Consider a valve position control' $($VPC$)$ system. Suppose the transfer function between cooling water flow and column pressure is given as Gm $($s$)=(0\mathrm{.}1$ psi$/$gpm$)/$s$($s $+1).$

The span of the pressure transmitter is $50$ psi. The control valve has linear installed characteristics and passes $600$ gpm when wide open. A proportional$-$only pressure controller is used B$1($s$)=$ K$1.$ An integral$-$only valve position controller is used B$2($s$)=$ K$2/$s$.$

i$.$ Draw a block diagram of the VPC closed loop system.

ii$.$ What is the closedloop characteristic equation of the system?

iii. Considering only the pressure control loop, determine the value of the pressure controller gain K$1$ that gives a closedloop damping coefficient of $0\mathrm{.}707$ for the pressure control loop. Sketch a root locus plot for the pressure control loop.

iv$.$ Using the value of K$1$ found above and with the pressure controller on automatic, determine the VPC tuning constant $($K$2)$ that makes the entire VPC$-$pressure controller system critically damped, i$.$e$.$ that gives a closedloop damping coefficient of $1.$ Sketch the root locus plot for the VPC controller with the pressure controller in automatic.