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 LMI optimization. The key ingredient in the design procedure resides in the choice of a central polynomial, or desired nominal closed-loop characteristic polynomial. Several numerical examples illustrate the relevance of the approach.