EE3530-5a

EE3530-5a - Chapter 5 The Root-Locus Design Method r 6 e...

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Chapter 5 The Root-Locus Design Method - - C ( s ) - P ( s ) - 6 r e u y - Closed-loop poles: 1 + P ( s ) C ( s ) = 0 . Denote L ( s ) := P ( s ) C ( s ) L ( s ) = K ( s - z 1 )( s - z 2 ) ··· ( s - z m ) ( s - p 1 )( s - p 2 ) ··· ( s - p n ) where z 1 ,...,z m are the open-loop zeros, p 1 ,...,p n are the open- loop poles, and K is a variable gain. Objective: study how the closed-loop poles change when K varies from 0 to . 1
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Find all points satisfying L ( s ) = - 1 . magnitude condition : | L ( s ) | = 1 phase condition : 6 L ( s ) = (2 k + 1)180 o , k = 0 , ± 1 ,... Magnitude condition can always be satisfied by a suitable K 0. Phase condition does not depend on the value of K (but does depend on the sign of K ): 6 L ( s ) = m X i =1 6 ( s - z i ) - n X j =1 6 ( s - p j ) = (2 k + 1)180 o . Thus the key is to find all those points that satisfy the phase condition. 2
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Example : Consider L ( s ) = K ( s - z 1 )( s - z 2 ) ( s - p 1 )( s - p 2 )( s - p 3 ) , p 2 = p 1 . a pole is represented by a “ × a zero is represented by a “ ”. The phase of L ( s ) at a point s in the complex plane is 6 L ( s ) = 6 ( s - z 1 ) + 6 ( s - z 2 ) - 6 ( s - p 1 ) - 6 ( s - p 2 ) - 6 ( s - p 3 ) = φ 1 + φ 2 - α 1 - α 2 - α 3 . - 6 * × ± O y × × ² 9 Y K M 0 z 2 z 1 p 3 p 1 p 2 s α 1 α 2 φ 1 α 3 φ 2 σ 3
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Root Locus Rules: 0 K ≤ ∞ 1 . The root locus is symmetric with respect to the real axis. 2 . The root loci start from n poles p i (when K = 0) and approach the n zeros ( m finite zeros z i and n - m infinite zeros when K → ∞ ). 3
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EE3530-5a - Chapter 5 The Root-Locus Design Method r 6 e...

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