Backpropagation Neural Networks (BPNNs) are widely used in fault diagnosis and parameter prediction due to their simple structure and strong universal approximation capabilities. However, BPNNs suffer from slow convergence and susceptibility to poor local minima under basic gradient descent settings. To address these issues, this paper proposes a Balanced Grey Wolf Optimizer (BGWO) as an alternative to gradient descent for training BPNNs. This paper proposes a novel stochastic position update formula and a novel nonlinear convergence factor to balance the local exploitation and global exploration of the traditional Grey Wolf Optimizer. After exploration, the optimal convergence coefficient is determined. The test results on the six benchmark functions demonstrate that BGWO achieves better objective function values under fixed iteration settings. Based on BGWO, this paper constructs a training method for BPNN. Finally, three public datasets are used to test the BPNN trained with BGWO (BGWO-BPNN), the BPNN trained with Levenberg–Marquardt, and the traditional BPNN. The relative error and mean absolute percentage error of BPNNs’ prediction results are used for comparison. The Wilcoxon test is also performed. The test results show that, under the experimental settings of this paper, BGWO-BPNN achieves superior predictive performance. This demonstrates certain advantages of BGWO-BPNN.

Permanent magnet synchronous motors (PMSM), with their high efficiency and power density, are widely used in industrial applications. For PMSM speed servo systems, fractional-order proportional-integral (FOPI) controllers demonstrate superior robustness and speed control performance compared to conventional PI controllers. However, FOPI controllers involve more parameters with insufficient tuning experience, making their parameter design more challenging. To address the aforementioned problems, this paper proposes a novel FOPI control strategy for PMSM that integrates an improved grey wolf optimizer (IGWO) with backpropagation neural networks (BPNNs). The conventional grey wolf optimizer (GWO) is enhanced in this study, and the test results demonstrate that the proposed IGWO exhibits improved convergence and robustness. BPNNs combined with IGWO are employed to fit the relationship between controller parameters and control performance indicators. IGWO employs the fitted values from BPNNs as evaluation criteria to perform online tuning of the FOPI controller parameters. The simulation results show that under the IGWO-BPNN-FOPI control strategy, the overshoot of the PMSM speed step response is only 1.958%, the settling time is reduced by 80%, and the load disturbance rejection performance of the PMSM is significantly improved.