# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach  
[PERSON]  
1 College of Mechanical Engineering, Dong  
\(x\) from the acquired signal \(y\), i.e., \(y=Ax+w\), where signal \(w\) represents additive noise. To estimate the fault signal \(x\), the optimization problem can be expressed by minimizing the objective cost function (OCF) \(F(x)\), i.e., \(\hat{x}==\underset{x}{\arg\min}\{\frac{1}{2}\|y-Ax\|_{2}^{2}+\lambda\theta(x)\}\), \(\lambda>0\), where \(\theta(x)\) is a penalty function that induces sparsity of the approximating value of \(x\) and \(\lambda\) is the regularization term parameter.  
Methodologies for SLMR can be roughly categorized as convex and non-convex penalty regularization approaches. Among convex penalty functions, L1-norm penalty function is commonly used to regularize linear inverse problems, i.e., penalty function \(\theta(x)=\|x\|_{1}\). In that case, the OCF \(F(x)\) is trivially convex and the global optimum can be obtained. Unfortunately, the L1-norm may underestimate the sparse coefficients of the underlying signal [16; 17]. In contrast, numerous non-convex penalties (e.g., L\({}_{\text{P}}\) pseudo-norm with \(p<1\)[18], reweighted L2/L1 norm [19]) and non-convex algorithms (e.g., orthogonal matching pursuit, OMP) have been proposed [20; 21; 22; 23; 24; 25], which estimate non-zero components more accurately than convex regularization. However, the previous non-convex penalties were considered based on separable penalty functions and thus suffer from several issues such as sensitivity to changes with initial data and regularization term parameters, etc., and the convexity of the OCF cannot be guaranteed simultaneously. Therefore, the core objective of SLMR is to determine how to maintain convexity of the OCF when the regularizer (or penalty function) is taken to be non-convex, so that the fault component can be estimated accurately from its noisy observation.  
To address such issues, a novel non-separable non-convex regularizer, namely augmented Huber penalty function, is developed in this paper, which overcomes limitations of separable non-convex regularization and ensures the strict convexity of the OCF. Specifically, the augmented penalty function is firstly defined by L1-norm and generalized Huber functions, the strict convexity and convexity condition of the proposed OCF is then derived and provided. Meanwhile, an iterative algorithm (e.g., forward-backward splitting, FBS) is presented as a solution to the proposed objective cost function. Lastly, both the simulated signal and engineering bearing data of larger reducers are employed to validate the effectiveness of the proposed approach in terms of diagnostic accuracy and signal-to-noise ratio, etc.  
The layout of this work is organized as follows: Section 2 introduces the algorithm and theoretical derivation of the augmented penalty optimization; Section 3 presents the simulation evaluation of the proposed augmented Huber non-convex penalty regularization (AHNPR) method; In Section 4, the practical diagnosis results and discussion for tapered bearing using the proposed approach with other methods are presented; and Conclusions are drawn in Section 5.