Stability Test for Complex Matrices Over the Complex Unit Circumference via LMIs and Applications in 2D Systems

This paper addresses the problem of establishing whether a matrix with rational dependence on a complex parameter and its conjugate is Hurwitz (i.e., has all eigenvalues with negative real part) over the complex unit circumference. A necessary and sufficient condition is proposed, which requires testing stability of a constant matrix and feasibility of a linear matrix inequality (LMI). Moreover, the numerical complexity of the proposed approach is investigated in terms of size and number of free scalar variables of the LMI by deriving their formulas as functions of the problem data. Also, it is shown that the numerical complexity may be significantly reduced without introducing approximations or conservatism whenever some symmetry properties are satisfied. Lastly, the extension of the proposed approach to the case of Schur stability (i.e., eigenvalues with magnitude smaller than one) and D-stability (i.e., eigenvalues on special regions of the complex plane) are presented. The proposed approach is illustrated by some numerical examples that also show its application to stability analysis of 2D systems with mixed signals.