Post on 08-Feb-2016
description
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 The022
02
11
222
sGssK
ssKsssssKsKGs
nn
Introduction
)2s(sK
+
-
x y
Ks2sK
)s(G1)s(G
)s(X)s(Y
2
Characteristic equation 0Ks2s2
1Kj1Roots
K1K11Roots
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:2s origin to thefrom vector theof magnitude the:
2
122
1
ss
ssK
ssK
ssK
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
qqpspszszssF
pspszszsK
sF
G(s)
H(s)
R +-
C
GH1G
RCF
01nn
01m
1mm
b.......bs)a......sas(K
)s(D)s(KNGH
nm
KNDGD
D/KN1GF
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
oK=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=14/K1
2K2s 2
1
4/K12
K2s 22
K)2K(ss
)1s(K)s(F 2
K=1
,5
K=1
,5
j
x-1-2
x
K= 0
K=0K=
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
K2K
1
00K
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=0K=
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)
mn180)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 z1
p1
XX
O OBreak-away Break-in
First method
Rules for plotting root loci/locaExample
)2s)(1s(sK)s(G
3 asymptotes: 60 °,180 ° and 300°
13
21
02
11
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(G1)s(v
)2s)(1s(sK)s(G
Example
Ks2s3s
K)2s)(1s(s)s(V
23
0K
2s6s3dsdV 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(sK)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 ?
KK6
31
00K2
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/locaExample
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
21
12
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