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Which transfer function ,open loop or close loop do we use/need as a pre req to perform root locus analysis?

I have also attached a snapshot of the relevant webpage of university of Michigan.

I have also highlighted in my snapshot both contradictory terms used in the webpage article.

enter image description here

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  • \$\begingroup\$ You need open loop transfer function. \$\endgroup\$ – ab29007 Nov 30 '19 at 6:34
  • \$\begingroup\$ No - the root locus shows the poles of the closed-loop function (roots of the denominator: 1+KH=0). The root locus test must not be confused with the Nyquist stability test which analyzes the loop gain (-KH) \$\endgroup\$ – LvW Nov 30 '19 at 16:19
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The answer is relatively simple:

  • From stability considerations we know that the poles of the closed-loop function H(s)=N(s)/D(s) must not enter the right side of the complex s-plane.

  • The poles of the closed-loop function are identical to the roots (zeros) of the denominator D(s) (which is identical to the characteristic polynominal of the system). Therefore, the root locus analysis is performed for the closed-loop system.

Comment (EDIT): I think, the wording of the quoted text is a bit confusing (..."root locus of an open-loop transfer function"...). For my opinion it should read: ...the root locus for a transfer function H(s) under full negative feedback for different proportional gain values K"...

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Any point in the root locus represents a pole of a closed-loop system. However the start and end points are poles/zeros of the open-loop system.

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