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Closed-Loop Transfer Function

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

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  1. 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)


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

    for a transfer function of

    D / (1 + DPH)


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


    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

    What am I missing here?
  2. 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.

  3. John Larkin

    John Larkin Guest

    The subtractor, namely


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

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


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

  5. Tim Wescott

    Tim Wescott Guest

    Aside from the already-critiqued format of your drawings, not much.

    If you want to know the behavior of your system from input to output
    then your first drawing is correct. Your second drawing gives the
    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
  6. Roy McCammon

    Roy McCammon Guest

    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. 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 Given

    Terry Given Guest

    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.

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