$($You are allowed to use MATLAB for this homework. All of the necessary m$-$files, functions and plots should be given in addition to the detailed analysis and comments in the report. You need to submit print$-$out report until the deadline. The solutions sent via e$-$mail will not be accepted.$)$

Consider an uncertain polynomial given as follows:

$P(s,q)=s^5+q_1{s}^4+\frac{288s^4}{11}+\frac{173q_1{s}^3}{14}+q_4{s}^3+\frac{2628s^3}{11}+\frac{2091q_1{s}^2}{19}+q_3{s}^2+\frac{63q_4{s}^2}{25}+\frac{10026s^2}{11}+\frac{6193q_{1}s}{8}+q_{2}s+$

$\frac{5q_{3}s}{3}+\frac{37q_{4}s}{9}+\frac{9379s}{7}+\frac{14645q_1}{6}+\frac{5q_2}{6}+\frac{7q_3}{10}+\frac{19q_4}{15}+\frac{8143}{3}$

where the uncertain parameters are bounded as $0q_{i}100($for $i=1,2,3,4).$

a$)(20$ points$)$ Draw the pole$-$spread of the polynomial.

b$)(20$ points$)$ Calculate Hurwitz matrices and the corresponding determinants. Examine the stability of the polynomial family using the $4$th Hurwitz determinant.

c$)(20$ points$)$ For $q_2=100,q_3=q_4=0,$ examine the stability of the polynomial by the help of Bialas Theorem.

d$)(40$ points$)$ Draw the Mikhailov Plot for the polynomial $(p(s,q))$ using gridding. Grid the frequency variable $(w)$ between $0$ and $50.($You can use line segments to join the points of the frequency plot for each qinQ $).$ By looking at this graph, what can you say about the stability of the polynomial family? $($Note that in order to comment on the graph you may have to draw the graphics at different scales. For this purpose draw the Mikhailov Plot for $0\mathrm{.}0$ and for $0.$