[From Bill Powers (950621.1415 MDT)]
Hans Blom (950621a)--
... The environment does not have the power to act on an
organism in such a way as to make it do any specific behavior.
No. Yes. Depends. At least upon what you consider "environment".
I was thinking of the inanimate environment, not other people who can
carry out intelligent strategies. One person can, of course, control
certain aspects of another person's actions, as long as the result is
not to cause error in the other's control systems which would lead to
resistance.
To say that we "filter out" perceptions implies, to me, the
wrong model of perception and the wrong epistemology. The brain
has to labor mightily to bring order into the field of
intensity-signals where all potential perceptions exist.
Maybe I ought to say "labor mightily" rather than "filter out", but
I doubt whether that would be clearer for most ;-).
I think these are two quite different concepts. To say that the nervous
system must carry out active processes in order to create perceptions
that would not otherwise exist is very different from saying that the
perceptions already exist, and it's just a question of filtering out the
ones you don't want.
"Filtering out" refers to pruning unwanted perceptual connections
from a perceptual input function rather than building up its con-
nections. Basically negative versus positive, certainly NOT a very
different paradigm.
I think it's extremely different, as the "pruning" concept assumes that
there are working connections already in existence.
Explicitly construct or explicitly prune, what's the difference?
Again, you can't prune something that isn't there. If a perceptual
function comes into being only through creating new connections, the
starting point is no perceptual function at all.
In a model based approach, connections initially have any value
(and a large variance), but after learning, that value has settled
down (and has a small variance). That's more like pruning away
uncertainty than it is building up certainty.
The mopdel-based approach as I have seen it so far gives the model a
great deal of advance information before it can even start working. I
know that some degree of that is necessary, but I would like to see an
approach that doesn't need such a head start on the final form of the
model.
···
-----------------------------------
The single "plant" of an organism is the earth, and there are a
great many commonalities between different local environments on
earth.
Well, that's a true statement all right, but rather useless in the
context. I was speaking about control sub-systems that might be in
furious use one second to type letters on a keyboard, and a fraction of
a second later might be used to brush a fly away, scratch a nose, or
pick up a cup of coffee with an unknown amount of liquid in it. The idea
of having to readapt the entire system for each new use of arms and legs
seems clumsy.
There are differences as well, of course, but humans fare equally
well in the Arctics, in the Kalahari and in New York (well, maybe
not New York :-).
The problem is time-scales. It may take days or weeks to adapt to a new
climate, especially if significant changes in altitude are involved. I'm
speaking of being able to switch contexts essentially instantly.
Try this: There is only one plant, but the plant's characteristics
can change, requiring renewed learning in a different environment.
This interpretation is at least as plausible to me.
Well of course it is; it's the idea you prefer. But when you think of
switching contexts on a time-scale of seconds, this view begins to be a
lot less attractive.
---------------------------------------------
Hans Blom (950621) --
But I am more interested in the analysis of the stable state. As a
thought experiment, set all disturbances to zero and check to what
state the system as a whole settles down to.
If there are no disturbances, including any natural tendencies of the
controlled variable to settle to a state other than the desired one,
then the error in a control system will become exactly zero, just as in
a compensating system. However, that tells us very little about the
capabilities of the control system, because this same equilibrium would
be reached by fast, tight control systems and slow loose control
systems. The whole test of effectiveness comes when there are
disturbances tending to change the controlled variable. How much is the
controlled variable changed by a disturbance of a given magnitude? That
is the real test.
If even when everything has settled down, non-zero "errors" are
required to keep the system in its stable state, it is difficult
for me to think of these "errors" as errors -- they can also be
interpreted as control variables that keep other variables at their
correct values. Please help me with a clearer definition of what
you consider to be error.
That is the case where the controlled variable's natural state is not
the desired state. In your model, you don't have any variable explicitly
representing the error signal -- that is, (xopt - x). You could easily
rearrange your equations to include one:
u = (xopt - ax - c - disturbance)/b; or
u = (xopt - x + x - ax - c - disturbance)/b, or
u = (xopt - x)/b + x(1-a) - c - disturbance)/b
error = e = xopt - x, so
u = e/b + (x(1-a) - c - disturbance)/b
When your system is working properly, it keeps the error at zero.
