Performance-Robustness Tradeoffs in Adversarially Robust Linear-Quadratic Control

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 from 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 stationary optimal controller that has a simple analytic form closely related to suboptimal $\mathcal{H}_\infty$ solutions. We then provide a quantitative performance-robustness tradeoff analysis, in which system-theoretic properties such as controllability and stability explicitly manifest in an interpretable manner. 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.