Randall's Experiment

[From Rick Marken (930330.1500)]

Bill Powers (930330.1330 MST) --

Thanks for the diagrams. I still have no response from Allan to
my questions about his experiment; but your diagram's will help
me describe what I think he's proposing.

My understanding was that he would do a simulation run with the
standard control system as below:

                     > ref sig (constant)
           p ----- comp ---- e = error
            > >
          inp funct integrator
           (i) (o)
          +| |+ |
           > ---------------

The disturbance would be, say, 1000 samples of filtered random
noise. Allan would then save the 1000 values of d, i and o
resulting from the run.

Now, in order to measure H(D) he injects a vector of i values
(the length of the vector going from 0 to 1000) in a run consisting
of 1000 iterations -- and computes the resulting 1000 outputs from
each run in an open loop preparation as below:

                     > ref sig (constant)
           p ----- comp ---- e = error
            > >
          inp funct integrator
            > >
           (a) (o)

The candidate i vectors enter through (a) and Allan compares
each resulting set of 1000 o values to the actual disturbance
vector. The length of the first candidate i vector that produces
o values that match d perfectly is H(D) (There is a problem here;
What if you don't get any perfect matches to d, Allan?.Nothing
but an infinite loop gain system could produce outputs that
match d perfectly anyway: how about doing this just until a
candidate i vector produces o values that match the disturbance
to the same degree as did the o values generated in the closed loop
case. So, if the closed loop correlation between disturbance and
output was .99967, pick the first i vector that produces o values
that correlate .99967 with the disturbance).

Assuming you can find H(D) using this decidedly peculiar technique
(why not just measure the variance of d?) the next step is to
find H(D|P). My understanding is that Allan would "play" the
original i vector (from the actual tracking run) into the open
loop system (injected at point a) ALONG WITH the candidate i
vectors that were used in the determination of H(D). Since
the shortest candidate i vectors are added in first, Allan is
assuming that the output resulting from the original i vector
along with the "null" (0 length) candidate i vector will produce an
output that perfectly matches the disturbance (or, at least, matches
the disturbance as well as the output did in the original run).

Is this a correct description of the experiment Allan?