However, your design is very strange from the standpoint of a control-
system engineer, in that the output is not based only on the error; it
is also based on the controlled variable, and both constant and variable
disturbances whose actual values are somehow fed into the system with
unlimited precision.
Because a
negative feedback control system always had to have some error
to produce a compensating output, it could never even in
principle control exactly. Ashby decided that the only "real"
control system was the compensator.
Isn't he right IN PRINCIPLE? In your scheme of things, a negative
feedback control system always has some error (1 in 100 in your
example) to produce a compensating output, even if the error is
small. If a compensator can avoid even this small error, wouldn't
it IN PRINCIPLE be better, particularly in those cases where we
want to have a control quality that exhausts all information in the
feedback signal down to the noise level?
Yes, but he's right in a sort of fairy-tale way. What always seems to
get lost in this comparison is the practical problem of implementing
systems of either kind. In the compensating system, perfect compensation
can be obtained with
1. Perfect linear infinite-bandwidth sensing of the state of the
disturbing variable with no changes in calibration.
2. A compensator that computes the exact negative of the link through
which the disturbing variable affects the controlled variable, including
all nonlinearities and changes with time (including changes in
environmental properties and spatial relationships).
3. An effector that can turn the computed output into exactly the right
physical effect on the controlled variable to cancel the effect of the
disturbing variable, with computed compensation for changes in effector
response, drift in DC response, calibration, and physical relationship
of the effector to the controlled variable.
4. A complete absence of disturbing variables affecting the controlled
quantity that can't be sensed.
When you think about actually building a compensating system, you come
up against all the reasons for which negative feedback control systems
were invented in the first place. A negative feedback control system
does not need to sense the disturbing variable or variables AT ALL, much
less with infinite precision. It does not need to do any computation,
precise or approximate, of the effect of the disturbing variable on the
controlled variable. And it does not need an actuator that will
precisely convert the compensation into an accurate physical effect on
the controlled variable; in fact the actuator properties can change over
an order of magnitude with no important effect on the accuracy of
control (try that with a compensating system!). Even the power supply
can vary its voltage over a factor of 10 with no important consequences.
The only critical requirement is that the control system be able to
_sense_ the state of the controlled variable accurately. The accuracy of
control depends directly on the accuracy of the sensor. Real negative
feedback control systems can control as precisely as our best sensors
can sense the controlled variable -- using components in the rest of the
system with +/- 20% tolerances or worse.
So in practical terms, a real control system will beat the performance
of any compensating system by a large margin, and do so using cheaper
components. The theoretical perfection of the compensating system is a
mathematical fiction, and anyone who believes such a system can
outperform a -- pardon me -- real control system is only betraying a
lack of practical experience.
The same holds for the models we use in tracking experiments.
The model can be adjusted to track far better than the person
does.
In some circumstances, but not in others. A long standing example
of this is the approximately zero response times that subjects soon
start to demonstrate when tracking a repeating (e.g. square wave)
pattern.
The idea that subjects soon produce approximately zero response time is
a myth. Normally, with a square-wave disturbance of constant frequency
and constant amplitude, subjects will correct the error after each
transition as rapidly as possible, meaning a time-delay of about 0.15 to
0.25 seconds, plus whatever settling time is involved. If urged to try
to eliminate the transient error by anticipating the transitions,
subjects will be able to do so, but their tracking errors will become
worse, since they are not able to maintain a constant frequency of
output transitions that is exactly timed. They will begin to show timing
errors such that their corrections come both before and after the time
of the actual transition; their amplitude errors will also become
larger.
An automatic negative feedback control system could be built which would
perform far better than human subjects could do, using a phase-locked
loop containing a variable-frequency square-wave oscillator as an output
function. I am surprised that you, an experienced control-system
engineer, did not remember that
I have yet to see a PCT-model that can do this well enough,
compared to a human subject..
I suspect that you have yet to see a human subject doing it as well as
you imagine it would.
To me, this example indicates that we ought to find out how this
type of prediction might come about.
But that's an old problem, solved long ago.
We adjust the model's parameters to make its simulated handle
movements match those of the real person as closely as possible;
then we use the resulting parameter values as measures of the
person's control properties.
That is exactly the problem. If you presuppose a PID-like organi-
zation in the human arm when the real organization is different,
you can adjust parameters until doomsday without finding a good
enough match.
