Notes on Root Locus Technique
(This handout is to complement the textbook (Ch4) and class notes)
Bulleted Items:
- The location of poles and zeros of a closed loop system have great importance on the
dynamic response of a system.
The roots of the characteristic equation (poles of the system) determine the absolute and
relative stability of a SISO system.
Remember, the transient response also depends on the zeros of the closed loop system.
- An Important study in linear control systems is the investigation of the trajectories of
the roots of the characteristic equation – simply put- the root loci.
- In this chapter (ch4) we are to study the simple rules governing the determination of
root loci.
- { by doing transformation s to z plane, we can also construct the root loci of a discrete
data system.}
-
Standardized notations:
o RL: Portion of the root loci when K is positive 0  K  
o CLR: complementary root loci : portion of the root loci when K is negative
  K 0
-
o RC: Root contour, the contour of the root loci when more than one
parameter varies.
o Root Loci: refers to the total root loci:    K  
Let us discuss the basic properties of a root loci:
o The transfer function is:

C ( s)
G ( s)
 GCL ( s) 
R( s )
1  G ( s) H ( s)

The characteristic equation of the closed loop systems is obtained by
setting the denominator polynomial of GCL(s) = 0
This results in:
1+ G(s)H(s) = 0
 Now, assume that the term GH contain a variable multiplying
parameter, K, then we have:
1+KGH = 0
G(s)H(s) =

1
K
(1)
To satisfy equation (1), we have the following two criteria:
G(s)H(s) 
1
K
  K  

Magnitude Criterion

Angle Criterion (to determine the trajectories on the RL on the splane.)
(a)
G(s)H(s)  (2i  1) (odd multiples of π
radians or 180o) 0  K  
(b) G(s)H(s)  2i (even multiples of of π radians or
i
180o)
 0,1,2,.... (any integer)
K 0
These two criteria play important roles in the construction of the root loci.
-
The construction of the root loci is basically a graphical and analytical problem.
The first thing to keep in mind is the location of poles and zeros of G(s)H(s)
o G(s)H(s) is written as:

( s  z1 )( s  z 2 ).....(s  z m )
( s  p1 )( s  p2 )......(s  pn )
 The zeros and poles are real and are in complex conjugate pairs.
o Applying the magnitude criteria, we get:
m

G (s) H (s) 
 s  zr
r 1
n
 s  pj

1
K
  K 
(2)
j 1
o Applying the angle criteria, we get:
0 K 

(RL)
m
n
G( s) H ( s)   ( s  z r )   ( s  p j )  (2i  1)  180 o
r 1

(3)
j 1
K 0
(CRL)
m
n
G( s) H (s)   ( s  zr )   ( s  p j )  2i  180 o
r 1
(4)
j 1
The graphical interpretation of equation (3) (RL) is:
The difference between the sum of the angles of the vectors drawn from the zeros and
those drawn from the poles of GH to s1 is an odd multiples of 180o . s1 is any point on
RL.
For the CRL (equation 4),
The difference between the sums of the vectors drawn from the zeros and those drawn
from the poles of G(s)H(s) to s1 is an even multiple of 180o , including zero degrees. s1 is
any point on the CRL.
-
Once the root loci are constructed, the values of K along the loci at any point s1 can
be determined by equation (2)
n
 s  pj
o
K 
j 1
m
(5)
 s  zr
r 1
Properties and Construction of Root Loci
These properties are useful for the construction and the understanding of the root loci.
These properties are developed based on equation (1):
AK =0, K = ± 
-
K = 0 points on the root loci are the poles of G(s)H(s)
K = ±  points on the root loci are the zeros of G(s)H(s)
Example:
Suppose:
1+KGH = 0
is:
s(s+2)(s+3)+K(s+1) = 0, yields:
( s  1)
1

