$(47\%)$ A process to be controlled has the transfer function

$G_P=\frac{1}{s^2+s+0\mathrm{.}25}$

The final control element has the transfer function

$G_v=\frac{1}{0\mathrm{.}5s+1}$

and the measurement element has the transfer function

$G_m=\frac{1}{0\mathrm{.}25s+1}$

a$)$ Please sketch a block diagram of the overall system, including the controller, the final control element, the measurement element, and the process to be controlled. Also include the setpoint; you do not need to include a disturbance.

b$)$ Please characterize the damping behavior of the process to be controlled $($defined by $G_P).$

Answer: $G_P$ is critically damped $(=1).$

c$)$ Please develop the overall transfer function from the setpoint $Y_{SP}$ to the process variable $Y$; write this in terms of the $G\text{'}$s$.$

Answer:

$\frac{Y}{Y_{SP}}=\frac{G_C{G}_V{G}_P}{1+G_C{G}_V{G}_P{G}_m}$

d$)$ It has been suggested using proportional control with $K_C=1\mathrm{.}5.$ Is this a good suggestion? Explain your answer, supported with appropriate calculations.

Answer: Suggestion is reasonable. Routh array indicated the

characteristic equation has no roots in the right half of the

complex plane.