But that is exactly what we do assume, and we get a match that is very
close. As I have mentioned several times, when you compute the RMS
difference between the simulated handle movements and the real ones,
even with a new pattern of smoothed random disturbances, you get an
error of about 3 to 5 percent of the range of the handle movements. Most
of the remaining error is high-frequency noise showing essaentially no
correlation (0.2 or less) with either the disturbance or the handle
movements.
Of course what you say is true: if the human arm control systems were
not organized as we assume, we could adjust parameters until doomsday
without getting good correlations. That is very reassuring; it suggests
that we have the right model. It we hadn't obtained good correlations,
we would not have used that form of model. We would have found a form
that worked. We didn't pick the PID model first and then look around for
behavior to fit it. I started with observations of tracking behavior
and, in fact, a _wrong_ model that didn't work very well (a straight
proportional model). It took me an embarrassingly long time to find the
simple model that we now use.
The amplifier in the higher system [in your diagram] assures
that only a small velocity error is sufficient to change the
reference force over its entire possible range. So the velocity
error is never allowed to get very large.
A small error in your diagram equals a small "nerve current". Due
to the discrete nature of "nerve currents" (quantized as action
potentials), one would expect that small currents result in rather
noisy control signals when amplified. Given your example of an
error of 1 in 100, the error signal would be around 3 action
potentials per second, assuming an upper rate of 300 a.p. per
second. This most likely would give Parkinson-like oscillations.
Does this correspond to the reality of a human arm?
You really haven't followed our postings or read our literature very
closely, have you? An arm control system that we draw in a simple way
actually, physiologically, consists of many control loops, many
hundreds, operating in parallel. What we show as "a" signal is really a
bundle of redundant signals, all derived from sensors of the same type
and near the same place, and all reaching a collection of motor neurons
in a small volume of the spinal cord. The outputs of these motor
neurons, the error signals, enter various fibers of the same muscle,
each activating several fibers but different fibers from those activated
by different motor neurons. I mentioned these considerations on pp. 93-
94 of B:CP. Twenty-three years ago.
So while your numbers are correct for a single nerve pathway, they are
too low by a factor of between 10 and 100 for the total composite
control system. In other words, the effective number of action
potentials per second in a muscle control system ranges not from 3 to
300 per second, but more like from 30 to 3000 per second and sometimes
as high as 300 to 30,000 per second. The dynamic range of the arm
control systems is at least 1000:1.
How do you think the orthogonalization comes about? Is it gene-
tically given or learned? Could it be related to the early death of
so many brain cells in the months immediately before and after
birth?
Yes, I think that is a reasonable guess. It is certainly learned, except
perhaps for the lowest-level control systems where the natural axes of
control are given by muscle placement and the kinematics of bones. I
prefer to take the conservative view that _everything_ must be assumed
to be learned until we have evidence to the contrary.
Have there been discussions about the importance of keeping
perceptions limited? I doubt that a subject who is happily tracking
away and hears the exclamation "fire!" coming from somewhere will
control his joy- stick well from then on. Even "coffee!" would do
it for me ;-).
I think that consideration arises from common sense.
If there are unpredictable variations in the reference level, no
variable will prove to be perfectly stabilized by the person's
actions. However, if we could roughly identify a number of such
controlled variables, we might be able to deduce the nature of a
higher-level system that is using them to control some higher-
level variable.
Watch out. Who sets the reference level? The subject, don't we
agree on that? Doesn't that imply that there are no unpredictable
variations in the reference level?
I don't consider predictability to be a property of a variable; it
reflects the ability of the observer to predict. An "unpredictable
variation in the reference level" is one that the observer-analyst can't
predict. Inside the behaving system there's no "prediction" of reference
signals; the reference signals are varied as the means of achieving
higher-level goals.
Moreover, it is difficult to learn if you are also controlling
well. In fact, if you are controlling perfectly, you cannot learn
at all (says theory) -- there is also no reason to learn (says
practice). The larger the mismatch between world and world-model,
the more rapidly learning takes place -- if learning CAN take
place, i.e. presupposed a model with a sufficient number of degrees
of freedom.
This is a self-cancelling paragraph. Naturally, if you're not
controlling well, you need to learn. Not controlling well is signified
by a large amount of error signal. The error signal can be used as the
basis for varying system parameters, an auxiliary control system varying
the parameters until the error signal is minimized. On the other hand,
if you are controlling well and the error signal is small, any variation
in parameters would be most likely to make control worse. So you stop
"learning" when error is small. What else?
