[Martin Taylor 931202 10:30]
(Bill Powers 931201.1950)
I don't know the mathematics of vortices, but I smell something
fishy in your diagram:
->-comparator-->- -<- energy source
> > >
fed back output
energy function
^ | |
> V V
^ | |
X--<--- power--<-- ----> energy sink
> to stable
> structure X
>
disturbance
In the first place this situation is not like that in a
neuromotor control system. Energy does not flow into the
neuromotor system; there is an energy cost in sensing variables
and transmitting signals. The analogy is flawed from the start.
Maybe the analogy is flawed, but you haven't hit the flaw, and I don't
see one yet. Do you think light quanta don't carry energy? The whole
business of receptor chemistry is to use that energy to modulate a greater
flow (amplify it), and so on up the line through the nervous system.
Same applies for acoustic energy, or its analogue, the pressure waves
corresponding to touch. The energy cost you are talking about is that
of the amplifier.
A stable vortex, as I understand it, is stable because any
deviation from the equilibrium form of the vortex creates
restoring forces.
You are thinking about something analogous to the marble in the bowl, aren't
you? I am talking about self-organizing structures in a high energy
far from equilibrium flow. In a sense, your wording could apply to
either, as it could apply to any control system's stabilized value(s).
But the connotations point to the potential-field situation. Your
further text enhances this interpretation:
As the vortex
dissipates energy, it will expand until the external energy input
is equal to the rate of dissipation. If there is a positive
disturbance, the disturbance injects more energy into the vortex,
and this energy produces a deviation from the former state of
equilibrium (an expansion). In the process of restoring
equilibrium, the injected energy is dissipated.
The vortex, left to itself, would dissipate into heat. Disturbances
are not really relevant to that. Do you remember the parody of (?)Dean
Swift's epigram:
Big whorls have little whorls
that feed on their velocity;
Little whorls have littler whorls
and so on to viscosity.
The self-organized structure depends on negative feedback to modulate the
high energy flow in such a way that the fed back energy (the perceptual
signal) remains stable. Such a structure may well have very little
stability against an external disturbance, though they often do. It
is precisely this structure that I see as the original basis of life,
and of control. Such systems occur in all sorts of places in nature.
You can hardly avoid them. "Vortex" is just a convenient example.
Control, to me, survives through evolutionary time because the next level
up permits a reference signal to exist in the negative feedback loop,
allowing the simple loop to contribute to enhanced stability of the
"collective." It's the next stage after the "mutuality" condition I
talked about a month or two ago.
An analogy would be a mass on a spring with some damping, being
driven by a loosely-coupled oscillatory source. The mass would
settle into an oscillation of an amplitude such that the power
lost through damping was equal to the injected power. If a pulse
of in-phase energy is injected arbitrarily, the amplitude of
oscillation will increase, then decay back to the original
amplitude. The decay will exactly dissipate the injected energy.
Loop gain is at most 1.
You describe well what will happen with the spring, but consider:
the energy source that drives the oscillation is, in the diagram,
the thing marked "energy source." If it is the only energy source,
elementary thermodynamics says that you can't get more than that
out of the spring oscillation. And there's no loop, so there's no
loop gain. Where the loop is is in the passing of energy back and
forth between kinetic and potential forms, each with one df, and with
some loss to heat on each interchange.
I'm not at all clear how one would treat the "gain" around such a loop.
But here's a hazard. Imagine that there has been a disturbance--someone
bumped the spring (in or out of phase, it doesn't matter). The total
energy stored in either potential or kinetic form is changed by deltaE,
the absolute value of which might be positive or negative. After some
time t, the oscillation has been to some degree restored toward its
original amplitude. The deviation from the original stored energy is
now delta1E. Using the usual loop equations, the effective gain
(in energy terms) is (deltaE/delta1E)-1. Over infinite time, for the
spring, this approaches infinity.
The initial increase represents the
energy-storage capacity of the system (the larger that capacity,
the less the increase), and is not resisted. All the power used
to restore equilibrium comes from dissipating the energy injected
by the disturbance.
But the same works when the disturbance extracts energy from the system.
Then where does the energy come from that restores the original state?
This is not a control system.
That's my opinion, too. It's the position I have been arguing for.
But both the spring and the vortex are negative feedback sytems with
potentially high gain. The spring isn't self-organized, so it's one
step closer to the marble in the bowl than the vortex is. The vortex
is one step closer to life and control than the spring is.
The kinds of dynamical systems you seem to be referring
to are the ones involved in chaos theory, I am guessing (since
you lapse into that language now and then).
They are, but only incidentally; some dynamic systems behave chaotically
when their coupling parameters take on certain values.
Chaos theory seems to
use models build mainly around some sort of damped nonlinear
oscillator or periodic function with damping, driven by an
external source. The damping makes it a "dissipative system."
