On Hurwitz stability for families of polynomials

The robustness of a linear system in the view of parametric variations requires a stability analysis of a family of polynomials. If the parameters vary in a compact set A$$ A $$, then obtaining necessary and sufficient conditions to determine stability of the family FA$$ {\mathfrak{F}}_A $$ is one of the most important tasks in the field of robust control. Three interesting classes of families arise when A$$ A $$ is a diamond, a box or a ball of dimension n+1$$ n+1 $$. These families will be denoted by FDn$$ {\mathfrak{F}}_{D_n} $$, FBn$$ {\mathfrak{F}}_{B_n} $$, and FSn$$ {\mathfrak{F}}_{S_n} $$, respectively. In this article, a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials FA$$ {\mathfrak{F}}_A $$. As a result of this study and the use of classical results found in the literature, it is shown the existence of an extremal polynomial f(alpha*,x)$$ f\left({\alpha}<^>{\ast },x\right) $$ whose stability determines the stability of the entire family FA$$ {\mathfrak{F}}_A $$. In this case f(alpha*,x)$$ f\left({\alpha}<^>{\ast },x\right) $$ comes from minimizing determinants and in some cases f(alpha*,x)$$ f\left({\alpha}<^>{\ast },x\right) $$ coincides with a Kharitonov's polynomial. Thus another extremal property of Kharitonov's polynomials has been found. To illustrate our approach, it is applied to families such as FDn$$ {\mathfrak{F}}_{D_n} $$, FBn$$ {\mathfrak{F}}_{B_n} $$, and FSn$$ {\mathfrak{F}}_{S_n} $$ with n <= 5$$ n\le 5 $$. The study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family FDn$$ {\mathfrak{F}}_{D_n} $$ is taken from the literature to compare and corroborate the effectiveness and the advantage of our perspective. Followed by two examples where the maximum robustness of the parameters of polynomials of degree 3 and 4 are obtained. Lastly, a family FB5$$ {\mathfrak{F}}_{B_5} $$ is proposed whose extreme polynomial is not necessarily a Kharitonov's polynomial. Finally, a family FS3$$ {\mathfrak{F}}_{S_3} $$ is used to exemplify that if the boundary of A$$ A $$ is given by a polynomial equation in several variables, the number of candidates to be an extremal polynomial is finite.