If you want to learn (say about higher-level systems), you might
want to consider doing this is a non-control context. Maybe a
stimulus-response type of context?
Why on earth would I want to do that?
In the hierarchical model, I think the whole system comes to a
state where error is more or less evenly distributed among all
the active systems. This seems to be about what you are saying.
Depends. According to the optimal control point of view, the system
stabilizes in a state where the optimality criterium (weighted sum
of squares of errors) is minimized. Differences with your
assumptions are: The errors are weighted; some low- level errors
contribute more to overall-error than others.
Those are not differences, they are details I didn't think needed
mentioning. The big difference is that in organisms, the criteria of
optimality are set inside the organism, while in the engineered system
they are set to suit the external user of the system.
When there would be no noise in the system, all errors would be
zero if there are no conflicts.
This isn't true in a negative feedback control system. If there are
systematic disturbances, there must be error to drive the output that
systematically opposes the disturbances. The error signal itself can be
small in comparison with the reference signal, but it must be amplified
so the result is sufficient output to handle the range of disturbances.
If there are conflicts, conflicting goals are realized equally well
(measured in terms of the error criterium).
Should I remind you that different weights may be given to different
goals, or is that a detail that don't need mentioning?
I think it's rather futile to speak of "optimal" or "best"
control -- the whole system simply controls all the variables of
concern to it as well as it can.
"As well as it can" equals "optimal", isn't it? The problem is,
however, that you cannot measure "good", "better", "best" if you do
not have a yardstick. Hence the ubiquitous use of an optimality
criterium in modern control theory.
You will find optimality criteria, as you call them, even in old-
fashioned perceptual control theory. Look up the 1960 articles by
myself, Bob Clark, and Bob McFarland (in Living Control Systems I).
There you will find a "negentropy" (now "reorganizing") system defined,
in which the criterion for starting reorganization is departure of a set
of critical variables from their intrinsic reference levels. The
"yardstick" of proper functioning in the living organism was assumed to
be a set of inherited specifications for the internal
physiological/biochemical state of the system. We didn't specify a mode
of operation for the reorganizing process because we didn't know of any
then, beyond the obvious one of trial-and-error change of system
parameters.
... If the brain were born knowing everything it might be called
upon to control, perhaps it could do somewhat better.
Could do WHAT better?
I can think of a few things I'd like to be able to do better, can't you?
Which goals would be realized better, and at the expense of which
other goals? There are many more bodily limitations, besides our
brains: due to a lack of actuators, it is generally only possible
to fully realize one (conscious) goal at the same time. More than
one goal at a time, and you're bound to find yourself in conflict
(another Zen lesson: realize your goals in succession, not
simultaneously). This suggests that at the highest level there can
only be one goal.
We commonly are working on achieving many goals simultaneously. I don't
think you have a broad enough picture of goal-seeking behavior,
confining the meaning only to higher levels of cognitive functioning. I
commonly maintain the goal of an upright position while I am mowing the
lawn, and I'm simultaneously getting some exercise and pleasing my wife,
not to mention satisfying my own (rather too-easily satisfied) goals for
a neat appearance of my lawn. At the highest levels, I want to be a good
husband, a good scientist, a good writer, a good father, a good ping-
pong player (alas), and a sex object. There is no reason I can't choose
behaviors that satisfy more than one of these goals at a time. As you
say, there is a risk of conflict, but with learning we can eliminate
that by orthogonalizing the dimensions in which we control. I think your
Zen master hasn't been paying attention.
Part of the environment is the other people in it. When _they_
are reorganizing, it becomes pretty difficult to predict how
that part of the environment should be modeled.
Unless you have a good model of HOW people reorganize, based on
extensive observation and experience.
This doesn't help much with reorganization that works as I have
proposed. If reorganization involves random trial and error hill-
climbing, nobody, not even the person doing the reorganizing, can
predict the final form of behavior that will be reached. And since the
criteria for stopping reorganization lie within the individual, I can't
see what anyone else's criteria are, so I can't even predict when
they'll stop reorganizing.
Note that the term "unmodelled dynamics" implies that there are
things "out there" that we cannot, will not, or do not model and
that we therefore assume to be "random". Not that part of what
happens in the world IS random (it may or may not be, we just don't
know), but that our knowledge is limited.
