[From Bill Powers (931223.0930 MST)]
Martin Taylor (931222.1800) --
To Rick:
If you happened to read my postings on vortices, you will note
that most of them carried the disclaimer that the topic had
nothing to do with PCT.
What they had to do with, as far as I am concerned, is the
meaning of negative feedback, which is the central topic in PCT.
You appear to be claiming that if it is possible to partition the
equations of a system in a way that separates forward from
reverse effects even though they lie in the same path, this makes
the physical system equivalent to another one in which the
feedback path is physically distinct. I am claiming that the two
physical systems are of fundamentally different types, even
though the equations describing them can be made to look the
same.
My claims are simple and checkable for any vortex-like system we
can analyze: The power or amplitude gain around the "loop" can't
be greater than unity, even incrementally ("modulation"). My
argument is not that the equations don't correctly predict the
outcome; it is that the resulting "control system" must have an
upper limit of performance which is below the useful range.
Useful control can't be achieved by a system in which forward and
feedback paths simply trace the same physical relationships
forward and backward again.
Consider any chain of physical processes in which an input energy
flux is transformed into an output energy flux. Regardless of any
nonlinearity or intermediate process involved, if we calculate
the partial of output energy flux with respect to the energy flux
at an arbitrary point in the chain, we will get a certain slope.
If the slope is greater than the average slope, there will be an
incremental gain between that point and the final output. If
feedback effects follow the same path but in reverse, we will
find at every point the negative of the same slope. If energy is
gained by a particle going in the forward direction, then when we
trace the apparent feedback effects, the same amount of energy
will be lost: we are simply looking at the same energy flow in
reverse. If any energy is dissipated in the forward direction,
the loop gain will be less than 1, because the output energy is
less than the input energy, but the feedback energy will still be
exactly equal to the forward energy. The loop energy gain will
always be 1 or less, regardless of nonlinearities. If it could be
greater, you would have a perpetual motion machine.
In a true feedback loop, the feedback path contains amplification
that is independent of the output energy flux being modulated. A
low-energy physical input becomes a higher-energy physical
signal, the extra power being supplied by a metabolic power
supply independent of the output power supply. The resulting
signal is compared with a reference, and the error signal is then
further amplified or integrated, the final amplification
occurring (in a nervous system) in the form of bifurcations in
output fibers that turn one signal into a multiplicity of signals
with the same magnitude. Now we have a large neural signal, but
still operating at power levels negligible in comparison with the
output power. Only at the final stage does "modulation" take
place, where a change in the low-power neural signal is converted
through a final power amplifier into a far higher-power change in
the output variable.
Now the partial derivative of feedback power (measured at a given
point) with respect to output power can be many times, thousands
of times, greater than the partial of output power with respect
to feedback power (measured at the same point). In a stable
control system this does not lead to immediate and violent
runaway, but to a reduction in the error signal that just brings
the feedback power to the level needed to offset losses in output
power. This is possible ONLY in a system where the feedback path
is physically distinct from the output path.
ยทยทยท
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Best,
Bill P.