Actually, chaos depends on feedback. There's no "building around
a function." There are relationships such as
x(t) = f(x(tau), x'(tau),..., y(tau), y'(tau),...)
y(t) = g(x(tau), x'(tau),..., y(tau), y'(tau),...)
That's a feedback system.
When overdriven to a certain degree, the oscillatory system shows
chaotic behavior that can be expressed as deviations from some
attractor point, curve, or surface.
Two misapprehensions. Firstly, overdriving has no intrinsic connection
with the approach to chaos, though operation at different amplitudes may
induce different aspects of the nonlinearities of the feedback structure
to become important (in the same way that harmonics show up in an
overdriven amplifier). The approach to chaos comes in changes of
coupling constants, normally, and of structural parameters, in general.
These parameters matter, even in control systems, once one permits
recursive connections. Secondly, chaotic behaviour defines an attractor,
rather than providing something that can be described as a deviation
from an attractor. The attractor has non-integer dimension (as a rule;
non-generically, it can have integer dimension by a fluke).
However, very little of human behavior shows any chaotic
characteristics; when they do show up, the result is likely to be
an emergency trip to the hospital. Human behavior is REGULAR, not
chaotic. However, to see the regularities you have to apply the
right theory, which is control theory. Most human behavior is not
even periodic, although some aspects of it are.
So far, I am in complete agreement. You persuaded me of this very
early in my acquaintance with PCT. Nevertheless, it is important
to see how control hierarchies can be structured and maintained so that
they avoid the chaos which normally occurs in large nonlinear tightly
coupled systems. In your basic structure, you avoid the problem by
permitting recursion only through the outer world. But as soon as
you allow imagination connections, and worse, the kind of within-level
connection among perceptual input functions that are needed for category
perception, THEN you have a problem with how on earth the biological
brain gets away without falling into chaotic oscillations. If, indeed,
it does (what are EEG signals?).
Applying the principles of chaos theory to human behavior is thus a wrong
development from a correct theory.
This I do not agree with. For one thing, the "principles" of chaos theory
are those of dynamics, not something special. Secondly, as mentioned above,
those principles are of potential value in discovering how evolution has
enabled us to operate in a non-chaotic manner.
···
================
Tracking note:
I find this whole discussion rather weird. My reconstruction of its
history is rather like this.
(1) Bill P asked me as an aside in a private posting whether I had any idea
where "the dynamicists" on the net were coming from, conceptually.
(2) I wrote Bill a rather casual answer, based on where I had been coming
from before learning about PCT.
(3) Bill, supported by Mary, urged me to make public my "wondeful essay"
after cleaning private references from it, as he/they thought it might help
to create a bridge between the dynamicists and the PCT core group. I did
so, with minimal editing to the original.
(4) Rick took me to task for trying to persuade PCT people of the correctness
of a selection of contributors to the Devil's Bibliography, none of whom
had figured in the original private discussion with Bill P. He also made
some comments about the intrinsic worthlessness of dynamic approaches.
(5) I tried to correct Rick's inversion of my intent. At the same time,
his comments made me wonder whether there might, in fact, be some real
use for taking a wider dynamic view on control systems, and I included
in my response(s) some musings to that effect. (Remember, I had used
that approach to the question of "language as artifact" in Durango, and
had received some approval from both Bill and Rick. This was at the
back of my mind as I wrote, and I forgot that I could be taken as
referring also to Tom's studies on multiple control systems).
(6) Rick, Tom, and Bill jumped on me for my "frequent" assertions that
dynamic analysis was better than control system approaches. I was
astounded, though from the sentences quoted back, I could see that I
had certainly written things that could give such an impression,
especially to Tom.
(7) More thought on the matter suggested to me that it would not be a
bad thing to put some emphasis on the energetics of control and of self-
organization, because I prefer to think of control as having purpose,
and I associate purpose with a reference variable. Variable means
not built-in and constant, so I proposed that control meant not just
negative feedback, which occurs in many natural situations, but negative
feedback with a reference signal input.
(8) I get jumped on more. When people criticize what I write, I re-evaluate
it. If I still believe what I said, I tend to continue the argument, to
support it in different ways. This can get out of hand. If after
re-evaluation I don't believe what I wrote, I say so, or leave it alone
to let the other person have the last word (sometimes it was a conceptual
typo, sometimes my prior failure to understand).
At present, I hold to my understandings (a) that control analysis is better
to employ than is the more general dynamic analysis, provided both are
appropriate and the subject of analysis is a control system, (b) that
general dynamic analysis can point out, at the very least, areas of
concern that might affect control--such as the likelihood of chaotic
operation of tightly coupled nonlinear systems, (c) it is worthwhile
arguing these points because PCT is at its heart a necessarily correct
theory, and the only basis for a sound science of life (which includes
but is not restricted to psychology).
And I retain the position I expressed to Bill P. in Durango; I hope I
never get to the stage of agreeing with him because he is usually right,
without convincing myself on each occasion that he IS right.
Martin