Assignment 5 Solutions 2009

# 46 448 ti 4112 2467 td 617

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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.

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