We show that artificial neural networks with rectifier units as activation functions can exactly represent the piecewise affine function that results from the formulation of model predictive control (MPC) of linear time-invariant systems. The choice of deep neural networks is particularly interesting as they can represent exponentially many more affine regions compared to networks with only one hidden layer. We provide theoretical bounds on the minimum number of hidden layers and neurons per layer that a neural network should have to exactly represent a given MPC law. The proposed approach has a strong potential as an approximation method of predictive control laws, leading to a better approximation quality and significantly smaller memory requirements than previous approaches, as we illustrate via simulation examples. We also suggest different alternatives to correct or quantify the approximation error. Since the online evaluation of neural networks is extremely simple, the approximated controllers can be deployed on low-power embedded devices with small storage capacity, enabling the implementation of advanced decision-making strategies for complex cyber-physical systems with limited computing capabilities.

中文翻译：

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通过深度学习有效表示和逼近模型预测控制律


我们证明，以整流单元作为激活函数的人工神经网络可以准确地表示由线性时不变系统的模型预测控制（MPC）公式得出的分段仿射函数。深度神经网络的选择特别有趣，因为与只有一个隐藏层的网络相比，它们可以代表指数级更多的仿射区域。我们提供了神经网络准确表示给定 MPC 定律所需的隐藏层和每层神经元的最小数量的理论界限。正如我们通过仿真示例所说明的，所提出的方法作为预测控制律的近似方法具有强大的潜力，与以前的方法相比，它具有更好的近似质量和显着更小的内存需求。我们还建议不同的替代方案来纠正或量化近似误差。由于神经网络的在线评估极其简单，近似控制器可以部署在存储容量较小的低功耗嵌入式设备上，从而能够为计算能力有限的复杂信息物理系统实施先进的决策策略。