A Sum-of-Squares Approach to Fixed-Order H_∞-Synthesis

Abstract

Recent improvements of semi-definite programming solvers and developments on polynomial optimization have resulted in a large increase of the research activity on the application of the so-called sum-of-squares (SOS) technique in control. In this approach non-convex polynomial optimization programs are approximated by a family of convex problems that are relaxations of the original program [4, 22]. These relaxations are based on decompositions of certain polynomials into a sum of squares. Using a theorem of Putinar [28] it can be shown (under suitable constraint qualifications) that the optimal values of these relaxed problems converge to the optimal value of the original problem. These relaxation schemes have recently been applied to various nonconvex problems in control such as Lyapunov stability of nonlinear dynamic systems [25, 5] and robust stability analysis [15].