[Martin Taylor 931222 11:00]
(Bill Powers 931221.0835)
I agree with Tom Bourbon that your exposition on dynamics was
clearer than most I have seen. I hope that my comments on it
don't reveal that my understanding is illusory.
You mean I wrote something clearly for once? Will miracles never cease?
I think your understanding and your extensions are fine, with one minor
problem that really has no effect on the rest.
In your descriptions, "orbits" seem to play a central part. If my
physical interpretation of phase space is right, the implication
is of an oscillatory or at least repetitive phenomenon.
No, there's no suggestion of repetition intended with the term "orbit."
Every point in a phase space lies on one and only one orbit, assuming that
the phase space is complete, in that it describes all of the motions that
matter in the system. Instead of a ball-in-a-bowl, imagine a ball on
a rugged terrain. Starting at lat-long (x,y), the ball can be given
an initial momentum vector (mx, my). There is exactly one orbit through
the point (x, y, mx, my) in the phase space. It traversed some path over
the terrain in the x-y plane, but that path is only a projection of the
orbit, which is defined in the 4-D space x-y-mx-my. That 4-D path is
the (one and only) orbit through the starting point (x, y, mx, my).
The same orbit goes, of course, through aleph-1 points in the phase space,
and there are aleph-1 orbits in the whole dynamic (assuming a continuous
space of location and momentum).
In a control system that affects the dynamic of some physical system:
The attractor is only detectable by an outside observer--the
analyst, who might deliberately impose disturbances (apply the
Test) to determine the nature of the dynamic and see whether it
has the characteristics of a controlled dynamic: orbits that
converge not to a point attractor or a limit cycle, but to a
noisy set of interweaving "orbits" that are essentially
parallel over some region of the phase space.One class of stable control systems conforms to this description,
but only in the sense that its behavior would be a particular
subset of all the behaviors that fit the description. The
"converging orbits" would consist of a single perhaps curved line
running from the perturbed position to the point attractor, and
the "noisy interweaving orbits" would exist in a tiny region in
the immediate vicinity of the point attractor, reflecting the
irreducible system noise. Any control system of this class -- a
controller of a scalar variable -- that does not exhibit this
behavior has something wrong with it.
I'm not sure what you intend to imply by "only in the sense that ..."
What I intended to describe was how an application of The Test looks
in the dynamic description. If you mean that there are non-control
systems that look the same, I'm not sure that there are. Possibly,
but I don't think so. Any ordinary system that is stabilized in a way
that can be described as minimizing some potential will not show the
"noisy interweaving orbits" but will converge to a simple infinitesimal
attractor.
The "irreducible system noise" is the limited information-gathering
rate of the perceptual system of the cotnrol system. The "noise" comes
from the divergence inherent in the dynamic of the physical system being
acted upon by the control system (if there were no divergence, there
would be no need for control other than a simple counterforce that
could be made infinitely precise over time). That divergence might
be caused by anything in the dynamic (you gave many examples in your
post on "abnormal physics). It may be caused by factors not included
in the phase space, such as the thermal noise of vibrating molecules,
or anything else, including unspecified external distubances. But the
control system compensates for the divergence within limits imposed by
its perceptual funtion, comparator, and output function. Those limits
determine the width of the "interweaving band" of orbits (actually a
projection onto this phase space from a larger phase space that would
describe all the disturbances, the control system, and all its reference
inputs).
In describing the "noisy interweaving orbits" I assume that the Tester
can measure the physical variable more precisely than the control system
can stabilize it. If not, the Tester can rely only on the shape of the
attractor dynamic and on exploration of the possibility that the supposed
control system senses and acts upon the physical variable. But I think
that the "noisy interweave" is likely to be diagnostic of control. A
counterexample would prove me wrong, and I'm not making any big bets that
I'm right.
Also, we would observe that the position of the point attractor
could be changed from time to time in the position dimension,
with all orbits consisting of smooth lines terminating at the
changed position, with zero velocity (except for noise).
