Robust Control Theory Based Stability Certificates for Neural Network Approximated Nonlinear Model Predictive Control

Model predictive control requires the real-time solution of an optimal control problem, which can be challenging on computationally limited systems. Approximating the solution such as by neural networks or series expansions, or deriving an explicit solution, can overcome this challenge. Using neural networks for approximation, a question arises as to how to guarantee closed-loop safety and stability. We use robust control theoretic tools to provide stability guarantees using a neural network trained to approximate a model predictive controller. Notably, the model predictive controller, which might offer desirable closed-loop performance, is not required to provide provable stability properties. To provide stability guarantees for the neural network approximated controller, the closed-loop system is reformulated as a diagonal nonlinear differential form, exploiting that the neural network activation functions are sector bounded and that their slopes are globally bounded. Based on this representation, we establish sufficient closed-loop stability conditions in form of linear matrix inequalities for the nominal and the disturbed system using the neural network approximated model predictive controller. Copyright (C) 2021 The Authors.