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Example Problems and Solutions 255
For example, if n=4, then
Thus it has been shown that the first column of the Routh array is given by
A–5–20. Show that the Routh’s stability criterion and Hurwitz stability criterion are equivalent.
Solution.If we write Hurwitz determinants in the triangular form
where the elements below the diagonal line are all zeros and the elements above the diagonal
line any numbers, then the Hurwitz conditions for asymptotic stability become
which are equivalent to the conditions
We shall show that these conditions are equivalent to
wherea 1 ,b 1 ,c 1 ,p, are the elements of the first column in the Routh array.
Consider, for example, the following Hurwitz determinant, which corresponds to i=4:
The determinant is unchanged if we subtract from the ith row ktimes the jth row. By subtracting
from the second row a 0 /a 1 times the first row, we obtain
¢ 4 = 4
a 11
0
0
0
a 3
a 22
a 1
a 0
a 5
a 23
a 3
a 2
a 7
a 24
a 5
a 4
4
¢ 4 = 4
a 1
a 0
0
0
a 3
a 2
a 1
a 0
a 5
a 4
a 3
a 2
a 7
a 6
a 5
a 4
4
a 17 0, b 17 0, c 17 0, p
a 117 0, a 227 0, p, ann 70
¢i=a 11 a 22 paii 7 0, (i=1, 2,p,n)
¢i= 6
a 11 *
a 22
0 aii
6 , (i=1, 2, p,n)
1, ¢ 1 ,
¢ 2
¢ 1
,
¢ 3
¢ 2
, p,
¢n
¢n- 1
¢ 4 = 4
a 1
a 3
a 5
a 7
1
a 2
a 4
a 6
0
a 1
a 3
a 5
0
1
a 2
a 4
4 = 4
a 1
a 3
0
0
1
a 2
a 4
0
0
a 1
a 3
0
0
1
a 2
a 4
4 =¢ 3 a 4