# Positive Feedback Equilibrium

[From Rick Marken (960115.1600)]

Bruce Abbott (960115.1215 EST) --

Me:

o := o + (k.2i-o)

Ye:

But doesn't this reduce to just

o := k.2i ?

Mathematically, yes. But with the replacement operation I think we just end
up adding an increment, k.2i-o, to o.

Me:

I can stabilize the loop with a loop gain of 1.0 or higher by making the
integrator "leak" more than it increments:

o := o + (k.2i-k.3o)

Ye:

Rick, you're going to have to explain your notation to me. Either I'm
reading it wrong or something is missing.

k.2 is the the loop gain constant (because I set k.1 to 1.0). When k.3 is 1
we have a simple leaky integrator; and the loop stabilizes only when k.2 <1.0
(as Bill claims in the LCS piece). ( I ran these simulations in Excel, by the
way).

When k.3 is > 1.0, the gain can be >1.0; when the values of k.2 and k.3 are
chosen appropriately the system will still stabilize. I don't know whether
this is legit or not or how it affects the notion that a positive feedback
loop is only dynamically stable when the loop gain < 1.0. I am not sure that
the loop gain (k.2) really is independent of the integration leak (k.3) in
the simulation; loop gain (k.2) and "leak rate" (k.3) appear to contribute
independently (additively) to the value of o in:

o := o + (k.2i-k.3o)

but k.3 is affecting the value of o itself. I think I'm in way over my head
(again).

My guess is that a positive feedback loop with "normal" integral leakage
(where the loss of output due to leakage would not take the output below
zero if there were no input) is stable only with gain < 1.0. By increasing
the output leakage beyond "normal" you can apparently reach equilibrium with
loop gain > 1.0.

Best

Rick

[From Rick Marken (960115.1930)]

Bruce Abbott (960115.2010 EST) --

Well, let's take an example. Assume o := 10, k.2 = 2.0, i = 3. Then

o := 10.0 + 2.0*3.0 - 10.0 = 2.0*3.0 = 6.0

Yes. But given the way I defined the input (o+c) the next value of i
(assuming c=0; no disturbance) is 6. So the next output is 6+12-6 =12
and the next is 12 + 24 -12 = 24 ...etc into exponential positive
feedback. Of course, it works the same if c <> 0.

Best

Rick