Enhanced Sparse Regularization Based on Logarithm Penalty and Its Application to Gearbox Compound Fault Diagnosis

A vibration signal observed from a gearbox is normally composed of the vibrations of bearing and gear components as well as the strong background noise. Sparse regularization performs as an effective method to deal with vibration signal denoising and compound fault diagnosis for the gearbox. In this article, a multivariate and non-convex logarithm penalty based on the generalized infimal convolution smoothing (GICS) is deduced. To guarantee the global minimum of the overall cost function, we also derive a convexity condition to prescribe the non-convex penalty. In this way, the optimal sparse solution to the cost function can be calculated by a convex algorithm. Multiplied by respective transform matrices, all fault components in a compound signal can be simultaneously reconstructed, and then each fault characteristic frequency can be extracted. Numerical and experimental analyses demonstrate the effectiveness of the proposed method. Compared with the classical convex $L1$ norm and newly developed non-convex generalized minimax-concave (GMC) penalties, the proposed method is superior in terms of inducing sparsity effectively and enhancing the estimation accuracy of signal components.