Generalizations of the Hermite–Biehler theorem

AbstractThe Hermite–Biehler theorem gives necessary and sufficient conditions for the Hurwitz stability of a polynomial in terms of certain interlacing conditions. In this paper, we generalize the Hermite–Biehler theorem to situations where the test polynomial is not necessarily Hurwitz. The generalization is given in terms of an analytical expression for the difference between the numbers of roots of the polynomial in the open left-half and open right-half planes. The result can be used to solve important stabilization problems in control theory and is, therefore, of both academic as well as practical interest