  Probabilistic models such as Gaussian processes (GPs) are powerful tools to
learn unknown dynamical systems from data for subsequent use in control design.
While learning-based control has the potential to yield superior performance in
demanding applications, robustness to uncertainty remains an important
challenge. Since Bayesian methods quantify uncertainty of the learning results,
it is natural to incorporate these uncertainties into a robust design. In
contrast to most state-of-the-art approaches that consider worst-case
estimates, we leverage the learning method's posterior distribution in the
controller synthesis. The result is a more informed and, thus, more efficient
trade-off between performance and robustness. We present a novel controller
synthesis for linearized GP dynamics that yields robust controllers with
respect to a probabilistic stability margin. The formulation is based on a
recently proposed algorithm for linear quadratic control synthesis, which we
extend by giving probabilistic robustness guarantees in the form of credibility
bounds for the system's stability.Comparisons to existing methods based on
worst-case and certainty-equivalence designs reveal superior performance and
robustness properties of the proposed method.
