$\mathcal{L}_1$-DRAC: Distributionally Robust Adaptive Control

Data-driven machine learning methodologies have attracted considerable attention for the control and estimation of dynamical systems. However, such implementations suffer from a lack of predictability and robustness. Thus, adoption of data-driven tools has been minimal for safety-aware applications despite their impressive empirical results. While classical tools like robust adaptive control can ensure predictable performance, their consolidation with data-driven methods remains a challenge and, when attempted, leads to conservative results. The difficulty of consolidation stems from the inherently different `spaces' that robust control and data-driven methods occupy. Data-driven methods suffer from the distribution-shift problem, which current robust adaptive controllers can only tackle if using over-simplified learning models and unverifiable assumptions. In this paper, we present $\mathcal{L}_1$ distributionally robust adaptive control ($\mathcal{L}_1$-DRAC): a control methodology for uncertain stochastic processes that guarantees robustness certificates in terms of uniform (finite-time) and maximal distributional deviation. We leverage the $\mathcal{L}_1$ adaptive control methodology to ensure the existence of Wasserstein ambiguity set around a nominal distribution, which is guaranteed to contain the true distribution. The uniform ambiguity set produces an ambiguity tube of distributions centered on the nominal temporally-varying nominal distribution. The designed controller generates the ambiguity tube in response to both epistemic (model uncertainties) and aleatoric (inherent randomness and disturbances) uncertainties.