A two-tank system containing a heated liquid has the model, where T() is the temperature of the fluid flowing into the first tank and T2 is the temperature of the liquid flowing out of the second tank. The block diagram model is shown in Figure DP9.9(b). The system of the two tanks has a heater in tank 1 with a controllable heat input Q. The time constants are n = 10s and T2 = 50 s.
(a) Determine T2(s) in terms of TQ(s) and T2d(s).
(b) If T2d(s), the desired output temperature, is changed instantaneously from T2d(s) = A/s to T2(T) - 2A/s, determine the transient response of T2(t) when Gc(s) = K = 500. Assume that, prior to the abrupt temperature change, the system is at steady state.
(c) Find the steady-state error ess for the system of part (b), where E(s) = T2d(s) - T2(s).
(d) Let Gc(s) = K/s and repeat parts (b) and (c). Use a gain K such that the percent overshoot is less than 10%.
(e) Design a controller that will result in a system with a settling time (with a 2% criterion) of Ts

(f) Prepare a table comparing the percent overshoot, settling time, and steady-state error for the designs of parts (b) through (e).

G(s) = Kp + K,