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Unformatted text preview: + 0. 9 3 s 3 y 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 s 2 + 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: 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 closedloop system remains second order with the inclusio n of 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 mia l beco me negat ive for k larger than 0.67 c
and they remain always posit ive for negat ive values of k c
Using sfb_Tool, we obtain two RootLocus plots shown in Figures 16 and 17 for a P controller depending on the sign o f the controller gain. In the case where the controller gain is posit ive, the system may beco me unstable for gains larger then 0.67. On the other hand, when a negat ive controller gain (reverse actio n) is introduced, one can observe no stabilit y limitat ions for the controller gain. Figure 16: RootLocus for PController (the dot at k c = 0.7 )....
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 Fall '08
 Hjortso,M

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