Power gain

[From Bill Powers (940528.0920 MDT)]

Tom Bourbon (940527.1325) --

When we talk about (and calculate) loop gain, aren't we talking
energy expended.

Yes. We define our systems in terms of magnitude, not energy,
measurements. Consider a limb position control system using muscles.
A change in a reference signal will cause a joint angle to change by
a certain amount. Suppose it moves an arm from a straight-ahead
position to a straight-up position, from _ to |.
The output function gain of the position control system is the ratio
of change in arm angle to change in position error signal. If a
neural amplifier follows the comparator, this ratio can be quite
large.

The actual conversion of error signal to arm angle comes about by
altering reference signals for lower control systems concerned with
muscle length and, at the lowest level, tension in tendons. If there
is a discrepancy between the actual muscle length and the reference
muscle length, the lower-level systems increase the average muscle
fiber contraction to eliminate the discrepancy. As a result, the
joint angle follows the reference signal for muscle length
regardless of the load, drawing as much energy from the body's power
supply as necessary to maintain the same magnitude (scalar)
relationships.

For light loads, a given movement entails a small amount of energy
usage; for heavy loads, a large amount. This makes the energy gain,
measured between the position error signal and the joint angle,
depend on the load. But the magnitude gain remains (more or less)
the same. The lower-level systems simply expend as much energy as
required to make the muscle length follow the reference signal that
is the amplified error signal from the position control system.

In living control systems there is no sensible way to define loop
gain in real physical energy terms. If you hold a 5-Kg weight
straight up in your hand, arm extended upward, you will be using
energy at a certain rate. As you slowly lower that weight to the
straight-ahead position, arm still extended, the weight will perform
work on your arm, putting energy INTO the muscles -- but the muscles
will be expending more and more energy as the angle from the
vertical increases. When you stop moving the weight, having reached
at the greatest rate of all, yet because the weight is now not
moving any more, you will be doing no physical work on the weight.
In terms of energy in versus energy out, the gain of the muscles at
that point is zero. Yet the magnitude gain is still what it must be
to maintain the weight in the desired position against the pull of
gravity.

Living control systems are highly dissipative systems: every
function requires the expenditure of energy, even in the "resting"
state. While it is possible to convert magnitude relationships to
"energy" relationships by using squared magnitude measures, the
resulting "power gain" is a fiction, having no relationship to the
actual, physical, energy relationships in the system. So whether you
express loop gain as a multiplying factor or in decibels is mostly a
matter of custom, and has no physical significance. The only basic
measure of loop gain that is physically defensible is the one based
on magnitudes of signals and variables.

In electronics, we speak of power gain in a control system, but this
is always in a situation where a specific control system is working
with a specific load. Living systems are not, however, permanently
coupled to specific loads. They are general-purpose devices, which
can operate through many different external loops having different
properties. We can't, therefore, characterize the output function in
terms of power gain, because the amount of power used in moving an
external object from point A to point B will depend on external
factors such as mass, friction, and independent disturbances. When
we characterize the output function in terms of magnitudes of
signals and physical effects, however, we can specify the
characteristics of the control system in a fixed way, with the loop
(magnitude) gain then depending on the external magnitude
relationships and the power gain being irrelevant. The only time
that the energy considerations become important is when the
controlling system runs out of energy supplies. As long as that
limit is not reached, the behavior of the system is essentially
independent of the amount of energy available and the amount
actually used.
Or, as they say in rule-of-thumb engineering, the performance of a
good control system is essentially independent of the power supply.

ยทยทยท

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Best,

Bill P.