Context: In the process of designing a phase-lead compensator for a system, I have used the design requirements to find the ideal locations for my closed-loop (CL) poles, have picked a place for a new open-loop (OL) zero, and am trying to use the angle criteria (angles to zeros - angles to poles = (2n+1)pi) to find the location for a new OL pole. I am hoping my method is correct for the moment, but feel free to query if you see a problem.

My question when using the angle criterion is how to treat the open-loop poles that are on the original (uncompensated) root locus that are not on the real axis.In the image below I've drawn in red the line from the location of the preferred CL location to an OL zero and the angle I would use in that case, then done the same in orange for the OL pole that is not on the real axis. Is the angle I've drawn the correct one to use in that case? (if you can't see the image: The angle between the line from OL pole to CL pole and a line parallel to real axis through the OL pole)

Image of root locus, design requirements, and angle I would use in case of OL pole on real axis (red) and


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