(30 points) Problem 3 You are asked to evaluate a proportional controller (Kc) that is d control the liquid level in a tank. The outlet flow rate qout is determined by Cvh where Cv is valve coefficient. The inlet flow rate qin depends on the difference between the measured liquid level h and the set point value of the h (i.e, hsp ). Adtdh=qinqoutqin=Kc(hsph)+100qout=Cvh 1) (10 points) Derive the differential equation where hsp is the input and h is the output. Linearize the model at the steady state condition with h=hsp=4 meters, Cv=50,A =5m2. Assume Kc is equal to 2.5 . 2) (10 points) Determine the transfer function from the input hsp to the output h. You can keep Kc in your solution. 3) (10 points) Assume the controller's Kc is equal to 2.5. Calculate the steady state value of h if hsp is changed from 4 to 5 meters at t=0. Is it equal to the set point value (i.e., 5 meters)? Hints: 1) plug in the term for qin and qout into the differential equation for dh/dt before the linearization; 2) use the gain to determine the new steady state value of h by using the gain of the transfer function. (30 points) Problem 3 You are asked to evaluate a proportional controller (Kc) that is d control the liquid level in a tank. The outlet flow rate qout is determined by Cvh where Cv is valve coefficient. The inlet flow rate qin depends on the difference between the measured liquid level h and the set point value of the h (i.e, hsp ). Adtdh=qinqoutqin=Kc(hsph)+100qout=Cvh 1) (10 points) Derive the differential equation where hsp is the input and h is the output. Linearize the model at the steady state condition with h=hsp=4 meters, Cv=50,A =5m2. Assume Kc is equal to 2.5 . 2) (10 points) Determine the transfer function from the input hsp to the output h. You can keep Kc in your solution. 3) (10 points) Assume the controller's Kc is equal to 2.5. Calculate the steady state value of h if hsp is changed from 4 to 5 meters at t=0. Is it equal to the set point value (i.e., 5 meters)? Hints: 1) plug in the term for qin and qout into the differential equation for dh/dt before the linearization; 2) use the gain to determine the new steady state value of h by using the gain of the transfer function