What is Root Locus
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Transcript What is Root Locus
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
ss 2
s s 2 2s K s 2 2 n s n2 0
T helocus of t heroot sas t hegain K is variedis found by requiring:
G s
K
1
ss 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 t hat cos-1
s 2 2s K 0
Characteristic equation
j
K>1
-2
x
K=0
x
K=1
K>1
0 K 1
Root s 1 1 K
1 K
Root s 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 : t hemagnit udeof t he vect orfrom t heorigin t os1
s1 2 : t hemagnit udeof t he vect orfrom - 2 t o 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 m1s m1 ......a 0 )
GH
Open loop transfer function
n
D(s)
s b n 1 .......b0
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
K0
• 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
2K
1 K2 / 4
2
s2
2K
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
K2
K
s2 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)
nm
Example
K
s(s 2)
m 0, n 2
G (s)
180
(L 0) 90
nm
3 *180
(L 1) 270
nm
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
nm 4
nm
j
pi pole
z i zero
Rules for plotting root loci/loca
• Example
K (s 2)
GH 2
s (s 4)
42
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
32 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) s 3 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( j1 )
1 K
0
D( j1 )
K
s(s 1)(s 2)
Characteristic equation
s3 3s 2 2s K 0
Rules for plotting root loci/loca
s3 3s 2 2s K 0
What do we get with Routh ?
s3
s1
1
3
6K
s0
K
s2
2
K
0
0
If K=6 then we have an pure imaginary solution
Rules for plotting root loci/loca
Example
s3 3s 2 2s K 0
s j
j3 32 2 j K (K 32 ) j(2 2) 0
2 2
K 32 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) K2 4K 4 4K K2
m 1
Rules for plotting root loci/loca
K=2
K=4
O
-2
=-3
K=2
K=0
XX
-1