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Homework Help: Nyquist Criteria Stability

  1. Dec 9, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-12-9_17-33-11.png



    2. Relevant equations
    Number of encirclements = Number of open loop poles - Number of Close loop poles on Right side of S plane.

    3. The attempt at a solution
    There is 1 open loop pole on RHS
    For Close loop poles I used Routh Herwitz method and got 1 pole on RHS. 1 sign change.
    So I get N = 0.
    Where am I wrong?
     

    Attached Files:

  2. jcsd
  3. Dec 9, 2017 #2

    rude man

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    Why are you trying to deal with closed-loop poles? Nyquist is strictly an open-loop stability criterion. G(s) is the open-loop transfer function.
     
  4. Dec 9, 2017 #3
    Nyquist criteria says encirclement of -1 + j0 is number of open loop poles - series of characteristic equation.
    Characteristic equation is 1 + G(s)
     
  5. Dec 9, 2017 #4

    rude man

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    What do you mean by "series of characteristic equation"?
    The Nyquist method does not involve closed-loop transfer functions.
    Nyquist determines whether the closed-loop transfer function is stable but its methodology does not involve any closed-loop transfer functions.
     
  6. Dec 9, 2017 #5

    rude man

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    I see from your attachment that in some cases they do consider closed-loop RHS poles, in others they stick to open-loop only.
    I have to admit I never heard of doing Nyquist analysis with anything other than open-loop transfer functions. Seems to me undesirable to have to compute 1 + G(s).
    So the only way I know to do this is
    (1) determine the Re and Im parts of G
    (2) draw polar plot of G
    (3) follow rules of Nyquist stability determination.
    Sorry that's all I can tell you.
     
  7. Dec 9, 2017 #6
    Sorry for the typo. It was 'zeroes' of characteristic equation and not 'series'. But yeah you're right. I read the question wrong. It says encircle the origin and not encircle -1
     
  8. Dec 10, 2017 #7

    rude man

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    OK. I have to admit I don't know on what basis the solution to ex. 39 is given.
    If G(s) is an open-loop transfer function then the thing that matters for determining stability of G(s)+1 is encirclement of G(s) of (-1,0), not (0,0). In other words, I guess I really don't understand their reasoning.
     
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