[From Bill Powers (931110.1745 MST)]
Martin Taylor (931110.1045) --
Consider a CEV that imposes a fixed transport delay. If the
rest of the loop (PIF, comparator, output function) is wide-
band, the loop will oscillate. That can be eliminated by
putting a comb filter somwhere in the loop, if it is well tuned
to the inverse of the delay.
I don't know about combh filters, but I know the oscillation can
also be eliminated by putting a simple integrative lag into the
error-correcting path. As I showed in the 1979 "spadework"
article and have demonstrated a number of times since, if the
right integrator factor is used, the response to a step reference
signal change will be complete (the error will be fully
corrected) in a total of two delay-times. Only one delay-time is
needed to correct the effects of a step-disturbance on the CEV.
But in either case the A() function can be wide-band if it is
outside the loop, not if it is inside.
Suppose it is wide-band: a step-change in the reference signal is
transmitted instantly to the output effector, which responds
instantly by applying a force to the delaying CEV. This step-
change in the reference signal is also applied to the comparator.
However, for one delay-time there is no corresponding change in
the perceptual signal, so the error signal, equal to the size of
the reference-signal's step, will start driving the closed-loop
part of the system to correct the error. When the first delay-
time is complete, the perceptual signal will now suddenly jump to
the same value as the reference signal, and stop the error
integration. That integration, however, has already produced a
corrective addition to the output equal in size (with optimal
adjustment) to the reference step, so a correction will take
place during the next delay. The CEV will just return to zero (or
its pre-step value) at the end of the next delay-time. Then, as
the reference signal is no longer changing, the error will be
corrected during the following delay time by the optimized
closed-loop system. Elapsed time: three delay times.
The only way to fix this would be to put a delay into the
connection between the reference signal and the comparator, set
to the same value as the delay in the response of the CEV. But
now we no longer have a simple system: the feedforward has
created a problem that must be fixed by adding another rather
complex component to the system. Adjustable time delays that do
not introduce attenuation are not neurally cheap or conceptually
elegant.
To sum up:
If you use a simple integrative-lag pure control system optimally
adjusted for the delay in the CEV, the controlled variable will
come exactly to the reference value two delay-times after the
reference-signal step, and one delay-time after a step-
disturbance. If you try to beat this record by adding a wide-band
feedforward link, you get a faster change in output (which we are
not concerned with controlling) but still a delayed effect on the
value of the CEV, and you create an oscillation and extend the
correction time to three delay-times. To eliminate the
oscillation, you have to introduce an adjustable time-delay
between the reference signal and the comparator -- and you still
need the optimally adjusted integrator in the control loop to
keep the system from oscillating. You have two additional
parameters to optimize, and for that you get to advance the
control of the CEV by one delay time. The correction time for
disturbances will remain at one delay-time.
I surmise that there would have to be a very great advantage in
improving response to reference signals by one delay-time to
warrant the complications of circuitry for optimizing three
parameters instead of one.
···
---------------------------------------------------------------
Best,
Bill P.