<Bob Clark (940415.17:33 EDT)>

Bill Powers & Martin Taylor:
Various posts from 4-7-94 through 4-13-94 -- and maybe more

These posts are a Classic Demonstration of a Classic Conflict!

The focus of the Conflict is the form of the Mathematical
Representation of the operation of a minimal control system.

The form of the representation is important because PCT is based on
the idealized representation of a minimal control system together
with its operation. PCT uses the idealized control system to
generate an idealized summary of the behavior of living beings.

This conflict can be resolved by a change of viewpoint. But first,
each viewpoint must be examined.

Bill's viewpoint.
Bill uses the standard analytic mathematical form with well-behaved
continuous functions of variables in general and time in particular.
He is familiar with alternative mathematical forms and uses
established procedures to reduce the alternatives to familiar
idealized algebraic form. He finds the alternatives unnecessary for
his purposes.

Martin's viewpoint.
Martin is familiar with mathematical analysis, but suggests that
there may be situations where these methods are misleading. He is
concerned that idealization may be carried too far -- that the
assumptions of continuity and omission of singular points may lead to
unsuitable results. He proposes some alternative mathematical
procedures that could be more inclusive and therefore more suitable
in some situations. In part, he seems to be more interested in an
accurate, complete representation than in an idealized form that
could be modified or extended where necessary. Martin thinks the
analytic algebraic form may over-look some important aspects of the

Observer's viewpoint.
They are each using the Observer's viewpoint to examine the minimal
system. Each would like to use some form of mathematics to represent
the system's operating characteristics. Each describes his
observations in his own terms. The system itself is incapable of
perceiving anything. It is an idealized abstraction, operating in
terms of the flow of signals from one location to another as they are
modified by specified functions that serve to combine and/or produce
other signals.

Alternative viewpoint -- The User.
The User's viewpoint is an interesting alternative. The User is the
set of higher level systems that achieve their goals by setting the
reference levels for the lowest systems, here the minimal system. --
This viewpoint uses a slower time scale in which the temporal details
of the lower level's operation are largely irrelevant. Sometimes the
operations can even be considered "instantaneous." The User is
satisfied as long as the perceptual variable(s) the User controls
remains within specified limits. If the lower level system is
inadequate, it may need to be evaluated on a faster time scale.

Under usual conditions, ordinary mathematical analysis is sufficient
for the User's needs. In this treatment, time is considered a
continuous and independent variable. Time scales are selected
according to the situation. This description is open to modification
for special situations.

A Surprising Example.
A thermostatic temperature control system is often used as an example
of a negative feedback control system. The system's operation is
usually described in terms of smooth, continuous changes in the
signals with time a continuous independent variable. The relevant
simultaneous equations are stated and solved. From the standpoint of
the ideal system, this is quite adequate.

HOWEVER the common residential heating system has an important
difference: the size of the error signal does not continuously adjust
the output of the furnace. This is quite acceptable from the
resident's viewpoint -- as long as his skin temperature is within
limits. No problem.

However the resident usually uses a fairly slow time scale that is
easily ignored. This time scale involves a small number of hours, so
variations in temperature involving only a few minutes are ignored.

If a faster time scale is used (of the order of a few minutes), it is
observed that this system uses a DISCONTINUOUS output (either ON or
OFF) and the algebraic representation simply does not fit.

For the User, this is important only when the cycle time becomes
excessive and he becomes aware of his varying skin temperature.


From the viewpoint of the User, the idealized minimal system can use

either mathematical formalism. If a more accurate representation is
needed, the formalism can be selected accordingly.

Regards, Bob Clark