Regions of chaos

CSGnet Archive CSG1995
1

I'm just a lurker, and I've been listening a bit.

This lurker happens to design electronic control systems for physical
devices -- much simpler than human beings.

But, I must admit some confusion. Perhaps someone can explain. The
confusion comes from a recent design I *tried* to complete in a
satisfactory way, but one where I didn't meet my own expectations.

The requirement was a feedback-system of some sort to minimize a
characteristic noise peak in a fiber-optic laser. The noise was related
to the relaxation time of the electron energy states. Pump an electron
up to a high enough state, and the electron will spontaneously collapse
into a lower-energy state, emitting a photon. If another photon happens
to breeze by, the collapse can be triggered by the second photon.
That's a crude description of an optical amplifier; one photon can
'gate' another photon -- or a whole bunch of photons.

You make a laser by adding mirrors, and reflecting the light back-and-
forth such that a photon is as likely to be absorbed by an electron/atom
(punching the electron to a higher energy state) as to be emitted by an
electron/atom. That takes an incredibly high light-intensity inside the
laser, but this point of equal-probability of absorbtion/emission pretty
much defines the point of 'lasing.' Once you get to that light
intensity, the stimulated-release of photons completely dominates the
system behavior, and you can pretty much ignore random emissions.

That's almost all we need to know about lasers for the time being --
except when it comes to sending power into the laser if we want to make
the output time-invarient. At least with fiber-lasers (spatially-large
compared to the wavelength of lasing-light) the behavior is chaotic,
with light output not well-defined on an instant-by-instant basis with
power input. (Simultaneity constraints are actually important.)

I found no satisfactory way of making a feedback loop around a chaotic
process. I could reduce the 'noise' output with some feedback, but
increased feedback resulted in increased noise -- not noise that would
be predicted by any linear feedback theory. (I was nowhere near
oscillation for any linearized lumped-element model.)

Now -- this may seem very far afield from what you're talking about
here. But it seems to me that a laser, even a laser operating in a
chaotic space, must have fewer degrees of freedom than a human brain.
And I can't help wondering what insights might come from an evaluation
of grossly nonlinear (chaotic) systems in the context of what's being
discussed here.

Please -- I acknowledge almost complete ignorance of what you're
discussing, and so I acknowledge that I may be introducing something
already dismissed as trivial, so feel free to ignore this intrusion.

-R

···

--
Richard Plourde (rplourde@scoot.netis.com)

2

[Martin Taylor 951027 10:30]

Richard Plourde, apparently Fri, 27 Oct 1995 04:00:56

Richard: could you head your messages with your name and a time-stamp,
as I have done, please? It makes cross-referencing a lot easier.

I found no satisfactory way of making a feedback loop around a chaotic
process.

Chaotic processes are not unpredictable processes. They are processes
in which after an observation, the uncertainty of the state grows at
some characteristic rate. After a while, you will know no more about
the state than you would have if the observation had never been made.
If you want to control a chaotic process without knowing the generating
equations (or their equivalent), you have to be able to affect the CEV
quickly enough after the observation.

Imagine a classical control loop, with a disturbance that is composed of
regularly timed impulses of known magnitude delta but unknown direction. If
the perceptual error is zero at time t0, it will be 1 impulse worth
immediately after a disturbance hit at t0+epsilon, and will recover before
the next impulse if the loop delay is short enough that the output can
reach nearly -delta before the next impulse. Let's suppose, though, that
the next impulse comes when the output has reached only -delta/2, and the
perception error is still +delta/2. If the next impulse is positive,
the error becomes +1.5delta; if it is negative, the error becomes -0.5delta,
but in either case, the output is, for a while, committed to going more
negative, even though half the time this would be increasing the error.
As the impulse rate increases, the observation of the error at time t
has less and less to do with what the actual error is when the output
affects the CEV. Correlation is lost.

The output of a perfect control system is correlated perfectly negatively
with the disturbance. If the disturbance has an autocorrelation function
that is appreciably less than unity for a lag equal to the equivalent
loop delay (the time constant of an integrator output function, for example),
then control will be poor. In a chaotic system, the autocorrelation function
declines to zero after some characteristic time, and you can't control the
CEV by applying counterforce unless you do it a lot quicker than that
characteristic time. I suspect your laser has a fairly short characteristic
time, and your electronics couldn't cope.

You have another problem, I think, and that is the positive feedback nature
of the lasing process. The "input" to any short piece of the photon path
is much more dependent on the stimulated photons in the rest of the path
than on the external input. When you reduce the input, the decay of the
effect must be somewhat slower than the speed with which the input is cut
off, and this adds to the equivalent loop delay of the control system.

I don't know the time constants concerned, so this is pure speculation.
But it's one place to look for the problem.

Controlling in a chaotic universe is what life is about, so the generic
problem is soluble.

Martin

3

<[Bill Leach 951027.05:36 U.S. Eastern Time Zone]

Message: 61909 on Fri, 27 Oct 1995 04:00:56 -0500
Author : Richard Plourde <rplourde@SCOOT.NETIS.COM>

Hi Richard and welcome.

...already dismissed as trivial, so feel free to ignore this intrusion.

I doubt that there are many (if any) here that would consider what you
are dealing with as trivial. I am not sure for myself if I see where
what you are talking about is relevant to PCT at the level where we
typically talk about and deal with our subject.

Some chaos theory issues may indeed be important in the (future)
understanding of what must be going on with respect to the random events
in reorganization. For now, only the concept that there are random
components in reorganization is particularly important to the theory.

As I see it, the system that you are describing is not being described in
classic PCT form (or I don't understand what you are saying well enough
to recognize the form). If I am understanding you correctly then what
you are calling the "feedback" is the pumped or reflected light.

Again, as I understand (or do not understand), what you are trying to
control is the noise and therefore the perception or feedback is this
noise value [(S+N)/N ?]. Your control system then is comparing the noise
figure to a reference and then adjusting some parameter of the laser as a
function of the difference signal (error). I would presume that this
"thing" that is adjusted is phase or intensity or both of the pumped
light.

If nothing else, it should be obvious that my knowledge of lasers is both
old and sketchy!

Basically as I see it, the honest evaluation including careful mapping of
parameters from the jargon of the discipline to that of PCT for ANY
closed loop negative feedback control system operation is relevant
discussion material for this forum. Whether it is "interesting" or not
would be a function of the potential for discovering some new
understanding of fundamental control behaviour that might be significant
in understanding living systems control behaviour. We are working in
such "virgin territory" that off hand dismissal of the experience of
engineered control systems experience would be short sighted indeed.

-bill