Design a digital controller for the liquid level in the storage system shown in Fig. E. Each tank is 2.5 ft in diameter. The piping between the tanks acts as a linear resistance to flow with R = 2 min/ft². The liquid level is sampled every 30 s. The digital controller also acts as a zero-order hold device for the signal sent to the control valve. The control valve and level transmitter have negligible dynamics. Their gains are K_v = 0.25 ft³/min/mA and K_m = 8 mA/ft, respectively. The nominal value of q₁ is 0.5 ft³/min.

(a) Derive Dahlin?s control algorithm based on a step change in set point.

(b) Does the controller output exhibit any oscillation?

(c) For what values of ? is the controller physically realizable?

(d) If you were to tune this controller online, what value of ? would you use as an initial guess? Justify your answer.

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LC 42 23