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Problem 5.21 PP
Consider the system in Fig..
(a) Using Routh’s stability criterion, determine all values of K for which the system is stable.
(b) Use Matlab to draw the root locus versus K and find the values of K at the Imaginary-axis 
crossings.
Figure Feedback system
E L h -
S tep -by-s tep s o lu tio n
step 1 of 6
Refer to Figure 5.59 in the textbook. 
Calculate the characteristic equation. 
1 + G ( j ) f f ( j ) = 0 
' K (s + 3 )
1 + m - ’
i ( j + l ) ( s ^ + 4 i + 5 ) + A : ( j + 3 ) 
j ( j + l ) ( i ^ + 4 i + s j 
+ j){ » ^ + 4 » + 5 )+ K s + 3 X = 0
tS s * + 9 j ^ + 5 j + * s +3A: = 0 
j * + 5 j ’ + 9 j ^ + (5 + A :)s + 3 X = 0 ( 1)
Step 2 of 6 ^
Calculate the range of using Routh’s stability criteria.
I 9 3K
5 5 ^ K
4 S -5 - K
3AT
4 0 -K 
5
3AT
From the Routh’s stability criteria, first column elements should be greater than zero. 
Therefore,
S
4 0 - K > 0
K 0
^ > 0
Therefore, the values of for which the system is unstable is |0 ^ AT ^
Step 3 of 6
(b)
Calculate the loop transfer function.
L (s ) = G (» )W (i)
 ̂ ^(^-^3) Y 1 j
[ i ( i ^ + 4 s + 5 ) J l i+ l
y ( » + 3 )
j ( j+ l ) ( j* + 4 s + S )
AT(j+3)
+ & ’ + 9 i* + 5 i
Consider that, =
Therefore,
(^ ^ 3 )
i ( s ) .
s * + J j ’ + 9 » " + 5 s
Step 4 of 6 ^
Write a MATLAB program to find the root locus of the system. 
» num=[1 3]:
» den=[1 5 9 5 0];
» sys=tf(num,den) 
sys = 
s + 3
sM + 5 s'^3 + 9 s'^2 + 5 s 
Continuous-time transfer function. 
» riocus(sys)
Draw the root locus.
Step 5 of 6
Figure 1
Step 6 of 6
The root locus is crossing the imaginary axis at ^ = ± J{ ,38. 
Therefore, the value of gain, is I S U