# Talkin' Positive Feedback Blues

[From Rick Marken (950116.2100)]

Bill Powers (960116.1300 MST)

Actually, I'm surprised that your simulation worked at all, unless you
had a slowing factor somewhere else in the loop. It should have run away
if the loop gain was less than -1 or greater than 1.

It ran away when the loop gain was greater than 1; this is what I thought
should happen so I didn't check to see if the loop ran away when the gain
was less than -1 (when there should be control) -- but I checked it and,
indeed, it does run away.

So I fixed up the simulation, added the slowing factor so that

o := o + slow * gain * i

and, sure enough, the loop only ran away when gain (loop gain)
was >= 1.0.

Anyone with a spreadsheet program that can deal with iteration (you
have to check the iteration selection in the calculation options in
early versions of Excel and Lotus) can use this simulations system;
here is the code:

A B
1 input =B4+B2
2 output =B2+B5*B4*B1
3 environment 2.0
4 gain 0.9
5 slowing 0.01

The environment, gain and slowing values shown are the last ones
I tried. You can enter new values to test the system to see how
they affect stability (stability can be seen when the loop variables
[input and output -- B1 and B2] stop changing.

You can think of this as the simulation of a sound amplification
system. The input to this system is the sum of independent
environmental influences (like the intensity of a singer's voice,
B4) and the feedback influence of the output of the system itself
(the intensity of the output of the speakers, B2). The output of
the system is the integral of the amplified input (amplified by B4,
which is equivalent to loop gain because all other multipliers in
the loop are 1.0). If the gain value is positive then the feedback
loop is positive; if the gain is negative the feedback loop is
negative.

The slowing factor can be used to stabilize the loop as the gain
increases; the larger the gain number, the smaller the slowing number
must be for stability; large (negative) gain produces the best control
but this control is purchased at the price of a slower approach to a
stable state. But no matter how small you make the slowing factor, you
will find that the loop never stabilizes when the gain is >= 1.0.

Note also that with sufficiently high negative gain the input variable
stabilizes at or near zero (the de facto reference value in a negative
feedback loop with no explicit reference input), regardless of the
value of the independent envionmental contribution to that input.

Best

Rick