A small water reservoir depicted in Fig. 1 has a cylindrical shape with the radius of...

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A small water reservoir depicted in Fig. 1 has a cylindrical shape with the radius of 0.1m and its output pipe has the radius of 0.005m. During exploitation, the reservoir water level Y changes in the range between 0.05m and 0.15m where the maximal volume flow rate is qin = 1.5-10-4m/s. The nonlinear model for the rate of change of the level is = gin S S - A (5 A 2AgY A)in +5 A where S = 0.1 m, A = (0.005) and g = 9.81kg. (a) If the reservoir level is at the constant level Y* = 0.1, what is the corre sponding constant flow rate q? Explain your result and use the above provided values to compute in (b) If u = din-qin is a small variation of din around in use din, Y* from (a) and the above provided values to find a linear differential equation for the corresponding small variation of y=Y-Y*. (c) Apply Laplace transform to the linear differential equation from (b) to obtain the transfer function from u to y and provide the plot of the cor- responding magnitude frequency characteristic. The vertical axis of the plot should be 20log10 and the horizontal one should be logio(w). gin 1 (jw) (d) Write a matlab script that computes an array din for an array of Y* levels that are in the range [0.05,0.15] (use 30 equally spaced levels from the range). Provide the plot of Y* values (x-axis) vs. din values (y-axis). [0 (1) (e) For all the 30 pairs (q,Y*) from (d), provide magnitude frequency char- y (jw) acteristics on a single plot. The vertical axis of the plot should be 20log10 ( and the horizontal one should be logio(w). Area S Figure 1: Area A V2 A small water reservoir depicted in Fig. 1 has a cylindrical shape with the radius of 0.1m and its output pipe has the radius of 0.005m. During exploitation, the reservoir water level Y changes in the range between 0.05m and 0.15m where the maximal volume flow rate is qin = 1.5-10-4m/s. The nonlinear model for the rate of change of the level is = gin S S - A (5 A 2AgY A)in +5 A where S = 0.1 m, A = (0.005) and g = 9.81kg. (a) If the reservoir level is at the constant level Y* = 0.1, what is the corre sponding constant flow rate q? Explain your result and use the above provided values to compute in (b) If u = din-qin is a small variation of din around in use din, Y* from (a) and the above provided values to find a linear differential equation for the corresponding small variation of y=Y-Y*. (c) Apply Laplace transform to the linear differential equation from (b) to obtain the transfer function from u to y and provide the plot of the cor- responding magnitude frequency characteristic. The vertical axis of the plot should be 20log10 and the horizontal one should be logio(w). gin 1 (jw) (d) Write a matlab script that computes an array din for an array of Y* levels that are in the range [0.05,0.15] (use 30 equally spaced levels from the range). Provide the plot of Y* values (x-axis) vs. din values (y-axis). [0 (1) (e) For all the 30 pairs (q,Y*) from (d), provide magnitude frequency char- y (jw) acteristics on a single plot. The vertical axis of the plot should be 20log10 ( and the horizontal one should be logio(w). Area S Figure 1: Area A V2