Robust stability of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H ∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.