EXERCISE

$8\mathrm{.}16\mathrm{.}33$: $($Chapter $8,$ Problem

$33$ in the $8$th Edition$).$

$($a$)$ You are given the unity$-$feedback system of the figure above, where

$G(s)=\frac{K(s+0\mathrm{.}02)}{s^2(s+4)(s+10)(s+25)}$

Use MATLAB to plot the root locus. Use a closeup of the locus $($from $-5$ to $0$ and $-1$ to $6)$ to find the gain, $K,$ that yields a closed$-$loop unit$-$step response, $c(t),$ with $20\mathrm{.}5\%$ overshoot and a settling time of $T_s=3$ seconds. Mark on the time response graph all other relevant characteristics, such as the peak time, rise time, and final steadystate value.

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