Root Locus Plot of Dynamic Systems
imtiaz.hussain@faculty.muet.edu.pk
1
We will cover........
Root Locus of LTI models
Root locus of a Transfer Function (TF) model
Root Locus of 1st order system
Root Locus of 2nd order system
Root Locus of higher order systems
Root locus of a Zero-pole-gain model (ZPK)
Root locus of a State-Space model (SS)
Gain Adjustment
Root Locus of 1st Order System
Consider the following unity feedback system
R(S )


Matlab Code
num=1;
den=[1 0];
G=tf(num,den);
rlocus(G)
sgrid
G (S ) 
K
S
C (S )
Root Locus of 1st Order System
Consider the following unity feedback system
R(S )

KS

C (S )
S 1
num=[1 0];
den=[1
1];
G=tf(num,den);
rlocus(G)
sgrid
Continued…..
Exercise#1
Plot the root locus of following first order systems.
R(S )

KS

R(S )

S 1
K (S  2)

C (S )
S 1
C (S )
Root Locus 2nd order systems
Consider the following unity feedback system
R(S )

K

C (S )
S ( S  3)
num=1;
den1=[1 0];
den2=[1 3];
den=conv(den1,den2);
G=tf(num,den);
rlocus(G)
sgrid
Determine the location of closed
loop poles that will modify the
damping ratio to 0.8 and natural
undapmed frequency to 1.7
r/sec. Also determine the gain K
at that point.
Exercise#2
Plot the root locus of following 2nd order systems.
R(S )
K ( S  3)

(1)
(2)

R(S )
C (S )
S ( S  1)
K ( S  2 )( S  3 )


S ( S  1)
C (S )
Root Locus of Higher Order Systems
Consider the following unity feedback system
R(S )

num=1;
K

C (S )
S ( S  1)( S  2 )
den1=[1 0];
den2=[1 1];
den3=[1 2];
den12=conv(den1,den2);
den=conv(den12,den3);
G=tf(num,den);
rlocus(G)
sgrid
Determine the closed loop gain
that would make the system
marginally stable.
Exercise#3
Plot the root locus of following systems.
R(S )
K ( S  3)


(1)
R(S )

(2)
S ( S  1)( S  2 )
K ( S  3 )( S  5 )

C (S )
S ( S  1)( S  2 )
C (S )
Root Locus of a Zero-Pole-Gain Model
R(S )
3( S  5 )


k=2;
z=-5;
p=[0 -1 -2];
G=zpk(z,p,k);
rlocus(G)
sgrid
S ( S  1)( S  2 )
C (S )
Root Locus of a State-Space Model
 x 1    5
 
 x 2   3
y ( t )  1
A=[-5 -1;3 -1];
B=[1;0];
C=[1 0];
D=0;
sys=ss(A,B,C,D);
rlocus(sys)
sgrid
 1   x1   1 
      u (t )
 1  x 2   0 
 x1 
0    D ,
 x2 
where D  0
Exercise#4: Plot the Root Locus for following LTI Models
(1)
(2)
(3)
G (S ) 
G (S ) 
S 1
and
S ( S  3 )( S  4 )
3 ( S  10 )
S ( S  1)( S  30 )
 x 1   0
  
x  0
 2 
 x 3    3
y ( t )  0
0
1
0
4
and
H (S )  (2  S )
H (S )  1
0   x1   0 
   
1
x  0 u (t )
 2   
 2   x 3   1 
 x1 
 
1 x 2  D ,
 
 x 3 
where D  0
Choosing Desired Gain
G (S ) 
num=[2
den=[3
2S  3
3S  4 S  1
2
3];
4
1];
G=tf(num,den);
[kd,poles]=rlocfind(G)
sgrid
Exercise#5
(1)
G( S ) 
4( S  3)
( S  1)( S  4)( S  6)
Plot the root Loci for the above ZPK model and find out the
location of closed loop poles for =0.505 and n=8.04 r/sec.
b=0.505;
wn=8.04;
sgrid(b, wn)
axis equal
Exercise#5:
(contd…)
(2) Consider the following unity feedback system
G( S ) 
i)
K
( S  1)( S  4 )
Plot the root Loci for the above transfer function
ii) Find the gain when both the roots are equal
iii) Also find the roots at that point
iv) Determine the settling of the system when two roots are
equal.
Exercise#5:
(contd…)
(3) Consider the following velocity feedback system
R(S )

K

C (S )
S ( S  3 )( S  5 )( S  7 )
3  5S
i)
Plot the root Loci for the above system
ii) Determine the gain K at which the the system produces
sustained oscillations with frequency 8 rad/sec.
End of tutorial
You can download this tutorial from:
http://imtiazhussainkalwar.weebly.com/