Oh, yes, I agree totally. There is no such thing as a random process;
there is only limited understanding of processes. But this does not say
that if we just try real hard, we will suddenly become able to predict
the weather a week ahead, and thus predict when someone will open an
umbrella. There will always be vastly more going on in the world than we
can predict; it's a good thing that living control systems don't depend
on predicting disturbances.
The problem is not that we cannot design algorithmic methods to
search for the optimal controller; that problem is akin to the
mechanical methods that exist to generate all theorems of first
order logic or all possible successions in a chess game.
What is the optimal controller if the input function is logarithmic and
the comparator is a multiplier? I doubt whether anyone knows how to
solve the control equations for such a system, and thus that anyone
knows whether that system would behave better or worse than any other
kind. When you say "controller" you're thinking of a particular design
of controller, not all possible designs. So the "optimum" of which you
speak is strictly local.
The problem is that WE have to specify what is optimal: what is the
goal (or what are the goals and what are their rela- tive
importances); what are the constraints. It is all too easy to
forget some goal or constraint.
Living organisms, fortunately, don't have that problem. The criteria of
optimal performance, all criteria that will ever have any effect, are
built into them. There's no chance of leaving any of them out or getting
any of them wrong. The problem is, of course, that we don't have any
conscious knowledge of what they are, except perhaps in a few sketchy
ways. We can be pretty sure, however, that they bear no resemblance to
the optimality criteria used in modern control theory. What's good for
the user is not necessarily good for the control system.
--------------------------------------------
What I found was that with an external disturbance of xt, the
values of the world-model parameters became very different from
those of the external system: _a_ became about 2*at, and _b_
about 1/2 bt.
With an external disturbance of xt, the model converges to para-
meter values that accomodate the disturbance as well as possible.
Actually, the world-model parameters change to minimize the effect of
the disturbance on the controlled variable as far as possible. The
difference between the world-model's parameters and those of the
external system is just what is needed to create a loop gain
considerably larger than 1, thus producing opposition to the disturbance
even though the disturbance is not modeled.
It's interesting that the EKF can do this; I'm sure this is an
unexpected side-effect.
--------------------------------
A compensator with incorrect parameters cannot control well, if at
all. That is why I advocate ADAPTATION of the parameters of the
compensator.
When you introduce adaptation, you have to add a sensor that detects the
actual state of the controlled variable (which does not exist in a pure
compensator). That would give an improvement over the compensation
model. However, since your adaptation model requires information about
disturbances, it still suffers the same limitation as the straight
compensation model: precise control requires precise sensing, precise
computations, and precise outputs. The negative feedback control system
requires only precise sensing.
---------------------------------
The reason your world-model behaves so perfectly is that the way
you calculate u automatically takes into account any variations
in the parameters; in fact the effects of the parameters cancel
out. If the world-model were any more complex, so you couldn't
find the value of u just by solving an algebraic equation, you'd
have to have an automatic inverse-finder as well as the system
for adjusting the forward parameters in the model.
I have no idea why you keep coming back to an "inverse-finder" time
after time. My demo did not incorporate one, and neither would a
more complex model necessarily need one.
Your world model, with provision for disturbances, has the form
x = ax + bu + c + disturbance
In other words, x = f(a,x,b,u,c,disturbance)
To find u, you substitute xopt for x and solve this equation for u,
which means taking the inverse of the world model with respect to u.
This inverse (which you don't seem to think your model needs) is used in
the expression for u:
u = (xopt - ax - c - disturbance)/b
Thus x is necessarily made equal to xopt; everything else cancels. If
your Kalmnan filter changes a parameter in the world-model, it must make
the same change where that parameter occurs in the computation of u.
This assures that x will equal xopt even while the parameters change.
I think that most control engineers would find this an extremely odd way
of finding the output function required for a control system.
Let me explain to you when an "inverse-finder" WOULD be required.
That would be in the situation where xopt (t) remains unspecified
in the formula above, because an optimal TRAJECTORY is not re-
quired, but only an optimal END POINT.
You've lost me here. If xopt suddenly changes to a new value, doesn't
the present system make x approach this new end-point? I get the
impression from your talk about trajectories that you're REALLY doing
things the hard way here.
-----------------------------------------
Assume that u has been zero for some time, and (therefore) x as
well.
