  Controller design faces a trade-off between robustness and performance, and
the reliability of linear controllers has caused many practitioners to focus on
the former. However, there is renewed interest in improving system performance
to deal with growing energy constraints. This paper describes a learning-based
model predictive control (LBMPC) scheme that provides deterministic guarantees
on robustness, while statistical identification tools are used to identify
richer models of the system in order to improve performance; the benefits of
this framework are that it handles state and input constraints, optimizes
system performance with respect to a cost function, and can be designed to use
a wide variety of parametric or nonparametric statistical tools. The main
insight of LBMPC is that safety and performance can be decoupled under
reasonable conditions in an optimization framework by maintaining two models of
the system. The first is an approximate model with bounds on its uncertainty,
and the second model is updated by statistical methods. LBMPC improves
performance by choosing inputs that minimize a cost subject to the learned
dynamics, and it ensures safety and robustness by checking whether these same
inputs keep the approximate model stable when it is subject to uncertainty.
Furthermore, we show that if the system is sufficiently excited, then the LBMPC
control action probabilistically converges to that of an MPC computed using the
true dynamics.
