Consider the characteristic polynomial qs s5 2s4 24s3 48

Consider the characteristic polynomial q(s) = s^5 + 2s^4 + 24s^3 + 48s^2 - 25s - 50 (a) Use the RH method to tell how many squareroots of q(s) are in the right half-plane, in the left half-plane, and on the jomega-axis.

Solution

Consider the quintic equation q(s) = 0

where q(s) is s⁵ + 2s⁴ + 24s³ + 48s² -25s 50.

The Routh array starts off as

s⁵ 1 24 25

s⁴ 2 48 50 auxiliary polynomial P(s)

s³ 0 0

The auxiliary polynomial P(s) is

P(s) = 2s⁴ + 48s² 50

which indicates that q(s) = 0 must have two pairs of roots of equal magnitude and opposite sign, which are also roots of the auxiliary polynomial equation P(s) = 0.

Taking the derivative of P(s) with respect to s we obtain

[dP(s)/ds] = 8s³ + 96s.

so the s³ row is as shown below and the Routh array is

s⁵ 1 24 25

s⁴    2 48 50

s³ 8 96 Coefficients of dP(s)/ds

s² 24 50

s¹ 112.7 0

s⁰ 50

There is a single change of sign in the first column of the resulting array, indicating that there q(s) = 0 has one root with positive real part. Solving the auxiliary polynomial equation,

2s⁴ + 48s² 50 = 0

yields the remaining roots, namely, from

s² = 1, s² = 25,

s = ±1, s = ±j5.

so the original equation can be factored as

(s + 1)(s 1)(s + j5)(s j5)(s + 2) = 0.