hwsln-421-s06-05

hwsln-421-s06-05 - ECE 421 Spring 2006 Solutions to HW...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 421, Spring 2006 Solutions to HW Assignment #5 Problem #B-5-23 From the given G ( s ) , the closed-loop transfer function and closed-loop characteristic equation are T CL ( s )= C ( s ) R ( s ) = G ( s ) 1+ G ( s ) = K s ( s +1)( s +2)+ K = K s 3 +3 s 2 +2 s + K (1) ( s s 3 s 2 s + K (2) and the Routh array is s 3 : 1 2 s 2 : 3 K s 1 : 6 K 3 s 0 : K For closed-loop stability, there must be no sign changes in the f rst column of the array. Checking the s 0 and s 1 rows (the only ones involving K ) produces the following limits on K for stability. s 0 : K> 0 ,s 1 :6 0 K< 6 (3) 0 <K< 6 (4) Problem #B-5-24 G ( s ) , the closed-loop transfer function and closed-loop characteristic equation are T ( s C ( s ) R ( s ) = G ( s ) G ( s ) = 10 s ( s 1) (2 s +3)+10 = 10 2 s 3 + s 2 3 s +10 (5) ( s )=2 s 3 + s 2 3 s (6) It is obvious by inspection of ( s ) that the closed-loop system is unstable since not all the coe cients have the same sign. Forming the Routh array to check this conclusion gives the following result. s 3 : 2 3 s 2 : 1 10 s 1 : 23 s 0 : 10 Since there are 2 sign changes in the f rst column of the array, there are 2 closed-loop poles in the right-half plane. Therefore, the closed-loop system is unstable. Problem #B-5-25 From the given closed-loop characteristic ( s s 4 s 3 +(4+ K ) s 2 +9 s +25=0 (7) the Routh array is 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
s 4 : 1 (4 + K ) 25 s 3 : 2 9 s 2 : 2 K 1 2 25 s 1 : 18 K 109 2 K 1 s 0 : 25 For closed-loop stability, there must be no sign changes in the f rst column, so the s 2 and s 1 rows must be checked. s 2 :2 K 1 > 0 K> 0 . 5 (8) s 1 :1 8 K 109 > 0 109 / 18 = 6 . 056 (9) Since the constraint from the s 1 row is more restrictive than the costraint from the s 2 row, the limit on K for closed-loop stability is 6 . 056 .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 8

hwsln-421-s06-05 - ECE 421 Spring 2006 Solutions to HW...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online