Problem $3$: Show that the block diagram $($Fig$.3$a$)$ where $\frac{C(s)}{U(s)}=\frac{K_m}{m{s}^2+bs}$ can be reduced to a unity feedback system $($Fig$.3$b$)$ where $G_c(s)=\frac{U(s)}{E(s)}=K_f{K}_c$ and $E(s)=R(s)-C(s)$ and $E(s)=R(s)-C(s)$ :

$R$

Derive the closed loop transfer function. Show that it can be expressed in the form of a standard $2^{nd}$ order system $\frac{C(s)}{R(s)}=\frac{{}_n^2}{s^2+2{}_{n}s+{}_n^2}.$ Determine $2{}_n$ and ${}_n^2$ in terms of the system parameters.

Assume a unit impulse reference input. Show that the solution $c(t)$ can be divided into four different cases depending only on the value of $.$ For each of the cases,

a$)$ determine the condition of $,$ and

b$)$ find the poles of $C\frac{s}{R}(s)$ in terms of $(,{}_n)$ and label them on the $s-$plane given below.

c$)$ For Case C $,$ find the magnitude M and angle $$ of the complex root.

Can the final value theorem $($FVT$)$ can be used to determine the steady$-$state solution for each of the cases? For the case where the FVT cannot be used, derive and sketch neatly the solution $c(t).$