Sparse Optimization of Vibration Signal by ADMM

Song Wanqing

doi:10.1155/2017/4612853

Abstract

In this paper, the alternating direction method of multipliers (ADMM) algorithm is applied to the compressed sensing theory to realize the sparse optimization of vibration signal. Solving the basis pursuit problem for minimizing theL1norm minimization under the equality constraints, the sparse matrix obtained by the ADMM algorithm can be reconstructed by inverse sparse orthogonal matrix inversion. This paper analyzes common sparse orthogonal basis on the reconstruction results, that is, discrete Fourier orthogonal basis, discrete cosine orthogonal basis, and discrete wavelet orthogonal basis. In particular, we will show that, from the point of view of central tendency, the discrete cosine orthogonal basis is more suitable, for instance, at the vibration signal data because its error is close to zero. Moreover, using the discrete wavelet transform in signal reconstruction there still are some outliers but the error is unstable. We also use the time complex degree and validity, for the analysis of the advantages and disadvantages of the ADMM algorithm applied to sparse signal optimization. The advantage of this method is that these abnormal values are limited in the control range.

Highlights

The monitoring and forecast technique of mechanical fault state is mainly applied to extract or separate the fault features which can reflect the development trend of the equipment fault
In this paper we study the application of alternating direction method of multipliers (ADMM) algorithm in the sparse reconstruction problem for the vibration signal of an equipment
ADMM possesses the good performance in sparse reconstruction compared with primal-dual interior point algorithm and orthogonal matching pursuit (OMP)

Summary

Introduction

The monitoring and forecast technique of mechanical fault state is mainly applied to extract or separate the fault features which can reflect the development trend of the equipment fault. In order to improve the accuracy of prediction, a large amount of vibration data must be collected from the equipment which runs for long time (e.g., half a year). The most popular algorithms in signal reconstruction are: the Minimum algorithm L0, the orthogonal matching pursuit (OMP) algorithm [2] and its improved type, the L1-magi algorithm [3, 4], the weighted algorithm L1 [5], the Homotopy algorithm [6], the Lq-FL algorithm [7]. These reconstructions, have to face some cumbersome problems both on the uncertainty arising in the mathematical inverse problem, and on the computational complexity due to the deficiency of a small quantity of sparse value. We will show that, comparing with the other current methods, the ADMM is more accurate and has low computational costs, being the more suitable method in engineering applications for the long-time monitoring of an equipment status

ADMM Reconstruction Compares with OPM and Dual Interior Point

Sparse Basis Selection during Reconstruction of ADMM

The Comparison between the ADMM and Other Algorithms

Conclusion