On the definiteness of univariate and bivariate two-side matrix polynomials over unitary complex numbers

Establishing whether a two-side matrix polynomial is definite over unitary complex numbers is an important problem in modeling, analysis and design of control systems. This paper addresses this problem for univariate and bivariate two-side matrix polynomials, i.e., matrix functions that are polynomial in the variables and their inverse. Sufficient and necessary conditions are proposed in terms of linear matrix inequality (LMI) feasibility tests by exploiting trigonometric transformations and the theory of positive polynomials. Some numerical examples illustrate the proposed conditions.