Root Locus Notes
• Writing the characteristic equation:
1+K
N (s)
= 0.
D(s)
• Expanded factored form:
"Q
#
m
(s
−
z
)
N (s)
i
1+K
= 1 + K Qni=1
= 0.
D(s)
j=1 (s − pj )
• The real-axis portion of the root locus is determined solely by the number and the
position of real-axis poles and zeros.
• If there are no poles or zeros on the real axis, then there will be no real-axis
portions of the root locus.
• For systems with more poles than zeros, the number of asymptotes is:
n − m.
• If there are no asymptotes, then the number of poles equals the number of zeros.
• Asymptotes are symmetric about the real axis.
• The centroid of the asymptotes is:
P
σA =
• Asymptote angles:
ϕA =
P
pi − zi
.
n−m
(2ℓ + 1)180◦
,
n−m
ℓ = 0, 1, 2, . . .
• Break points occur when two or more loci join and then diverge.
Root Locus General Notes
• Break points may occur on or off the real axis.
• A point is a break point if K is positive and real when:
dK
= 0.
ds
• There will always be an even number of loci around a break point since one locus enters
and another leaves.
1
• To compute a break point explicitly, require:
d
(1 + GH) = 0.
ds
• Angle criterion determines the direction of root movement as K increases.
• Angle of departure/arrival is computed at complex poles/zeros.
• Single real-axis poles or zeros always have departure/arrival angles equal
to 0◦ or 180◦ .
• Imaginary-axis intersections indicate the value of K at which the system
becomes marginally stable.
• The root locus may cross back and forth over the imaginary axis depending on system
order.
• If the system has more than two asymptotes, the root locus will definitely cross the
imaginary axis.
• Imaginary-axis crossings are likely if poles/zeros are close to the imaginary axis or if
angle departure/arrival points toward it.
• Overall locus shape depends heavily on pole-zero proximity.
Table 7.1: Seven Steps to Sketch the Root Locus
Step
Procedure
Related Equation or Rule
1.
(a) Write the characteristic equation so that
K appears explicitly.
(b) Factor the open-loop transfer function in
terms of poles and zeros.
1+K
N (s)
=0
D(s)
m
Y
(s − zi )
1 + K i=1
n
Y
(s − pj )
j=1
2.
Locate all open-loop poles and zeros on the
s-plane.
2
Root locus starts at open-loop
poles and ends at open-loop zeros.
3.
Determine the number of separate loci:
Equals the number of asymptotes
when n > m.
SL = n − m.
4.
Use symmetry: the root locus is symmetric Complex poles and zeros occur in
about the real axis.
conjugate pairs.
5.
Determine real-axis segments of the locus. Real-axis root-locus rule.
The root locus lies on any real-axis interval
that is to the left of an odd number of realaxis poles and zeros.
6.
Determine asymptote intersection and angles
(when n > m).
7.
Find imaginary-axis crossings to determine Use the Routh–Hurwitz criterion
the value(s) of K for marginal stability.
applied to the characteristic equation.
8.
Compute breakaway and break-in points along
the real axis.
P
P
pi − zi
σA =
n−m
(2q + 1)180◦
, q = 0, 1, 2, . . .
ϕA =
n−m
• Express K in terms of s from the
characteristic equation.
• Solve
9.
10.
Compute departure and arrival angles at complex poles and zeros using the angle criterion.
Sketch the final root locus using all information above (poles, zeros, symmetry, asymptotes, break points, crossings, and angles).
3
dK
= 0 for real s.
ds
∠P (s) = 180◦ ± 360◦ k,
k∈Z