INC341
Root Locus
Lecture 7
INC 341
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Rectangular vs. polar
jω
s = 4 + j3
3
σ
4
Rectangular form:
Polar form
INC 341
4 + j3
magnitude=5, angle = 37
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Rectangular form
Add, Subtraction
(4  j3)  (1  j )  5  j 4
Polar form
Multiplication
537  2  12
 5  2(37  12 )  1025
Division
INC 341
537  2  12
5
 49
2
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a  jb
r  
2
b
  arct an 
a
a  r cos
b
r
θ
a
INC 341
r  a b
2
b  r sin 
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Vector representation of a transfer function
m
F (s) 
 ( s  zi )
i 1
n
 ( s  pi )
i 1
m
s  zi
 zerolengths i
M
 n1
 polelengths
 s  pi
i 1

 zeroangles  poleangles
m

 (s  z )  (s  p )
i
i 1
INC 341
n
i
i 1
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Vector s
(s+a)
INC 341
(s+a)
(s+7)
s = 5+j2
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Example
( s  1)
F ( s) 
s( s  2)
Find F(s) at s = -3+j4
F ( s) 
( s  1)
s( s  2)
 2  j4
(3  j 4)(1  j 4)
20116.57

5126.87  17104.04
 0.217  114.34

INC 341
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What is root locus and why is it needed?
• Fact I: poles of closed-loop system are an
important key to describe a performance of the
system (transient response, i.e. peak time,
%overshoot, rise time), and stability of the
system.
• Fact II: closed-loop poles are changed when
varying gain.
• Implication: Root locus = paths of closed-loop
poles as gain is varied.
INC 341
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Cameraman
Object Tracking
using infrared
INC 341
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Varying gain (K)
Varying K, closed-loop poles are moving!!!
INC 341
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Transient:
• K<25  overdamped
• K=25  critically damped
• K>25  underdamped
• Settling time remains the
same under underdamped
responses.
Stability:
• Root locus never crosses
over into the RHP, system is
always stable.
INC 341
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Concept of Root Locus
INC 341
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Closed-loop transfer function
KG ( s)
T ( s) 
1  KG ( s) H ( s)
Characteristic equation
KG(s) H (s)  1  1180

magnitude
phase
INC 341
KG(s) H (s)  1
KG( s) H (s)  (2k  1)180
k  1,2,3,...
PT & BP
If there is any point on the root locus,
its magnitude and phase will be
consistant with the follows:
magnitude
phase
KG(s) H (s)  1
KG( s) H (s)  (2k  1)180
k  1,2,3,...
Note that: phase is an odd multiple of
180
INC 341
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Example
Is the point -2+3j a closed-loop pole for some value
of gain? Or is the point on the root locus?
INC 341
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KG ( s) H ( s) 
K ( s  3)(s  4)
( s  1)(s  2)
1   2  3   4  56.31  71.57  90 108.43  70.55
-2+3j is not on the root locus!!!
What about  2  j( 2 / 2) ?
INC 341
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The angles do add up to 180!!!
 2  j ( 2 / 2) is a point on the root locus
What is the corresponding K?
KG ( s ) H ( s )  1
1
K
G ( s) H ( s)
K 
L3 L4
L1 L2
0.7 0 7 1.2 2
2 .1 2  1 .2 2
 0 .3 3

L1L2
G( s) 
L3 L4
INC 341
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Sketching Root Locus
1. Number of branches
2. Symmetry
3. Real-axis segment
4. Starting and ending points
5. Behavior at infinity
INC 341
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1. Number of branches
Number of branches = number of closed-loop poles
INC 341
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2. Symmetry
Root locus is symmetrical about the real axis
INC 341
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3. Real-axis segment
On the real axis, the root locus exists to
the left of an odd number of real-axis
INC 341
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KG(s) H (s)  (2k  1)180

• Sum of angles on the real axis is either 0 or
180 (complex poles and zeroes give a zero
contribution).
• Left hand side of even number of poles/zeros
on the real axis give 180 (path of root locus)
INC 341
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Example
root locus on the real axis
INC 341
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4. Starting and ending points
Root locus starts at finite/infinite poles of G(s)H(s)
And ends at finite/infinite zeros of G(s)H(s)
closed-loop transfer function
KG ( s)
T ( s) 
1  KG ( s) H ( s)
N G ( s)
G(s) 
DG ( s )
INC 341
N H ( s)
H ( s) 
DH ( s)
PT & BP
T ( s) 
KN G ( s) DH ( s)
DG ( s) DH ( s)  KN G ( s) N H ( s)
K=0 (beginning) poles of T(s) are DG ( s) DH ( s)
K=∞ (ending) poles of T(s) are KNG (s) N H (s)
INC 341
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Example
INC 341
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5. Behavior at infinity
Root locus approaches asymptote as the
Locus approaches ∞, the asymptotes is given by
a
finite poles  finite zeros


# finite poles# finite zeros
(2k  1)
# finite poles# finite zeros
k  0,  1,  2, ...
a 
INC 341
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Rule of thumb
# of poles = # of zeroes
K
KG ( s) H ( s) 
s( s  1)(s  2)
has 3 finite poles at 0 -1 -2, and 3
infinite zeroes at infinity
INC 341
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Example
Sketch root locus
0 
(1  2  4)  (3)
4

4 1
3
(2k  1)
# finitepoles# finitezeros
  / 3 , for k  0

 ,
for k  1
 5 / 3 , for k  2
INC 341
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INC 341
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