Back to Basics II

[From Rick Marken (930317.1530)]

Here are some replies to a private post from Martin that he
said I could reply to on the net.
Rick,

You always post the equation as p(t) = o(t) + d(t), but in discussion
everyone, including you most of the time acknowledge that there is a
not-well-known function relating the output to the effects on the CEV.
If there weren't, wouldn't the cognitive outflow people be right?

This might be my fault; the letters p,o and d refer to variables,
not functions, and the t in parenthesis is an index, not an operand.
This equation just decribes physical reality in a compensatory
tracking task (if we think of p as the number representing the
CEV -- position of a line -- rather than the perception of that
line; as I said in another post, there is very good reason to
suspect that the perception of the CEV (in a tracking task) is
linearly proportional to the physical measure of the CEV).

The output function is really the function that transforms the
error signal into the output variable, o. So the output function, O,
is o(t) = O(e(t)) and O is likely to be highly non-linear. This is
indeed one reason why cognitive outflow models can't work -- but
another reason is that the intended result of the cognitive outflow
(the CEV) also depends on disturbances (actually, on their effects,
d(t)).

I said:

This is only a problem for those who think of control loops
sequentially. In the formula CEV(t) = d(t) + o(t) it is
always the current value of the output (occuring at time t) that
is combined with the current value of the disturbance. The fact
that o(t) might be the result of processes occuring earlier in time
is of absolutely no consequence. The physical fact of the matter
is that the current state of the CEV is determined, simultaneously,
by the current value of the disturbance and the current value of the
output.

Martin replies:

This cannot be a correct interpretation of the equation. The CEV does
not react instantaneously to output. The equation talks about signals
which add to form a perceptual signal, or else it talks about physical
effects that add to change the CEV and therefore the perceptual signal.
Either way, the three terms in the equation must be of the same form,
and that form is not the form of the output signal of the ECS.

The equation should read p(t) = P sub t (F(o) + D(d)) where P sub t is
function P evaluated at time t. F is some function of o, where o is the
history of all output, D is some function of d, where d is the history
of all disturbance. One can simplify this by saying simply that the
"disturbance" is D, rather than D(d), since perceptual control doesn't
care about the distinction. And if we reference p(t) to the CEV rather
than to the perceptual signal, we can eliminate the function P, just
evaluating F at time t. But we can't ignore F, as you and Bill have
both pointed out today. Allan is saying that F is uncertain. So do you.
He says past actions have side effects. That's not controversial in any
version of PCT that I know.

This confusion may all result from my notation. Think of p(t) as
the sequence of numbers (over time) that represent the position of
the cursor on a computer screen, o(t) are the numbers coming from
the joystick, d(t) are just smoothly varying random numbers. At any
time t, p(t) = o(t) + d(t); that is just the way the physical
situation is set up. I agree that o(t) at a particular time might
be the response to a perceptual input from time t-tau. But what
is currently on the screen, p(t) is always the simultaneous result
of the current disturbance number, d(t) and output number, o(t).

So time delays certainly do exist in control loops -- but they
are not pertinant to the question of whether or not there is
information about d(t) in p(t). The fact of the matter is that
p(t) is always a JOINT result of o(t) and d(t) (see Bill's earlier
post today on disturbances). This is a VERY important point to
understand. If we can't agree on this (that the perceptual input
to a control system is at all times the simultaneous, joint result
of both disturbance and output) then we are really thermo-
dynamically isolated from each other.

Best

Rick