disturbances

[From Bill Powers (960917.0545 MDT)]

Chris Cherpas (960913.1839 PT)

I would have thought that any error coincident with a change in the
input signal would also be called a disturbance, whether the environment
be inside the boundaries of an intact organism or not, whereas an error
accompanying a change in the reference signal would not be called a
disturbance...

This term "disturbance" has caused almost as much trouble as the term
"control." The reason seems to be that some people consistently use it to
mean "perturbation" -- a change in a variable that is disturbed -- which is
not the PCT meaning of the term. A disturbance in PCT is one physical
variable that affects the state of another, and it can be measured quite
independently of any effect it may have on the other variable. This is why,
when confusion enters the discussion, I have taken to using the term
"disturbing variable" -- that is, a variable that disturbs something else,
or would do so if no other influences opposed its effect.

The most readily available example is the rubber band demo. Here the
controlled variable is the position of a knot relative to a spot. We can
ignore the spot for the time being. There are two variables on which the
position of the knot depends: (1) the position of the experimenter's end of
one rubber band, and (2) the position of the controller's end of the other
rubber band. If e is the position of the experimenter's end on a ruler
(visible only to us -- and you'll need a yardstick, not a foot ruler), and c
is the position of the controller's end on the same ruler, then (given equal
rubber bands) the position of the knot k on the ruler is

k = (e + c)/2.

The output or action of the controlling system is measured as c; the
magnitude of the disturbance is measured as e. This is an equilibrium
system; the knot is always at the point that represents an equilibrium
between the forces generated by the two rubber bands, and that is the point
given by the above equation. If you try this yourself, using a ruler and two
equal rubber bands, you'll see that the equation is always very nearly true
while there is any tension in the rubber bands. This is, of course, a
one-dimensional example.

Suppose the disturbance e has a value of 1 inch. What will its effect be on
the position of the knot, k? If you've been following me, you'll see that I
haven't given you enough information to compute the position of the knot.
The position of the knot depends equally on c, the position of the
controller's end, and I haven't said what that is. If I tell you that c is 9
inches, then k is 5 inches, and if I say that c is 19 inches, then k is 10
inches -- both with the same value of the disturbance, e = 1 inch.

So the actual effect of a given magnitude of disturbance on the position of
the knot depends on the magnitude of the controller's output. You must know
BOTH the value of the disturbance AND the value of the output to compute the
state of the knot. Notice that we're not talking about how the controller
works; we're just talking about physical properties of the environment.

Now suppose that the controller is basing the output on the position of the
knot relative to a spot on a piece of paper under it. The controlled
variable is what the controller perceives, which is the distance between the
knot's position k and the spot's position s, which we can measure on the
same ruler. The controlled perception p is then p = k - s (omitting the
factor that converts from physical units in the environment to neural-signal
units inside the perceiving person). Let us say that the desired distance
between knot and spot is 1 inch: the reference signal r = 1. This means k is
to be 1 inch to the right of the position s of the spot as measured on the
ruler (with numbers ascending to the right), since r is a positive number.

Suppose, as in the real case, that the controller is an integrating system:
this means that the output keeps changing until the error r - p comes
exactly to zero. If the distance is too great ( p > r), the output position
c will start going negative, toward smaller values of position, slowing down
as p approaches r and coming to rest when p = r. This approach to a final
value can occur rapidly or slowly, depending on the amplification of the
error signal as it enters the output integrator inside the controller. In a
real person it takes a few tenths of a second. The position of the knot
relative to the spot will change until k - s = 1 inch. At that time, k will
be 1 inch to the right of s, as perceived.

When the error has been corrected, where will the knot be on the scale? It
will be 1 inch to the right of the spot. If the spot is at 7 inches on the
ruler, the knot will be at 8 inches. Note that we haven't mentioned the
magnitude of the experimenter-set disturbance e or of the controller's
output, c. That's because they aren't determined by the reference condition:
whatever the disturbance e is, c will come to the value needed to make the
knot move to 8 inches on the ruler.

Suppose we know that the disturbance is 1 inch. Then, knowing or observing
that the knot will come to the position at 8 inches, we can calculate what
the output c will be: c will be 15 inches. We can calculate c, which
measures the behavior of the controller, knowing only what the reference
condition is and what the disturbance is, and of course knowing the
properties of the two rubber bands. This is what Rick Marken (and every
other PCTer) means by saying that the behavior of the controlling system
reveals only properties of the environment. If we know that the behaving
system is an accurate controller, we don't need to know anything else about
it except the reference condition it is seeking, which we can determine from
observation of the knot.