Quite so. This is the situation facing a Tester trying to determine
whether a variable is being controlled when the control system's reference
is changing as fast as the tester's application of a disturbance. The
two effects can, in principle, be disentangled, but you are looking now
at a superdynamic in which the variable parameter is the control system's
reference level. That's much harder to trace through experiment than is
a fixed dynamic that is assumed to be reasonably smooth over almost all
of its extent. That particular superdynamic is a description of the
dynamic of a two-level control hierarchy, in the larger phase space
that includes the new variables implied by the higher level's control
parameters.
There is a second class, a controller of cyclic variables such as
would be found in walking, singing in a rhythm, beating a drum in
rhythm, and so forth. The orbit for this kind of control system
would run from its perturbed position along a line that curves
and quickly joins an attractor in the form of a stable and
repeatable limit cycle, with a small region of "noisy
interweaving orbits" in a narrow band around the attractor. Any
cyclical controller that did not behave in this way would also
have something wrong with it.
Exactly so.
This second type or orbit, however, is hard to take seriously as
a relevant description of a controller of time-dependent
variables. The reason is that the observed behavior could be
brought about in two completely different ways, one involving
physical dynamics and the other not.It could be created by an inherently oscillatory system with the
correct amount of damping and nonlinear positive feedback: two
stages of nonlinear integral feedback and the proper amount of
negative rate feedback, for example. This would make the whole
system oscillatory and make the control of static variables
impossible.Or the same pattern could be brought about by scanning a stored
table of reference signal values in which the successive values
of reference signals for a scalar controlled variable are issued
one after the other. In the latter case, the only real attractor
is the point-attractor for the scalar control system of the first
class described above, at a lower level.
Right. There's a real problem for the external observer/analyst in this.
I think, but with no assurance, that you can see these two different
modes in figure skaters. My impression of very young (and thus physically
small) figure skaters is that they control in the second way, continually
adjusting the positions of their arms and legs. My hypothesis has been
that they do this because the natural pendulum dynamics of their limbs
has the wrong period. As they grow, arms and legs swing more slowly
(and they have more practice in control), and what they control is the
phase of the oscillations rather than the positions of the limbs. This
is all just a feeling I get from watching them, not a serious analysis,
but it might be relevant. How could one tell whether I am right? Put
weights on the wrists and ankles of junior and senior skaters, expecting
the juniors to be less affected than the seniors?
From your description, most of the behaviors that can be
described in terms of limit cycles, attractor basins, and
repulsor edges, do not occur in the behavior of real organisms.
Remember that the dynamics I emphasized were those of the physical
variables. When the variable is acted on as part of a control loop
with a fixed reference level, all of these phenomena can be seen,
although limit cycles are probably as rare as you say. Attractor
basins are a part of any dynamic with attractors, and control implies
that the dynamic has at least one attractor. That the range of control
is usually limited implies that the dynamic has more than one basin
of attraction, and therefore there is at least one repellor. So these,
at least, are a normal part of the description of the behaviour of normal
organisms, as seen by an outside observer (that last is important).
Remember that the outside observer see only orbits, and at that the
orbits are projected onto a space defined by the observer's perceptual
functions, not those of the organism. A dynamic (or superdynamic) has
to be inferred from examination of orbits. The Test involves setting
the physical variable in some new place in the phase space and seeing
where the orbit from that place leads. The behaviour of organisms
is exhibited in actual orbits. The potential behaviour is described
by the whole dynamic.
Plotting position against velocity is an arbitrary way of
representing data.
That is a phase space only for kinematic data. What is important is
that all aspects of the behaviour of the system be represented in the
phase space. And that is not arbitrary.
Patterns may appear in such plots, but that
mode of perception has little application in ordinary behavior.
The perception is that of the analyst, not of the behaving system, so
whether it has application depends on whether it tells you anything
about the system being analyzed. To me, it does. Maybe to you it doesn't.
The only way to tell whether it is worthwhile is to discover whether I
learn stuff about behaving systems that I otherwise would not have done,
even though you may have known the same things by other means. If I can
learn something you didn't find by other means, that's even better. If
I don't learn anything from it that's useful and that I didn't learn
otherwise, then I would say it "has little application in ordinary
behavior."
As in control theory, when you view data through odd
transformations, you tend to lose contact with the actual
phenomenon. You may learn something new, but what you learn may
not have much importance (as opposed to fascination, which such
plots obviously produce).
Right. But then again, it may.
Overall, I think you understood very well what I was saying.
Martin