============
>>- . || ^^
>> Perceptual signal ==========. sensing c.v. || ||
<======== p =========== H .<================= ||
==========. ||
....................... independent effect ||
Disturbance ====================
Now G is the output gain (FNO) and H is the perceptual function (FNI). The
dots separate the control system from its environment; everything to the
right and below the dots is environment. The disturbance acts on the
controlled variable, not the comparator. The role of sensing is now
explicit, and there's a place for the generic disturbance to act on the
controlled variable outside the system.
Suppose both G and H are simple multipliers, and that the state of the
controlled variable (c.v.) is K times the measure of output. Then the
steady-state loop gain is just GKH. The optimum slowing factor to put in G
is S = 1 + GKH. So the output variable o would be computed on each
iteration as
o = o + (G*error - o)/(1 + GKH)
You can use a larger value of slowing factor to get a gradual approach to
steady state.
Note that "loop gain" means the product of all multipliers encountered in
one trip around the loop. Starting at the comparator you encounter G on the
way to the output, K associated with the controlled variable, and H in the
perceptual or input function: G*K*H. The loop gain is actually negative
because the perceptual signal is subtracted at the comparator, putting in a
hidden factor of -1. For computing the optimum S, ignore the -1.
In your servo texts, G and H are probably meant to be complex functions --
Laplace transforms. Ignore all that -- our way is easier and works just as
well in simple cases. If you analyse most systems properly, they're all
simple cases. Only the environment is complicated.
···
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Ed Ford and David Goldstein (920525) --
Ed, that was a beautiful exposition of what I think of as HPCT therapy. If
you understand how control systems work, you don't need any formal
"method." Even the method of levels isn't a formal method -- it's just a
way of bringing out the fact that one level of goals serves another of
higher level. When you really SEE your client as organized to control at
many levels, the way into the case, I should think, would practically
define itself.
I can say, in defense of David, that there is merit in following some
systematic approach, too. This is particularly true if you hope to teach
this concept of therapy to others, particularly beginners. Not all people
who do therapy are going to be as creative as Ed and David (and David is a
lot more creative than Ed gives him credit for). I'm a firm believer (well,
not totally soft anyway) in bottom-up teaching: do it this way and get
right results; understand it later. A set procedure can provide a focus;
later on, you will see (if the teaching is done right) that this is only
one of many ways to get the same result. Of course too much teaching stops
with the set procedure, but we know that principles have to be learned,
too, don't we?
The moral: there is more than one style of HPCT therapy that will work.
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Greg Williams (920524++) --
To change the relationship of "in" and "out" curves, play with the
integration factors in the "S" list of parameters for Shoulder y and Elbow.
Thanks much for the nice word about Arm v2. I will take you up on the
editing offer -- if you can pull together a coherent account of the arm
from past posts, that will give us the nucleus of the article. I will then
add intro and other material, and make some diagrams with PictureThis that
you can check and edit. I'm hoping that you will take it upon yourself to
provide links to the literature with commentaries. This will be, of course,
a joint article.
Your OK of the Schwartz letter means it will go in the mail tomorrow.
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Best,
Bill P.