An interval polynomial is of the form Ps a0 a1s a2s 2 a3s

Chapter 6, Problem 47

(choose chapter or problem)

An interval polynomial is of the form

\(P(s)=a_0+a_1 s+a_2 s^2+a_3 s^3+a_4 s^4+a_5 s^5+\cdots\)

with its coefficients belonging to intervals \(x_i \leq a_i \leq y_i\), where \(x_i, y_i\) are prescribed constants. Kharitonov's theorem says that an interval polynomial has all its roots in the left half-plane if each one of the following four polynomials has its roots in the left half-plane (Minichelli, 1989):

\(\begin{aligned} & K_1(s)=x_0+x_1 s+y_2 s^2+y_3 s^3+x_4 s^4+x_5 s^5+y_6 s^6+\cdots \\ & K_2(s)=x_0+y_1 s+y_2 s^2+x_3 s^3+x_4 s^4+y_5 s^5+y_6 s^6+\cdots \\ & K_3(s)=y_0+x_1 s+x_2 s^2+y_3 s^3+y_4 s^4+x_5 s^5+x_6 s^6+\cdots \\ & K_4(s)=y_0+y_1 s+x_2 s^2+x_3 s^3+y_4 s^4+y_5 s^5+x_6 s^6+\cdots \end{aligned}\)

Use Kharitonov's theorem and the Routh-Hurwitz criterion to find if the following polynomial has any zeros in the right half-plane.

\(\begin{aligned} & P(s)=a_0+a_1 s+a_2 s^2+a_3 s^3 \\ & 2 \leq a_0 \leq 4 ; \quad 1 \leq a_1 \leq 2 ; \quad 4 \leq a_2 \leq 6 ; \quad a_3=1 \end{aligned}\)