  While $\mathcal{H}_\infty$ methods can introduce robustness against
worst-case perturbations, their nominal performance under conventional
stochastic disturbances is often drastically reduced. Though this fundamental
tradeoff between nominal performance and robustness is known to exist, it is
not well-characterized in quantitative terms. Toward addressing this issue, we
borrow the increasingly ubiquitous notion of adversarial training from machine
learning to construct a class of controllers which are optimized for
disturbances consisting of mixed stochastic and worst-case components. We find
that this problem admits a linear time invariant optimal controller that has a
form closely related to suboptimal $\mathcal{H}_\infty$ solutions. We then
provide a quantitative performance-robustness tradeoff analysis in two
analytically tractable cases: state feedback control, and state estimation. In
these special cases, we demonstrate that the severity of the tradeoff depends
in an interpretable manner upon system-theoretic properties such as the
spectrum of the controllability gramian, the spectrum of the observability
gramian, and the stability of the system. This provides practitioners with
general guidance for determining how much robustness to incorporate based on a
priori system knowledge. We empirically validate our results by comparing the
performance of our controller against standard baselines, and plotting tradeoff
curves.
