[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)
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
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,
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.