The Root Locus Diagram
By
Dr. M.S.Rao
Outline of presentation
• The Root Locus Concept
• The Root Locus Procedure
• Generalized root locus or Parameter RL
• Parameter design by root locus method
• PID controllers and RL method
• Examples and simulation by MATLAB
• Summary
Introduction
• RL is first developed by W.R.Evans
• Graphical method to study the movement
of roots in the s-plane
– One system parameter (Kc) is changed.
– In general Kc is varied between 0 to infinity.
• Definition: The root locus is the path of the
roots of the characteristic equation traced
out in the s-plane as a system parameter i
s varied.
The root locus concept
• Powerful tool to design and analyze feed b
ack control systems
• System performance analyzed using RL
– Stability
– Dynamic performance
– Steady-state error
Root-locus plots
• Angle and Magnitude conditions
– Consider the closed loop transfer function
– Characteristic equation is obtained by setting
denominator equal to zero.
– 1+GH=0; GH=-1
– Split the characteristic equation into two
separate equations
Root locus equation
• Relationship between the open-loop
and closed-loop poles and zeros
• Root locus equation:
m n
( s z j ) ( s pi ) ( 2k 1) ( k 0, 1, 2 )
j 1 i 1
n
K i 1
s pi
m
j 1
s zj
Basic task of root locus
• How to determine the closed-loop poles
from the known open-loop poles and zeros
and gain by root locus equation.
• Angle requirement for root locus
• Magnitude requirement for root locus
Necessary and sufficient condition for root locus plot
Gain evaluation for specific point of root locus
The Root Locus Procedure
• Step 1:Write the characteristic equation as
1 F ( s) 0
• Step 2: Rewrite preceding equation into
the form of poles and zeros as follows:
m
(s z )
j 1
j
1 K n
0
(s p )
i 1
i
The Root Locus Procedure
• Step 3: Locate the poles and zeros with speci
fic symbols
• The root locus begins at the open-loop poles
and ends at the open-loop zeros as K increas
es from 0 to infinity.
• If open-loop system has n-m zeros at infinity,
there will be n-m branches of the root locus a
pproaching the n-m zeros at infinity.
The Root Locus Procedure
• Step 4: The root locus on the real axis lies in a
section of the real axis to the left of an odd
number of real poles and zeros.
• Step 5: The number of separate loci is equal to
the number of open-loop poles.
• Step 6: The root loci must be continuous and
symmetrical with respect to the horizontal real
axis.
The Root Locus Procedure
• Step 7: The loci proceed to zeros at
infinity along asymptotes centered a at
a
and with angles :
n m
p zi j
a i 1 j 1
nm
( 2k 1)
a ( k 0,1, 2, n m 1)
nm
The Root Locus Procedure
• Step 8: The actual point at which the ro
ot locus crosses the imaginary axis is re
adily evaluated by using Routh criterion.
• Step 9: Determine the breakaway point
d (usually on the real axis):
m n
1 1
j 1 d z j
i 1 d pi
The Root Locus Procedure
• Step 10: Determine the angle ofp
i
departure of locus from a pole and
zi
the angle of arrival of the locus at a
zero by using phase angle criterion.
m n
p 180 ( z
i
0
j pi
p j pi )
j 1 j 1, j i
m n
z 1800 (
i
j 1, j i
z j zi p j zi )
j 1
The Root Locus Procedure
• Step 11: Plot the root locus that satisfy
the phase criterion.
P ( s ) ( 2k 1) k 1, 2,
• Step 12: Determine the parameter value
K1 at a specific root s1 using the
magnitude criterion.
n
(s p )
i
K1 i 1
m
(s z )
j 1
j
s s1
