[From Bill Powers (950328.0815 MST)]
RE: irrelevant side effects
Bruce Abbott (950327.1950 EST)--
Rick Marken (950327.1430) --
I think that the original intent of the term "irrelevant side effects"
has become detached from context and floated off across the landscape.
When I first used the term, if I remember correctly, I was saying that
the only effect of a control system's behavioral outputs that is
relevant to the control process generating them is the effect on the
input quantity being perceived. The output actions might have many other
effects on the world, but to the control system in question they are
necessarily irrelevant, as the control system can't sense them. These
other effects are side-effects of the control behavior and play no part
in the way that one control system works.
This seems plain enough to me: some effects of control actions are part
of the loop in question, and some aren't. However, as soon as this came
up on the net, the lawyers got into the act. The first lawyer said, I
believe, that if your control system for steering a car takes the car
over a cliff (or some such example), that was certainly "relevant" to
the control system at the wheel. Of course that entails viewing the
entire person driving the car as a single control system, and changes
the point of view from that of the one simple steering control system to
that of an omniscient external observer. Naturally, for the external
observer or for other control systems inside the driver, the fact that
the car goes over the cliff is very relevant because it affects
variables under control by those systems: it is relevant to other
control systems. However, it is not relevant to the steering control
system, which continues to work to make the picture in the windshield
match the picture given to it as a reference signal until that control
system abruptly ceases to function at all.
By getting onto the sidetrack of _objective_ relevance, the lawyers
managed to miss the main point, which is that a PCT-naive external
observer has no way to decide whether a given environmental effect of a
behavioral action is intended. The hostess offers you a plate with
macaroons and pieces of chocolate cake on it, and when you choose the
chocolate cake because that's what you feel like eating, she says
"What's wrong with my macaroons?" The apparent rejection of the
macaroons, of course, is a side-effect on the perceptions of an external
observer, and is irrelevant to the control system controlling for eating
chocolate cake. On the other hand, if you dislike both chocolate cake
and macaroons equally but feel you must take one or the other to satisfy
your desire to please the hostess, you may choose a small macaroon just
to avoid hurting the hostess's feelings, only to find that the next time
you visit you are offered only macaroons. In that case, choosing the
macaroons OR the chocolate cake was an irrelevant side-effect of
resolving a conflict concerning quite different variables. The hostess
in both cases mistook a side-effect of your action for the intended
effect.
The point of all this was to show why it is necessary to apply the Test
before you can judge what a person is really "doing." Simply "observing
behavior" is insufficient, because the behavior you observe may be quite
irrelevant to the behavior that is actually going on -- that is, to the
effects of motor behavior that the other system is actually trying to
control.
Now we seem to have still another version of "relevant" creeping into
the discussion. Rick seems to be saying that a discriminative stimulus
is an irrelevant side-effect. In fact, a discriminative stimulus in its
role as a disturbance of a controlled variable is a HIGHLY relevant
variable (to the controlling system), because the actions of the system
are specifically organized to counteract the effect of the
discriminative stimulus on the perception of the controlled variable.
And of course if the SD is actually an input to a logical control system
(a possible model but by no means the only possible model) it is even
more relevant to that control system: it is part of the definition of
the controlled variable, whatever symbol we use to refer to it -- D or
SD or P.
···
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Bruce Abbott (950327.1750 EST) --
Period k sd k hi k/lo k rms sd rms
1200 0.13 0.04 1.6 3.5 0.4
600 0.12 0.02 1.4 3.9 0.3
300 0.12 0.01 1.6 6.9 0.8
150 0.16 0.02 1.3 12.4 1.0
75 0.16 0.08 3.9 33.4 7.2
What you are seeing here, somewhat dimly, is (as you guessed) called the
"frequency response" of the control system. It would be even clearer if
you picked one value of k such as 0.14 and ran the model with all the
various frequencies of disturbance.
