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I want to understand how in process of design of a controller I have to meet requirements using root locus method. I know that for the given damping ratio I need to estimate the arccos of it and everything below that angle is the desired region of my poles and zeros. But how about the settling time when it's said that it should be less than 1 second, how do I find the region (the vertical line) that corresponds to that requirement?

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The settling time $T_s$ is related to the damping ratio $\zeta$ and the natural frequency $\omega$ by $$ T_s = \frac{4}{\zeta \omega}$$

If $T_s<1$ this means that $\zeta \omega >4$. To plot it on the complex plane we need to first solve for $\zeta$ and $\omega$ in terms of $x$ and $y$.

This relationship is $$ x=-\zeta \omega,y=\omega \sqrt{1-\zeta ^2}$$

Once we solve for $\zeta$ and $\omega$, we need to find the region where $\zeta \omega >4$. These calculations are tedious, so I used Mathematica.

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