An extension of the generalized Hermite-Biehler theorem: relaxation of earlier assumptions

A generalization of the classical Hermite-Biehler theorem was derived by the authors (1997) and shown to be useful for solving a number of fixed order and structure stabilization problems. This generalization, though adequate for solving these stabilization problems, required the assumption that the polynomial in question have no roots on the imaginary axis except for possibly a simple root at the origin. In this note, one result is extended to also allow roots on the imaginary axis: the main conclusion is that the roots, if any, at the origin modify the earlier theorem statement only very slightly while the other imaginary axis roots leave it unchanged. The extension presented here permits a clearer exposition of the stabilization results previously obtained.