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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:
,...540,180 and 12
sG
:requiringby found is variedisK gain theas roots theof locus The
022
02
11
222
sGss
K
ssKsss
ss
KsKGs
nn
Introduction
)2s(s
K
+
-
x y
Ks2s
K
)s(G1
)s(G
)s(X
)s(Y2
Characteristic equation 0Ks2s2
1Kj1Roots
K1
K11Roots
1K0
S plane
xxK=0 K=0
-2
K=1
K>1
K>1
j
1-
221
cos that know we1For
1,
nnss
The Root Locus Method (cont.)• Example:
– As shown below, at a root s1, the angles are:
11
11
11
11
s to2- from vector theof magnitude the:2
s origin to thefrom vector theof magnitude the:
2
122
1
s
s
ssK
ss
K
ss
K
ss
The Root Locus Method (cont.)• The magnitude and angle requirements for the root locus are:
• 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.
integeran :
360180......
1...
...
2121
21
21
q
qpspszszssF
psps
zszsKsF
G(s)
H(s)
R +-
C
GH1
G
R
CF
01nn
01m
1mm
b.......bs
)a......sas(K
)s(D
)s(KNGH
nm
KND
GD
D/KN1
GF
The poles of the closed loop are the roots ofThe characteristic equation
0)s(KN)s(D
Root locus
Closed loop transfer function
Open loop transfer function
Root locus (Evans)
0K 0)s(KN)s(D)s(P
oK=
x x
o
K=0 K=0
K= j
s plane
• 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 looppoles
Open loopzeros
K=0 K=
Root locus example
)2s(s
)1s(KGH
H=1
4/K12
K2s 2
1
4/K12
K2s 2
2
K)2K(ss
)1s(K)s(F
2
K=
1,5
K=
1,5
j
x-1-2
x
K=
0
K=
0
K=
oo
K=
1s2sPoleZero
K>0
Any information from Rooth ?
Root locus example
K)2K(ss
)1s(K)s(F
2
K)2K(ss2
0
1
2
s
s
s
K
2K
1
0
0
K
As K>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)
)4s(s
)2s(KGH
2
Three loci (branches)
XXX
-4
O
-2
Poles
Zero
•Rule 2:Each locus starts at an open-loop pole when K=0 and finishesEither at an open-loop zero or infinity when k= infinity
Problem: three poles and one zero ?
Rules for plotting root loci/loca
• Rule 3Loci either move along the real axis or occur as complex Conjugate pairs of loci
S plane
xxK=0 K=0
-2
K=1
K>1
K>1
j
Rules for plotting root loci/loca
• Rule 4A point on the real axis is part of the locus if the number ofPoles and zeros to the right of the point concerned is odd for K>0
K=
1,5
K=
1,5
j
x-1-2
xK
=0
K=
0
K=
oo
K=
1s2sPoleZero
K>0
Rules for plotting root loci/loca
)4s(s
)2s(KGH
2
Three loci (branches)
XXX
-4
O
-2
Poles
Zero
Locus on the real axis No locus
Example
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 axisGiven by: (n poles, m zeros of open-loop)
mn
180)1L2(
There are n-m asymptotes
90)0L(mn
180
270)1L(mn
180*3
L=0,1,2,3…..,(n-m-1)
Example
2n,0m
)2s(s
K)s(G
Rules for plotting root loci/loca
S plane
xxK=0 K=0
-2
K=1
K>1
K>1
j
Rules for plotting root loci/loca• Rule 6 :Intersection of asymptotes with the real axisThe asymptote intersect the real axis at a point given by
j
4mn
mn
zpn
1i
m
1iii
polepi
zerozi
Rules for plotting root loci/loca
)4s(s
)2s(KGH
2
XXX
-4
O
-2
12
24
-1
• Example
Rules for plotting root loci/loca• Rule 7The break-away point between two poles, or break-in pointBetween two zero is given by:
X X XO O
m
1i i
n
1i i z
1
p
1
XX
O OBreak-away Break-in
First method
Rules for plotting root loci/locaExample
)2s)(1s(s
K)s(G
3 asymptotes: 60 °,180 ° and 300°
13
21
02
1
1
11
Break-away point
0263 2
577.1,423.0
Part of real
axis excluded
xxx-1-2
-0.423
Rules for plotting root loci/loca
Second method
The break-away point is found by differentiating V(s) withRespect to s and equate to zero
)s(G
1)s(v
)2s)(1s(s
K)s(G
Example
K
s2s3s
K
)2s)(1s(s)s(V
23
0K
2s6s3
ds
dV 2
423.0,577.1s
Rules for plotting root loci/loca
• Rule 8: Intersection of root locus with the imaginary axisThe limiting value of K for instability may be found using the Routh criterion and hence the value of the loci at theIntersection with the imaginary axis is determined
0)j(D
)j(NK1
1
1
Characteristic equation
)2s)(1s(s
K)s(G
Example
Characteristic equation 0Ks2s3s 23
Rules for plotting root loci/loca
0Ks2s3s 23
0
1
2
3
s
s
s
s
What do we get with Routh ?
K
K6
3
1
0
0
K
2
If K=6 then we have an pure imaginary solution
Rules for plotting root loci/loca
0Ks2s3s 23
js
0)2(j)3K(Kj23j 2223
63K
22
2
Example
xxx-1-2
-0.423
2j
2j
K=6K=0
Rules for plotting root loci/loca• Rule 9Tangents to complex starting pole is given by
)'GHarg(180S
GH’ is the GH(starting p) when removing starting p
Example )j1s)(j1s(
)2s(KGH
)j2(
)j1(K)j1s('GH
459045'ArgGH
13545180S
j
-1-j
X
X
Rules for plotting root loci/loca• Rule 9`Tangents to complex terminal zero is given by
)''GHarg(180T
GH’ is the GH(terminal zero) when removing terminal zero
Example)1s(s
)js)(js(KGH
)j1(j
)j2(K)js('GH
45'ArgGH
225)45(180S
j
-j
O
O
Rules for plotting root loci/loca
Example
2)1s(
)2s(K)s(GH
Two poles at –1One zero at –2One asymptote at 180°Break-in point at -3
XX-1
O-2
=-3
K=2
K=2
K=4 K=0
2
1
1
2
142
Rules for plotting root loci/loca
Example Why a circle ?
Characteristic equation 01K2)K2(ss2
For K<4
2
)K4(Kj)K2(s 2,1
For K>4
2
)4K(K)K2(s 2,1
2
)K4(Kj)K2(2s 2,1
Change of origin
222 KK44K4K)K4(K)2K(m4
1m
Rules for plotting root loci/loca
XX-1
O-2
=-3
K=2
K=2
K=4 K=0