用物理约束神经网络精准建模同步电机磁特性，支持实时控制。
Gradient Networks for Universal Magnetic Modeling of Synchronous Machines
- 将梯度网络嵌入电机方程，直接学习磁能梯度以保证物理一致性。
- 仅需少量数据即可实现高精度建模，且输出平滑单调。
- 适合电机控制中的模型逆推与轨迹优化，已实现在嵌入式平台运行。
本文提出一种物理约束的神经网络框架，用于饱和同步电机的动态建模，包含空间谐波。该架构将梯度网络直接嵌入基本电机方程中，通过学习磁能梯度，天然满足互易性与能量守恒约束。模型可通用逼近任意物理可行的磁特性，相比查表法和传统黑箱网络具有单调性、输出平滑及小样本下更好泛化能力。这些特性也支持鲁棒的模型逆推与轨迹优化，适用于控制。方法在一台5.6-kW永磁同步磁阻电机的实测与有限元法（FEM）数据上验证，并在嵌入式控制平台实现闭环实时运行。结果表明，即使训练数据有限，仍能保持高精度与物理一致性。
原文摘要 · Abstract (English)
This paper presents a physics-constrained neural network framework for dynamic modeling of saturable synchronous machines, including spatial harmonics. The proposed architecture embeds gradient networks directly into the fundamental machine equations to model nonlinear, coupled electromagnetic behavior. By learning the gradient of magnetic field energy, the model satisfies reciprocity and energy-balance constraints by construction. The approach can universally approximate any physically feasible magnetic characteristics while offering key advantages over lookup tables and conventional black-box networks: monotonicity, smooth outputs, and improved generalization from limited data. These properties also support robust model inversion and trajectory optimization for control. The method is validated using measured and finite-element-method (FEM) data from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine and is further demonstrated experimentally in real-time closed-loop operation on an embedded control platform. The results show accurate and physically consistent modeling performance, even with limited training data.
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