  We propose and analyze a real-time model predictive control (MPC) scheme that
utilizes stored data to improve its performance by learning the value function
online with stability guarantees. For linear and nonlinear systems, a learning
method is presented that makes use of basic analytic properties of the cost
function and is proven to learn the MPC control law and the value function on
the limit set of the closed-loop state trajectory. The main idea is to generate
a smart warm start based on historical data that improves future data points
and thus future warm starts. We show that these warm starts are asymptotically
exact and converge to the solution of the MPC optimization problem. Thereby,
the suboptimality of the applied control input resulting from the real-time
requirements vanishes over time. Simulative examples show that existing
real-time MPC schemes can be improved by storing data and the proposed learning
scheme.
