[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.

But that doesn't happen; instead, you reduce your pull and let your friend

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.

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