Consider cascade control as shown below: with

$G_v=\frac{5}{s+1}$;$G_{p1}=\frac{4}{(4s+1)(2s+1)}$

$G_{p2}=1$;$G_{d2}=1$;$G_{m1}=0\mathrm{.}05$

$G_{m2}=0\mathrm{.}2$;$G_{d1}=\frac{1}{3s+1}$

Derive the closed$-$loop transfer functions for $\frac{Y_1}{D_1}$ and $\frac{Y_1}{D_z}$ as a function of $K_{c1}$ and $K_{c2}$

Examine the effect of $K_{c2}$ on the critical gain of $K_{c1}$ by varying $K_{c2}$ from $1$ to $20.$ Do a comparative analysis on what values of $K_{c2}$ do the benefits of cascade control seem to be less important?

Suppose a unit step change in secondary disturbance variable $D_2$ occurred, calculate the resulting offset assuming $K_{c2}=4$

Suppose if instead of a proportional controller, PI controller is used in $G_{c1}$ with ${}_{I1}=5$ mins, calculate the ultimate controller gain. Provide your analysis on whether the offset with a unit step change in $D_2$ be avoided with PI controller

I need appropriate results please find the value of ultimate gain and offset for PI