**5. Conclusions**

In this research, aiming to address the difficulty of the low precision of fault diagnosis methods for industrial systems, a new fault diagnosis methodology, named WMPSO-LSSVM, is proposed. Based on the decomposition of fault signals for feature extraction, the gearbox and bearings derived from the composable components are taken as the specific objects, and the vibration can be decomposed without information loss based on WPT. By comparing the proposed method with the existing pattern recognition methods, the results show that the WMPSO-LSSVM method can achieve higher classification accuracy for multiple fault modes in industrial systems.

In addition, PSO optimized by the wavelet mutation is combined with the LSSVM algorithm to realize the further optimization of the regularization parameter and kernel

function in the LSSVM, thereby improving the fault diagnosis accuracy. Particles that jump out of the local extreme value through the wavelet mutation algorithm will seek the optimal solution of parameters in the global space, so the optimal hyperplane of the LSSVM model can be established. As demonstrated via the comparative experiments, the accuracy of the WMPSO-LSSVM is almost 12% higher than that of the LSSVM, and is 9% higher than the ELM; moreover, the average error of the regression is 0.365 less than that of the traditional linear regression model, implying the potency of this scheme.

However, how to better select the parameters in the wavelet mutation function adaptively is not ye<sup>t</sup> resolved in this work. Further research on the optimization of parameters in wavelet mutation is warranted.

In summary, the WMPSO-LSSVM proposed in this paper can significantly improve the fault diagnosis accuracy for complex industrial systems, and therefore, it offers better operability and scalability in the actual industrial environment.

**Author Contributions:** S.G. (Shuyue Guan): writing-original draft, methodology, investigation; D.H.: funding acquisition, supervision; S.G. (Shenghui Guo): writing—review & edtiting, methodology; L.Z.: methodology; H.C.: writing—review & editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the National Natural Science Foundation of P.R. China, under Grants 61663008 and 62073051, the Chong-qing Technology Innovation, Application Special Key Project, under Grant cstc2019jscx-mbdxX0015, and the 2018 Reliable control and safety maintenance of dynamic system under Grant JDDSTD2018001.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data is provided by the Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis of China for providing the dataset of the rolling bearings.

**Acknowledgments:** The authors would like to thank the Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis of China for providing the dataset of the rolling bearings.

**Conflicts of Interest:** The authors declare no conflict of interest.
