1. No category

What is Root Locus ?
The characteristic equation of the closed-loop system is
1 + K G(s) = 0
The root locus is essentially the trajectories of roots of the
characteristic equation as the parameter K is varied from 0
to infinity.
A simple example
A camera control system:
How the dynamics of the camera changes as K is varied ?
A simple example (cont.) : pole locations
A simple example (cont.) : Root Locus
(a) Pole plots from the table.
(b) Root locus.
The Root Locus Method (cont.)
• Consider the second-order system
• The characteristic equation is:
s   1  KGs   1 
K
0
s s  2
s   s 2  2 s  K  s 2  2 n s  n2  0
The locus of the roots as the gain K is varied is found by requiring :
K
G s  
1
s s  2 
and
G s   180,540,...
Introduction
+

x
Y(s)
G (s)
K

 2
X(s) 1  G (s) s  2s  K
K
s(s  2)
y
s1 , s2   n  n  2  1
For   1 we know that   cos -1
s 2  2s  K  0
Characteristic equation
j
K>1
-2
x
K=0

x
K=1
K>1
0  K 1
Roots  1  1  K
1 K  
Roots  1  j K  1
K=0
S plane
The Root Locus Method (cont.)
• Example:
– As shown below, at a root s1, the angles are:
 K 
K



1
 ss  2  s  s1 s1 s1  2
 K  s1 s1  2
s1 : the magnitude of the vector from the origin to s1
s1  2 : the magnitude of the vector from - 2 to s1
The Root Locus Method (cont.)
• The magnitude and angle requirements for the root locus are:
F s  
K s  z1 s  z 2 ...
s  p1 s  p2 ...
1
F s   s  z1   s  z 2   ...  s  p1   s  p2   ...  180  q360
q : an integer
• The magnitude requirement enables us to determine the value of K
for a given root location s1.
• All angles are measured in a counterclockwise direction from a
horizontal line.
Root locus
R
C
+
-
G(s)
C
G
F 
R 1  GH
H(s)
KN(s) K(s m  a m1s m 1  ......a 0 )
GH 

Open loop transfer function
n
D(s)
s  b n 1  .......b 0
mn
G
GD
F

1  KN / D D  KN
Closed loop transfer function
The poles of the closed loop are the roots of
The characteristic equation
D(s)  KN(s)  0
Root locus (Evans)
P(s)  D(s)  KN(s)  0
K0
• Root locus in the s plane are dependent on K
• If K=0 then the roots of P(s) are those of D(s): Poles of GH(s)
• If K= then the roots of P(s) are those of N(s): Zeros of GH(s)
Open loop
poles
K=
K=0
K=
K=
j
o
xK=0
K=0x
s plane
o
Open loop
zeros

Root locus example
K (s  1)
GH 
s(s  2)
H=1
K (s  1)
F(s)  2
s  s(K  2)  K
s1  
2K
 1 K2 / 4
2
s2  
2K
 1 K2 / 4
2
K>0
o
K=1,5
K=1,5
j
o
-1
x
s2
-2
Zero
x

s1
Pole
Root locus example
Any information from Rooth ?
F(s) 
K (s  1)
s 2  s(K  2)  K
s2
s1
s0
As K>0
1
K2
K
s 2  s(K  2)  K
K
0
0
Rules for plotting root loci/loca
•Rule 1:
Number of loci: number of poles of the open loop transfer
Function (the order of the characteristic equation)
Poles
K (s  2)
GH  2
X
O
XX
s (s  4)
-2
-4
Three loci (branches)
Zero
•Rule 2:
Each locus starts at an open-loop pole when K=0 and finishes
Either at an open-loop zero or infinity when k= infinity
Problem: three poles and one zero ?
Rules for plotting root loci/loca
• Rule 3
Loci either move along the real axis or occur as complex
Conjugate pairs of loci
j
K>1
-2
x
K=0

x
K=1
K>1
K=0
S plane
Rules for plotting root loci/loca
• Rule 4
A point on the real axis is part of the locus if the number of
Poles and zeros to the right of the point concerned is odd for K>0
K>0
o
K=1,5
K=1,5
j
o
-1
x
s2
-2
Zero
x

s1
Pole
Rules for plotting root loci/loca
Example
GH 
Poles
K (s  2)
s 2 (s  4)
X
Three loci (branches)
O
-2
-4
Zero
Locus on the real axis
XX
No locus
Rules for plotting root loci/loca
•Rule 5:
When the locus is far enough from the open-loop poles and zeros,
It becomes asymptotic to lines making angles to the real axis
Given by: (n poles, m zeros of open-loop)
There are n-m asymptotes
(2L  1)180
L=0,1,2,3…..,(n-m-1)