s( s  2)( s  3)
K
When K =0, the roots of the equation are: 0,-2,-3 (poles)
When K approaches  , the roots of the equation are : -1, +  , -  !!!
Important: keep in mind that the infinity in the s-plane is a point of concept. Imagine the
s-plane is a small portion of a sphere with infinite radius. Then, infinity in the s-plane is
a point on the opposite side of the sphere that we face.
BThe number of branches of the root loci.
A branch of the root loci is the locus of one root as K varies from -  to + 
The number of branches on the root loci is equal to the order of the polynomial.
The branches of the root loci are continuous curves that start at each of the poles of GH
(k=0) and approach the zeros(k=  .) The root loci for excess poles extends infinitely
far from the origin (towards infinity). Whereas for excess zeros, the loci extends from
infinity.
Example: For the equations; s(s+4)(s+8)+k(s+2)=0, there are three branches (3rd
order equation) Poles at { 0,-4,-8} and zeros at {-2, infinity, infinity} - keep in mind !
C-
Symmetry of the root loci:
The root loci are symmetrical with respect to the real axis of the s-plane. In
general, the root loci is symmetrical with respect to the axis of symmetry of the pole –
zero configuration of G(s)H(s)
Example:
For the equation: s(s+1)(s+2)+k=0
k
s( s  1)( s  2)
poles ate 0,-1,2 and three zeros at inf, +inf, - inf
poles at k=0, zeros at k= 
G( s) H ( s) 
For the symmetry:
Consider:
s(s+2)(s+1+j1)(s+1-j1) +k=0
 1  j1
X
X
X
  1
-2
X
 1  j1
D-
Angles of the asymptotes of the root loci:
The properties of the root loci near infinity in the s-plane are described by the asymptotes
of the loci when s   .
Generally, when n  m, there will be 2 n  m asymptotes to describe the behavior of
the root loci. The angles of the asymptotes and their intersection with the real axis of the
s-plane is as follows:
Where, n: number of finite poles, m: number of finite zeros.
i 
2i  1
 180 o n  m where i = 0,1,2,3,.... n  m  1 (RL)
nm
and
i 
E-
2i
 180 o n  m where i = 0,1,2,3,.... n  m  1 (CRL)
nm
Intersection of the asymptotes:
The intersection of the 2 n  m asymptotes of the root loci lies on the real axis of
the s-plane.
i 
 real
parts of the poles of G ( s) H ( s) 
nm
 real
parts of the zeros of G ( s) H ( s)
 is the center of the root loci and is always a real number.
Let us apply D and E to the following example
G( s) H ( s) 
k ( s  1)
s( s  4)( s 2  2s  2)
@ k=0 , { 0,-4,-1+j1,-1-j1}
@ k=   , {-1 , , ,  }
Number of branches = 4
The root loci is symmetrical to the real axis
The angles of asymptotes are: (there are 2 n  m  2(4  1) =3 asymptotes for the RL
and + 3 more for the complementary root locus (CRL).
i  0 :0 
180
 60 o
3
540
 180 o
3
900
i  2 :2 
 300 o
3
i  1 : 1 
The intersection of the asymptotes is:

(4  1  1)  (1)
5

4 1
3
The drawing for the above results is shown below:
60o
-5/3
60o
FF:
Angles of departure and angles of arrival of the root loci
The angle of departure or arrival of a root locus at a pole or zero, respectively, of
G(s)H(s) denotes the angle of the tangent to the locus near the point.
Consider the following example
s(s+3)(s2+2s+2)+k=0
-
The root locus plot is shown below.
At this point, we are interested in the angle of departure at the pole –1+j1. So we located
a point on the root loci near the complex pole of interest. The angles (  1, 2, 3, 4) are
measured as positive CCW rotation from the positive real axis.
The angle criteria we get:
   other GH pole angles 
GH zero angles  180
  (90  26.6  135)  0  180  71.6 o
This is illustrated in the graph for THETA2
The Matlab code to generate the root locus plot is also given.
The Matlab Code is:
% to plot the root locus
clear all
k=[-60:0.01:60];
num=1;
den=[conv(conv([1 0],[1 3]),[1 2 2])]
sys=tf(num, den);
sys
pause % to show the GH(s) equation
P=pole(sys);
Z=zero(sys);
rlocus(sys,k)
Fun information: You can move the cursor on the plot and notice the values for k,s, w,
damping ratio. Here is an example:
We can use this relationship to determine the value of k (gain) and the frequency of
oscillation when the root loci crosses the imaginary axis.
Another note: On the Matlab Code, I could have added the following:
[r,k]=rlocus(sys,k)
This would have given me two vectors, the poles of the GH system and the corresponding
k (gain) values. Try this
G-
The intersection of the root loci with the imaginary axis
This is found as was done earlier (using the Routh -Hurwitz Criterion)
H- Breakaway / Entry points:
Refer to the figures illustrated in your textbook in table 4-2.
The breakaway points on the root loci of an equation do correspond to the multiple-order
roots of the equation.
The breakaway points on the root loci of 1+KG(s)H(s)=0 must satisfy
dG ( s ) H ( s )
 0 It is important to note that all the breakaways points on the root locus
ds
must satisfy the equation above, but not all solutions to the equation above are breakaway
points.
I- Angle of arrival and departure at the breakaway points:
In general;
n root loci arrive or depart a breakaway point at 180/n degrees apart.
Example:
For the system:
s(s+2)+k(s+4)=0
s4
s ( s  2)
dG ( s) H ( s )  s ( s  2)  2( s  1)( s  4)  0
ds
s 2 ( s  2) 2
G(s)H(s)=
or
s2+8s+8=0
Which results in s=-1.172 and –6.828
Here is helpful Matlab Code ….
%------------------clc;clf;clear all
% this program code demonstrates some of the features related to Root Locus
% For the demonstration, we will use the following example:
% s(s+3)(s^2 + 2s + 2) + K =0
% construct the GH TF
den1=[1 3 0];
den2=[1 2 2];
den=conv(den1,den2)
num=[1];
GH=tf(num,den)
pause
pzmap(GH) % this command returns the plot of the poles and zeros of GH
%-----------------------------% note, we could have assigned the values to vectors P and Z using any of
% the following two options ... your choice -- no preference.
%[P,Z]=pzmap(GH)
% or
% P=pole(GH) and
% Z = zero(GH)
% -------------------------------% Now, we can map the poles and zeros
% how about drawing the root locus:
rlocus(GH) % Note, by default, this command will always yield RL due to K>=0
pause
% we can specify the K vector
K=-200:.02:200;
rlocus(GH,K)
grid % you should now see an axis for the dampming -- how nice !
% After you are done with all, play with the following .....
% this is the beginning of control ... -- TAKE CONTROL of the system.