[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