Closed loops and filters

CSGnet Archive CSG1996
1

[From Rick Marken (960117.1500)]

I said:

Now I can see that the equation above not only _behaves_ like a closed loop,
it _is_ a closed loop. I was wrong to say that "it is still not a closed
loop". It is a closed loop and I am pond scum for having suggested
otherwise;-)

On re-thinking this, I realized that I'm not that bad after all. In order for
Bruce's equation

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

to behave like a closed loop system, it MUST be true that

X(n-1) = X(n)

That is, it MUST be true that the inputs to the equation, X(n-1), ARE the
outputs of the equation, X(n). But it is not necessarily true that the inputs
to this equation are the outputs and, indeed, in most applications of
equations like

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

X(n-1) is viewed as an independent, time varying input waveform and X(n) is
the time varying, now filtered output of the equation.

So I'm going to go back to my pre-pond scum self and argue that the
equation:

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

does NOT behave like a closed loop system because it is NOT a closed loop
system unless there is _another_ equation that explicitly closes the loop by
defining a relationship between the output, X(n), and the input, X(n-1).

So I argue that a closed loop must be described by at least two simultaneous
equations: one describing output as a function of input and the other
describing input as a function of output.

Best

Rick

2

[Martin Taylor 960118 13:20]

Rick Marken (960117.1500)

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

does NOT behave like a closed loop system because it is NOT a closed loop
system unless there is _another_ equation that explicitly closes the loop by
defining a relationship between the output, X(n), and the input, X(n-1).

So I argue that a closed loop must be described by at least two simultaneous
equations: one describing output as a function of input and the other
describing input as a function of output.

One can always sustain arguments by insisting that apples are not oranges,
but why would one want to?

A single equation that describes the dynamic behaviour of a variable as
a function that includes its own past can be the description of the dynamics
of a point in a closed loop, adn often is. It cannot be a complete
description of a closed loop, nor even of the dynamics of the complete
loop. To do that requires the description of ALL the variables in the loop,
not just the two that Rick wants.

A loop is a common way in which a variable can be influenced by its own
history. There may be other ways, but when you see an equation like that,
it's a reasonable bet that if you look for a loop you are likely to be
rewarded.

Bruce (and I) present descriptions of what happens at a point in the loop.
Rick quite correctly observes that this is not a complete description of the
loop. So what? Methinks he doth protest too much.

Martin