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Unformatted text preview: real and imaginary terms:  625 2 + 1 + 1. c = 0 w 2 k  1500 + 70  6 c w = 0 w3
w k Solving these Equations simultaneously for kc and w, we get: kc = 7.61 (this is the ult imate kc) and wco = 0.127 2p 2 p
=
= 49 3 . wco 0 127 . Again, the next step is to determine the ZN controller settings using the ult imate gain and the ult imate period: Pu = kc P PI PID 3.80 3.46 4.48 t I
41.12 24.67 t d
6.17 To validate the results using the graphical approach, we implement the same setup (plant) within the APC_Tool environment. Using a proportional control, we increase the gain unt il we reach the sustained oscillat ions. Figure 4 illustrates this condit ion and the value o f the gain is 7. Consequent ly, fro m graphical approach k cu = 7 and fro m the same figure Pu = 50 . From these two values, the ZN controller settings for a PI controller are calculated as: k c = 3.15 and t I = 41 66 . These are very close to the values obtained using the analyt ical approach. Figure 4: Closedloop response with a P controller using k c = 7 . Figures 5 and 6, show the closedloop responses using the designed controller for both setpoint and disturbance changes. Figure 5: Closedloop setpoint response with a PI controller using k c = 3.15 and t I = 41 66 . . Figure 6: Closedloop disturbance response with a PI controller using k c = 3.15 and t I = 41 66 . . 2. For the bioreactor (Exercise 4, Assignment 2), we would like to control the bio mass concentration ( x1 ) using the dilution rate (D). 1. Sketch the feedback control loop 2. Find the closedloop transfer funct ion between the process output and the setpoint for a P controllers 3. Find the characterist ic equat ion 1. Analyze the stabilit y o f the closedloop system using RootLocus 2. Design a PI controller using ZN tuning rules. (Hint: introduce a small delay, co mpared to the process time constant, in the tran...
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This note was uploaded on 01/12/2014 for the course CHE 4198 taught by Professor Hjortso,m during the Fall '08 term at LSU.
 Fall '08
 Hjortso,M

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