# Intelligent Diagnosis of Compound Faults of Gearboxes Based on Periodical Group Sparse Model
## 1 Introduction  
With the increasing demand for energy in social development, wind turbines, gas turbines and other high-end equipment will be the main body of new energy supply under the carbon neutral strategy. As a key component of energy conversion, gearboxes are susceptible to irregular alternating loads and impact loads during operation, thus generating failures [1]. Timely and reliable condition monitoring and fault diagnosis technology is important in reducing operation and maintenance costs, improving production management capability, and providing \"foresight\" predictive maintenance solutions. Traditional time-frequency analysis methods, such as wavelet transform [2, 3], empirical modal decomposition [4] and variational modal decomposition [5], have achieved satisfactory results in single-fault diagnosis. In the actual operating conditions, a single fault will have a chain reaction with each component of the transmission chain, and multiple faults will occur in successive cascades to form a compound fault. At the same time, various components in the signal are affected by the background noise, so that the vibration signal has strong interference, nonlinearity, weakened fault characteristics, coupling modulation characteristics and other characteristics, resulting in the weakening of the signal fault characteristics. Therefore, it is difficult to separate the composite fault components using traditional signal processing methods. Recently, sparse representation has attracted attention as a novel signal processing method in many fields, such as target detection [6], face recognition [7], and speech signal processing [8]. Sparse representation is essentially the use of a small number of atoms todescribe the internal structure and feature information of a signal without preprocessing operations such as denoising, the optimal selection of fault features and the stripping of working condition information, which enables a flexible, concise and adaptive representation of the signal. The focus of sparse representation is on the construction of the dictionary. The parametric dictionary consists of predefined basis functions to match the periodic impulse features in the fault signal. [PERSON] et al. [PERSON] et al. (2018) designed a parametric dictionary that can highly match the bearing fault features and improve the iterative stopping criterion of the orthogonal matching tracking algorithm to successfully diagnose the rolling bearing fault signal. [PERSON] et al. [PERSON] et al. (2018) constructed an adaptive adjustment Gabor dictionary with the residual signal, which solved the shortcomings of the traditional method with large calculation and improved the signal sparsity while ensuring the accuracy of the fault features. [PERSON] et al. (2018) optimized the quality factor in the resonance-based sparse decomposition (RSSD) using the squirrel algorithm, and the obtained optimal resonance components effectively improved the separation accuracy of the bearing fault features. However, due to the variability and unpredictability of the pulse components in the actual vibration signal, the performance of the parametric dictionary is limited by the atomic structure [PERSON] et al. (2018), and the atoms obtained using explicit mathematical formulas are no longer adaptively adjusted to the changing characteristics of the signal. For this reason, learning dictionaries such as the optimal direction method and K-singular value decomposition (K-SVD) provide new inspiring ideas for fault diagnosis. [PERSON] et al. [PERSON] et al. (2018) improved the objective function and constraints in the K-SVD model to construct a dictionary matching fault features without considering sparsity, and achieved a feature enhancement of weak bearing faults. [PERSON] et al. [PERSON] et al. (2018) used VMD to select the optimal components to form the training set, and then used K-SVD for dictionary learning.