Given the unity feedback system of Figure P$9\mathrm{.}1$ with

$G(s)=\frac{K}{(s+2)(s+4)(s+6)(s+8)}$

Find the transfer function of a lag$-$lead compensator that will yield a settling time $0\mathrm{.}5$ second shorter than that of the uncompensated system. The compensated system also will have a damping ratio of $0\mathrm{.}5,$ and improve the steady$-$state error by a factor of $30.$ The compensator zero is at $-5.$ Also, find the compensated system's gain. Justify any second$-$order approximations or verify the design through simulation. $[$Section: $9\mathrm{.}4]$