Deep Learning-Based Approximation of Model Predictive Control Laws Using Mixture Networks

In recent years, researchers have proposed the approximation of model predictive control (MPC) using deep neural networks (DNNs). However, a limitation arises as DNNs inherently offer one-to-one mappings, posing a challenge when multiple optimal control inputs correspond to each system state, thereby leading to one-to-many mappings. Therefore, we propose an alternative scheme using mixture networks (MNs) with components of probability (density) distributions in the output layer. This method uses conditional probabilities offered by combining several estimated probability distributions, enabling generating multiple control inputs with the highest probabilities. Notably, this approach is applicable to several problems by choosing a suitable probability distribution, such as using a Gaussian distribution for nonlinear problems and a Bernoulli distribution for mixed-integer linear programming (MILP) problems. We investigate two case studies illustrating that the mixture network-based approximation outperforms the DNN-based approximation.
Note to Practitioners
—This article proposes using mixture (density) networks, a type of machine learning technique, to enable the online implementation of model predictive control. The prohibitive computation cost associated with model predictive control poses a significant challenge when implementing it in complex nonlinear and MILP problems. While approximation methods of control laws using deep neural networks have been studied to address this issue, it is unsuitable for problems where each state has multiple optimal control inputs. In contrast, our proposed approach can accurately approximate the control laws for these problems. This notable feature can facilitate the online implementation of model predictive control in a wide range of nonlinear and MILP problems.