Value Function Approximation for Nonlinear MPC: Learning a Terminal Cost Function with a Descent Property

We present a novel method to synthesize a terminal cost function for a nonlinear model predictive controller (MPC) through value function approximation using supervised learning. Existing methods enforce a descent property on the terminal cost function by construction, thereby restricting the class of terminal cost functions, which in turn can limit the performance and applicability of the MPC. We present a method to approximate the true cost-to-go with a general function approximator that is convex in its parameters, and impose the descent condition on a finite number of states. Through the scenario approach, we provide probabilistic guarantees on the descent condition of the terminal cost function over the continuous state space. We demonstrate and empirically verify our method in a numerical example. By learning a terminal cost function, the prediction horizon of the MPC can be significantly reduced, resulting in reduced online computational complexity while maintaining good closed-loop performance.