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Assignment 5 Solutions 2009

# 5 0 79 4 s 5 2 s 1 6 0 9 3 s 3

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Unformatted text preview: sfer funct ion) 4. Analyze the set­point responses. 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 feedback s) ) control loop shown in Figure 7. y sp u g c ( = k s) c g p ( s = ) - (1. 5 + 0. 79) 4 s 5 2 s + 1. 6 + 0. 9 3 s 3 Figure 7: Closed­loop block diagram for Exercise IV.1. The closed­loop transfer funct ion between the output and set­point is given by, g p g c g sp = 1 + g p g c Subst ituting for the values in our example, - (1 45 + 0 579 . s . ) k c 2 s + 1 36 + 0 39 . s . g sp = - (1 45 + 0 579 . s . ) 1 + 2 k c s + 1 36 + 0 39 . s . g sp = g sp = g sp = - (1 45 + 0 579 c . s . )k 2 ( + 1 36 + 0 39 - (1 45 + 0 579 c s . s . ) . s . ) k - (1 45 + 0 579 c . s . )k 2 s + 1 36 + 0 39 - 1 45 c s + 0 579 c . s . . k . k - (1 45 + 0 579 c . s . )k 2 s + (1 36 - 1 45 c ) + (0 39 - 0 579 c ) . . k . . k The characteristic equation is then given by: y s 2 + (1. 6 - 1. 5 c ) + ( . 9 - 0. 79 c ) = 0 3 4 k 0 3 5 k The fo llowing can be observed: · The clo sed­loop system remains second order with the inclusio n o f a proportiona l controller · The two closed­loop system po les are now funct ion of the controller parameter k c · The independent term o f the po lyno mial beco me negative for k larger than 0.67 and the y c remain always posit ive for negative values of k c Using sfb_Tool, we obtain two Root­Locus plots shown in Figures 8 and 9 for a P­controller depending on the sign of the controller gain. In the case where the controller gain is posit ive, the system may become unstable for gains larger than 0.67. On the other hand, when a negative controller gain (reverse act ion) is introduced, one can observe no stabilit y limitat ions for the controller gain. Figure 8: Root­Locus for P­Controller (the dot at k c = 0.7 ). Figure 9: Root­Locus for P­Controller (the dot at k c = -1.2 ). Next, integral action is introduced. In the case o f the PI controller, different plots are produced for different controller parameters and are given in Figures 10 to 12. Figure 10:...
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