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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . +- R ( s ) C ( s ) K G ( s ) G ( s ) c o n t r o l l e r p l a n t c p H ( s ) Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 28 1 Reading: Nise: Chapter 8 1 Root Locus Development (contd. from Lecture 27) 1.1 Behavior of the Root Locus as the Gain K Becomes Large (contd.) For a closed-loop system with open-loop transfer function G ( s ) = KG c ( s ) G p ( s ) H ( s ), we saw in Lecture 27 that the asymptotic angles are summarized in the following table: n m 1 2 3 4 Asymptote Angles 180 90 , 270 60 , 180 , 300 45 , 135 , 225 , 315 1 copyright c D.Rowell 2008 281 n - m = 1 1 8 0 o n - m = 2 2 7 0 o 9 0 o 1 8 0 o 3 0 0 o 6 0 o n - m = 3 n - m = 4 4 5 o 1 3 5 o 2 2 5 o 3 1 5 o We now refine this property a little further. Let s m + b m 1 s m 1 + b m 1 s m 2 + ... + b ( s z 1 )( s z 2 ) ... ( s z m ) G ( s ) = K = K . s n + a n 1 s n 1 + a n 1 s n 2 + ... + a ( s p 1 )( s p 2 ) ... ( s p n ) For any polynomial of degree k , the coecient of the term in s k 1 is the sum of the roots of the polynomial , so that b m 1 = ( z 1 + z 2 + ... + z m ) a n 1 = ( p 1 + p 2 + ... + p n ) ....
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lecture_28 - MIT OpenCourseWare http://ocw.mit.edu 2.004...

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