Consider the level control system shown in Fig T$3\mathrm{.}1,$ implemented with a computer whose inputs and outputs are calibrated in terms of full range $(100\%).$ The tank is $1m$ in diameter, while the valve on the exit line acts as a linear resistance with $R=6\mathrm{.}37mi\frac{n}{m^2}.$ The level transmitter has a span of $2\mathrm{.}0m$ and an output range of $0$ to $100\%.$ The valve characteristic $f$ of the equal percentage control valve is related to the fraction of lift $C_V$ by the relation $f=(30{)}^{CV-1}.$ The air$-$to$-$open control valve receives a $3$ to $15$ psi signal from an $\frac{\text{I}}{P}$ transducer, which, in turn, receives a $0$ to $100\%$ signal from the computer$-$implemented proportional$-$only controller. When the control valve is fully open $)=(1,$ the flow rate through the valve is $0\mathrm{.}2\frac{m^3}{m}in.$ At the nominal operating condition, the control valve is half$-$open $)=(0\mathrm{.}5.$ Using the dynamic model in the block diagram in Fig T$3\mathrm{.}2,$ calculate the following:

a$)$ Process transfer function

b$)K_m,K_{IP}$ and $K_V$

$\backslash$table$[[Kc,{}_{\text{I}},K_{\text{I}}$