  Despite decades of research and recent progress in adaptive control and
reinforcement learning, there remains a fundamental lack of understanding in
designing controllers that provide robustness to inherent non-asymptotic
uncertainties arising from models estimated with finite, noisy data. We propose
a robust adaptive control algorithm that explicitly incorporates such
non-asymptotic uncertainties into the control design. The algorithm has three
components: (1) a least-squares nominal model estimator; (2) a bootstrap
resampling method that quantifies non-asymptotic variance of the nominal model
estimate; and (3) a non-conventional robust control design method using an
optimal linear quadratic regulator (LQR) with multiplicative noise. A key
advantage of the proposed approach is that the system identification and robust
control design procedures both use stochastic uncertainty representations, so
that the actual inherent statistical estimation uncertainty directly aligns
with the uncertainty the robust controller is being designed against. We show
through numerical experiments that the proposed robust adaptive controller can
significantly outperform the certainty equivalent controller on both expected
regret and measures of regret risk.
