<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

situation.

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.

Resolution.

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