[From Bill Powers (950313.1540 MST)]
Bruce Abbott (950313.1325 EST) --
Not to gang up on you, but ...
An open-loop perceptual "translation" from color to selected target
would be very efficient of resources and speedier.
This is one of the myths I have been battling for years: that open-loop
reactions are more efficient, simpler, and faster than closed-loop
control. This is simply not true. The closed-loop system is always
simpler and faster, as well as being more reliable (less sensitive to
changes in system properties or external disturbances).
The only reason that the open-loop design seems simpler is that those
who promote it are looking only at a small part of what would be
required to make it work: namely, the initial cause and the final
effect. If you just look at the input, the stimulus, and the final
result, the movements that create the desired effect, it looks very
simple indeed. But if you had to simulate such a system, you would find
that between the stimulus and the final observable effect is a large
slithery slimy tangle of wiggling worms.
Consider reacting to a blip of light by moving a cursor from one target
to another. That sounds simple enough.
Without feedback what's necessary is to connect the signal from the blip
of light to a circuit that will generate the motor output signals needed
to move a hand on the end of a multiply-linked structure with at least
four main degrees of freedom, accelerating it and decelerating it not
just to reach a designated point in space, but to make a visual display
depending on the arm position come to a preselected state. And this has
to be accomplished from every possible starting position and velocity of
the arm-hand system and in the presence of all possible disturbing
forces -- without any knowledge of the current state of affairs during
the action.
What we're talking about here is a computational problem of staggering
magnitude.
Suppose we reduce the problem simply to generating a single motor output
signal on the basis of a single sensory input signal. At first glance,
it might seem simplest just to run the input through an amplifier to
produce an output. If you want 10 units of output per unit of input, the
amplifier has a gain of 10. What could be simpler or faster?
The answer is that by adding a negative feedback connection from output
to input, you can use the same kinds of components to get a FAR faster
response at the output of the system, still with an overall gain of 10.
The reason stems from the fact that any real amplifier will have a
finite rise-time. Suppose the amplifier has a rise-time of 1 second.
This means that presented with an input of 1 signal unit, the output
will rise in a negative exponential curve that reaches about 6.3 units
in 1 second, about 8.6 units in 2 sec, and about 9.5 units in 3 seconds.
The final level is 10 units, which is the gain of 10 that we wanted.
Let's start again, only this time we use an amplifier with a gain of 100
units. We connect a negative feedback path so that the feedback is 10
percent of the output. In approximate terms this will give us an overall
gain of 10 between input and output. The loop gain of this system is 10.
The amplifier, although it has higher gain, still has the same one-
second time constant.
Now if we apply one unit of input signal, the output will begin a
negative exponential rise not toward 10 units but toward 100 units. As a
result, the output will rise to about 6.3 units in less than 0.1 sec,
and to 9.5 units in less than 0.3 seconds (remember that the error
signal driving the output is dropping very rapidly as the input climbs,
because of the negative feedback). We have reduced the effective time
constant of the amplifier by a factor of 10, simply by adding the
feedback connection. In general, the closed-loop time constant is
approximately the open-loop time constant divided by the loop gain.
Total cost: a simple proportional feedback link.
Everybody says "Oh, but feedback has a one-quarter second time lag in
it, so you can't use strong feedback." That, too, is nonsense. The time-
delay in the first-order control systems is about 9 milliseconds. In the
second-order loop it is around 50 milliseconds. It's not until you get
to the midbrain, and specifically involve vision, that you get a delay
of about 200 milliseconds. And that delay is only for sudden jumps of
the eye. Furthermore, the visual control systems have high gain and are
almost ideally stable. The dreaded "quarter-second delay" is a figment
of the uninformed imagination, and even in systems complex enough to
make such delays visible, stabilizing the system and maintaining tight
control is no trick at all. Provided you didn't get the feedback theory
you're using from reading what psychologists or cyberneticists have
written.
That takes care of "faster." What about "simpler?" Suppose you were
determined to build an open-loop system with a gain of 10 and a rise-
time of 0.1 second using an amplifier with whatever gain you need but
with an irreducible time constant of 1 second.
It could be done. To do it you would need to add a filter that responds
to a one-unit input step by generating an output that rises to 1 unit
instantly, then decays back to 0.1 unit. The output of this filter would
enter an amplifier with a gain of 100 and a 1-second time constant. The
overall response could be adjusted, by adjusting the time-constant of
the filter, so that it would have a net gain of ten and a time constant
of 0.1 second. Same result as we obtained with the feedback connection.
But look what we have to add: not just a simple proportional feedback
link, but a dynamic filter with a specific mix of proportional and
first-derivative response. Such a filter would actually add some time-
delay of its own, as all physical systems involve lags, so to match the
performance of the closed-loop system we would have to jack the
amplifier gain up and make the filter time constant shorter. The open-
loop system is clearly more complex than the closed-loop system, given
that the same performance is reached.
As a bonus, the closed-loop system gives us relative immunity to changes
in the amplifier. If the gain of the amplifier changes from 100 to 20,
the overall gain of the closed-loop system will drop from 10 to 9.5. The
gain of the open-loop system will drop from 10 to 2.
Even further, the closed-loop system protects the output against
disturbances. If you add a 1-unit disturbance directly to the output of
the amplifier, the open-loop system's output will change by 1 unit,
while the output of the control system will change by 0.1 unit.
The real payoff of negative feedback comes when there are complex
relationships between input and output. By feeding back from the final
result to intermediate stages of the forward process, we can simply
eliminate the complex computations that otherwise would have to be done.
The feedback accomplishes with a few simple connections what would
otherwise have to be done by computing the inverses of all the forward
functions that lie between input and the final result. That's how the
Little Man arm model can stabilize a 3-df arm in real time using only
about 30 or 40 lines of code, where the equivalent open-loop performance
using inverse kinematics and inverse dynamics would take a Cray computer
to run as fast, and would use hundreds of lines of code.
So remember: the simplest and fastest kind of system is the negative
feedback control system. Open-loop systems are always slower and more
complex, given that the same performance is achieved.
That's as true as most rules of thumb are.
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Best,
Bill P.