Assignment_4_2007_Solutions

# 67 c andtheyremainalwaysposit ivefornegat ivevaluesof

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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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