aa
Example Problems and Solutions 257
From this analysis, we see that
The Hurwitz conditions for asymptotic stability
reduce to the conditions
The Routh array for the polynomial
wherea 0 >0 andn=4, is given by
From this Routh array, we see that
(The last equation is obtained using the fact that ) Hence the
Hurwitz conditions for asymptotic stability become
Thus we have demonstrated that Hurwitz conditions for asymptotic stability can be reduced to
Routh’s conditions for asymptotic stability. The same argument can be extended to Hurwitz
determinants of any order, and the equivalence of Routh’s stability criterion and Hurwitz stabil-
ity criterion can be established.
A–5–21. Consider the characteristic equation
Using the Hurwitz stability criterion, determine the range of Kfor stability.
Solution.Comparing the given characteristic equation
s^4 +2s^3 +(4+K)s^2 +9s+ 25 = 0
s^4 +2s^3 +(4+K)s^2 +9s+ 25 = 0
a 17 0, b 17 0, c 17 0, d 170
a 34 =0, aˆ 44 =a 4 , and a 4 =b 2 =d 1.
a 44 =aˆ 44 -
aˆ 43
a 33
a 34 =a 4 =d 1
a 33 =a 3 -
a 1
a 22
a 23 =
a 3 b 1 - a 1 b 2
b 1
=c 1
a 22 =a 2 -
a 0
a 1
a 3 =b 1
a 11 =a 1
a 0
a 1
b 1
c 1
d 1
a 2
a 3
b 2
a 4
a 0 s^4 +a 1 s^3 +a 2 s^2 +a 3 s+a 4 = 0
a 117 0, a 227 0, a 337 0, a 447 0, p
¢ 17 0, ¢ 27 0, ¢ 37 0, ¢ 47 0, p
¢ 1 =a 11
¢ 2 =a 11 a 22
¢ 3 =a 11 a 22 a 33
¢ 4 =a 11 a 22 a 33 a 44