In this paper, we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty, applicable to multistage robust optimization and stochastic programming. We introduce a hierarchy of near-optimal polynomial disturbance-feedback control policies, and show how these can be computed by solving a single semidefinite programming problem. The approach yields a hierarchy parameterized by a single variable (the degree of the polynomial policies), which controls the trade-off between the optimality gap and the computational requirements. We evaluate our framework in the context of three classical applications—two in inventory management, and one in robust regulation of an active suspension system—in which very strong numerical performance is exhibited, at relatively modest computational expense.