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$23.$ For the system of Figure P$8\mathrm{.}8($a$),$ sketch the root locus and find the following: $[$Section: $8\mathrm{.}7]$

a$.$ Asymptotes

b$.$ Breakaway points

c$.$ The range of $\backslash ($ K $\backslash )$ for stability

d$.$ The value of $\backslash ($ K $\backslash )$ to yield a $0\mathrm{.}7$ damping ratio for the dominant second$-$order pair

To improve stability, we desire the root locus to cross the $\backslash ($ j $\backslash$omega $\backslash )-$axis at $\backslash ($ j $5\mathrm{.}5\backslash ).$ To accomplish this, the open$-$loop function is cascaded with a zero, as shown in Figure P$8\mathrm{.}8($b$).$

e$.$ Find the value of $\backslash (\backslash$alpha $\backslash )$ and sketch the new root locus.

f$.$ Repeat Part $\backslash (\backslash$mathbf$\{$c$\}\backslash )$ for the new locus.

g$.$ Compare the results of Part $\backslash (\backslash$mathbf$\{$c$\}\backslash )$ and Part $\backslash (\backslash$mathbf$\{$f$\}\backslash ).$ What improvement in transient response do you notice?