Disturbances; one-way control; briefly, the meeting

[From Bill Powers (940801.0900 MDT)]

Chuck Tucker --

Delete the file called "adconfig" and start the program. This time,
answer the question about "slow or fast computer" with "f" for fast.
Pick the mouse as the pointing device, and select 200 as the
sensitivity. This should fix your problem. If it doesn't we have a
hardware compatibility problem and will have to look further.

···

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Bob Clark (940731.1500) --
RE: Disturbances

(Bill Leach:)

In this case I still disagree. The definition of the term itself
has nothing to do with its magnitude.

(Bob)

You seem to prefer the more general definition, that can include any
magnitude, even to the destruction of the system of interest!

There is a confusion here that arises from not distinguishing the cause
of a disturbance from the effect of a disturbance. In our canonical
diagram, the disturbance is shown as a separate variable, d, which
contributes via a function Fd(d) to the state of the controlled
quantity. This is REQUIRED if we are to be able to manipulate
disturbances independently of the control system, while the control
system itself maintains its integrity.

If you were to define the disturbance as a change in the controlled
quantity itself, this could not be thought of as an arbitrary change,
because doing so would make that change into an independent variable.
What we mean by "independent" variable is a variable we can set
arbitrarily to any value, or cause to change in any arbitrary pattern,
independently of what the affected variables are doing.

In a control system, the state of the controlled quantity is a
_dependent_ variable, determined jointly by the system's output and some
disturbing variable, the one we represent as d. It is possible to vary d
in any way the experimenter desires without breaking the feedback loop.
But it is not possible to establish an arbitrary pattern of changes in
the controlled variable itself while the loop is intact. To do so would
require the experimenter to overcome the effects that the system's own
output is having, preventing the output from altering the desired change
in the controlled variable. And then, of course, we would no longer have
a complete control loop.

Your comment that we treat disturbances normally as small or even
infinitesimal applies only to the calculated net effect of the
disturbance on the controlled variable, not to d itself. It applies, in
other words, to the difference between the way the controlled variable
behaves when the disturbance is zero and when it is nonzero. If you
stick to defining the disturbance as d, then clearly the disturbance is
not small or infinitesimal; it can be large enough to call for the
maximum output that the system can produce, without going outside the
normal range of operation of the system. When I speak of disturbances, I
ALWAYS mean the separate variable d, and I NEVER mean the observed
change in the controlled variable (unless I add qualifiers to say so
explicitly -- I hope). The observed change in the controlled variable is
always the sum of effects of a changing d and a changing output of the
system itself.

To forget these details is to fall into the same trap that has claimed
Martin Taylor: you forget that the output, too, is always having a
changing effect on the controlled variable. You can't (that is, you

shouldn't) ignore those changes and pretend that you can see the effects
of the disturbance (d) alone reflected in changes in the controlled
variable. Not, that is, without stepping into the bog of confusion.

I didn't understand the role of disturbances myself until I realized
that a disturbance could NOT be treated as if it had a known or
predetermined effect on the controlled variable. The consequence of
applying a disturbance to a controlled variable can't be known without
knowing the static and dynamic properties of the entire control loop, as
well as the behavior of the reference signal. If you treat the
controlled variable IN ANY WAY as an independent variable, you fall back
into S-R thinking and all the problems that come with it.

This may seem like quibbling over an unimportant point, but it is really
central to the whole issue of how control systems relate to their
environments -- whether in some way the environment determines the
actions of the control system, or whether the control system is
fundamentally autonomous. We must realize that disturbances are always
applied _by some means_, and that they do not have an exclusive or
separable effect on controlled variables.

My way of drawing the disturbance d with a function connecting it to the
controlled variable reminds us always to consider the means by which the
disturbance is applied, whether it be by adding a number invisibly
inside a computer or altering the position of one end of a rubber band
and thus stretching the rubber band to create an applied force. Without
knowing how much the position of the knot will change as we move our end
of the rubber band, we can't even predict the amount of force that will
be applied due to a change in position of our end of the rubber band.
When you consider the means of applying a disturbance, the idea of
predicting the effect of doing so (without considering the whole loop)
is revealed as untenable.

We are not talking about effects of disturbances but influences of
disturbances. The very same disturbance applied to the very same
controlled variable may have a large influence or an almost invisibly
small influence, depending on the gain in the control system's output
function. The effect that a change in a disturbance appears to have on a
controlled variable can change radically as the internal properties of
the control loop change.

