HOMEWORK 6

Question 1A block diagram for internal model control, a control technique that we will see again in the near future, is shown in the Figure below. Transfer function denotes the process model while Gp denotes the actual process transfer function. It has been assumed that Gv=Gm=1 for simplicity. Derive closed-loop transfer functions for the disturbance and set point changes.

Question 2A control system has the following transfer functions in its block diagram shown below: Gc=1, Gv=2, Gd=Gp=, Gm=1. For a unit step change in Ysp determine:(a) Y(s)/Ysp(s)(b) y()(c) offset (note that proportional controller is being used)(d) y(0.5)(e) if closed-loop response is oscillatory

Question 3A temperature control system for a distillation column is shown in the figure below. The temperature T of a tray near the top of the column is controlled by adjusting the reflux flow rate R. Draw a block diagram for this feedback control system. You may assume that both feed flow rate F and feed composition xF are disturbance variables and that all of the instrumentation, including the controller, is pneumatic.

Question 4Consider the liquid-level PI control system in the Figure below with the following parameter values: A=3 ft2, R=1.0 min/ft2, Kv=0.2 ft3/min psi, Km=4mA/ft, Kc=5.33, KIP=0.75 psi/mA and I=3 min. Suppose that the system is initially at the nominal steady state with a liquid level of 2 ft. If the set point is suddenly changed from 2 to 3 ft, how long will it take for the system to reach (a) 2.5 ft and (b) 3 ft?

Question 5Consider proportional-only control of the stirred-tank heater control system whose block diagram is shown below. For this control loop, Km=0.32 mA/F. The controller has a gain of 5, while the gains for the control valve and current-to-pressure transducer are Kv=1.2 (dimensionless) and KIP=0.75 psi/mA, respectively. The time constant for the tank is =5 min. The control valve and the transmitter dynamics are negligible. After the set point is changed from 80 to 85 F the tank temperature eventually reaches a new steady-state value of 84.14 F, which was measured with a highly accurate thermometer.(a) What is the offset?(b) What is the process gain K2?(c) What is the pressure signal pt to the control valve at the final steady-state?

Question 6The block diagram of a special feedback control system is shown in the figure below. Derive an expression for the closed-loop transfer function Y(s)/D(s).