# Intelligent Diagnosis of Compound Faults of Gearboxes Based on Periodical Group Sparse Model
## 2 Materials and Methods
### Proposed Fault Diagnosis Method  
#### 2.2.1 Compound Fault Diagnosis Method Based on Period Group Sparse Model  
Suppose that the fault signal \(x\) is recovered from the noisy signal \(y\), where \(x\) is with the group periodicity property. However, the large magnitude coefficients in the signal do not exist independently of each other, but are composed in group clusters. The l1 parametric and other sparse models cannot capture the clustering trend of the coefficients, so [PERSON] et al. [19] proposed an overlapping group sparse model to recover the fault signal through constructing an objective function containing a nonconvex penalty function. However, in practical fault diagnosis, the pulse lengths of various types of fault information cannot be determined in advance, so the penalty function is improved by assuming that each group has the same size and maximum overlap [18] and defined as shown in Equation (16). Define the index sets \(\Omega\) and \(\psi\) as shown in Equation (17). \[ \begin{array}{ll}P(x)&=\lambda\sum\limits_{i\in\Omega}\left([\sum\limits_{ j\in\psi}|x(i+j)|^{2}]^{\frac{1}{2}}+\rho\sum\limits_{j\in\psi}|x(i+j)|\right)\\ &=\lambda\sum\limits_{i\in\Omega}\left(\|x_{i,K}\|_{2}+\rho\|x_{i,K}\|_{1} \right)\\ &\Omega=[1,\ldots N]\ \ \ \psi=[0,1,\ldots K-1]\end{array}  \tag{16}\]  
where \(\lambda\) and \(\rho\) are the regular term parameters. The length of the signal is \(N\) and each group has the same size \(K\).\(i\) is the group index and \(j\) is the coefficient index. \(x_{i,K}=[x(i),\ldots,x(i+K-1)]\in R^{K}\) indicates that the size of the \(i\)th group is \(K\). To decouple multiple faults from a composite fault signal, the penalty term of the overlapping group sparse group model is modified by the feature frequencies of the faults. A binary sequence is used to make the penalty term concise and highly structured. \[ \begin{array}{ll}\text{b}=\underbrace{[1,1,\ldots,1]}_{N_{1}},\underbrace {0,0,\ldots 0}_{N_{0}},\underbrace{1,1,\ldots,1}_{N_{1}},\ldots,\underbrace{0,0, \ldots 0}_{N_{0}},\underbrace{1,1,\ldots,1}_{N_{1}}]\\ &\\ \hskip 142.26378 ptN_{0}+N_{1}\approx F_{s}/f_{c},N_{0}+N_{1}=(K+N_{0})/M\end{array}  \tag{17}\]  
where \(N_{1}\) is the estimated length of the oscillation wave and \(N_{0}\) is the time interval between each pulse. \(K\) is the size of the group. \(F_{s}\) and \(f_{c}\) denote the sampling frequency and fault characteristic frequency, respectively. \(M\) is the number of \(N_{1}\) in each group. The three parameters satisfy the relation in Equation (17). The penalty term and the corresponding term in the binary sequence are element-wise multiplied so that inter-group sparsity is maintained between different faults, while intra-group sparsity is formed by maintaining a certain degree of sparsity in the feature information of individual faults. Where the first part of the penalty term representsintergroup sparsity and the second part represents intra-group sparsity. \(\odot\) denotes the multiplication between corresponding elements in the matrix. \[P(\mathrm{x};\mathrm{b})=\sum_{i\in\Omega}\phi(\left\|\mathrm{b}\odot\mathrm{x}_ {i,k}\right\|_{2})+\rho MN1\left\|\mathrm{x}\right\|_{1,\mathrm{w}} \tag{20}\]  
\(\phi\) is a non-convex penalty function, such as an absolute value function. The nature of the function is proved in the literature [20]. The convexity of the model is maintained while promoting sparsity. The expression of the group sparse model is formed as follows:  
\[\min_{\mathrm{x}_{1},\sim\infty}\frac{1}{2}\left\|\mathrm{y}-\sum_{l=1}^{L} \mathrm{x}_{l}\right\|_{2}^{2}+\sum_{l=1}^{L}\lambda_{l}P(\mathrm{x}_{l}; \mathrm{b}_{l}) \tag{21}\]  
The basic idea of the MM algorithm is to define an easily solvable auxiliary function to replace the optimization problem to be solved when the optimization problem is difficult to solve, and to indirectly achieve the solution of the original optimization problem by iteratively optimizing the auxiliary function several times, thus accelerating the convergence speed and reducing the time of the computation [21].