Assignment_4_2007_Solutions

67 c andtheyremainalwaysposit ivefornegat ivevaluesof

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )....
View Full Document

Ask a homework question - tutors are online