"Random" vs "Arbitrary" disturbances; Blom's model

[From Bill Powers (950504.1315 MDT)]

Martin Taylor, Hans Blom, others in modeling game:

Having studied Hans Blom's model some more, I realize now that we PCTers
made a terminological mistake way back in the beginning. The mistake was
to say that we use "random disturbances" in our experiments and models.

As I look back on arguments concerning the disturbance, and see how
Hans' model deals with disturbances, I realize that the term "random"
should have been "arbitrary." When we speak of a random disturbance,
what comes to mind for those used to dealing with statistical processes
is a variable that jumps from one unpredictable value to another, but
which has some mean value, some spectral distribution, and (usually) a
normal or Gaussian distribution. The image that comes to mind is
"noise." But this is not what is meant in PCT models where a disturbance
acts directly on the controlled variable.

An arbitrary disturbance is one which arises from ordinary regular
physical processes in the environment, but which is independent of the
actions of the control system, in the sense of not being produced by any
effect of the output of the control system. It is not necessarily
independent in the sense of having a low correlation with the output;
whether it does or not is a matter of coincidence, and a matter of how
the control system acts in the presence of the disturbance. The concept
of randomness, in the sense of an inherently unpredictable variation
with no systematic relationship to other environmental variables (like
radioactive decay or thermal noise), is not and was not intended.

Here is an example of an arbitrary disturbance waveform:

···

*
* *
* * * *
* **** * *
* * * * **********
* * *
***** * *
* *

I don't think anyone would try to deal with this waveform statistically;
it would fail all tests for belonging to any standard random
distribution, and while unpredictable it is far from unsystematic.
Furthermore, its average value would be meaningless because the
variations are clearly not uniformly distributed about any mean. The
most we can say is that this variable varies due to processes we know
nothing about, but most likely in a physically lawful way.

A proper control system of the kind we find in the behavior of living
systems will, if it has a constant reference signal, produce an output
that is the mirror image of the above waveform (vertically). To
accomplish this it needs only to know the value of the controlled
variable, the variable that is influenced by the above waveform and by
its own output. It does not need any signal indicating the state of the
disturbing waveform. It does not need to compute the mean or variance of
the disturbance.
-----------------------------------------
My preliminary look at Hans' model shows that for the loop involving the
model's model, the loop gain is 1. In other words, the "control" u
(which we call the output) is computed so as to bring the model's
controlled quantity x exactly to the value of the reference signal on
the next iteration. In a proper control system, with a loop gain in the
tens or hundreds, the value of u would become far larger than actually
needed to correct the error, but the system dynamics would see to it
that only part of the error was corrected on each iteration. The result
would be tight control of the model's output.

However, this would be relatively pointless in this model because the
true system output, xt, is not being compared with the reference signal;
only the model's output is. Thus the system is inherently incapable of
counteracting ARBITRARY disturbances of the system's output state. Only
for very slowly changing arbitrary disturbances would the model be able
to oppose effect of the disturbance on the real output xt, by going
through the model-correcting process and changing the value of the
"constant" variable c, and thus the state of the "control" u. So control
of the real system could occur only within the bandwidth of the very
complex computations that keep matching the model's output to that of
the real system.
----------------------------------
The Blom model is interesting because it is able to keep the movements
of control going even during periods when the perception of the real
system is interrupted. Also, the "Kalman filter" approach may be
interesting. However, I do not consider this model a candidate for a
model of how a living system does these things. As already noted, it is
not able to protect the real controlled variable from arbitrary
disturbances as the living system can do. And there are too many
calculations, of kinds that I think are highly unlikely to occur in a
nervous system (would anyone believe that the nervous system is
calculating covariance matrices and variances?). If Hans' program is
really supposed to be a model of how the nervous system accomplishes
adaptive control, then it is claiming that every operation performed by
the program is performed by the nervous system, a claim that I would
most sincerely doubt.

The concept of model-based control should be considered for inclusion in
a model of living control systems. I have, in fact, had it in mind for
two or three decades. However, there is a problem in making this a
_full-time_ method of control, because of the impossibility of dealing
with arbitrary disturbances. I can see this method of control being
implemented at higher levels, where disturbances change slowly, or being
switched in when normal control is lost. But the switching itself would
be hard to model, as well as the processes by which the models are kept
updated. If it takes a PhD in mathematics to compute the model
parameters, then few nervous systems would be able to do it. Whatever
the modeling process is, it has to be basically simple. I don't think we
have the right one yet.
----------------------------------------------------------------------
Best,

Bill P.

[Martin Taylor 950505 18:30]

Bill Powers (950504.1315 MDT)

Martin Taylor, Hans Blom, others in modeling game:

Can't speak for Hans or others, but for me...

As I look back on arguments concerning the disturbance, and see how
Hans' model deals with disturbances, I realize that the term "random"
should have been "arbitrary."

Probably a good thought. It's worth while to keep these ideas differentiated.
"Random" does have the connotation of "white noise." The disturbances
that have interested me in our discussions have been "arbitrary" while
any noise sources associated with the operations of the loop have been
"random."

[Picture of a waveform omitted]

I don't think anyone would try to deal with this waveform statistically;

Don't bet on it

it would fail all tests for belonging to any standard random
distribution, and while unpredictable it is far from unsystematic.

True (informally). That's why you should withold your bet. (More formally,
"far from unsystematic" is a near synonym for "partially predictable").

A proper control system of the kind we find in the behavior of living
systems will, if it has a constant reference signal, produce an output
that is the mirror image of the above waveform (vertically).

But that depends on the time scale of your drawing as compared to the
properties of the loop. The waveform in your figure started with an
explicit step, and no control system will produce an exact mirror of
that at its output, no matter how slow the rest of the waveform. ("Exact"
includes, of course, temporal synchronization).

There are several places in any control loop where the properties of the
loop affect just what will appear at the output when that particular
disturbance appears at the input at a particular speed. And for a
particular set of properties what appears at the output will be different
at different speeds of the waveform. What changes about the disturbance
when its speed changes? Only a shift of its spectrum upward or downward
on the frequency scale.

You know all this, and you are making extreme statements simply for effect.
Nothing wrong with that; I do it often enough myself, to provoke discussion.

Yes, let's try to be a little more precise about terms like "random."

No, let's not assume that statistical parameters are irrelevant when dealing
with waveforms better termed "arbitrary."

Martin