Modern Control Engineering

aa

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