I argued against the principles of this control theory method
on mathmatical grounds and, like others, was unconvinced by the
arguments of some of the PICLIST subscribers.  It is not easy to
argue.  I promised I would investigate this stuff.

I read (or tried to read):

  "Neural Networks and Fuzzy Systems" by Bart Kosko ISBN 0-13-611435-0

 Actually, rather than read, I bent my itellectual reed to the book's
intense wind of analysis. This material seems almost desperate in its
excitement to sate the hunger of skeptics to criticize it.  I agree.
Kosko has approached his argument from signals and systems,
engineering mechanics, stochastic processes, statistics and logic to
form arguments for the adoption of adaptive systems in the solution
of real world problems.

  I now know how hard it is for people to describe the theory in
words that do not sound ambiguous; Not surprising since fuzzy systems
deal with event abmiguity.  We naturally see geometries, sets,
frequency distributions and creative solutions as "second nature".
The accuracy and logistic mapping of the memorized data figure in the
accuracy and precision of an inferred decision. Kosko has made a very
verbose argument for machine inference in this book.  The difficulty
in dismissing the lack of mathematical rigor, is the unwillingness to
accept anything but absolute truth in a solution.

 Fuzzy systems are not required to be linear or time-invariant.
Actually, the stochastic, predictive nature of these systems implies
time-variance.  I was reminded that models are just models from one
PICLIST subscriber.  Kosko describes these systems as model-free
estimators. You can't freeze the time axis on a complex fuzzy system
and use the usual tool suite of system analysis to describe the
system behavior.  The system itself is a fluid entity. You can't even
take multiple time snapshots and analyze them, because if there is
any non-zero random noise component to any of the system inputs, it
will show up in the system's impulse model and your experiment will
fail range and repeatability. If you are going to be anal about fuzzy
system analysis, then your going to be as unhappy as Einstien was
about Heisenberg uncertainty.

  My primary argument against this theory was the concept of
singularities and runaway solutions which would lead to oscillatory
and unstable conditions. The "fuzzy" in fuzzy logic shines here.
Fuzzy systems contain a level of intensity and decisive granularity
in each system factor. This allows you to decide the level and
precision of control interpolation or inference from control decision
to the next.  This naturally desensitizes the fuzzy system to control
singularities.

  You can use a signal correlator which does a linear sweep of
frequencies to pluck out signals buried in noise, or you can use a
fuzzy correlator which will statistically tune in to the
non-randomness of a signal and accomplish the same thing. Fuzzy
systems are just another tool in the engineer's toolbox to deal with
problems.

  Thanks to the PICer's who beat on me to look into this further.
Wonderfull stuff.

                                    Mike