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Problem 3.54PP
Use Routh’s stability criterion to determine how many roots with positive real parts the following 
equations have;
(a) s4 + 8s3 + 32s2 + 80s + 100 = 0
(b) s5 + 10s4 + 30s3 + 80s2 + 344s + 480 = 0
(c) s4 + 2s3 + 7s2 - 2s + 8 = 0
(d) S3 + s2 + 20s + 78 = 0
(d) S3 + s2 + 20s + 78 = 0
(e) s4 + 6s2 + 25 = 0
Step-by-step solution
step 1 of 5
(a)
The characteristic equation is,
* * + &t’ + 3 2 i’ + 80s + 100 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial. 
s*: 1 32 100
s ’ : 8 80
22 100 
s' : 43.6 
s": 100
Since there are no sign changes in the first column of Routh array, the number of roots with the 
positive real parts is zero.
Step 2 of 5 ^
(b)
The characteristic equation is,
s* +1 Oj ^ + 30s’ + 80s^ + 3 4 4 s+ 4 8 0 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial.
344 
480
1 30
10 80
s’ : 22 296
s’ : -5 4 .5 480
s': 490
s’ : 480
Since there are two sign changes in the first column of Routh array, the number of roots with the 
positive real parts is two.
Step 3 of 5
(c)
The characteristic equation is,
s * + 2 s ’ + 7 s ’ - 2 s + 8 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial.
1 7 8
2 - 2
j ’ ; 8 8
»>: - 4
j* : 8
Since there are two sign changes in the first column of Routh array, the number of roots with the 
positive real parts is two.
Step 4 of 5
(d)
The characteristic equation is,
*’ + i ’ + 20* + 78 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial.
1* : 1 20 
1 78 
-5 8 
78
Since there are two si'^n Ghan*^es in the first c-oiui 
positive real parts a two.
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Step 5 of 5 ^
(e)
The characteristic equation is, 
sV 6 j “ + 2 5 - 0 
5 * + Os ’ + + O r+ 2 5 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial.
t ’ : 1 6 25
i * : 4 12
3
100
25
i * : 1 2 - -2 13
i * : 25
From the characteristic equation, it is clear that there two coefficients missing so there are roots 
outside of the LHP. Since there are two sign changes in the first column of Routh array, the 
number of roots with the positive real parts is two.