On Positivity of Linear Combinations of Polynomials

Abstract

Since the Kharitonov's theorem for the robust stability of interval polynomials was appreciated, the robustness of varied characteristics of uncertain systems has been extensively investigated. Though positivity of real polynomials is a fundamental condition in network theory and adaptive control systems, little results are known about its robustness. We study the robust positivity of real polynomials with uncertain coefficient parameters which are expressed by linear combinations of polynomials. “Positivity margins” are obtained by means of the discriminants and the inners of the matrices composed of coefficients of polynomials. Both of them are sufficient for the linear combinations of polynomials to be positive. They can be easily obtained by calculating mainly eigenvalues of matrices.