[From Bruce Abbott (960903.1220 EST)]
Hans Blom, 960903 --
Some questions, Bruce, where the two situations are maybe more than
just superficially the same. Some sadism in the experimenters must be
taken for granted ;-).Situation 1: An orthopedic surgeon mechanically fixes the angle
between the upper and the lower arm of a patient (victim?).Situation 2: A neurosurgeon cleverly modifies and/or reroutes the
peripheral nervous system in the patient (victim?)'s arm in such a
way that the musculature controls the angle between the upper and the
lower arm at a fixed angle.
If you don't mind, I'd like to substitute an equivalent comparison that
sounds a bit less diabolical. Two engineers design an adjustable platform
consisting of a heavy, wide base, a jointed arm, and a sturdy rectangular
plate, which is kept horizontal at all positions via some linkages.
Engineer A designs his platform with a locking screw; once the platform has
been positioned, the screw is tightened, freezing the platform at that
height. Engineer B designs his platform with an electrical
servo-motor/controller; the height of the platform is set by adjusting the
controller's set-point, which is compared to a sensor readout of the
platform's position. I think you'll agree that these systems are the
equivalents of your two "frozen arms." Now, on to your questions.
Question 1: How fundamentally different are these two situations? Can
you tell the difference if you are not allowed to look inside the arm
in any way?
Both will support a weight (within limits) and both platforms will sag
increasingly as more weight is added. The passive (equilibrium) system will
sag purely due to the stiffness of its components as the downward force
exerted by the added weight bends the arm. The active (control) system will
be affected by the same bending forces; let us assume that the
position-sensor picks up these movements. Force that affects the sensed
position will be resisted by the servo in proportion to the gain; with high
gain there will be little movement, with low gain there will be more
movement before sufficient power develops to resist further movement.
In operation the active system will seem to be stiffer than the passive
system built according to the same plan, of the same materials. (Any
bending will be nearly compensated for by additional counter-forceIf the
active system is well-designed it may even be able to damp out oscillations
of the platform height due to the springiness of the arm that would appear
in the passive system. If poorly designed it may go into forced
oscillations not characteristic of the passive system. You may be able to
tell what kind of system you are dealing with by its dynamic response to a
change in load.
The active system will be continuously expending energy to support the
weight; the passive system will be resisting further movement by storing
energy in the arm that came from the work expended in bending it. If the
active system's power supply peters out, its platform will fall; the passive
system's platform will fall only if there is a catostrophic failure (or
someone loosens the set-screw). And of course, the active system will be
much more expensive than a set-screw.
Whether you can tell the difference between the two systems without being
able to look inside depends partly on what you mean by "look inside." Is
hearing the servo-motor's whine equivalent to looking inside? What about
observing the power requirements? Dynamic response to disturbance? If you
mean just watching them both sitting there bearing a load, without being
about to see whether bending is being compensated for, then probably not.
Question 2: Which is the more clever solution, assuming that the
rigidity of the joint is the same in both situations?
Depends on what you mean by "clever." If you need accurate positioning even
though the structure deforms significantly under load, the active system is
best. If you just want to support a load about "here," the set-screw system
is best, and in fact the complex and energy-wasteful servo-system is plain
foolish.
Question 3: More generally, how can we tell whether some "black box"
is a control system or not, if control (if it is control) takes place
entirely within that black box?
In a control system, the energy used to resist disturbances comes from a
source other than the disturbance itself, so the dynamic response of the
control system will be different from (the response can be more energetic)
that of the passive system, whose energy must always equal (or be less than,
admitting frictional losses) that of the disturbance.
Question 4: When is a controller a better solution than a mechanical
fixture?
A passive system of equivalent resistance to disturbing forces must be built
more massively than one that can actively compensate. The active system may
be able to stablize against disturbance patterns that would send the passive
system into oscillation. If changes in applied force are involved, the
response of the active system may be better (less overshoot, quicker
damping, etc.) than that of the passive system. And if you want accurate
positioning in an environment in which frictional or other forces are
somewhat unpredictable, you'd want the active system over a passive one,
whose final position would vary depending on the current values of those
effects.
Regards,
Bruce