If I seem to be saying the same thing six different ways, that's because
I'm running through all the considerations that finally led me to my
present way of treating disturbances, the one way that I think is
consistent under all circumstances. Other ways of thinking about
disturbances are consistent with _some_ special situations, but I think
the way represented in the canonical diagram is the only one that
remains consistent under _all_ conditions.
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Bill Leach (940731.1241 EDT], replying to
Avery Andrews (940730.1750)
RE: one-way control (power steering example)

If you examine the component control loops then you see that there are
really two 'independent' loops. One has the ability to move the tires
to the left and the other has the ability to move the tires to the
right. When you turn the steering wheel to the right, the reference for
the portion of the hydralic system that can drive the tires to the left
has a zero reference (that is, any position is OK), but the portion of
the system that operates the portion of the hydralic cylinder that
moves the tires to the right has some non-zero reference.

This non-zero reference is causing the output portion of the control
system to admit additional hydralic fluid to the cylinder which
increase pressure, force, etc.

If you should set the hydralic control valve to its' center and keep it
there, then as far as the control system is concerned, reference is for
zero in both loops and any position of the tires is "OK".

Think about that, Bill. You're saying that if you center the steering
wheel, the front wheels will wobble left and right in an uncontrolled
way with every bump in the road. You know that isn't what happens. The
power steering works equally well with the steering wheel in any
position. What's the matter with this argument?

The key is what happens when you lose the hydraulic pressure to the
power steering mechanism. When that happens, the driver can still steer
the car, but a very much larger effort is required to turn the wheels.
So clearly the wheel is connected to the rack-and-pinion mechanism
through a torque spring that is rather stiff. This is like feed-forward,
in that the steering wheel mechanically affects the output even without
the control system operating.

The control system is designed to reduce the difference in angle between
the steering wheel and the position of the linkage after the power
steering unit. It senses the difference through a valve, the body of
which is moved by the steering-wheel end of the torque spring and the
gate of which is moved by the output end of the torque spring. The
position of the steering-wheel end could be considered the reference
signal, the position of the gate the perceptual signal, and the valve
action itself as the comparator. The hydraulic fluid pressure passed by
the valve is the error signal, and the piston that converts the pressure
to an aiding torque at the output end is the output function.

Turning the steering wheel to the right moves the body of the valve past
the gate in a direction that creates a positive error signal (output
hydraulic pressure), producing a force to the right at the output of the
power steering unit in the same direction that the wheel is turned. This
reduces the difference in angle between the input shaft and the output
shaft by making the output shaft turn more in the direction of the input
shaft movement. The reflected force, which is caused by the twist in the
torque spring connecting the input and output shafts, is thus reduced as
close to zero as will still permit getting some "road feel".

The reason that effort is necessary to turn the steering wheel of a
rolling car is that when the wheel is turned to the right and the front
wheels begin to cock to the right, the caster of the steering mechanism
(which raises the front end of the car) produces a counter-twist to the
left, against the direction of the steering wheel angle (if the car is
stationary, the twist of the rubber in the tires causes the restoring
force). From this we can deduce that with the steering wheel in _any_
position, any force tending to twist the front wheels to the left of the
position indicated by the steering wheel will produce an output from the
power steering device as a twisting force to the right. This will be
true whether the steering wheel is twisted to the right or centered.
Anything that causes the output shaft to be twisted less to the right
than the input shaft is twisted will open the valve and cause a
hydraulic torque to the right. A bump in the road or a crosswind will do
just fine.

When we consider two of these systems, one for rightward turns and the
other for leftward turns, we find the same situation on each side. With
the steering wheel in any position, the output shaft of the power
steering unit will be zero in either direction if the front wheels are
cocked in just the way that leads to zero difference between input and
output shaft angles. With the car on a straight level bump-free windless
road and the steering wheel centered, this will be the situation.

If the steering wheel moves to the left, OR a bump tends to cock the
front wheels to the right, the leftward-force comparator valve will open
and the steering unit will produce a force to the left. Ditto, the other
way, for a twist of the steering wheel to the right OR bumps that tend
to twist the output shaft to the left. Provided that on each side the
valve closes only when the input and output angles are exactly the same,
there will be no dead zone and all external forces to either side will
be resisted by the power steering unit, regardless of steering wheel
angle. Note that if you are steering to the left around a long curve and
there is just the right amount of crosswind from the right and leftward
bank of the road, you can negotiate the curve with zero steering force
applied to the steering wheel. The wheels will be cocked by the correct
amount to match the slight leftward turn of the steering wheel and to
match the circular track of the car's path, but there will be no
reflected force and both valves will be closed.

