Mathematically, what is the source of the lines of constant damping factors that are generated by ZGRID in Control System Toolbox 9.4 (R2012b)?

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I am using the ZGRID command in Control System Toolbox 9.4 (R2012b). I see that this command plots lines of constant damping ratio zeta in the z-plane. I am curious to know the source of these curves, from a mathematical standpoint.

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MathWorks Support Team
MathWorks Support Team on 1 Feb 2013
The shapes of the curves of constant damping factors are a result from control theory. Essentially, the equivalent characteristics in the z-plane are related to those in the s-plane by z = exp(s*T), with T being the sample period, and so the diagonal lines of constant zeta in the s-plane get mapped to spirals starting from (1,0) in the z-plane. A control theory textbook would be a good reference for the mathematical details.
For example, see pp.602-603 of Feedback Control of Dynamic Systems, by Franklin, Powell, and Emami-Naeini.

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