Sum Of Squares Relaxations For Robust Polynomial Semi-Definite Programs

Many robust analysis and synthesis problems can beformulated as robust Semi-Definite Programs (SDPs), i.e. SDPs withdata matrices that are functions of an uncertain parameter whichis only known to be contained in some set. We consider uncertaintysets described by general polynomial semi-definite constraints,which allows to represent non-convex compact uncertainty sets. As themain novel result we present a family of LMI relaxations based on sum-of-squares (sos)decompositions of polynomial matrices whose optimal valuesconverge to the optimal value of the robust SDP. The number ofvariables and constraints in the LMI relaxations grow onlyquadratically in the dimension of the underlying data matrices. Wedemonstrate the benefit of this a priori complexity bound by anexample and apply the method in order to asses the stability of a4th order LPV model of a helicopter.