Tests for Hurwitz and Schur properties of convex combination of complex polynomials

A test of Hurwitz (Schur) stability of a convex combination of Hurwitz (Schur) polynomials that requires only the checking for absence of zeros in the interval, 0<λ<1, of a polynomial in λ having complex coefficients is studied. From this polynomial Δ(λ) a polynomial Δ*(λ) can be constructed by complex conjugating the coefficients of Δ(λ) in order to form a polynomial which has real coefficients. When the coefficients are restricted to real coefficients, a simplification is provided that is particularly attractive since the Hurwitz stability of a convex combination of strict Hurwitz nth-degree polynomials requires the testing for the absence of zeros in the real interval (0, 1) of a polynomial of degree (n-1). A similar statement applies to a specialization of the results pertaining to Schur stability when the polynomial coefficients are real. This study, therefore, provides a unified approach for testing the Hurwitz or Schur stability of a convex combination of polynomials and generalizes earlier results to the complex coefficient case