Now: the stretch and tendon reflexes.

In the stretch reflex we have a mechanical comparator. Gamma efferent
signals, which are the reference signals for this system, shorten the
muscles at the ends or poles of the muscle spindle organ, stretching the
spiral length-sensing nerve wrapped around the middle and producing a
signal proportional to the stretch (plus some first derivative, but
that's just for damping and we can ignore it). A contraction of the main
muscle (with a movable load) will shorten the main muscle and also the
muscle spindle attached by its ends in parallel to the main muscle. That
will counteract the effect of the contraction of the muscles at the ends
or poles of the spindle, in terms of the stretching of the length-
sensing spiral sensory nerve. The signal from the spiral sensory nerve
is like the output of the comparator and is an error signal, which
enters the spinal motor neuron in the excitatory sense.

So the analogy to power steering can be made exactly. The gamma efferent
reference signal is the steering wheel, and its effects on the length of
the poles of the muscle spindle are like the effects of the steering
wheel in moving the body of the valve. The contraction of the main
muscle is like the reflected effect of the front wheels on the position
of the valve gate. The error signal in the neuromotor system enters the
spinal motoneuron to cause a large physical torque at the joint, which
bends the arm and allows the main muscle to shorten. This is like the
hydraulic pressure (maintained by energy obtained from the car's engine)
which is converted by a piston to a mechanical force that moves the
front wheels.

We have also the same situation with respect to disturbances in opposite
directions. Any disturbance that makes the length of the muscle spindle
greater than the length set by the gamma efferent signal produces an
error signal that causes a macroscopic force tending to oppose the
effect of the disturbance. This happens on both sides of a balanced pair
of systems. If the forces from both systems decline proportionally to
zero just as the error signals go to zero, all disturbing effects will
be resisted regardless of the angle at the joint.

In the muscle system the force is nonlinearly related to muscle length,
but the tendon reflex strongly linearizes this response as far as the
connection between the length error signal and the resulting tendon
force is concerned. Especially with some muscle tone present (equal
signals on both sides), there is no dead zone. The effective output
sensitivity depends on the common-mode muscle-tone signal, however, so
when the length reference signals on both sides are both zero, the
reference muscle length is longer than the maximum possible muscle
length, and both comparators produce zero error signal. Then the arm can
be pushed around like a limp pendulum. This doesn't happen with the
power steering unit, which is roughly linear right down to zero.

In order for the arm to really go limp in the presence of large
disturbances, a zero reference signal for muscle contraction would have
to bis the spindle far outside the normal length range.

The best way to learn about these things is to fire up Simcon 4.5 and do
some simulations. That is left as an exercise for the student.

Oh, yes: "feedforward." I let that term drop when speaking of the effect
of the steering wheel when hydraulic pressure to the steering unit is
lost (as when the engine dies). Here the mechanical reference signal
feeds through the comparator to have some small effect on the steering.

In neural control systems having an excitatory reference signal and an
inhibitory perceptual signal, loss of feedback effects still leaves a
connection from the reference signal to the error signal via the
comparator. Raising the reference signal will cause a positive error
signal, even more so than with feedback working. So there is a sort of
fail-safe condition, in which loss of sensory information at low levels
does not prevent control actions by higher systems. However, these
control actions will start out at wildly too sensitive and control will
be poor. With learning, however, the higher systems can reduce their
output gain, and some semblance of control, at a slower speed, can be
relearned.

The analogy to the power steering is not exact, because reference and
error signals in neural systems can't by themselves produce large
physical outputs. Loss of power to the _output_ devices would prevent
the neural control systems from working.
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Cliff Joslyn (direct) -- I'll get back to you soon.
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The conference was excellent with many smart newcomers and a lot of
emphasis on applications. Even the experimenters and modelers, who were
kept pretty much in the background, were pleased. There were some
developments concerning funding which can't be laid out now, but look
most promising. I'll let others go into more detail about the subjects
covered at the meeting. It was right up there with our best conferences.
Dick Robertson confessed that he had been wondering when we're going to
run out of steam, but it clearly wasn't this year, our tenth annual
meeting.
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Best to all,

Bill P.