Abstract
Studies on stability robustness issue in these years have been carried out towards two directions: researches to cope with unstructured bounded perturbations and those to tackle structured uncertainties. The latter approach was first triggered by the celebrated theorem by Kharitonov appeared in a Russian literature in 1978. The theorem claims that the strict Hurwitz property of real polynomials whose coeffcients fall into a hyperbox in the coefficient space is equivalent to that of polynomials corresponding to only four edges of the hyperbox. Since then, many attempts have been made to extend the theorem to various directions, together with its alternative proofs and applications, relating to stability robustness problems.
This paper also addresses an extension of the theorem, in the sense that the assumptions of the theorem are weakened or generalized. Namely, the case is considered when the strict Hurwitz property is replaced by “weak” one in the theorem, that is, one allows the roots of the polynomials within the closed left half complex plane. It turns out that the theorem still holds in an exact parallel way as expected. It is furthermore demonstrated that several additional results take place in this limiting case. Before stating the main results, an extension of Mikhailov's theorem, a classical stability criterion, is first made. With the aid of this extension, the results are proved and summarized in the form of two theorems.
