A sum-of-squares approach to fixed-order H∞-synthesis

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]. In this work we apply these techniques to the fixed order or structured H∞-synthesis problem. H∞-controller synthesis is an attractive model-based control design tool which allows incorporation of modeling uncertainties in control design. We concentrate on H∞-synthesis although the method can be applied to other performance specifications that admit a representation in terms of Linear Matrix Inequalities (LMI’s). It is well-known that an H∞optimal full order controller can be computed by solving two algebraic Riccati equations [7]. However, the fixed order H∞-synthesis problem is much more difficult. In fact it is one of the most important open problems in control engineering, in the sense that until now there do not yet exist fast and reliable methods to compute optimal fixed order controllers. As the basic setup we consider the closed-loop interconnection as shown below, where the linear system P is the generalized plant and K is a linear controller.