# A Useful Insight at last

[Gabriel to NET 921220 10:01 CST]

Bill P and I have had an offline discussion that generated lots of
light for me. So I want to share. We have two rather different views
of Control Theory and hence of BCP. I think this serves to unify
the two positions in exactly the technical sense of being the least
general theory that includes both.

I'm going to avoid the word "Control" because it has so many different
meanings. This does not signify any quarrel with BCP - just that there's
a mathematical theory including BCP as a special case that I'm comfortable
with.

The term "Reference Signal" is replaced by "Desired State (of the observed
environment)" for similar reasons. The term "Reference" has Rock of
Gibraltar like connotations, but I think we can all agree it's possible
to change our desires from time to time, just as the BCP signal topology
allows and encourages - i.e. the "desire" comes from within, not from
without.

Thus we begin with two states of the external environment, Desired (D),
and Perceived (P). For the unified theory it is sufficient for D and P
to have "representations" in the mathematical sense, as points in a
"metric space". This is a space where "distance" has a meaning, of
one or more dimensions, each of which may be discrete or continuous.
This allows us to talk about the distance between D and P, and
because the space is a metric space, we are guaranteed not to reach
any absurd conclusions.

Arm and Man live in 6N dimensional metric spaces - 3 spatial coordinates
for each degree of freedom, and 3 momenta for each degree of freedom.
If you are doing rigid body kinematics, the 6 becomes a 12, but no matter.

Most of my concepts live in many dimensional spaces where the component
of state in one of the dimensions has two or three possible values -
TRUE/FALSE or TRUE/UNKNOWN/FALSE. If we increase the number of possible
state values in a single dimension to say 256 or 1024 or 8192 ....
we can approach a continuous state space as closely as we wish, we
may exactly represent the digital computational models of Man and Arm,
and in fact we may represent any physical system where there is noise,
either thermal or quantum, within the precision of experimental observation.

A metric space has one other imporatnt property. Given two points say
D and P we have a well defined distance between them. Distances MUST
obey the triangular inequality, if A B and C are in the space, then

AB + BC >= AC

Now the central thesis of BCP can be stated.

People behave to move P closer to D, where this is possible
at an acceptable cost.

Notice this is a constrained minimisation process. Also unlike the
error P-R of BCP, DP is always positive, and may not always attain
the desired minimum of zero. But the version of control theory used
in BCP does in fact minimise DP for the cases considered in pratcice.

Notice also that D and P also have representations by data streams,
i.e. by samples in the metric space at time intervals appropriate to
the Nyquist criterion for lack of aliasing. P type data comes from
sensors. D type data comes from desires.

In this case Shannon's Theory allows a metric space to be constructed
(with a weighted Hamming metric) pretty much regardless of the details
of the data streams, provided they are sequences of arrivals of symbols
(including pulses of depolarisation across nerve membranes).

I think for the moment I need say little more. In both the continuous
and discrete cases orthogonal function expansions are possible
with arbitrary positive definite metric for the scalar products.
These metrics have to do with power limits, or value systems.

This is why the detailed machinery of Control Theory does the job.
Essentially the metric space allows Fourier Analysis and Laplace
Transforms, which lead to the control formalism of BCP. But
in some cases minimisation algorithms, taking account of the
possible may be closer to the real world. All this also includes
a lot of other things, like population genetics if you replace
the machinery moving D around with Darwinian Selection. The story
gets a little different but not very much.

Over and out. I'm exhausted.

John