Sum of Squares Relaxations for Robust Polynomial Semi-definite Programs

A whole variety of robust analysis and synthesis problems can be formulated as robust Semi-Definite Programs (SDPs), i.e. SDPs with data matrices that are functions of an uncertain parameter which is only known to be contained in some set. We consider uncertainty sets described by general polynomial semi-definite constraints, which allows to represent norm-bounded and structured uncertainties as encountered in μ-analysis, polytopes and various other possibly non-convex compact uncertainty sets. As the main novel result we present a family of Linear Matrix Inequalities (LMI) relaxations based on sum-of-squares (sos) decompositions of polynomial matrices whose optimal values converge to the optimal value of the robust SDP. The number of variables and constraints in the LMI relaxations grow only quadratically in the dimension of the underlying data matrices. We demonstrate the benefit of this a priori complexity bound by an example and apply the method in order to asses the stability of a fourth order LPV model of the longitudinal dynamics of a helicopter. Copyright c 2005 IFAC