Now set u=1 henceforth and see how x changes:
x (t+1) = 0.9 * x (t) + 1
x (0) = 0
x (1) = 1
x (2) = 1.9
x (3) = 2.7
Interesting. I hadn't noticed that. So if you apply a step-function as
xopt, x will follow it in one iteration, but the real system will
approach the value of xopt exponentially? This would be an interesting
relationship between the world-model and the real system.
--------------------------------------------
This surprising finding [world-model <> world] needs to be
explained, doesn't it? When there is a disturbance, the world-
model constants come to values different from those of the real
system (I have observe this, too).
Basically there are two causes: 1) incorrect initial assumptions
and 2) learning in the face of noise is a stochastic process whose
outcome is guaranteed to be correct only ON AVERAGE. If you choose
initial assumptions with very large variances (pxx, paa, etc) you
eliminate the first cause, but at the expense of a longer time to
convergence.
I think you answered this off the top of your head instead of actually
working it out. The initial assumptions and the stochastic effects can't
explain this result, because the world-model comes to the SAME final
(wrong) values EVERY TIME.
If you run a large number of trials with different noise sequences,
you will find that the converged model-parameters have some
probability distribution around the world-parameter value.
No. That is not what happens. Try it your self and see.
---------------------------------------------------------------------
RE: Anecdote about adaptive blood pressure control system
I was astonished by this tale. I think you gave up far too soon on a
non-adaptive controller, or at least on finding a method of adaptation
that would work. I'm glad you report this as an _initial_ set of trials,
although it seems to me that using a live patient before you knew that
the system would work was inexcusable. When I worked in medical physics,
I NEVER allowed a doctor to use a system with live patients until I was
sure, through extensive testing, that it would function properly.
... when the controller started to infuse SNP, the blood pressure
went rapidly and significantly UP. So the model decided that the b-
parameter was positive, i.e. that SNP _raises_ the blood pressure
rather than lowering it. So the controller decreased the SNP flow
rate. Then the pressure went down. But too far, below the setpoint.
So more SNP was infused. The blood now pressure went up again. By
now, the model was pretty certain that SNP is a drug that RAISES
the blood pressure.
I can't think of a better illustration of the inadequacies of this kind
of control model. A properly stabilized negative feedback control system
would NEVER have created this kind of dangerous problem.
Just out of curiosity, what other kinds of control systems have you
designed and operated? I am interested in how this approach works in
other situations.
I have had only one somewhat similar problem, controlling the movements
of a diffraction-grating ruling engine using a laser interferometer. The
basic control loop involved modulation of one of the interferometer
mirror positions by a little less than 1/4 wavelength, and using a
synchronous detector to derive a low-frequency error signal that drove
the error-correction mechanisms (a motor for coarse errors, a
piezoelectrical crystal stack for fine adjustment).
The adaptation problem arose because the fringe intensity varied with
carriage position over a factor of about one million, changing the
sensitivity of the control loop by the same amount. My solution was to
add another control system that varied the gain of the amplifiers (at
the modulation frequency only) to keep the signal from the photocell at
a constant amplitude. This also, as a side-effect, kept the slope of the
signal with respect to the sine-wave modulation constant. The engine
thus performed with equal control accuracy over the whole range of
carriage positions, with interferometer arms varying from equality in
length to a length-difference of about 10 inches.
This was all done with perfectly conventional negative feedback control
systems, the only slightly unconventional aspect being that one of the
control systems acted by varying the amplification of the input function
of the other control system. I didn't even think of this as an
"adaptive" control system at the time. It was just a way of
counteracting the various kinds of disturbances that could occur. If
anything, I would have called it an AGC -- automatic gain control --
device, like the ones I had learned about at the age of 17 in the Navy,
learning to troubleshoot radios 50 years ago.
Well, you get the picture. A sudden noise burst at the wrong moment
may be explained as significant. And when it repeats, as very
significant. This leads to superstition. There are solutions to the
problem of building up incorrect knowledge, of course, but these
require more sensors, a more complex world-model, and/or built-in
limits on how far the values of the parameters can vary. We chose a
combination of the latter two approaches.
Yes, I get the picture. When you use an overcomplicated model based
largely on the wrong conception of control, you get poor results and
have to make the system even more complicated to get it to work at all.
-----------------------------------------------------------------------
Best,
Bill P.