nm
Example
K
s(s  2)
m  0, n  2
G (s) 
180
(L  0)  90
nm
3 *180
(L  1)  270
nm
Rules for plotting root loci/loca
j
K>1
-2
x
K=0

x
K=1
K>1
K=0
S plane
Rules for plotting root loci/loca
• Rule 6 :Intersection of asymptotes with the real axis
The asymptote intersect the real axis at a point  given by
n

m
p z
i 1
i
i 1
i
nm  4
nm
j
pi  pole
zi  zero


Rules for plotting root loci/loca
• Example
K (s  2)
GH  2
s (s  4)
42

 1
2
X
-4
O -1 XX
-2
Rules for plotting root loci/loca
• Rule 7
The break-away point between two poles, or break-in point
Between two zero  is given by:
X
X
O
O
Break-in
Break-away
X
X
O
O


n
First method
X
m
1
1


i 1   p i
i 1   z i
Rules for plotting root loci/loca
Example
K
G (s) 
s(s  1)(s  2)
-2
x
3 asymptotes: 60 °,180 ° and 300°
1  2

 1
3
Break-away point
1
1
1


0
  1   2
-1
x
-0.423
Part of real
axis excluded
32  6  2  0
  0.423,1.577
x
Rules for plotting root loci/loca
Second method
The break-away point is found by differentiating V(s) with
Respect to s and equate to zero
1
v(s) 
G (s)
Example
K
G (s) 
s(s  1)(s  2)
s(s  1)(s  2) s3  3s 2  2s
V(s) 

K
K
dV 3s 2  6s  2

0
ds
K
s  1.577,0.423
Rules for plotting root loci/loca
• Rule 8: Intersection of root locus with the imaginary axis
The limiting value of K for instability may be found using
the Routh criterion and hence the value of the loci at the
Intersection with the imaginary axis is determined
Characteristic equation
Example
G (s) 
N ( j1 )
1 K
0
D( j1 )
K
s(s  1)(s  2)
Characteristic equation
s 3  3s 2  2s  K  0
Rules for plotting root loci/loca
s 3  3s 2  2s  K  0
What do we get with Routh ?
s3
1
2
s2
3
K
s1
6K
0
s0
K
0
If K=6 then we have an pure imaginary solution
Rules for plotting root loci/loca
Example
s 3  3s 2  2s  K  0
s  j
 j3  32  2 j  K  (K  32 )  j(2  2)  0
2  2
K  32  6
j 2
K=6
K=0
-2
x
-1
x
x
-0.423
j 2
Rules for plotting root loci/loca
• Rule 9
Tangents to complex starting pole is given by
 S  180  arg( GH ' )
GH’ is the GH(starting p) when removing starting p
Example
GH 
K (s  2)
(s  1  j)(s  1  j)
K (1  j)
GH ' (s  1  j) 
(2 j)
ArgGH '  45  90  45
X
 S  180  45  135
-1
X
j
-j
Rules for plotting root loci/loca
• Rule 9`
Tangents to complex terminal zero is given by
 T  180  arg( GH ' ' )
GH’ is the GH(terminal zero) when removing terminal zero
Example
GH 
K (s  j)(s  j)
s(s  1)
K (2 j)
GH ' (s  j) 
j(1  j)
ArgGH '  45
O j
 S  180  (45 )  225
O -j
Rules for plotting root loci/loca
Example
Two poles at –1
One zero at –2
One asymptote at 180°
Break-in point at -3
K (s  2)
GH (s) 
(s  1) 2
2
1

 1   2
K=2
2  4    1
K=4
O
-2
=-3
K=2
K=0
XX
-1
Rules for plotting root loci/loca
Example Why a circle ?
Characteristic equation
s 2  s(2  K)  2K  1  0
For K>4
For K<4
s1, 2
 (2  K )  j K (4  K )

2
Change of origin
s1, 2
 (2  K )  K (K  4)

2
 (2  K )  j K (4  K )
s1, 2  2 
2
4m  (K  2) 2  K(4  K)  K 2  4K  4  4K  K 2
m 1
Rules for plotting root loci/loca
K=2
K=4
O
-2
=-3
K=2
K=0
XX
-1