b$)$ Consider a process with the following $($forward$)$ transfer function of a unity feedback system:

$G(s)=\frac{1}{(s+2)(s+3)(s+4)}$

and the controller:

$G_c(s)=K_d(s+\frac{K_p}{K_d}),$

If the system is designed to satisfy the damping ratio $0\mathrm{.}6,$ use MATLAB to obtain the time constants, $T($not the time period$),$ undamped natural frequencies, ${}_n,$ damped natural frequencies, ${}_d,$ derivative gains, $K_d,$ and proportional gains, $K_p,$ for the values of $\frac{K_p}{K_d}$ proposed in Table $1.$ Show the detailed steps of obtaining the values. Use illustrations as much as you can.

Fill in the table cells.

Briefly discuss the effect of $\frac{K_p}{K_d}($zero location$)$ on the speed of response.

$[6$ marks$]$

Table $1.$ System and controller parameters

$\backslash$table$[[\frac{K_p}{K_d},0\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}5,3\mathrm{.}5],[T,,,,],[{}_n,,,,],[{}_d,,,,],[K_d,,,,],[K_p,,,,]]$