challenge comparisons

CSGnet Archive CSG1995
1

[From Bill Powers (950928.1545 MDT)]

Hans Blom (950928) --

... you aren't gaining much from the adaptation of k.

     Yes, there are cases when adaptation doesn't help much because it
     may not be required in the first place.

It looks more to me as though the algorithm has difficulty in finding
values of k that track the actual values, at least when k is varying
rapidly. The model seems to do nearly as well when a constant value of k
is used rather than attempting to reproduce the actual changes in k.

     Generally, a system does not know the form of the equation. It
     might be first order in the variables, or second order or even more
     complex. Strange enough, first order usually already works very
     well, and almost never do you have to go beyond second order.

This is an important finding, because it says that reorganization
doesn't have as hard a job to do at the lower levels. At higher levels
of organization I don't know if we could count on that generalization --
but again, maybe the higher levels are not as complex as we would
flatter ourselves into thinking they are. Remember the giant computer in
the Hitchhiker's Guide to the Galaxy. It was asked, "What is the answer
to life, the universe, and everything?" The answer it came back with,
after considerable computing, was "43."

I note that in practice, your model does not seem to work for the
reasons you give in the mathematical analysis. That is, it works even
though the values of k are not correct for the world model. While this
is a pleasant result, it requires an explanation. I think there is more
going on in the Kalman Filter approach than meets the eye, and that some
of it has to do with hidden negative feedback loops that are not
apparent because of the way the equations are written.

     [The match of models] is surprising, because with very smooth
     changes, i.e. if d[i] is almost equal to d[i+1], the noise model
     ought to perform best. I must have done something else wrong.

Here are the results for a wider range of slowing factors (they vary as
the square of the iteration number):

slowing HANS BILL
0.0010 4.717 7.284
0.0040 8.921 7.965
0.0090 4.661 5.880
0.0160 4.282 4.508
0.0250 8.140 9.033
0.0360 14.601 12.656
0.0490 16.298 17.028
0.0640 19.157 22.521
0.0810 22.324 23.184
0.1000 29.131 29.295
0.1210 38.995 41.822
0.1440 52.491 45.864
0.1690 67.106 58.845
0.1960 65.453 65.499
0.2250 85.219 80.210
0.2560 95.741 90.774
0.2890 110.745 104.947
0.3240 127.006 121.636
0.3610 143.927 136.596
0.4000 162.748 159.215

The little differences are probably due to differences in rounding
errors in the two models (especially yours which contains so many more
computations per iteration), coupled with the fluctuations in the
various disturbances and reference signals. I would guess that
differences of 3-5 units between models are insignificant. My model used
a value of 0.42 for b -- slightly different values would lead to one
model or the other being best for a given slowing factor, by a small
amount.

Don't forget that in an integrating control system, control also gets
better as the curves become smoother.

···

-----------------------------------------------------------------------
Best,

Bill P.

2

[Hans Blom, 951002]

(Bill Powers (950928.1545 MDT))

Hans Blom (950928) --

It looks more to me as though the algorithm has difficulty in
finding values of k that track the actual values, at least when k is
varying rapidly.

Yes. I'll give the reason below.

I note that in practice, your model does not seem to work for the
reasons you give in the mathematical analysis. That is, it works
even though the values of k are not correct for the world model.
While this is a pleasant result, it requires an explanation. I think
there is more going on in the Kalman Filter approach than meets the
eye ...

Very perceptive. But the reason that you assume is incorrect:

that some of it has to do with hidden negative feedback loops that
are not apparent because of the way the equations are written.

The reason is a basic uncertainty relation, very much like in quantum
mechanics you cannot measure an electron's position and velocity
simultaneously with infinite precision. The explanation is this: The
system's unknown k must be estimated from some number of most recent
samples. The faster k moves, the fewer samples must be used in order
to keep k's value up to date. Those samples are kind of "averaged" to
obtain the k. The fewer samples are processed, the more k is up to
date, but also the larger the uncertainty in k: "averaging" over a
long period gives a high certainty, but if k changes rapidly, the
"averaged" value will lag too far behind.

There is a solution (of sorts): create a better / higher order model
of how k changes, including one or more derivatives or their equiva-
lents. But the basic problem of prediction remains: an eternal com-
promise between accuracy and up-to-date-ness.

Here are the results for a wider range of slowing factors (they vary
as the square of the iteration number): ...

You're a pretty good coli!

My model used a value of 0.42 for b -- slightly different values
would lead to one model or the other being best for a given slowing
factor, by a small amount.

One way in which you might improve _my_ controller -- if you are
interested -- is by changing the time constants of the disturbance
estimators, dependent on the amount of slowing. You did this before,
already, in order to remove an oscillation. A better procedure would
be to make this time constant adaptive as well: a larger value with
smooth disturbances, a smaller value with high frequency variations.

Greetings,

Hans