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 , then obtaining necessary and sufficient conditions to determine stability on the family F A is one of the most important tasks in the field of robust control. Two interesting classes of families arise when A is a diamond or a box of dimension n +1. These families will be denoted by F D n and F B n , respectively. In this paper a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials 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 ( α ∗ , x ) whose stability determines the stability of the entire family F A . In this case f ( α ∗ , x ) comes from minimizing determinants and sometimes f ( α ∗ , x ) coincides with a Kharitonov’s polynomial. Thus another extremal property of Kharitonov’s polynomials has been found. To illustrate the versatility/generality of our approach, this is addressed to families such as F D n and F B n , when n ≤5. Furthermore, the study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family 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 F B 5 is proposed whose extreme polynomial is not necessarily a Kharitonov’s polynomial.