Summary of Rules for Plotting Root Locus
Rule #1 The Starting Points and the End Points
of the Root Locus (根轨迹的起点和终点 ）
Rule #2 The Segments of the Root Locus on the
Real Axis (实轴上的根轨迹 ）
Rule #3 The Symmetry and the Asymptotes of
the Root Locus (根轨迹的对称性和渐近线 ）
Rule #4 The Real Axis intercept of the
Asymptotes (渐近线和实轴的交点 ）
Rule #5 The Angle of Emergence from Complex
Poles and The Angle of Entry into Complex Zeros (根轨迹的出射角和入射角）
Rule #6 The Root Locus Crossing with the Imaginary
Axis (根轨迹与虚轴的交点）
Rule #7 The Breakaway Point of the Root Locus (根轨迹的分离点）
Rule #8 The angle between the direction of emergence
(or entry) of q coincident poles (or zeros) on the real axis
(根轨迹离开或进入实轴上 q重极点（或零点）方向之间的夹角）
Rule #9 The gain at a selected point s1 on the locus
(在某特定点上的根轨迹增益 K）
Rule #10 The sum of the closed-loop poles (闭环极点之和）
Rule #11 The number of branches of the root locus
(根轨迹的分支数）
Instructional objectives:
At the end of this lecture students
should be able to
Find the directions of the pole asymptotes
Find the values of K for which the poled
cross the imaginary axis
Determine the root locus and select,good”
values for K
Module 11
System Design Using the Root Locus
(2 hours)
)1(
2
1
)()
2
11
(
2
1
2
11
)()(
2
1
1
2
1
2
1
)1
34
10
34
1)(1
2
1(
2
1
)3410)(2(
34
)()(
2
22
tetC
ssss
sCc
ssssssss
sGb




)
5
4(K
)2;0;35( 132,1 zpjp
SP.4
1)(;
)10)(2(
)(?

sH
ss
KsG
What value of K make the system stable?
1)()(0)()(1 sHsGsHsG

3 6 00)()(
1)()(
isHsG
sHsG
Problem 1 The root locus of a system is shown below,Fromthe figure,estimate the lowest value of the loop
gain K,for which the system becomes unstable.
Problem 2
A system is expressed by the following transfer
function
)0(1)(;)106()( 2 EE KsHsss KsG
(1) Sketch the root-locus of the system.
(2) Find the value so that the closed-loop system has the fastest
response time,and sketch the unit step response curve in this case.
)816.2 183.13 660]106[( 23

dsssds
d
ds
dK
ds
Fig.1
Problem 3
Fig.1 shows the open-loop poles and zeros of a feed back
control system.
(1) Will the un-overshoot step response appear in the
system? (You can answer this question according to the
root-locus of the system.)
(2) Determine the
range of values of
which makes the
system stable.
Instructional objectives:
At the end of this session students should
be able to solve Simple problems on
Shaping the transient response of a second order
system
Meeting stability requirements using Routh array
Draw approximate root locus and select
appropriate gain.