t06 control system

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K
0 0
Σ
Curtin University of Technology 
Department of Electrical and Computer Engineering 
 
CS301/CS603 
  
 
Tutorial 6 
 
1. Find necessary and sufficient conditions for all the roots of the equation 
 
s4 + a1s3 + a2s2 + a3s + a4 = 0 
to lie in the open left half plane. 
Answer: 
The entries in the first column of the Routh array are 
 
 
1, a1, a1a2 − a3 , 
a1 
a3(a1a2 − a3) 
a1 
 
− a1a4, a4 
 
By Routh’s criterion, all the roots of the equation lie in the open left half plane if and only if 
all the above entries are positive. 
 
 
 
    
2. Consider the closed-loop system shown in Figure 1. Determine the range of K for stability. 
Assume that K > 0. 
 
+ 
R s− 2 Y 
(s+1)(s2 +6s+25) 
− 
     
Figure 1 
  
Answer: 
The closed-loop transfer function is 
 
Y (s) 
= 
    
K (s − 2) 
  
From the Routh array 
R(s) s3 + 7s2 + (31 + K )s + 25 − 2K 
 
 
1 31 + K 0 
7 25 − 2K 0 
192+9K 
7 
25 − 2K 0 0 
 
it is seen that the range of K for ensuring closed-loop stability is 0 < K < 12.5. 
3. Consider the servo system with tachometer feedback shown in Figure 2. Determine the ranges
of stability for K and Kh. Note that Kh must be positive.
Σ
+
−
Σ
+
−
K 20
(s+1)(s+4)
Kh
1
s
R Y
Figure 2. Servo system with tachometer feedback.
Answer:
The closed-loop transfer function is
Y (s)
R(s)
=
20K
s3 + 5s2 + (4 + 20KKh)s+ 20K
From the Routh array
1 4 + 20KKh 0
5 20K 0
4 + 20KKh − 4K 0 0
20K 0 0
the stability conditions are found to be
K > 0 and Kh >
K − 1
5K
As Kh > 0, we are led to
K > 0 and Kh > max
(
0,
K − 1
5K
)