  We propose a new method to obtain feedback controllers of an unknown
dynamical system directly from noisy input/state data. The key ingredient of
our design is a new matrix S-lemma that will be proven in this paper. We
provide both strict and non-strict versions of this S-lemma, that are of
interest in their own right. Thereafter, we will apply these results to
data-driven control. In particular, we will derive non-conservative design
methods for quadratic stabilization, H_2 and H_inf control, all in terms of
data-based linear matrix inequalities. In contrast to previous work, the
dimensions of our decision variables are independent of the time horizon of the
experiment. Our approach thus enables control design from large data sets.
