ENT385-CONTROL ENGINEERING
SESSION 2012/13
TUTORIAL 5
CHAPTER: ROOT LOCUS
1.
An open-loop system has a transfer function given by:
GO ( s) 
K
s( s  2)( s  8)
Find the maximum value of the gain K before the system response begins to
oscillate. Determine the closed-loop system poles and the corresponding
transfer function for the condition above.
Solution:
From the open-loop TF, we obtain the poles on the s-plane. We have 3 poles on
the real-axis.
The locus starting from poles at 0 and -2 will meet and
breakaway into the complex s-plane. When this happens, the closed-loop poles
will have complex values, and this will result in oscillatory process response.
This means that the maximum value of K just before the system oscillates is at
the breakaway point.
We can find this by differentiating the characteristic equation with respect to s
and equate it to zero.
For the system, the characteristic equation is given by:
1
K
0
s( s  2)( s  8)
s 3  10s 2  16s  K  0
Rearranging gives :
K   s 3  10s 2  16s
Obtain the differential of K with respect to s and determine its max value:
dK
 3s 2  20s  16  0
ds
This gives:
s1, 2 
 20  20 2  4(3)(16)
 0.93,5.74
2(3)
Another approach is a non-differentiation method. It is given that :
m
n
1
1

1   z 1   p
i
i
where zi and pi are the negatives of the zeros and poles values respectively of
G(s)H(s). Then we have:
0
1


1
1

 2  8
0  3 2  20  16
This will give
2.
 = -0.93 and -5.74 as before.
Sketch the root locus and find the range of K for stability for the unity feedback
system shown in figure P8.3 for the following conditions:
Figure P8.3
a)
G( s) 
K ( s  1)
( s  1)( s  2)( s  3)
b)
G( s) 
K ( s 2  2s  2)
s( s  1)( s  2)
Solution:
a)
b)
2
3.
For the unity feedback system the transfer function G(s) is given by:
G( s) 
1
s( s  6)( s  9)
Plot the root locus and get the critical points such as breakaways, asymptotes,
jω-axis crossing etc.
Solution:
4.
The forward-path transfer function of a unity feedback system is given by:
G( s) 
K ( s  4)
s ( s  4 s  4)( s  5)( s  6)
2
Construct the root locus for K0. Find the value of K that makes the damping ratio of
the closed-loop system (measured by the dominant complex characteristic equation
roots) equal to 0.707 if such a solution exists.
Asymptotes:
K > 0:
o
o
o
45 , 135 , 225 , 315
Intersect of Asymptotes:
1 
Breakaway-point Equation:
o
2  2  5  6  ( 4 )
51
5
4
 2.75
3
2
4 s  65 s  396 s  1100 s  1312 s  480  0
Breakaway Points:
0.6325,
When   0.707 , K = 13.07
5.511
(on the RL)
5.
The forward-path transfer function of a unity feedback system is given by:
K
s ( s  2)( s  5)( s  10)
G(s) 
Construct the root locus for K0. Find the value of K that makes the damping ratio of
the closed-loop system (measured by the dominant complex characteristic equation
roots) equal to 0.707 if such a solution exists.
o
o
o
o
45 , 135 , 225 , 315
Asymptotes: K > 0:
Intersect of Asymptotes:
0  2  5  10
1 
 4.25
4
Breakaway-point Equation:
When   0.707 , K = 61.5
3
2
4 s  51s  160 s  100  0