How to measure the angle between the poles and the x-axis using the Root Locus?

I became recently aware of the matlab commands rlocus and sisotool which draws out the root locus and allows us to perform certain operations. I wish to ask if there is a command or a way that I can measure the angle between the poles and the x-axis as done when deriving a lead-lag compensator using what is called the angle criterion. I hope for some assistance on this matter and thank you.

Answer 1

[William Rose]

@Tarek Hajj Shehadi,

Suppose you have a real-axis pole , or a zero at , on the real axis at position . And suppose you have a pair of conjugate poles on the root locus at . Draw a line from the real axis pole or zero to the upper pole on the root locus. The angle betwee this line and the x-axis is .