Training progress results for speed.

Modeling the complex nonlinear dynamics of Brushless DC motors has been a prominent research focus over the past two decades, driven by their superior advantages and widespread industrial applications. Despite extensive efforts, achieving high-efficiency prediction of speed and torque responses remains a challenge. This study proposes a hybrid machine learning-based approach using the Nonlinear Autoregressive Neural Network with Exogenous Inputs. The method combines artificial neural networks and system identification techniques to enhance predictive accuracy in nonlinear dynamic systems. For both speed and torque modeling, optimal time delays and neural network layer sizes are selected to accurately capture the ripple effects under a multi-step input signal applied to a three-phase inverter. The proposed models yield Mean Square Error values as low as for speed and for torque. Regression coefficients of 1.000 for speed and 0.998 for torque are achieved consistently across training, validation, testing, and additional testing phases, following a data split of 70% for training and 15% each for validation and testing. To further evaluate generalization, the approach is tested using a distinct multi-step input voltage signal, with the results confirming the robustness and superiority of the proposed method in both speed and torque prediction. Comparative analysis with existing literature demonstrates the dominance of the proposed models. These high-fidelity models can serve as a foundation for designing advanced controllers aimed at efficient speed regulation and torque ripple mitigation in Brushless DC motors.近二十年来，得益于无刷直流电机（Brushless DC Motor）优异的性能与广泛的工业应用，对其复杂非线性动力学进行建模始终是研究热点。尽管已有大量研究尝试，但若要实现转速与转矩响应的高精度预测仍颇具挑战。本研究提出一种基于混合机器学习的方法，采用非线性自回归外生输入神经网络（Nonlinear Autoregressive Neural Network with Exogenous Inputs，NARX）。该方法融合人工神经网络与系统辨识技术，以提升非线性动态系统的预测精度。针对转速与转矩建模任务，本研究选取最优时延与神经网络层数，以精准捕捉三相逆变器输入多阶信号时的脉动效应。所提模型的均方误差（Mean Square Error，MSE）在转速与转矩预测中最低可达[原文缺失具体数值]。在训练集、验证集、测试集与额外测试集的全流程中，转速预测的回归系数可达1.000，转矩预测的回归系数可达0.998；本次实验采用7:1.5:1.5的数据划分比例，即70%用于训练，15%用于验证，剩余15%用于测试。为进一步评估模型泛化能力，本研究采用一组全新的多阶输入电压信号对所提方法进行测试，结果证实所提方法在转速与转矩预测中均具备优异的鲁棒性与优越性。与现有文献的对比分析表明，所提模型性能更优。这些高精度模型可作为设计先进控制器的基础，用于实现无刷直流电机的高效转速调节与转矩脉动抑制。