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

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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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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