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Material Type: Notes; Professor: Zhu; Class: Control Systems; Subject: Mechanical Engineering; University: Michigan State University; Term: Spring 2009;
Typology: Study notes
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2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Root locus, sketching algorithm
-^
Root locus, examples
-^
Root locus, proofs
-^
Root locus, control examples
-^
Root locus, influence of zero and pole
-^
Root locus, lead lag controller design
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
W. R. Evans developed in 1948.
-^
Pole location
characterizes the feedback system
stability
and
transient properties
.
-^
Consider a feedback system that has one parameter (gain) K
> 0 to be designed.
-^
Root locus
graphically shows how poles of CL system varies
as
K
varies from 0 to infinity.
L^ L
( s ( s
):):
openopen
2009 Spring ME451 - GGZ
Page 4
Week 10-11: Root Locus
-^
Characteristic eq.
-^
It is hard to solve this analytically for each
K
.
-^
Is there some way to
sketch a rough
root locus by hand?
(In Matlab, you may use command “
rlocus.m”.
)
s
s s
s
s K
s
s s
s
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Root locus is symmetric w.r.t. the real axis.
-^
The number of branches = order of
L
( s
)
-^
Mark poles of L with “x” and zeros of L with “o”.
Re^ Re
Im^ Im
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Number of asymptotes = relative degree
(
r )
of
L
:
-^
Angles of asymptotes are
r
k
k
r
{
{
)
deg(
deg(den)
:^
m num
n
r
−
=
2009 Spring ME451 - GGZ
Page 8
Week 10-11: Root Locus
-^
Intersections of asymptotes
Asymptotes^ Asymptotes (Not root locus)^ (Not root locus)
Re^ Re
Im^ Im
(cont(cont
’d)’d)
s
s s
s
s L
r
zero
pole
2
2
) 1 ( )) 3 ( ) 2 ( (^0) (
zero
pole
r
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
Quotient rule:
(cont(cont
’d’d
1)1) 2
)) ( (
s D
ds
s D d s N s D
ds
s N d
ds
s D
s N
d
2
2
3
2 2
2
s
s s
s
s
s
s
s s
s s s s s s
ds
s
s s
s
d
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
Re^ Re
Im^ Im
Breakaway point^ Breakaway point
(cont(cont
’d’d
2)2)
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Asymptotes– Relative degree 2– Asymptote intersection
-^
Breakaway point
Re^ Re
Im^ Im
2
s s
s
ds
s
dL
1
2
) 2 (
0
−
s
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Root locus– What is root locus– How to roughly sketch root locus
-^
Sketching root locus relies heavily on experience. PRACTICE!
-^
To accurately draw root locus, use Matlab.
-^
Next, more examples
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Consider a feedback system that has one parameter (gain) K
> 0 to be designed.
-^
Root locus
graphically shows how poles of a CL system
varies as
K
varies from 0 to infinity.
L^ L
( s ( s
):):
openopen
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
Re^ Re
Im^ Im
-^
Root locus is symmetric w.r.t. the real axis.
-^
The number of branches = order of L
( s
)
-^
Mark poles of L with “x” and zeros of L with “o”.
2009 Spring ME451 - GGZ
Week 10-11: Root Locus
-^
Number of asymptotes = relative degree
(
r )
of L:
-^
Angles of asymptotes are
r
k
k
r
{
{
)
deg(
deg(den)
:^
m num
n
r
−
=
2009 Spring ME451 - GGZ
Page 20
Week 10-11: Root Locus
Asymptote (Not^ Asymptote (Not
root locus) root locus)
Re Re
Im Im
(cont(cont
’d)’d)
-^
Intersections of asymptotes
s
s s
s L
r
zero
pole
2
3
) 0 ( )) 5 ( ) 1 ( (^0) (
zero
pole
r