Speaking of a disturbance as a constant magnitude will sound strange to
anyone who is used to thinking of behavior in terms of "events." An event is
a pattern of change from one condition to another, not a constant situation.
Unfortunately, psychologists got into the habit of seeing behavior with
their event-level perceptual systems, and forgot that there are other ways
of perceiving the world.

We can easily convert our basic environmental equation, k = (e + c)/2, into
an equation that describes changes:

delta-k = (delta-e + delta-c)/2

This is not the best way to describe changes, because it omits any statement
about how fast the changes occur. A better way is to speak in terms of time
derivatives:

dk/dt = (de/dt + dc/dt)/2.

This says that the rate of change of the position of the knot is the average
of the rates of change of position of the experimenter's and controller's
ends of the rubber bands.

You'll notice that even in terms of rates of change, a change in the
disturbance does not determine changes in the position of the knot. The rate
of change of position of the knot always depends on BOTH the change in the
disturbance AND the change in the output of the controller that is occurring
at the same time.

The subject of delays in the controller is irrelevant here. The above
differential equation always holds exactly true. The present-time rate of
change of the controller's output may be due in part to a change of the
perception (or reference signal) at some time in the past, but the equation
deals only with what is happening right now. It is the PRESENT rate of
change of output (no matter when its cause happened) that is added to the
PRESENT rate of change of the disturbance to create the PRESENT rate of
change of the position of the knot.

Speaking only in terms of rates of change has the decided disadvantage that
it can't tell you the actual values of the variables. To predict the actual
measurements at a given time, you have to sum or integrate all the rates of
change over time, and even that won't do you any good if you can't say what
all the starting values were. In mathematical language, you have to
determine the constants of integration for each independent term. It is much
better to work with the magnitudes of the variables if you can, and in the
case of the rubber-band experiment we obviously can.

The equation k = (e + c)/2 tells us the relations among the variables at all
times in terms of their magnitudes, as long as changes don't occur so fast
that dynamic considerations are important.

If the reference condition requires the knot to be at 8 inches, and the
disturbance is at 1 inch, then we know that the controller's output will be
at 15 inches. If the disturbance now changes to 2 inches, we can immediately
calculate that the controller's output will change to 14 inches, because k
must always be equal to (e + c)/2: 8 = (2 + 14)/2 just as 8 = (1 + 15)/2.

Similarly, if the disturbance e is described by a constant plus a sine wave,

e = 1 + sin(at),

then we know that the output will be described by the sine-wave

c = 15 - sin(at).

Check it out: 8 = [1 + sin(at) + 15 - sin(at)]/2

If we make the frequency (set by _a_) too high, the dynamic characteristics
of the controller will start to make this prediction somewhat inaccurate,
but for slow changes the equation will tell us very nearly the truth.

Do you want a "response" to a "stimulus?" Let the disturbance start at zero
inches, change instantly to 2 inches for 1 second, then return instantly to
zero inches. The controller's output will start at 16 inches [8 = (0 +
16)/2], fall to 14 inches [8 = (2 + 14)/2] and then return to 16 inches
again. Of course because the controller is a real physical system and not an
ideal one, it can't change its output instantly and there's a little delay
in its action, so the output changes won't follow the input changes
perfectly during this event. But they will come as close to doing so as
possible. By making the disturbance into an "event" we create a response
"event." Underlying these "events", however, is the same continuous
relationship which holds true all through the transitions that make up the
event, give or take some lags due to the physical nature of the system.

Does a disturbance cause a error? Not necessarily. It might actually reduce
the error, since disturbances of both signs can exist. To understand this
it's necessary to think of controllers that aren't perfect integrators that
always reduce errors exactly to zero, given enough time. If the integration
is imperfect -- a "leaky integrator" -- then a steady disturbance will leave
the system with a small amount of error between perception and reference
signal, this being the small error needed to drive the output (through the
output amplifier) to balance out the disturbance. Now we can imagine a
second disturbance that acts in a direction that tends to reduce this small
residual error. Reducing the error will reduce the output, meaning that the
output changes in the direction opposed to the second disturbance, too! Even
the effect of an _aiding_ disturbance is opposed by the necessary change in
the output.

