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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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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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