Open and closed loop systems

[From Rick Marken (960117.0800)]

Here's a correction to last night's post [Rick Marken (950116.2100)] --

Better change the equation in cell B2 (the output equation):

=B2+B5*B4*B1

to

=B2+B5*(B4*B1-B2)

(a leaky integrator).

Bruce Abbott (960117.0845) --

It's worse that that, Rick. Without the slowing factor it isn't even a
simulation of a closed loop. All you've done is iterate the same pair of
equations I presented and, as you pointed out, this does not give a
closed-loop solution.

No. This is not true. The simultaneous equations in my simulation define a
closed loop. By omitting the slowing factor, I caused the variables in the
loop to be dynamically unstable -- but the variables were still in a closed
loop relationship.

In effect, you've made the same mistake I did.

No. I think it was a different mistake; not a _better_ mistake. Just
different;-)

Me:

1. X(n) = g X(n-k) does not describe a closed loop system

Bruce:

(1) X(n) = g*X(n-k) by itself is a spiral, not a closed loop. It can be
made to behave like a closed loop by adding an integrating output with a
slowing factor s such that 0 < s < 1.0:

    X(n) = X(n-1) + s*[(g*X(n-1) - X(n-1)]

No, this is still not a closed loop. You've defined an output variable, X(n)
as a function of an input variable, X(n-1). This equation is just _half_ of
what is needed to specify a closed loop; it is open loop as it stands. To
close the loop you need another equation that describes the _simultaneous_
effect of output X(n) on input (X(n-1).

Of course, in the sequential system you describe, the current output, X(n),
cannot have an effect on an earlier input, (X(n-1). But in a continuous
closed loop system, the output does have an effect on the input _while_ that
same input is having an effect on the output.

Letting X(t) be the input variable and Y(t) be the output, a closed loop is
defined by the following pair of _simultaneous_ equations:

Y(t) = g [X(t)]

X(t) = h [Y(t)]

In our computer simulations, where the computation of X(t) from Y(t) and Y(t)
from X(t) must occur in _sequence_, we "simulate" simultanaeity by letting
the variables (X and Y) change by only a fraction of amount by which the
algebraic equations say they should change on each iteration. The smaller the
change in the variable values on each iteration (the smaller the "slowing"
factor) the closer the behavior of the loop approximates what happens in the
continuous case.

Me:

2. When loop gain in a positive feedback system is >= 1.0 the system becomes
unstable.

Bruce:

(2) No one here has disagreed with statement #2.

I'm sorry. Then I misunderstood you [Bruce Abbott (960112.2015)] when you
said:

To be in equilibrium the loop gain has to be 1.0, which results from
the balance of opposing forces.

Maybe we just have a terminology problem. When loop gain is 1.0, I would
call the behavior of the variables in the loop "runaway", not "equilibrium".

Best

Rick