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Lec_11

# Lec_11 - Lecture 11 Root Locus Analysis Root Locus Analysis...

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1 Lecture 11: Root Locus Analysis Root Locus Analysis Reading: Chapter 7.6 Root Locus Method: Problem Formulation C ( s ) H ( s ) G ( s ) R ( s ) + ¡ T K Closed-loop poles are the solutions to the characteristic equation 1 + K ¢ GH ( z ) = 0 Typical open-loop transfer function GH ( z ) = ( z ¡ z 1 ) ¢¢¢ ( z ¡ z m ) ( z ¡ p 1 ) ¢¢¢ ( z ¡ p n ) ( m · n ) with m open-loop zeros z 1 ,…, z m and n open-loop poles p 1 ,…, p n How do the n closed-loop poles change as K varies from 0 to 1 ?

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2 Continuous Systems Case C ( s ) H ( s ) G ( s ) R ( s ) + ¡ K 1 + KG ( s ) H ( s ) = 1 + K ( s ¡ z 1 ) ¢¢¢ ( s ¡ z m ) ( s ¡ p 1 ) ¢¢¢ ( s ¡ p n ) = 0 Closed-loop poles are the solutions to the characteristic equation z 1 ; : : : ; z m p 1 ; : : : ; p n How do the n closed-loop poles change as K varies from 0 to 1 ? open-loop zeros open-loop poles Root Locus Overview The root locus of a characteristic equation consists of n branches Start from the n open-loop poles p 1 ,…, p n at K =0 m of them converge to the m open-loop zeros z 1 ,…, z m The other n - m diverge to 1
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