# Intelligent Diagnosis of Compound Faults of Gearboxes Based on Periodical Group Sparse Model
## 1 Introduction
Compared with the parametric dictionary, the method applied to pipeline leakage vibration signals shows better compressive reconfigurability and sparsity. When extracting fault features using sparse representation, most studies use 10 or 11 norm and its deformers, which suffer from algorithmic degradation leading to amplitude underestimation when approximating the sparsity of the characterized pulse. For this reason, it has been proposed to embed the structural information of the signal into SR and exploit the sparsity within and across groups (SWAG) of the features [PERSON] et al. (2018) to improve the extraction accuracy of fault features. [PERSON] et al. [PERSON] et al. (2018) constructed a periodic convolutional sparse representation using a learned dictionary and a Fourier parameter dictionary to extract fault classification and harmonic components so that the sparse coefficients have periodicity and group sparsity. The method performs well in extracting bearing fault features. [PERSON] et al. [PERSON] et al. (2018) combined a sparse fidelity term and a generalized minimum-maximum concave penalty term in the sparse representation model to effectively prevent the signal amplitude from being underestimated and improve the reconstruction accuracy of gear and bearing fault features. [PERSON] et al. [PERSON] et al. (2018) introduced an enhanced sparse group lasso to improve the SWAG property, and then used an optimized minimization algorithm (Majorize-Minimization, MM) to iteratively solve the model to effectively extract the fault features of the inner and outer rings of the bearings while maintaining the magnitude and sparsity of the reconstructed signals. In summary, the vibration signal-based gearbox fault diagnosis technology is becoming increasingly mature, but the following problems still exist:  
1. Most of the existing research focuses on single faults or localized compound faults, such as gear local faults, and bearing inner and outer ring compound faults. There is less research on composite faults formed in different positions of bearings and gears, etc. 2. Multiple faults in the transmission path due to mutual interference between fault features and energy loss and other effects of modulation coupling, decoupling separation that lead to an underestimation of the amplitude, thus leading to the signal reconstruction effect. 3. The traditional fault diagnosis using the fault feature frequency detection method; the detection performance is easily deteriorated due to factors such as part manufacturingerrors and shaft misalignment. The engagement vibration and background noise can submerge or interrupt the impulse features of the fault. According to the above analysis, this paper proposes an intelligent diagnosis method for compound faults based on the periodic group sparse model. Different from the existing diagnostic methods, firstly, a binary periodic sequence is constructed to provide periodic prior knowledge for the fault signal, and a sparse model for handling compound faults is constructed by combining with a group sparse model with overlapping properties. Then, the MM algorithm is used to decouple and separate the fault signals; compared with other conventional methods, the proposed algorithm shows a higher discrimination and robustness in practical applications. Finally, the energy features of wavelet packets with the decoupled different fault states are extracted and imported into the sparse dictionary classification model as the data input to achieve fault feature classification and identification with the self-driven property of data. So far, the intelligent diagnosis of compound faults in gearboxes is realized. The effectiveness of the proposed method is verified with the analysis of the simulation signal and the experimental signal; the overall technology roadmap is shown in Figure 1.  
The arrangement of the paper is as follows: Section 1 is the introduction of the article.