Assignment_4_2007_Solutions

S consequent ly 1 5 0 79 4 s 5 gps 2 s

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Unformatted text preview: s: Models were obtained before 4. For the bioreactor introduced in (Exercise 2, Assignment 2), we would like to contro l the bio mass concentration ( x ) using the dilut ion rate (D). 2 1. Sketch the feedback control loop 2. Find the clo sed­loop transfer funct ion between the process output and the set­ point for a P controllers 3. Find the characterist ic equat ion 4. Analyze the stabilit y o f the closed­loop system using Root­Locus 5. Design a PI controller using ZN tuning rules. (Hint: introduce a small delay, compared to the process time constant, in the transfer funct ion) 6. Analyze the set­point response. Solution: From Exercise II.7, we have b s + ( a - b a ) b y ( ) s = g p ( = 1 2 2 12 1 22 s) u ( s) s - a 2 s - a 2 a 1 2 12 Subst ituting the numerical values for the constants: y ( s) - (1 45 + 0 579 . s . ) = g p ( = 2 s) u ( s) s + 1 36 + 0 39 . s . Consequent ly, - (1. 5 + 0. 79) 4 s 5 g p ( s = 2 ) s + 1. 6 + 0. 9 3 s 3 We have g c ( = k c for a P­controller and assuming g f ( s = g m = 1 , we obtain the s) ) feedback control loop shown in the figure. y sp g c ( = k s) c u g p ( s = ) - (1. 5 + 0. 79) 4 s 5 2 s + 1. 6...
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