If that sounds complex, it is really simple. Suppose you've holding a heavy
suitcase up against the disturbance due to gravity. Your upward effort is
balancing out the pull of gravity. Now a friend comes along and lends a
hand, inserting his hand around yours and helping you to pull upward. That
is a disturbance being applied to the controlled variable, the position of
the suitcase. What will happen to the upward pull you're exerting? If it
stayed the same, the total upward pull would exceed the pull of gravity, and
the unbalance of forces would accelerate the mass of the suitcase upward.
make up the difference, and the suitcase stays where it is. So a helping
disturbance is opposed just as much as an opposing disturbance.

It's hard to overcome a lifetime of perceptual habits and word associations,
but the PCT meaning of "disturbance" requires doing just that. I hope this
rather wordy discussion will help.

Best,

Bill P.

ยทยทยท

--------------------------------------------------------------------------

[Hans Blom, 960918e]

(Bill Powers (960917.0545 MDT))

This term "disturbance" has caused almost as much trouble as the
term "control." The reason seems to be that some people consistently
use it to mean "perturbation" -- a change in a variable that is
disturbed -- which is not the PCT meaning of the term. A disturbance
in PCT is one physical variable that affects the state of another,
and it can be measured quite independently of any effect it may have
on the other variable. This is why, when confusion enters the
discussion, I have taken to using the term "disturbing variable" --
that is, a variable that disturbs something else, or would do so if
no other influences opposed its effect.

"A disturbance ... can be measured quite independently". That makes
it a perception if it _is_ measured and otherwise an unmeasured but
potentially measurable perception.

So could we say that a disturbance is, in principle, something that
_could be_ but _is not_ measured? If that is the case, the PCT and
MCT views suddenly coincide...

Apart from this, your statement "A disturbance is [a] physical
variable" invites the question of how we could access this "physical
variable" except through a perception. By hypothesizing/modeling it?
our model of it? Occam's razor seems to suggest that we'd better not
talk about the things that are hidden (one step or more) behind our
perceptions if we can equally well talk in terms of perceptions only.

One more thing. Those familiar with the feedforward control
literature know that in feedforward controllers one installs extra
sensors to measure what would otherwise be disturbances. If _all_
disturbances could be measured, one would not need closed loop
control anymore. Yet, even if lots of sensors are installed (which
has its problems too, because sensors can fail), feedback is usually
employed to remove the residual error.

Extra sensors allow more of the control to be feedforward (model
based). Because we cannot sense everything, feedback is indicated.
The question is now: why do humans have so many sensors?

Greetings,

Hans

[From Bill Powers (981130.1100 MST)]

Bruce Gregory (981130) --

You expressed surprise that "the disturbance" in the PCT diagram includes
all of the environment, and commented that you were just joking.

Actually, what we represent as a single disturbing quantity and a single
disturbance function is simply the sum of all influences on the controlled
quantity that do not come from the control system's own output. When you
add the effect of "the" disturbance to the effect of the system's actions
on the controlled variable, you have accounted for all possible influences
on that variable, and thus have determined its state completely -- or at
least have provided the means for doing so. Remember that in PCT, complex
quantities are broken down into sets of one-dimensional variables --
scalars -- each under independent control. What we lose in compactness of
representation we get back in the simplicity of each control system. All
disturbances affect a one-dimensional input quantity, which can only
increase or decrease; so we can speak rigorously of an equivalent single
net disturbance.

Best,

Bill P.

[From Bruce Gregory (981130.1500 EDT)]

Bill Powers (981130.1100 MST)

Bruce Gregory (981130) --

You expressed surprise that "the disturbance" in the PCT
diagram includes
all of the environment, and commented that you were just joking.

For the record, my reference was to Rick's comment that I might as well
be dead and my response which was to ask him if this was an observation
or a suggestion. I promise never to be this subtle again.

Actually, what we represent as a single disturbing quantity
and a single
disturbance function is simply the sum of all influences on
the controlled
quantity that do not come from the control system's own
output. When you
add the effect of "the" disturbance to the effect of the
system's actions
on the controlled variable, you have accounted for all
possible influences
on that variable, and thus have determined its state
completely -- or at
least have provided the means for doing so. Remember that in
PCT, complex
quantities are broken down into sets of one-dimensional variables --
scalars -- each under independent control. What we lose in
compactness of
representation we get back in the simplicity of each control
system. All
disturbances affect a one-dimensional input quantity, which can only
increase or decrease; so we can speak rigorously of an
equivalent single
net disturbance.

Thanks, this makes the point very clear.

Bruce Gregory