Taylor stuff

[From Bill Powers (920330.1600)]

Martin Taylor (920330.1320)--

I promise not to open any new subjects -- I know you're trying to get
unhooked so you can do all those things you REALLY HAVE TO DO now. Don't
even reply to this ---

Re: perceptual distortion in arm model.

Actually, there isn't a "perceptual" problem of this kind. The idea >that

there is stems from an insufficient adherence to the ideas of PCT.

Oh, yes there is -- in MY arm model. I don't have any reorganizing
abilities in the model, or any way of mapping from one space to another
(except Cartesian to r-theta-phi, which is trivial). I've been sorely
tempted to cheat and say "Oh, well, the brain will adapt itself and provide
x,y,z signals anyway, so why not just use the external coordinates for the
perceptual signals?" Actually it's much more instructive to go through all
this and see just what you can get away with without any adaptation at all
-- and what jumps up and bites you. In principle, the perceptual space
doesn't have to be modified: all that has to change is the set of points
that's called "a straight line." If the reference signals trace out the
correct set of points, the result will be a straight line in external
space. Right now I'm tinkering with the innards of the program, trying to
find out what's working well enough to leave alone and what has to be
changed to introduce the higher levels. Hand me that wrench, will you?


I guess we agree on entropy. When you get back we'll see. An entropy
measure can indicate the success of the control system. Sort of like error
as a fraction of the total range...?

Significant disturbances:

As for whether a disturbance causes a "significant" departure from the
undisturbed state--significance is in the eye of the beholder.

The rule of thumb I use is based on the total range of the controlled
variable, which is the range of the reference signal. A sort of nominal
control system can counteract any disturbance large enough (when unopposed
except by the passive dynamics of the controlled variable) to drive the
variable to its limits (if it has enough output capacity to cancel such a
disturbance). A "large" disturbance of a pendulum would be one that pushes
it 90 degrees from vertical (less force is needed to push it further).
Control is "good" if that size of disturbance is kept from varying the
controlled variable more than 10 percent of the total range. That is, the
same force that would push the pendulum 90 degrees without control now
moves it only 9 degrees. This is obviously an arbitrary measure. It's
conservative, though, in that most compentent control systems I come across
in behavior can do better than this.

If one disturbs a dynamic structure to a point near the edge of its
attractor basin, and it returns (i.e. negative feedback), has the
disturbance been "significant?"

By my definition, yes. One of the natural ways to define a "large"
disturbance is in terms of the range of the affected variable, which in
complex systems is usually finite. If a measure of the state of the system
is driven all the way to a limit, the disturbance is large. Also, the
resistance of the system to disturbances is weak.

This is a somewhat tricky question because "disturbance" is ambiguous. It
can mean either the cause or the effect. In CT we almost always mean the
cause, the independent variable that INFLUENCES the controlled variable. If
you grab the controlled variable and force a change, this breaks the loop.
So I translate your disturbance to mean "a change in an independent
variable that results in the dynamic structure moving to the edge of its
attractor basin." This means that a control system could vary a second
independent variable that has the opposite effect, and thus maintain the
dynamic structure at the center of its basin even in the presence of the
other disturbing influence. It could also keep the dynamic structure at any
arbitary distance from the center of its basin, in the presence of abitrary
disturbing influences.

This is the very question that Normal Packard, at the U of IL, copped out
on many a moon ago. I had asked him what the effect on one of his dynamic
systems would be if a constant disturbance were applied. He said he would
answer "after the holidays" and never did.

Re: reference signals


                    ...High gain negative feedback (with or without a
variable reference signal, with or without an explicit comparator)
creates the kind of behavior we see in living systems at all levels
from biochemistry to control of system concepts.


Yep, but not without reference signals, I think. As to whether
comparators are explicit, I'm not sure what that would mean except in a

All you need for a control system, beside external feedback and dynamic
stabilization, is a system in which output = -K(input), assuming an
external feedback connection with a positive constant. This system doesn't
need a comparator and it has no provision for a reference signal. It will
keep its input near zero. It is, of course, equivalent to a control system
with a comparator and a reference signal set permanently to zero. It
couldn't be a subsystem in a hierarchy of control, however. But it's a
control system, if the loop gain is high and negative.

Reference signals can be added into perceptual functions or output
functions (the latter is how some brainstem systems seem to work). There's
no need for a separate circuit that does the subtracting. It's just easier
to understand the system if there is one.

something in the loop must insert averaging of such an amount that the
system would behave no differently if the actual lag were zero.
Otherwise the loop can't be stable.

That's too strict a criterion. There are lots of ways to stabilize
systems with delay. It's bandwidth and phase response that count in a
linear system, and who knows what in particular kinds of non-linear

My criterion is equivalent to yours, and applies in the nonlinear case as
well. If the Laplace transform has a delay of tau seconds, it contains a
term exp(-tau*s). The system is stabilized by a single leaky integrator
with a time constant of tau. The result is that the system behaves like
another system with no delay and no leaky integrator. Probably I should
have mentioned that it's the low-frequency behavior that's the same -- of
course with no delay and no filtering, a system could respond infinitely
fast. I just meant that if the system is stable, its delay can be ignored
in computing its behavior over the frequency range of the stabilized system
(as long as you don't get too close to the limits of performance).

Re: chaos

Once again, no-one is (yet) suggesting chaos for reorganization. The
pending claim is that it is required for the kind of rapid response at
high levels that you were talking about the other day.

You'll have to develop that thought a bit further before I see the

Go away. Have fun in Paris.

Bill P.