Positive polynomials and robust stabilization with fixed-order controllers

Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixed-order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed, based on linear matrix inequality optimization. The key ingredient in the design procedure resides in the choice of the central polynomial. Several numerical examples illustrate the relevance of the approach.