On moment relaxations for linear state feedback controller synthesis with non-convex quadratic costs and constraints

We present a simple and effective way to account for non-convex costs and constraints in state feedback synthesis, and an interpretation for the variables in which state feedback synthesis is typically convex. We achieve this by deriving the controller design using moment matrices of state and input. It turns out that this approach allows the consideration of non-convex constraints by relaxing them as constraints on the expected value of quadratic functions, and that the variables in which state feedback synthesis is typically convexified can be identified with blocks of these moment matrices. The employed relaxation can lead to constraint violation along sample paths.