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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- Spring '08
- evens
- 1 m, 1J, Breakway, 0381K, 386K
-
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