A simple test for Schur stability of a diamond of complex polynomials

Motivated by the fact that the theorem of Kharitonov on Hurwitz stability of a rectangle of polynomials does not hold for discrete-time systems, the author studies the Schur stability of a so-called diamond of polynomials. The results obtained can be stated as follows: a polynomial p(z,α,β) (α₀≠0, β₀≠0) with complex coefficients varying in a diamond in Schur stable if and only if four vertices of the diamond are Schur stable. Consequently, assuming that the zero-th order coefficient of the polynomial is complex, the `magic number´ four, computed by Kharitonov for Hurwitz stability of a rectangle of polynomials still holds for Schur stability of a diamond of polynomials