Question I:

$(2\mathrm{.}5$ Marks$)$

Consider the following block diagram

With:

$G(s)=\frac{1}{s(s+2)(s+4)(s+6)}$

$F(s)=K($Proportional control action$)$ and $H(s)=1$

Find the Open Loop Transfer Function $F_1(s).$

Find Static Position Error $(E_p),$ Static Velocity Error $(E_v),$ and Static Acceleration Error $(E_a).$

Find the closed loop transfer function CLTF and Prove that the characteristic equation of the system is $s^4+12s^3+44s^2+48s+K=0.$

Using Routh's criterion, give the condition on the gain $K$ yield a stable system?

Which values of the gain $K$ that makes the system critically stable.I hope the solution is on paper with details.