Chapter 6, Problem 47P

An interval polynomial is of the form

$P\left(s\right)=a_0+a_{1}s+a_2{s}^2+a_3{s}^3+a_4{s}^4+a_5{s}^5+\mathrm{...}$

with its coefficients belonging to intervals $x_1\le a_i\le y_i$ , where x_i, y_iare 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):

$\begin{array}{l}{k}_1\left(s\right)=x_0+x_{1}s+y_2{s}^2+y_3{s}^3+x_4{s}^4+x_5{s}^5+y_6{s}^6+\mathrm{...}\\ k_2\left(s\right)=x_0+y_{1}s+y_2{s}^2+x_3{s}^3+x_4{s}^4+y_5{s}^5+y_6{s}^6+\mathrm{...}\\ k_3\left(s\right)=y_0+x_{1}s+x_2{s}^2+y_3{s}^3+y_4{s}^4+x_5{s}^5+x_6{s}^6+\mathrm{...}\\ k_4\left(s\right)=y_0+y_{1}s+x_2{s}^2+x_3{s}^3+y_4{s}^4+y_5{s}^5+x_6{s}^6+\mathrm{...}\end{array}$

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

$\begin{array}{l}P\left(s\right)=a_0+a_{1}s+a_2{s}^2+a_3{s}^3\\ 2\le a_0\le 4;1\le a_1\le 2;4\le a_2\le 6;a_3=1\end{array}$