  In this paper we develop new data informativity based controller synthesis
methods that extend existing frameworks in two relevant directions: a more
general noise characterization in terms of cross-covariance bounds and
informativity conditions for control based on input-output data. Previous works
have derived necessary and sufficient informativity conditions for noisy
input-state data with quadratic noise bounds via an S-procedure. Although these
bounds do not capture cross-covariance bounds in general, we show that the
S-procedure is still applicable for obtaining non-conservative conditions on
the data. Informativity-conditions for stability, $\mathcal{H}_\infty$ and
$\mathcal{H}_2$ control are developed, which are sufficient for input-output
data and also necessary for input-state data. Simulation experiments illustrate
that cross-covariance bounds can be less conservative for informativity,
compared to norm bounds typically employed in the literature.
