A negative unity feedback system has a plant transfer function given by:

$G_p(s)=\frac{1}{s(s+2)}$

A system response to a step input is specified as follows:

Percent overshoot P$.$ O$.=4\mathrm{.}32\%$

Settling time $T_s=1sec.$

a$.$ Determine whether both specifications can be met simultaneously by using proportional controller.

b$.$ Design a cascade proportional derivative $($PD$)$ compensator $G_c(s)=K(s+z_0)$ so that the above specifications are met.

c$.$ Design a cascade "lead compensator" $G_c(s)=\frac{K(s+z_0)}{s+p_0}$ so that the above specifications are met $($let $z_0=4).$

d$.$ Using the concept of system "type number", determine for the PD compensated system $($b$),$ the steady$-$state error for $(1)$ unit step input and $(2)$ unit ramp input

Given information:

Percent overshoot P$.$O$.=10e^{-\sqrt[\phantom{2}]{1-{}^2}}$

Settling time $T_s=\frac{4}{{}_h}$

where ${}_h$ and $$ is the system natural frequency and system damping ratio, respectively.