Question $4(15$ pts$)$: For the block flow diagram:

a$)$ Derive the "closed loop set output function for a set point change to the process variable

and determine the characteristic equation. $(10$ pts$)$

b$)$ Assuming the controller transfer function $G_c(s)=K_c.$ Further, assume the process

transfer function is for a first order system that has very little deadtime and a process gain

of $1\mathrm{.}0$ and a time constant of $1\mathrm{.}0.$ Further, assume the sensor has very fast measurment

dynamics. Determine the characteristic equation and find values of $K_c$ where the process

is stable. $(5pts$