Figure P9.6(a) shows a heat-exchanger process whose purpose is to maintain the temperature of a liquid at a prescribed temperature. The temperature is measured using a sensor and a transmitter, TT 22, that sends the measurement to a corresponding controller,TC22, that compares the actual temperature with a desired temperature set point, SP. The controller automatically opens or closes a valve to allow or prevent the flow of steam to change the temperature in the tank. The corresponding block diagram for this system is shown in Figure P9.6(b) (Smith, 2002). Assume the following transfer functions:

a. Assuming G_c(s) = K, find the value of K that will result in a dominant pole with ζ = 0.7. Obtain the corresponding T_s.

b. Design a PD controller to obtain the same damping factor as Part a but with a settling time 20% smaller.

c. Verify your results through MATLAB simulation.

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0.02 70 G,(s) G(s): H(s) = 4s +1 50s + 1 12s +1