Every control system has a maximum frequency at which it can deliver
good performance. The lowest-frequency behavior, extrapolated to zero
frequency, is the steady-state behavior (behavior in the presence of a
constant disturbance). As the frequency of the disturbance begins to
rise, at first the integrating control system is able to keep the error
very close to the steady-state error, zero. But as the frequency
continues to rise, the output begins to fall and the error begins to
increase. The phase of the output action starts to lag the phase of the
sinusoidal disturbance. The "bandwidth" of the control system is
customarily taken to be the frequency at which the output has dropped to
0.707 (square root of two) of the value of the zero-frequency output for
the same amplitude of disturbance.
In the model we're using, the output function is a pure integrator. A
sine-wave input to an integrator yields a cosine-wave output, a sine
wave that lags by 90 degrees in phase. If you keep the amplitude of the
input sine-wave constant and vary its frequency, you will find that the
output sine-wave amplitude is inversely related to the input sine-wave
amplitude. As the input _frequency_ goes toward zero, the output
_amplitude_ goes toward infinity. This is why it is often said that the
loop gain of a control system with an integrating output is infinite at
zero frequency.
With an integrating output, the loop gain of a control system depends on
the frequency of the disturbance. In fact, the loop gain can be shown to
decrease at "3 db per octave", meaning simply that when the disturbance
frequency doubles, the loop gain halves.
The output response of a control system to a disturbance is ideally
equal and opposite to the disturbance. In terms of loop gain G, the
output is actually G/(1+G) of the disturbance -- not quite equal, but
near enough when G is large, say 100 or 1000. As G begins to fall, in
our case due to the falloff of the output amplitude response with
increasing frequency, the loop gain falls off in the same way, but
because of the way it appears in the factor G/(1+G) there is little
effect on the output amplitude until the gain has become very low. In
fact, the effective output amplitude will reach 0.707 when G has fallen
to about 2. At this point the output phase lags the disturbance phase by
45 degrees (even though the lag in the output function itself is always
90 degrees in a pure integrator -- the feedback changes the phase
relationships).
The foregoing is very rough because I'm not taking phase lags
numerically into account, but the general idea is right. The negative
feedback, which is responsible for that factor G/(1+G), keeps the output
matching the disturbance while the output gain falls from a very high
value at low frequencies to nearly zero at high frequencies; then the
quality of the control begins to drop rapidly. The frequency we call
"the bandwidth" is arbitrary; it's actually defined in a mathematically
convenient way more than a practical way. If you define "good control"
as cancelling 99% of the error, then you would say that the bandwidth
limit is at a lower frequency than if you called only 90% or 70%
cancellation "good" control.
When you use pure sine-waves as disturbances, you're doing a "frequency-
domain" analysis. When you use step-disturbances, you're doing a "time-
domain" analysis. In a time-domain analysis with a step disturbance, the
disturbance begins at zero and jumps instantly to some constant value
which is maintained. The output starts integrating at a fast rate,
because the initial error is the full size of the step-disturbance, and
then integrates at a lower and lower rate as the error signal decreases.
The result is an output that rises along a 1 - exp(-kt) curve to an
asymptote which is the same as the response to a "zero-frequency"
(constant) sine-wave disturbance.
The time-domain analysis is related to the frequency domain analysis
through a Fourier transformation of the step-disturbance. The step-
disturbance is equal to a set of superimposed sine-wave disturbances
including all odd harmonics of frequencies from 0.5/(duration of the
step) to infinity. The magnitude of the components decreases as the
frequency increases. The step itself, if it has a finite rise-time as
all real steps do, defines the upper frequency limit. So if the control
system is linear, we are seeing the superimposed cosine-wave responses
to every odd harmonic of the lowest frequency sine-wave into which the
disturbance wave-form can be mathematically decomposed.
The point of all this seat-of-the-pants engineering talk is to make
clear that the behavior of a control system _even with constant
parameters_ will vary with the frequency or speed of the disturbance
waveform. For each specific design of the control system we will expect
to see a characteristic change in the output and error with changes in
the frequency content of the disturbing waveform.
In your table above, in which I presume you re-matched the model for
each new frequency of disturbance, we do not have constant parameters,
because the parameter k was re-computed for each run. What we really
want is a model with one _fixed_ parameter that will behave like the
person over the entire frequency range. This is a much more difficult
requirement than asking only that one value of k be found for one
condition. Actually, you found quite a constant value of k, considering
the problem: it ranged only from 0.12 to 0.16 over a 16:1 range of
frequency. That's only +/- 17% of the mean value of k, 0.14. That's
comparable to the scatter in determinations of k.
