what is root locus ? the characteristic equation of the closed-loop system is 1 + k g(s) = 0 the...
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![Page 1: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/1.jpg)
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.
![Page 2: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/2.jpg)
A simple example
A camera control system:
How the dynamics of the camera changes as K is varied ?
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A simple example (cont.) : pole locations
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A simple example (cont.) : Root Locus
(a) Pole plots from the table. (b) Root locus.
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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
![Page 6: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/10.jpg)
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=
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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
![Page 12: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/13.jpg)
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 ?
![Page 14: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/15.jpg)
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
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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
![Page 17: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/18.jpg)
Rules for plotting root loci/loca
S plane
xxK=0 K=0
-2
K=1
K>1
K>1
j
![Page 19: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/20.jpg)
Rules for plotting root loci/loca
)4s(s
)2s(KGH
2
XXX
-4
O
-2
12
24
-1
• Example
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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
![Page 22: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/27.jpg)
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
![Page 28: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/28.jpg)
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
![Page 29: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d435503460f94a200b7/html5/thumbnails/31.jpg)
Rules for plotting root loci/loca
XX-1
O-2
=-3
K=2
K=2
K=4 K=0