[Martin Taylor 920304 17:30]
I'm still have a problem with the zeros in a perfectly controlling hierarchy.
Rick has sent me a SYLK version of his spreadsheet that I have not yet
downloaded, so I will hold off more comment until I have tried to analyze
better its behaviour. But Bill triggered something a couple of days ago
with his comment that to stably control something like walking around a circle
involved a lot of changing references that resulted in a lot of actions.
What this triggered was a thought about what it means to control continuously
for a sequence. In simple terms, suppose one is controlling for a sequence
of values between 0 and 1, and the percept strays from the desired values
in a consistent way. Then an error signal is presumably generated, which
affects lower-level control systems in such a way as to reduce that error,
but it does so only for future parts of the sequence. If they stay close
to the desired values, the error signal should likewise stay close to zero.
But the sequence, when it is finished, was not the one desired. Shouldn't
that result in an error signal? Or, are there no error signals during the
acquisition of the sequence, and only a *multi-degree-of-freedom* error
signal at the end, which could affect subsequent attempts to perceive the
desired sequence? Neither view seems satisfactory.
Looking at it from the more abstract view, the sequence can be described
by a z-transform of the time series. The behaviour of the (sampled) controller
can be described in the same terms. The bheaviour of the desired percept
(variable according to factors outside the ECS) is entirely merged with the
intrinsic characteristics of the controller-environment loop. Most such cases
are dealt with by methods such as inverse filtering, turning the time-spread
signal into a series of point events, but I cannot see that this approach
would be useful here. It would be akin, I think, to the second possibility
listed above, to report only a whole-sequence error after the fact.
In respect of the zeros problem, sequence provides a distinct glitch, in that
the reference sequence is not a simple representation of a one degree-of-freedom
(scalar) error signal. It provides a set of arbitrarily changing references
based on some internal memory to the lower ECSs. These then are subject to
the transients that I excluded in my original questions about the hierarchy
in close control of slowly changin disturbances.
Rick's spreadsheet doesn't have sequences, if I remember correctly, and if
it nevertheless shows non-zero percepts and references throughout the hierarchy
after the initial transient has decayed, then I have to rethink my intuitions
(if that concept makes sense). Until then, I don't want to proceed with
the degrees of freedom part of the discussion.
I may not have time to do it anyway. I have three chapters for different books
to complete before April 1, only two of them so far drafted and none completed.
Then I will be away for about 2 months (as yet indeterminate) and out of touch
with CSG-L. So we may have to postpone the debate until summer.
Martin