model-based control, the challenge

William_T_Powers4 · September 7, 1995, 6:09pm

[From Bill Powers (950907.1044 MDT)]

Hans Blom (950907) --

Hans has made statements that sound to me like saying that NO
control system can work under these conditions (totally unmodelled
dynamics). I'm willing to meet that challenge, if it was meant.

     It was meant. And it is easy to do. We can do it two ways: theore-
     tical (math) or practical (through simulation). I guess the second
     way would appeal more to you. Then let me formulate the challenge
     more precisely as follows: _I_ know a world and its "laws of
     nature", i.e. the way in which it reacts when you push it (the
     environment's transfer function). _You_ don't know those laws.
     _You_ attempt to design a "controller" that controls well in this
     world that is unknown to you. And I will show you that your
     "controller" cannot control.

I can't meet that challenge; it isn't the one I offered. Perhaps the
misunderstanding is due to my misuse of the term "unmodeled dynamics,"
which I had taken (from context) to mean only lack of a model of
arbitrary independent disturbances. You're bringing in the problem of
system identification (again, I'm using the term as I understand it from
context), which is far more difficult. Even in your Kalman Filter model
you began with a system having a pre-identified form, with only the
values of a few coefficients remaining to be determined. Moreover, the
perceptual signal (y) was simply proportional (noise aside) to the
variable that was the output of the plant; the model already knew the
nature of the variable that was to be controlled. But I'm not
complaining about that; it's a problem for any modeler of adaptive
control systems.

The focus of my challenge was intended to be much narrower; that is why
I was willing even to allow the world-model to be pre-adapted by hand
before the test -- because I would have to design the non-adaptive PCT
model to suit the plant, and didn't want that to create an unfair
advantage for me.

The only point of my challenge was to show that a PCT-type control
system (which is really a misnomer, since a world-model-based control
system can be part of a PCT model where it's shown to be needed) --

Let me start over.

The point was that a non-world-model-based negative feedback control
system of the simple design we've used for modeling tracking behavior
can deal with totally unknown disturbances of reasonable magnitude and
bandwidth, maintaining accurate control for an indefinite period of
time. A model-based control system, as you have presented it, requires a
model of the disturbance waveform in order to do the same thing.

You have said, and have cited papers saying, that there exist methods of
deducing the nature of invisible disturbances and predicting their
future course, methods which an adaptive model could incorporate to
provide an adequate model of the disturbance using only the available
input information. This claim was the focus of my challenge. You have
appeared to say that unless disturbances CAN be modeled, control _by any
kind of control system_ is impossible. Perhaps I have misunderstood you;
if so, a clarification of this apparent claim might make my challenge
irrelevant. However, assuming the contrary:

Consider the following diagram:

ref | signal
v

···

Assume that the designer knows the nature of the plant and of the input
variable (output of the plant) that is to be controlled, but not the
nature of the additive disturbance that may nor may not affect the
output of the plant. The problem is to design the control system box so
that the input variable will match the reference signal as closely as
possible. The line labeled "additive disturbance" is what I took, from
context, to be your meaning of "unmodeled dynamics", since you used that
term to refer to my addition of an independent arbitrary disturbance of
low frequency to your model.

It doesn't matter here whether the control system is adaptive or is
simply designed with full knowledge of the plant. That may matter for
other reasons, but isn't the point here. In a world-model-based control
system of the design you have described, control of the plant output
would require that the world-model (inside the Control System box) be
able to generate an internal waveform like that of the additive
disturbance, in order to compensate for the effects of that disturbance
on the output of the plant. In the model I have been using, it is NOT
necessary for any internal model of the disturbance to be present in
order for the control system to keep the input signal close to the
reference signal.

If the world-model-based control system is to be considered the general
case as you prefer to do, and the simple negative feedback control
system only a special case, then it would be necessary to demonstrate
that the world-model-based system could in fact handle the situation in
the diagram. Real organisms can handle it, and the simple negative
feedback control system can handle it. The model that is considered the
more general should be able to handle it, too.

Demonstrating that would in turn require showing that either a design
engineer or an adaptive control scheme (I don't care which you use)
could in fact design a system that could compensate for any possible
waveform of the additive disturbance, AND DO SO AT LEAST AS WELL AS THE
SIMPLE NEGATIVE FEEDBACK CONTROL SYSTEM COULD DO IT. That is the essence
of my challenge, because I don't believe that such a design is possible.
Or perhaps I should say that if you did solve this problem, I would be
extremely interested in how you did it because I can't at present
imagine how it could be done.

To make the conditions clear, the additive disturbance can have any
waveform, with an amplitude limit less than or equal to the maximum
output of the control system, and a bandwidth from 0 to some frequency
within the control-system's overall bandwidth. Other than that, there
are no restrictions on the waveform of the disturbance. To make the
problem even simpler, let's make the plant elementary:

x(t) = k*u(t) + d(t), where
x = output of plant (input to control system)
k = constant (for example, 1)
d = disturbance waveform.
u = output of control system

Alternatively, if you like, we can use the plant in the program you
published.
----------------------------------------
I emphasize that the point of the challenge concerns how disturbances
are handled. Obviously, your model can do something that the simple
negative feedback control model can't do: continue producing
approximately appropriate outputs when the input is temporarily lost.
Until some alternative way of doing that is found, your model is the
only candidate for a behavioral model when the real behavior we observe
shows the same capability. Of course when the real behavior is _unable_
to continue when inputs are lost, the world-model-based model would be
the wrong one. It would not, for example, be the correct model of
tracking behavior, where control falls apart immediately when visual
input is lost. And where arbitrary disturbance can be accurately
resisted, only (as far as I know, and this is the point of the
challenge) a negative feedback control system of the conventional design
can behave properly.

My purpose in offering this challenge is not to prove that one model is
better than the other. It is to test my suspicion that we need two kinds
of models which may be appropriate at different levels of organization.
A world-model-based model would clearly be inappropriate for the spinal
control loops where there is simply no neural machinery for doing the
necessary computations, and where a simple feedback control model works
perfectly well. It, or something much like it, would clearly be
appropriate at the levels where we make plans and carry them out.
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Best,

Bill P.