  We propose a learning-based robust predictive control algorithm that can
handle large uncertainty in the dynamics for a class of discrete-time systems
that are nominally linear with an additive nonlinear dynamics component. Such
systems commonly model the nonlinear effects of an unknown environment on a
nominal system. Motivated by an inability of existing learning-based predictive
control algorithms to achieve safety guarantees in the presence of
uncertainties of large magnitude in this setting, we achieve significant
performance improvements by optimizing over a novel class of nonlinear feedback
policies inspired by certainty equivalent "estimate-and-cancel" control laws
pioneered in classical adaptive control. In contrast with previous work in
robust adaptive MPC, this allows us to take advantage of the structure in the a
priori unknown dynamics that are learned online through function approximation.
Our approach also extends typical nonlinear adaptive control methods to systems
with state and input constraints even when an additive uncertain function
cannot directly be canceled from the dynamics. Moreover, our approach allows us
to apply contemporary statistical estimation techniques to certify the safety
of the system through persistent constraint satisfaction with high probability.
We show that our method allows us to consider larger unknown terms in the
dynamics than existing methods through simulated examples.
