Module-4-A-Graphical Tools 1 ECB 3123 CONTROL SYSTEMS Dr Irraivan Elamvazuthi MODULE 4-A GRAPHICAL TOOLS Root Locus Techniques
Feb 08, 2016
Module-4-A-Graphical Tools 1
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
MODULE 4-A
GRAPHICAL TOOLS
Root Locus Techniques
Module-4-A-Graphical Tools 2
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
In this chapter, you will be able to:
• Sketch a root locus
• Refine the sketch of root locus
• Use root locus to find poles of a closed-loop system
• Use root locus to describe transient response and stability
when system parameter, K is varied
Module-4-A-Graphical Tools 3
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Root Locus: Graphical presentation of closed-loop poles
as system parameter is varied
Module-4-A-Graphical Tools 4
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Root Locus: Graphical presentation of closed-loop poles
as system parameter is varied
Module-4-A-Graphical Tools 5
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Review: Closed-Loop Systems
Closed-loop poles:characteristics equation
0)()()()( sNsKNsDsD HGHG
)()()()(
)()()(
)()(1
)()(
)(
)()(
)(
)()(
sNsKNsDsD
sDsKNsT
sHsG
sKGsT
sD
sNsH
sD
sNsG
HGHG
HG
H
H
G
G
Module-4-A-Graphical Tools 6
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Review: Complex Numbers
1
22
tan
M
MMe
jsj
Module-4-A-Graphical Tools 7
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Review: Complex Numbers
jasF
ajsF
assF
js
)()(
)()(
)(
jsas
jsasM
js
to connecting line
tohorizontal from angle
to from distance
Module-4-A-Graphical Tools 8
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Example 1: Given F(s) = s+7, find F(s) at point s = 5+j2
43.6312
2tan
16.12212
212)(
7)25()(
25
7)(
1
22
25
M
jsF
jsF
js
ssF
js
Module-4-A-Graphical Tools 9
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Review: Complex Numbers
js
)())((
)())((
)(
)()(
21
21
n
m
pspsps
zszszs
sD
sNsF
n
j
j
m
i
i
ps
zs
sF
1
1
)(
)(
)(
magnitude
angle
lengthspole
lengthszero
ps
zs
Mn
j
j
m
i
i
1
1
)(
)(
anglespoleangleszero
pszsn
jj
m
ii
11
)()(
Module-4-A-Graphical Tools 10
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Example 2: Evaluation of a complex function via vectors
Given )2(
)1()(
ss
ssF , find F(s) at s = -3+j4
SOLUTION
6.11643.632
4tan
204)2(
42)(
1)43()(
1)(
1
22
43
1
M
jsF
jsF
ssF
js
9.12613.533
4tan
5204)3(
43)(
)(
1
22
43
2
M
jsF
ssF
js
10496.751
4tan
174)1(
41)(
2)(
1
22
43
3
M
jsF
ssF
js
0 -1 -2
j
Module-4-A-Graphical Tools 11
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
3.114217.
)0.1049.1266.116(175
20M
6.11620
4.10417
9.1265
Module-4-A-Graphical Tools 12
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
)()(1
)()(
sHsKG
sKGsT
Properties of the Root Locus
Root locus:
0)()(1 sHsKG
conditions:
1)()( sHsKG
180)12()()( ksHsKG
)()(
1
sHsGK
Odd multiples of 180
Module-4-A-Graphical Tools 13
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
)2)(1(
)4)(3()()(
ss
ssKsHsKG
Test point 1: s = -2+j3
Example 3: Given open loop transfer function as follows, determine whether
the test point is located on the root locus.
55.7043.1089057.7131.56
4321 Not an odd multiple of 180, the
test point is not a point on the root locus.
Module-4-A-Graphical Tools 14
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
magnitude:
33.0)22.1)(12.2(
)22.1)(707.0(
zeros from distance
poles from distance
)()(
1
21
43
LL
LLK
sHsGK
Test point 2: )2/2(2 j
Angles do add up to odd multiples of 180,
the test point is a point on the root locus.
Thus, we can say that:
• Given poles and zeros of the open-loop transfer function, KG(s)H(s), a point
in the s-plane is on the root locus for some value of K, if
180)12( kanglespoleangleszero
• Gain K at which angles add up to (2k+1) 180 is
zeros from distance
poles from distance
)()(
1
sHsGK
180
74.1449026.3547.194321
Module-4-A-Graphical Tools 15
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Sketching the Root Locus
1. Write the characteristic equations
0)()(1 sHsKG
2. Open Loop:
Factor and locate poles and zeros of G(s)H(s)
0)())((
)())((1
21
21
n
n
pspsps
zszszsK
3. Real axis:
On the real axis, for K > 0, the root
locus exists on the left of
odd numbered real roots
(poles/zeros).
Module-4-A-Graphical Tools 16
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
4. Branches
The number of branches equals
the number of open loop poles.
*Loci always start, K = 0, at a pole
and terminate at a zero, K = .
*Loci that do not terminate at a zero
will approach a point (zero) at infinity .
Root Locus Construction Steps
5. Symmetry
The locus is always symmetrical wrt the real
(horizontal) axis.
Module-4-A-Graphical Tools 17
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
6. Asymptotes
Loci proceeding to zeros at will
follow the asymptotic lines with
centroid
angle
zp
ii
Ann
zp
zeros#poles#
zerospoles
3
4
14
)3()421(
1,03
)12(
kforkfor
nn
k
zp
A
Module-4-A-Graphical Tools 18
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
7. j crossing
Determine j crossing by finding gain K
that will ensure stability using Routh-Hurwitz
table
KsKsss
sKsT
ssss
sKsHsKG
)8(147
)3()(
)3)(2)(1(
)3()()(
234
s4 1 14 3K
s3 7 8+K
s2 90-K 21K
s1 K
KK
90
720652
s0 21K 0 0
59.1
07.20235.8021)90(
65.9
072065
22
2
js
sKsK
K
KK
Module-4-A-Graphical Tools 19
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
82.3,45.1
0)158(
612611
)158(
)23(
1)23(
)158()()(
22
2
2
2
2
2
s
ss
ss
ds
dK
ss
ssK
ss
ssKsHsKG
8. Breakaway and Breakin points
Determine breakaway and breakin points
by determining the maximum/minimum
change in K wrt changes in s
0
1)()(
sds
dK
sHsKG
Breakaway point
Breakin point
Module-4-A-Graphical Tools 20
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Example: Given the following system
1. Find the exact point and gain where the locus crosses
the j-axis
S2
K+1
8+20K
S1
6-4K
0
S0
8+20K
0
6-4K = 0
K = 1.5
(K+1)s2 + (8+20K) = 0
s = j3.9
2. Find the breakaway point on the real axis
204
)86(2
2
ss
ssK
)208()46()1(
)204()(
2
2
KsKsK
ssKsT
dK/ds = 0, b/away point: -2.88
-4 -2
2+j4
2-j4
Module-4-A-Graphical Tools 21
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
Thus, sketch of root locus:
Breakaway point,
s = -2.88
j-axis crossing,
s = j3.9
Module-4-A-Graphical Tools 22
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
9. Angle of Departure and Arrival
Determine angle of arrival for
complex zeros and angle of
departure for complex poles
180polesother from angle
zerosother from angledep -
180zerosother from angle
polesother from anglearr -
Module-4-A-Graphical Tools 23
ECB 3123 CONTROL SYSTEMS
Dr Irraivan Elamvazuthi
180polesother from angle
zerosother from angledep -
4.1086.251
180)2
1tan90(
1
1tan
180)(
11
423dep
Example: Angle of departure from a complex pole