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The same control system shown in Fig. 13.k is to be considered. This time the set point remains fixed; that is, Ti = 0. We am interested in the response of the system to a change in the inlet stream temperature, i.e., to a load change. Using the methods of Chap. 12, the overall transfer function becomes 1 _ = AA- /(TS + 1) T 1 + K,A/(m + 1) = TS + 1 + KJ Ti This may be arranged in the form of the first-order lag; thus T A2 -=Ti where A2 = 1 1 + K,A 71s + 1 (13.6) (13.5)
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l = 1 + K,A As for the case of set-point change, we have an overall response that is first-order. The overall time constant ri is the same as for set-point changes. The response of the system to a unit-step change in inlet temperature Ti is shown in Fig. 13.3. It may be seen that T approaches l/( 1 + K,A). To demonstrate the benefit of control, we have shown the response of the tank temperature (open-loop response) to a unit-step change in inlet temperature if no control were present; that is, K, = 0. In this case, the major advantage of control is in reduction of offset. From E5q. (13.3), the offset becomes Offset = T;(m) - T (m) = 0 - 1 + K A c 1 =1 + K,A
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(13.7)
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As for the case of a step change in set point, the offset is reduced as controller gain K, is increased.
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FIGURE 13-3 Unit-step response for load change (P control).
TRANSIENT RESPONSE OF SIMPLE CONTROL SYSTEMS