[Martin Taylor 960621 14:30]

Bill Powers (960621.0900 MDT)

Where is the "influence" of which you speak?

You ask about where is the "disturbing influence" on a shaft

influenced by several torque variables, when the perceived variable

is shaft angular velocity.

A CEV is defined by a perceptual function that has all sorts of inputs

through sensors, but ultimately has one value. Likewise the CEV thus

defined has a single time-varying value in the (assumed to exist)

real world.

To me, a disturbing influence has to have the same dimensionality as the

influence of the control system's output on the CEV. One could, in this

case, talk of the disturbing influence being torque, since the control

system specifically provides torque to the shaft, but I think that would

require an explicit knowledge of the physics. It would be better to deal

exclusively in the physical variable defined by the perceptual input

function--shaft velocity. (In more abstract situations, there may well

be no "physical variable" measurable by an instrument, but there will still

be some function of sensory and --perhaps-- imagination variables that

defines a CEV in the world. It is its effect on the value of the CEV

that determines the "influence" in question).

Let us suppose that torques combine additively, for notational convenience

(it doesn't matter whether they do or not). Then angular acceleration is

proportional to the sum of all torques, and angular velocity to the integral

of that sum. So we have:

V(t) = integral(brake torque+bearing torque+load torque + output torque)dt

Â Â Â which can be written

V(t) = integral (disturbance torques + output torque) dt

The "disturbing influence" in this case is then

dist(t) = integral (disturbance torques) dt,

Â Â Â since the sum and integral operators can be interchanged. If they

couldn't be interchanged, as would be the case if the disturbance were

non-additive, the formula becomes more complex, but the end point is

nevertheless of the form

dist(t) = CEV.with.output(t) - CEV.without.output(t)

If the physics of the CEV are unknown (as is ordinarily the case), one

can't actually use this formula, but that's a problem for the analyst.

The situation for the controller doesn't depend on whether there's an

analyst who understands the physics.

If the disturbance doesn't influence the CEV directly, but affects the

influence of the output on the CEV, it is a disturbance to the operation

of the control system, but not an "influence" on the CEV. It may affect

the output's influence on the CEV. I think we probably need a different

word for this kind of disturbance influence. It is "influence" but it

is of a different kind.

Of course, in actual control, dist(t) is never computed explicitly within

the control system, as (I hope) we all agree.

Martin