DYnamical systems, control systems, and stability

Cliff Joslyn, 931119 replying to Bill Powers:

Your definition of a dynamical system is interesting, because it
seems to say that living control systems are not dynamical
systems, at least not in any practical terms.

Well, probably not in any practical terms. Maybe when they're
straight-jacketted into Skinner boxes, but that's neither practical for
real organisms nor interesting in general.

The first thing to keep in mind is that a "dynamical system" is NOT a kind
of physical thing, it is rather a kind of MODEL. An object can be living or
not, a control system or not, etc. But it makes no sense to say that an
object "is" a dynamical system. Instead, sometimes we can succesfully model
a given object USING a dynamical system. Since the model is a homomorphism,
it necessarily represents only a part of the object. So the dynamical model
only accurately reflects CERTAIN ASPECTS of the object: properties at a
certain level of analysis, or "granularity" of measurement. And further,
since generally there are MANY possible complementary models available of
any object at any given time, therefore something may be SIMULTANEOUSLY
regarded as a dynamical system or not depending on which perspective of it
is taken.

Consider a simple control system coupled with a reorganizing
system. The reorganizing system finds least-error parameters for
the control system using a process of randomly varying them at
intervals inversely proportional to the absolute rate of change
of error and by amounts proportional to the mean absolute error.

Of course for the random reorganizing effects, there is no
deterministic equation (we could use radioactive decay as the
basis), so the future state of the system can't be predicted
deterministically from its present state even if the system is
interacting with a deterministic environment.

This is a good example. Consider a function f(x,y) = z. This is (generally,
not technically) a dynamical system, in that ONCE x AND y ARE FIXED, then z
is determined. This is not to say that x or y will not vary. Indeed, they
may vary in a completely random, unkown manner. But that's not the
question. The point is that once they "enter" the system f, they are
"measured" and BECOME known AT THAT TIME, and THEN z can be predicted
unambiguously.

Now consider the situation where x becomes fixed (no variation) but y does
not (some variation). This determines a family of functions f_x(y) = z
(where _ is subscript), one for each x, and each one a new dynamical
system. Now from the NEW perspective of f_x only, variation in x will
produce a random, non-deterministic effect in z, because we no longer "see"
x if it changes. Its just that all of a sudden everything acts differently.
But in the global perspective (the higher level) where the full function
f(x,y) is known, there is no nondeterminism in f.

Translating to the control system now, x is the effect of reorganization,
while f is the action of the CS. The perspective of the control system
PROPER (absent knowledge of HOW it is reorganized), is that of f_x(y). So I
would agree that now the control system per se is not a dynamical system.
But if we know the state of the reorganizing system, then we take the
perspective of f(x,y), and the ENTIRE control meta-system (control system
plus reorganizer) again becomes a dynamical system.

To repeat: whether something "is" a dynamical system or not depends
entirely on how it is represented, and there are usually multiple,
complementary representations available.

Another problem is the reference signal in a hierarchical system.
Each reference signal, as an output from a higher system, is
determined partly by perceptual signals and partly by higher-
level reference signals reaching the higher system.

This is exactly the same issue. Once the input reference level to a CS at a
given hierarchical level is fixed, THEN the CS can be regarded as a
dynamical system. If the activity or presence of the RL cannot be
determined, then the CS cannot be regarded as a dynamical system, since its
behavior (meaning its change in state in time, not its "behavior" in the
BCP sense) cannot be determined.

And finally we have to consider disturbances.

Same principle: once the disturbance is fixed as it "enters" the perceptual
system, THEN the CS can be regarded as a dynamical system.

It's really very simple: what are the sources of input to a CS?
Environmental disturbance, reference level, and reorganizational "tuning".
In a system with three inputs f(x,y,z) = w, then all three x, y, and z must
be known to predict w. Lacking knowledge of the effects of ANY of them, the
CS cannot be taken as a dynamical system.

Now map this to the REAL behavior of organisms. How many parameters are
involved in modeling the control behavior of a person? What hope do we have
of predicting all their effects? In general, organisms cannot practically
be regarded as DS's. But simplify their range of inputs and actions (say in
a Skinner box), and some predictability MAY be recovered. Or, try this:
drop me out a window, and I'm going to act like a dynamical system; give me
a hang-glider, and I probably won't.

