[From Bill Powers (2002.09.09.0701 MDT)]
Bruce Nevin (2002.09.09 20:48) --
I described a while back a phenomenon that could be called dithering due to
indecision. ...This is a form of conflict in which the lower order systems
do not control a median state of the conflicted variable, but rather
alternate in
controlling a variable (hand position, leg position) which subserves
control of the conflicted variable.
I agree. I have described this as an oscillation, but dithering is just as
good a term. Another term I've used is vacillation. However, it's not
necessary to think of this as switching control back and forth between two
controlling systems: the oscillations can occur when _both_ controlling
systems are active at the same time. All that's needed is for the gain in
the opposing systems to become high enough to exceed the limit of
stability. When that happens, spontaneous oscillations will occur and you
will see the controlled variable changing back and forth between the
goal-states of the two opposed systems, with their output actions rising
and falling accordingly, 180 degrees out of phase. This can give the
appearance of switching control systems on and off, but while that is
possible, it is not the most likely explanation.
Typically, I believe, human control systems are nonlinear in the sense that
when errors are small, gain tends to be small, and when disturbances (like
those created by conflict) cause errors to increase, gain increases.
Muscles definitely work this way. If you press your hands together in front
of you as hard as you can, oscillations will soon start. Muscles have an
exponential force-tension characteristic, so the greater the effort is, the
higher is the spring constant of the muscle -- which is one factor in
determining loop gain. When the spring constant is high enough, the loop
gain will be too high for the inherent lags in the spinal systems and
oscillations will begin at three or four cycles per second. Also, of
course, at extreme efforts the response of the muscle to further neural
drive signals actually decreases -- but I promised not to get into the
Universal Error Curve ;-)>
I think this may be a more common occurrence than is obvious. It seems to
be not the sort of thing that we attend to (and remember). I have not yet
imagined a way to induce the phenomenon experimentally. Perhaps someone can
who is more clever than I and more experienced in experimental methods.
Another inducement to oscillation, dithering, or vacillation, is the
nonlinearity that occurs when an output function hits an abrupt limit to
its action. Conflict is generally limited by the maximum effort a system
can generate -- either the maximum or minimum frequency a neural signal can
attain, or the limit of force that a muscle can generate. When conflict
causes one system to hit a limit, the effect is to create a discontinuity
in the error-output curve, and discontinuities typically produce
high-frequency components not present in normal variations of the signal or
force. These high frequencies can result in oscillations.
An example of a discontinuity (of a different kind) can be see by gently
touching your upper and lower front teeth together to produce a very small
contact force. Most people I know can't do it, because the relationship
between position and force involves, at the point of contact, almost an
infinite slope, which raises the loop gain far beyond what can be tolerated
in a stable system. So your teeth oscillate between just touching and just
not touching, at a fairly high frequency. Of course you can press them
together hard enough to stop the oscillations, but it's hard to stop the
oscillations when the touch is very light.
When conflict is so severe that both conflicted systems are being driven
"against the stops", neither one can change its output and any disturbance
will cause the controlled variable to drift, uncontrolled, over some dead
band. If an external disturbance is strong enough, it can aid one of the
conflicted systems, reducing its error and thus reducing its output action.
This creates an unbalance, since the other system is still producing its
maximum output. Kent McClelland simulated this and other conflict
situations very neatly.
The point here is that a discontinuity can be created by bringing the
output of a system _below_ an abrupt limit just as easily as by raising the
output _to_ an abrupt limit. If you/'re pressing your upper and lower front
teeth together hard enough to prevent an oscillation, _reducing_ the
pressure can also cause oscillations just as the teeth are about to
part. So dithering or vacillation can occur when a conflict is unbalanced
by external disturbances and one of the conflicted systems just starts to
come back into the normal range of operation.
I know that higher-order control is not like touching teeth together, but
the same principles would apply. There is always a limit on loop gain due
to lags in the control system. Lags in higher systems are relatively long,
so gain must decrease more rapidly than at lower levels as the frequency
content of signals increases. Gain can still be very high, and control can
be accurate, for steady or slowly-varying disturbances, but if something
like a discontinuity causes the loop gain to rise too much, oscillations
have to occur. You can also find discontinuities in many ordinary control
processes if the environment develops kink of some sort, but generally we
learn to avoid those situations or find ways to move rapidly through the
region of instability.
So -- your explorations of dithering are well-founded, though they take us
into regions of control theory that are difficult to discuss in a
non-technical way. I don't mean to imply that I've discussed or even
considered all the circumstances in which dithering is expected to occur.
I'm saying only that your basic idea is supported by control theory.
Best,
Bill P.
···
/Bruce Nevin
But this is not really taking place in the error _signal_