? ?
Handout 4: Root-Locus Review
Eric Feron
Feb 17, 2004
Summary of Guidelines for plotting a root-locus
1. Mark Poles X and Zeros O.
2. Draw the locus on the real axis to the left of an odd number of real
poles plus zeros.
3. Draw n ? m asymptotes (n is the number of poles, m the number of
zeros). The asymptotes are centered at α and leave at angles Φ
l
, where
z
i
α =
p
i
?
=
?a
1
+ b
1
,
n ? m n ? m
180
o
+ l360
o
φ
l
= , l = 0, 1, 2, . . . n ? m ? 1.
n ? m
4. Compute the loci departure angles from the poles and arrival angles
at the zeros.
5. Assume s
0
= jω
0
and compute the point(s) where the locus crosses
the imaginary axis for positive K.
6. The equation has multiple roots at points on the locus where
da db
b
ds
? a
ds
= 0.
If s
0
is on the real axis, these points are points of breakaway or break-
in. Compute the angles of arrival and the angles of departure for any
points of multiple roots.
7. Complete the locus, using the previous steps and your experience.
1
s + 1
G(s) =
s
2
(s + 4)
2
s + 1
G(s) =
s
2
(s + 12)
3
(s + 0.1)
2
+ 16
G(s) =
s((s + 0.1)
2
+ 25)
4