#$9$

$(50$ points$)$

The unity gain feedback control system $(H(s)=1)$ has proportional control $K_p$ and a plant transfer function equal to:

$G(s)=\frac{s+1}{s(s-1)(s+6)}$

Verify the root$-$locus plot shown by identifying on the diagram:

a$)$ Number of branches of the root loci

b$)$ Real axis portions of the root loci

c$)$ Starting and terminating points of the root loci

d$)$ Angles of the asymptotes of the root loci

e$)$ Real$-$axis intercept of the asymptotes