Performance-Robustness Tradeoffs in Adversarially Robust Control and Estimation

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.