# Closed-Loop Transfer Function

Discussion in 'Electronic Design' started by Gregory L. Hansen, Jun 29, 2004.

1. ### Gregory L. HansenGuest

There's something I don't understand about the closed-loop transfer
function. Suppose you have a system like

---- D ---- P ----
| |
----- H ------

where D is your controller, P the plant, H the transducer. And the
transfer function would look something like

DP / (1 + DPH)

But the only thing special about the line extending from P is that you
care about what the plant produces. Physically, it's just a loop that
could as well be drawn like

---- D ---- P ---- H ----
| |
---------------------

for a transfer function of

DPH / (1 + DPH)

or

-------- D --------
| |
--- H --- P ---

for a transfer function of

D / (1 + DPH)

or

---------------------
| |
-- H -- P -- D --

for

1 / (1 + DPH)

The poles are the same in each case, but it changes the zeroes, which
would change the behavior of the system, including the behavior of the
plant.

What am I missing here?

2. ### Charles DH WilliamsGuest

Saying what's bothering you perhaps?

You may be failing to distinguish between 'loop gain' (the product of
all the gains acting around a loop) and 'closed loop transfer
gain' which differs depending on which pairs of nodes it refers to.

You are also living dangerously by not explicitly showing the
summing term in your diagrams. The last one is ambiguous IMHO.

Charles

3. ### John LarkinGuest

The subtractor, namely

This---
|
|
v

in ---(+-)-- D ---- P -------> out
| |
----- H ------

What makes these loops different is where you inject the input, and
where you measure the output.

John

4. ### Rene TschaggelarGuest

You're interested that D*P is optimized for speed, accuracy, whatever.
The whole loop has to be stable, though.

Rene

5. ### Tim WescottGuest

If you want to know the behavior of your system from input to output
system from the point of view of the controller (it can't "see" the
plant, only what the transducer tells it -- kinda like upper
management). The third drawing tells you how you have to drive the
plant for a given input. The fourth drawing, of course, um, uh, well,
proves that you know your permutations!

The system behavior never changes, only your view of it. It is very
comforting to see that the poles never change -- it would be odd to
think that you could make an oscillation go away just by looking at it
right.

6. ### Roy McCammonGuest

you are right on track. What you are missing is that
its all the same system, with outputs from different nodes.

That would be: plant output, transducer output, controller output and
controller input respectively.

7. ### Gregory L. HansenGuest

Wow, five replies in a few hours, each with another peice of insight. I
can't reply to all of them without just being repetitive...

But yeah, that makes sense. Tim and I are both comforted that the poles
don't change, but it didn't really sink in that the output of one section
could be different from the output of another section. I mean, that's
what they're for! We wouldn't expect a step function input to look like a
step function on the other side of a filter.

And that probably affects my application, since the control power is what
we measure and get the physics from, and that should be smoothed out for a
cleaner measurement. I'll have to explore that angle.

8. ### Terry GivenGuest

another way of looking at it: If you augment your model with a load
disturbance, you can calculate two useful transfer functions viz:

1) reference-to-output transfer function Fout(s)/Fref(s)
2) load disturbance-to-output transfer function Fout(s)/dFout(s)

again these will all have the same poles, but in general the zeroes are
different. A so-called "regulator" is interested in the latter (regulating
out a disturbance), cf a "controller" which is concerned with following a
reference ie the former transfer function.

Cheers
Terry