[From Bill Powers (2006.07.08.0747 MDET)]
Martin Taylor 2006.07.08.08.47 –
I agree with your comment on the relation between gain and error. Let me
try to communicate the same point from a slightly different angle, and
consider some added points:
First, gain in the context of PCT is a physical property of a component
in a (linear) control system: it is the ratio of the output of the
component to the input. “Loop gain” is the product of all the
component gains in a closed loop.
If G stands for gain, the output of a component is G times the input to
the component, or output/input = G. It is also possible that higher
systems can adjust G in a lower system, by any of several means. That
case is not considered here.
Second, error is a variable in a control system that is affected by
disturbances, the system’s own actions, and the setting of the reference
signal.
For a given loop gain, disturbances will have a calculable amount of
effect on the controlled quantity, and for given reference-signal
values the perceptual signal will be a calculable fraction of the
reference signal (if the loop gain is 100, for example, the reference
signal will be 100/101 of the reference signal: p = 0.990099r to 6
places). The error signal will therefore be 1/101 of the reference
signal.
As the loop gain increases, the effect of a disturbance on the controlled
variable decreases and the error signal resulting from a given
disturbance or a given setting of the reference signal decreases. As loop
gain increases, the effect of the action on the controlled variable
becomes more nearly equal and opposite to the effect of the disturbance
on the same variable.
The input and output functions in a control system involve not only a
gain factor but a change of units from physical units outside the system
to nervous system units (NSU) inside the system, or the other way around.
The change of units can be used to make the gain factor of the modeled
input function equal to 1. Suppose that bending the elbow through an
angle of 90 degrees changes the composite perceptual signal representing
elbow angle from 3000 to 60000 impulses per second (this is a summation
over all sensory nerves affected by that joint angle). From this
measurement we find that a change of 57,000 impulses per second
represents a change of 90 degrees. For this control system, then we can
say that one NSU of perceptual signal equals one degree of elbow
angle, where one NSU is 633.33… impulses per second. If we measure
elbow angle in degrees and the perceptual signal in NSU, the input gain
factor will be 1. p (NSU) = angle (degrees).
This of coure is a linear approximation, but it is useful over some range
of applications.
Once the value of an NSU is established, that same scaling factor must be
used for measuring all signals inside the control system. For example, to
make the elbow angle be 10 degrees, the reference signal must be set to
10 NSU, or 6333 impulses per second (summed over all redundant control
systems). If we observe a disturbance-induced angle error of 2 degrees,
we can say that the error signal’s magnitude has changed by 2 NSU, or
1267 impulses per second.
At the output function, the conversion factor is from NSU to physical
units. Suppose that by measuring the effects of disturbances we calculate
that the loop gain of the elbow angle control system is 50. Let the
external feedback gain be 1 (it’s observable). The gains of the input
function and comparator are 1 and -1 respectively, so the gain of the
output function must be 50 units of angle change per NSU of error signal.
The loop gain is then input gain times comparator gain times output gain
times feedback gain or 1 * -1 * 50 * 1 which equals -50. The minus sign
shows that we have negative feedback.
We can see that the loop gain in this case is set by the gain of the
output function, since the magnitudes (disregarding sign) of all the
other gains are 1. This gain shows the amount of action that will be
produced by each unit of error signal: the ratio of effort to error. I
have proposed this ratio as a measure of the “importance” of a
given controlled variable, since it measures how hard we will work to
keep that controlled variable at any given reference level when
disturbances occur. From a different angle, it is a measure of how much
error we will tolerate in the course of opposing a given disturbance.
With high loop gain, the output rises very quickly to oppose the
disturbance before the error has become significant. A low loop gain
means that the output will not reach full opposition to the disturbance
until the error signal has become large (which is what I mean by
“tolerating” error).
These descriptions imply some slowing factor, so we are really speaking
of steady-state gains; the relationships above all apply to
steady-state conditions, not to the dynamic processes during changes. A
more complete picture requires differential equations or simulations. The
results in terms of steady states, however, will agree with all the
above.
Best,
Bill P.