EE3530-5

# EE3530-5 - Chapter 5 The Root-Locus Design Method r 6 e C(s...

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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
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 × × 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 . The root locus includes all points on the real axis to the left of an odd number of open loop real poles and zeros. 4 . As K → ∞ , n - m branches of the root locus approach asymp- totically to n - m straight lines (called asymptotes) with angles θ = (2 k + 1)180 o n - m , k = 0 , ± 1 , ± 2 , . . . and the starting point of all asymptotes is on the real axis at σ = n X i =1 p i - m X j =1 z j n - m = X poles - X zeros n - m . 5. The breakaway points (where the root loci meet and split away, usually on real axis) and the breakin points (where the root loci meet and enter the real axis) are among the roots of the equation: dL ( s ) ds = 0. (On real axis only those roots satisfy Rule 3 are breakaway or breakin points.) 4
6. The departure angle φ k (from a pole, p k ) is given by φ k = m X i =1 6 ( p k - z i ) - n X j =1 ,j 6 = k 6 ( p k - p j ) ± 180 o . (In the case p k is l repeated poles, the departure angle becomes φ k /‘ .) The arrival angle ψ k (at a zero, z k ) is given by ψ k = - m X i =1 ,i 6 = k 6 ( z k - z i ) + n X j =1 6 ( z k - p j ) ± 180 o . (In the case z k is l repeated zeros, the arrival angle becomes ψ k /‘ .) 5

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Example : Asymptotes and breakaway points are illustrated. Con- sider an open loop transfer function L ( s ) = K s ( s + 4)( s + 5) . - 6 × × × - 7 w ² I N 0 60 - 60 180 - 5 - 4 - 3 breakaway point the angles of the three asymptotes: θ = (2 k + 1)180 o 3 = 60 o , - 60 o , 180 o for k = 0 , - 1, and 1 and the intersection of the asymptotes with the real axis is given by σ = 0 - 4 - 5 3 = - 3 . Note that one could have set k = 0 , 1 , 2 to get θ = 60 o , 180 o , 300 o which are the same angles.
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