  We propose a learning-based robust predictive control algorithm that
compensates for significant uncertainty in the dynamics for a class of
discrete-time systems that are nominally linear with an additive nonlinear
component. Such systems commonly model the nonlinear effects of an unknown
environment on a nominal system. We optimize over a class of nonlinear feedback
policies inspired by certainty equivalent "estimate-and-cancel" control laws
pioneered in classical adaptive control to achieve significant performance
improvements in the presence of uncertainties of large magnitude, a setting in
which existing learning-based predictive control algorithms often struggle to
guarantee safety. In contrast to previous work in robust adaptive MPC, our
approach allows us to take advantage of structure (i.e., the numerical
predictions) in the a priori unknown dynamics learned online through function
approximation. Our approach also extends typical nonlinear adaptive control
methods to systems with state and input constraints even when we cannot
directly cancel the additive uncertain function from the dynamics. We apply
contemporary statistical estimation techniques to certify the system's safety
through persistent constraint satisfaction with high probability. Moreover, we
propose using Bayesian meta-learning algorithms that learn calibrated model
priors to help satisfy the assumptions of the control design in challenging
settings. Finally, we show in simulation that our method can accommodate more
significant unknown dynamics terms than existing methods and that the use of
Bayesian meta-learning allows us to adapt to the test environments more
rapidly.