Of course we would like to get closer to the right model. The question
is, what is the difference in behavior between a model with a _constant_
value of k and the real behavior, evaluated at all frequencies? If we
can find any systematic dependence of the difference on frequency, we
can then look for ways to change the model to reduce the difference over
the frequency range. Also, it would be wise to do the same thing over an
_amplitude_ range, because there could be nonlinear effects, and those
nonlinear effects might turn out to account for the frequency effects,
too (the error grows with frequency, and perhaps if it grows nonlinearly
it would account for the difference in measured k).
Ideally, we would like for the same model to match behavior perfectly
for all disturbances independently of both frequency content and
amplitude of the disturbance (within the physical limits of the system).
Practically, we have to decide what degree of fit is good enough to go
on with. Accounting for 95% of the variance can be done with a very
simple model. The next 2% can perhaps be attained by putting a
perceptual delay in the model. And the next 2% might take us 20 years to
achieve, still leaving us with 1% unaccounted for.
I'm wondering if performance under the sine-wave disturbance might
be analogous to a "forced oscillation" situation. Picture a spring
whose top end is being moved vertically in a sine-wave motion
across a given distance. Attached to the bottom end of the spring
is a mass; the spring and mass constitute an oscillating system
with its own natural frequency, one which may differ from the
driving frequency.
Yes, it is analogous. The differential equations that describe a simple
control system are identical to those that describe a mass on a spring
with damping. Of course the physical situation is very different, but
the same form applies. In fact, in human kinesthetic control systems,
there is a slight resonance at about 2.5 Hz, which would correspond to
the "natural frequency" of the mass on a spring with rather strong
damping (because the resonance is not very great). The basic equations
describe a damped harmonic system with the disturbance being the driving
function. The exact behavior depends on the waveform of the driving
function, which is arbitrary (a sine-wave is just one possibility).
-----------------------
Accounting for the mass of the arm implies including lower levels of
control in the model. You need to add a two-level system. The output of
the lowest level applies a force to a mass. The integral of force/mass
gives you the velocity of the mass, which is the sensory input to the
lowest-level system. This system can be a simple proportional controller
(output proportional to error). Then the velocity is integrated again to
give the position of the mass (i.e., the hand and cursor), and position
is fed back to the second-level system. The second-level system, too,
can be a simple proportional controller. By adjusting the output gains
of the two systems you can adjust damping (velocity-control system) and
speed of position correction (position control system). The reference
input to the position control system is the output of the integrator of
the model we are presently using.
The best way to adjust this model is to turn off the level-three system
and play with the gains of the other two systems, using a square-wave
driving function as the reference signal of the second-level system. You
might find my "setparam" unit helpful here, although it's a pain to set
up, because it lets you adjust parameters while the system is running
continuously. Use the "loadfrm.inc" and "loadpar.inc" files, which I
think you have, as templates. When you have the lowest two levels
responding to changes in the reference signal the way you want, you can
turn on the third level and see what happens.
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Martin Taylor (950327 11:10)--
The word-category /bank/ is _associated_ with the concept-category
/place of money/, and with /a type of billiard shot/, and with
/land beside a river/ and with ... The word-category is not part
of those concept categories, for me, and as far as I can
introspect, the word-category does not participate in any thoughts
I have when dealing logically with those concept-categories.
I don't know what you mean by a "word-category." And it's sort of
unconvincing to think of a "place of money" category, a term that would
never have occurred to me. On the other hand, perhaps all this is an
excellent illustration of the fact that different people think of
categories differently, and that we're really not ready to model that
level.
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Chuck Tucker --
Got your data from the spring class. What do you suggest we do with it?
My feeling is that these statements call for some more questions, like
what is it that is disturbed when you say something is disturbing?
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Ken Hacker --
Sorry for the delay. Yes, I'd be happy to get that paper from you and
comment on it.
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Best to all,
Bill P.