[From Bill Powers (931106.1115 MST)]
John Gardner (931105) --
Hi, John, I thought you'd given up on us.
A hybrid feedforward/feedback control system.
The idea of feedforward with feedback acting as a trim is
feasible. I've tried to think of how to do this before, but this
time the request was honored. As usual, the solution is
embarrassingly simple. It comes in two stages.
First stage:
ref sig r
>
>----------
> e |
V V
p ----->[COMP]-*g--> OUTPUT
> FUNCTION
input function |
> >
etc etc
Now we have a reference signal that enters the output function
directly, driving it to produce an output in a calibrated
feedforward way. The same reference signal enters a comparator
where it is compared with the perceptual signal p as usual. The
error signal adds to and subtracts from the direct effect of the
reference signal on the output function.
With a linear output function the output signal is thus
k*(r + g*(r-p)) instead of just g*(r-p). The constant k is
adjusted to produce an amount of output that will make p = r
(approximately) in the absence of transient disturbances.
When a disturbance comes along, it affects p but not r. As a
result, the error signal changes to alter the output from what it
would have been with no error, by exactly the amount that the
feedforward component of output is wrong. So the system responds
to disturbances very much as it would with a simple (canonical
PCT) control system. The only difference is that if there is some
steady-state disturbance, the feedforward component can be
calibrated roughly to cancel its average effect, leaving the
feedback part of the system to take care of deviations of the
environment from the average condition.
If the feedforward component is nonlinear, or differently
nonlinear from the effects that are supposed to be created, the
feedback component of the system will eliminate the
nonlinearities in the same way it would eliminate the effect of
any disturbance.
This system has the same problem as the canonical one when the
feedback path is severed and p goes to zero. The error signal
becomes, in general, very large and the output is thrown far off
the value it should have, so the feedforward calibration is
ineffective.
Second stage:
There is one simple way to fix this problem (there are many
complicated ways) based on a fundamental property of neural
analog computers.
In the nervous system, all comparators are one-way and all
perceptual signals are positive. Two-way action always requires a
pair of systems, each controlling in only one direction. In order
for lack of a perceptual signal to be interpreted as zero error
regardless of the setting of the reference signal, the reference
signal must be inhibitory and the perceptual signal must be
excitatory at the comparator. With this arrangement, a 1-unit
reference signal results in 0 error signal until the perceptual
signal just reaches 1 unit. Any higher value of perceptual signal
creates an output from the comparator, an error signal. In a
high-gain control system, the error signal is always held near
zero. With the inhibitory reference signal, therefore,
disappearance of the excitatory perceptual signal will simply
make the comparator output zero, as if the error were exactly
zero.
Combining this feature of neural comparators with the hybrid
feedforward/feedback system above, we now have a system that will
convert a reference signal to a standard calibrated feedforward
output when the feedback signal disappears. A two-way system made
of two of these one-way systems will then not produce an extreme
output upon loss of the feedback information; it will simply
create a precalibrated output that is approximately right for a
disturbance-free environment.
This design has the great advantage that no switching of
functions is required. The system is always set up the same way,
and nothing else has to tell it when to use feedback and when to
rely on feedforward. It simply uses feedback control when the
perceptions are available. This is the solution I was looking
for, and didn't find, some years ago.
In some real neural control systems, the feedback signal is
inhibitory and the reference signal is excitatory, so the above
arrangement will not work (example, the tendon reflex). In
others, the reference signal and comparator are mechanical and
the error signal is excitatory (stretch reflex) with the needed
sign reversal occurring elsewhere in the loop, so the above
arrangement still will not work.
There are, however, some cases where it seems that the necessary
conditions are fulfilled. In the cerebellum, for example, the
outputs of the Purkinje cells reach brainstem motor nuclei as
inhibitory signals, and "recurrent collaterals" from sensory
nuclei at the same level reach the same motor nuclei as
excitatory signals (the comparators seem to be a special outer
layer of the motor nuclei where these two sets of signals
converge). It is possible that the same reference signals which
inhibit at the point where they converge with excitatory sensory
signals send processes elsewhere that have excitatory effects in
the motor nucleus, directly or through sign-reversing
interneurons. If that is true, then we have the conditions needed
for a hybrid feedforward/feedback control system that will
automatically default to calibrated feedforward when the
perceptual signals are interrupted.
We really need systems of this kind to explain examples that have
come up regarding higher levels of behavior. I know of no
evidence for them, but at least we have what looks like an
existence theorem, making it relatively safe to conjecture that
such arrangements also exist at higher levels.
Even without the feedforward connection directly from the
reference signal to the output function, a somewhat similar
effect is obtained by combining pairs of systems having
inhibitory reference signals. If the output function is an
integrator, zeroing the error signal will leave the output signal
constant, because it changes only when the error signal is
nonzero. Here we have to suppose that both comparator-halves
drive the same output function, one upward and the other
downward. If all this holds true, then loss of both perceptual
signals will result in zero error signal and the output will hold
its value for some time, until the perceptual signals return.
This should be distinguishable from the previous case, because
loss of the perceptual signal will freeze the output at whatever
value it had when the lights went out, even if the reference
signal continues to change. In the previous case, the changing
reference signal would cause the output to continue changing even
without the perceptual signal present, because the reference
signal is connected directly to the output function as well as to
the comparator. So I think we could determine experimentally
which kind of connection exists (assuming we don't observe a
blow-up when the perceptual information is blanked out).
Note that in ordinary behavior with continuous feedback the
hybrid system behaves just like the ordinary canonical model.
···
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Best,
Bill P.