a$.$ Design a cascade controller for the following loop transfer functions and draw the biock diagram, $G_1(s)=\frac{100}{(s+1)(1+2)}$ and $G_7(s)=\frac{3}{(s+10)}$

Justify the selection of inner loop and outer loop transfer function. $($S$)$

$6.$ Design a proportional controller for the Inner loop with the gain of $1.$ Find out p$-1$ control sethings for outer loop. $($Note: Use Z$-$N closed loop tuning procedure$)(4)$

C$.$ Find out the characteristics polynomial of cascade controller for $4(a)$ and comment on its stabily $(4)$

d$.$ Design a feed forward cascade controller for the above process w th the known disturbance transfer function $G_d(s)=\frac{10}{0\mathrm{.}5s+1},$ Draw the block diagram. Explain the working principle as well as write down necessary conditions to be followed for implementing the above control schere. $(7)$