This paper presents a model predictive control (MPC) for dynamic systems
whose nonlinearity and uncertainty are modelled by deep neural networks (NNs),
under input and state constraints. Since the NN output contains a high-order
complex nonlinearity of the system state and control input, the MPC problem is
nonlinear and challenging to solve for real-time control. This paper proposes
two types of methods for solving the MPC problem: the mixed integer programming
(MIP) method which produces an exact solution to the nonlinear MPC, and linear
relaxation (LR) methods which generally give suboptimal solutions but are much
computationally cheaper. Extensive numerical simulation for an inverted
pendulum system modelled by ReLU NNs of various sizes is used to demonstrate
and compare performance of the MIP and LR methods.

该研究论文提出了一种用于动态系统的模型预测控制（MPC）方法，该方法利用深度神经网络（NNs）对系统的非线性和不确定性进行建模，并提出了两种用于解决 MPC 问题的方法：混合整数规划（MIP）方法和线性松弛（LR）方法。进行了广泛的数值模拟来演示和比较 MIP 和 LR 方法的性能。

非线性深度神经网络建模的高效模型预测控制

Efficient model predictive control for nonlinear systems modelled by  deep neural networks

The use of Recurrent Neural Networks (RNNs) for system identification has
recently gathered increasing attention, thanks to their black-box modeling
capabilities.Albeit RNNs have been fruitfully adopted in many applications,
only few works are devoted to provide rigorous theoretical foundations that
justify their use for control purposes. The aim of this paper is to describe
how stable Gated Recurrent Units (GRUs), a particular RNN architecture, can be
trained and employed in a Nonlinear MPC framework to perform offset-free
tracking of constant references with guaranteed closed-loop stability. The
proposed approach is tested on a pH neutralization process benchmark, showing
remarkable performances.

通过使用稳定的门控循环单元（GRUs）的递归神经网络（RNNs），在非线性模型预测控制（NMPC）框架下实现无偏移跟踪的研究，其结果表明在不降低系统稳定性的情况下取得了显著的性能提升。