$v(t)=Ri(t)+L\frac{d}{dt}i(t)+e_b(t),e_b(t)=k(t)$

$J\frac{d}{dt}(t)=(t),(t)=ki(t),J=J_m+J_h+J_d$

$u_1(t)=K_a\{r(t)-(t)\},u_2(t)=K_r\{u_1(t)-(t)\},v(t)=K_{\text{I}}\{u_2(t)-i(t)\}$

The $3$ grains are Ka $,$ Ki and Kr

Obtain the closed$-$loop transfer function from $R(s)$ to $(s)$ by hand from the given ODEs $(1),(2)$

and the control equation $(3).$ Discuss the roles of three gains in the transient response. How

about the steady$-$state error? Does any gain affect the steady$-$state error with respect to a step

input and a ramp input?