So if it's
impossible to predict disturbances into the indefinite future
(for example, the effects of weather on controlled variables like
staying dry), then it's not possible to predict the actions of
the control system, either. The "butterfly effect" then makes the
behavior of the system as a whole, environment plus organism,
non-deterministic. This butterfly effect, by the way, no doubt
exists inside the control system as well as outside it.

The presence of chaos introduces a levels problem of a different sort. In a
truly chaotic system (that is, a DYNAMICAL system with sensitivity to
initial conditions) there are actually TWO complementary modeling levels of
analysis available. At the micro level, it is truly a dynamical system,
with each state uniquely determining the next (once the input is known). It
is only over the "long run" that a macro level emerges, since initially
very close trajectories diverge arbitrarly far over an arbitrarily long
time. So yes, in a CS which is so wired up as to be chaotic, it would in
the macro sense not be a DS.

There's potentially yet a third levels issue involved here. Let's say
f(x,y) = z is NOT a dynamical system, in that there is some INTERNAL factor
w which affects z in a random manner, even once x and y are known. It may
still be possible to characterize w, say as a normal or uniform
distribution with known parameters. Then again while the micro behavior of
z will not be predictable, macroscopically it WILL be predictable. Given x
and y, z will not be restricted to a single, unique state, but it may be
restricted to a single, unique META-STATE, for example a certain kind of
distribution with certain parameters.

I think that what's really in doubt here is the concept of a
deterministic system. Considering all the above points, can we
really believe in such things any more as a general philosophical
principle of science? The universe itself is deterministic only
in a relative way, with the degree of determinism rapidly
declining as projections go farther into the future and as one
examines even present behavior in finer and finer detail.

Well yes, this really is the question: what is the domain over which
dynamical models are accurate? Obviously, they are accurate to a certain
extent, but not completely. They are approximations; they are MODELS.

What living control systems do is to force their local
environments to be far more deterministic over far longer
stretches of time than would be possible without the presence of
the control systems. Control has
the effect of making at least significant parts of the local
universe predictable, where otherwise they would be random or
chaotic.

I agree: negative feedback has the property that it "squeezes" the
dynamical behavior of a system into a very small domain, increasing its
stability. In the limit, this leads to ABSOLUTE determinism (lack of
variation of the system state, not just our ability to predict what
variation there is).

"Stability" is a qualitative term when used this way.

"Stability" is ALWAYS a qualitative term.

What makes the difference between negative feedback
control systems and passively stable systems is the DEGREE of
stability against perturbations, and the range of types of
perturbations against which the outcome is stable.

I agree: the presence of control produces at least a vast QUANTITATIVE
improvement in the level of stability, and at most introduces stability
where there would not otherwise be any. Once a quantitative change is large
enough, the demmand to make a qualitative distinction becomes
overwhelming.

There is an analogy to catalysis. If a favored reaction will reach
equilibrium in 10 years, and the presence of a catalyst will reduce it to
.0001 second, then mathematically each is "stable" around that equilibrium,
but it also makes perfect sense to say that the prior system is unchanging,
while the latter changes instantly to the stable state.

I mentioned the other day getting up, walking upstairs,
and bringing a cup of coffee back down to my computer table. In
open-loop terms that is a completely impossible action. When I hear the
word "stability," that's the sort of thing I think of. Not
marbles in bowls.

Well, honestly, both are "stability". It's just that control systems are a
hell of a lot more interesting. My only point is that there are MANY ways
to produce stability. I make no argument for open-loop systems or marbles
in bowls over control. But what about this: "natural" stability occurs
where it wants, and in generally simple systems; and "open-loop" stability
is not very good; but "closed-loop" stability can be made virtually
anywhere we choose, in systems of virtually any complexity, and is orders
of magnitude better than anything else. Do you have any argument with that?

O----------------------------------------------------------------------------->

Cliff Joslyn, Cybernetician at Large, 327 Spring St #2 Portland ME 04102 USA
Systems Science, SUNY Binghamton NASA Goddard Space Flight Center
cjoslyn@bingsuns.cc.binghamton.edu joslyn@kong.gsfc.nasa.gov

V All the world is biscuit shaped. . .