  This work presents a novel loss function for learning nonlinear Model
Predictive Control policies via Imitation Learning. Standard approaches to
Imitation Learning neglect information about the expert and generally adopt a
loss function based on the distance between expert and learned controls. In
this work, we present a loss based on the Q-function directly embedding the
performance objectives and constraint satisfaction of the associated Optimal
Control Problem (OCP). However, training a Neural Network with the Q-loss
requires solving the associated OCP for each new sample. To alleviate the
computational burden, we derive a second Q-loss based on the Gauss-Newton
approximation of the OCP resulting in a faster training time. We validate our
losses against Behavioral Cloning, the standard approach to Imitation Learning,
on the control of a nonlinear system with constraints. The final results show
that the Q-function-based losses significantly reduce the amount of constraint
violations while achieving comparable or better closed-loop costs.
