Robust D-stability analysis of uncertain polynomial matrices via polynomial-type multipliers

This paper addresses robust D-stability analysis problems of uncertain polynomial matrices. The underlying idea we follow is that a given polynomial matrix is D-stable if and only if there exist polynomial-type multipliers that render the resulting polynomial matrices to be strictly positive over a specific region on the complex plane. By applying the generalized S-procedure technique, we show that those positivity analysis problems can be reduced into feasibility tests of linear matrix inequalities (LMIs). Thus we can obtain varieties of LMI conditions for (robust) D-stability analysis of polynomial matrices according to the degree/structure of the multipliers to be employed. in particular, we show that existing LMI conditions for robust D-stability analysis can be viewed as particular cases of the proposed conditions, where the degree of the multipliers chosen to be the same as those of the polynomial matrices to be examined. It turns out that, by increasing the degree of the multipliers, we can readily obtain less conservative LMI conditions than the one found in the literature.