This paper shows how to train Model Predictive Control (MPC) so it adapts quickly to new tasks. The authors combine gradient-based tuning of MPC with Model‑Agnostic Meta‑Learning (MAML). They also add online system identification to keep the predictive model up to date. In hardware tests on a Ball‑on‑Plate system, the method adapts faster to different trajectory‑tracking tasks than retraining from scratch.

The basic idea is to treat the MPC controller as a policy with trainable pieces. The MPC uses a predictive model f_theta(x,u) and cost matrices that depend on a parameter p. During operation the MPC predicts future states by linearizing the model along the previous MPC solution. The controller can relax state constraints with slack variables so the optimization stays feasible on real hardware. The authors use gradient descent to tune the MPC parameters and the model parameters together, but they embed that tuning inside a meta‑learning loop so the controller learns a single generalized parameter p_bar. From p_bar, a task‑specific parameter p_r for a new reference can be obtained with only one or a few gradient steps.

At a higher level they adapt the MAML idea to MPC. During training they solve an inner problem that simulates or runs the closed loop for a given reference trajectory r and produces a cost Cfull(r,p,theta_r). The meta step then adjusts p_bar so that a single gradient update from p_bar gives a good task‑specific p_r. Because exact computation of the meta‑gradient would require costly second derivatives of the MPC mapping, the authors use approximations and they update the predictive model parameters theta_r with online system identification. Practically, estimating the meta‑gradient requires two interactions with the real system: one to form the gradient used for adaptation and one to evaluate the adapted controller.