[From Rick Marken (930327.2030)]
I hate to do this but I have to eat crow; I WAS WRONG.
I said (Rick Marken (930327.0900))
Martin's claim here [that perfect control means zero disturbance in the
perceptual signal] turns out to be FALSE
This statement was based on a mathematically incorrect analysis of the
following equation:
p = (k.e k.o r + k.d d)/(1+k.e k.o)
I thought that even if k.e k.o (loop gain) went to infinity there
was still a little, teensy weensey amount of disturbance left
over. But Bill Powers (930327.1600) set me straight:
You can show that 1/(1+g) is less than or
equal to any small epsilon for g greater than some amount.
Therefore the limit of the function as g goes to infinity is
zero.
So infinity was a bad choice of values on my part -- and when the
loop gain goes to infinity, p = r and there is no disturbance
represented in the perceptual signal. So MARTIN WAS RIGHT: I WAS WRONG;
if control were perfect (loop gain = infinity) there would unquestionably
be no information about the disturbance in p because there would be NO
effect of the disturbance AT ALL on p. So we are back to where we
were: Martin claiming that there is no information about the disturbance
ONLY when loop gain is infinity; me claiming that there is no information
about the disturbance EVER.
I tell you; admitting that you are wrong is NOT EASY. I hate this.
But I did it, not only because my error was made in public so I was
caught "read" -handed, but also to show that I am willing to admit
when I am wrong (NB. Ken Hacker -- is this arrogance?). So Martin and Allan
can be assured that if they can persuade me (with evidence) that there IS
information about the disturbance in the perceptual input to a control system
then I will (eventually) admit that they are right and I am wrong. I'll hate
doing that; just as I imagine they would hate admitting that Bill and
I are right. I know admitting error is really tough -- but it's not the
end of the world. Now I know that infinite loop gain does produce perfect
control -- ie. p = r. Now I just have to figure out whether such a
system is physically possible (it probably isn't); if it is, then I'm
going to have to do some figuring just like Martin and Allan might
have to do some figuring if they are able to convince themselves that
there actually is no information about disturbances in perception.
Anyway, it is a far far better thing I do ....
Back to the fray.
Best
Rick