A feedback controller can be used to control the level of a fluid $(L)$ in a tank. The controller adjusts the feed flow rate $(W^{}),$ when the level setpoint is changed $(L_{sp}^{\text{'}})$ by a process control technician. The level deviation $(L^{\text{'}})$ in response to set point changes for three different feedback control loop designs is described by:

$\frac{L^{\text{'}}}{L_{sp}^{\text{'}}}=G_1=\frac{1\mathrm{.}1(15s+1)}{(10s+1)(2\mathrm{.}5s+1)}$

$\frac{L^{\text{'}}}{L_{sp}^{\text{'}}}=G_2=\frac{0\mathrm{.}9e^{-10s}}{8s+1}$

$\frac{L^{\text{'}}}{L_{sp}^{\text{'}}}=G_3=\frac{1}{16s^2+2s+1}$

When a process control technician changes the set point by $+5$ degrees:

$L_{sp}^{\text{'}}(s)=\frac{5}{s}$

$($a$)$ Plot the level deviation, and the responses of all three designs $($in the same plot$).$ Use a time vector over the domain $t=[0,50],$ containing at least $101$ data points.

$($b$)$ Discuss one advantage and one disadvantage of each control scheme relative to the others. You might refer to the speed of the response, oscillatory behavior, and the ability of the level to reach the set point at long time.

$($c$)$ Discuss the features of the transfer function that you observe in the responses that you discussed in time $b,$ above.