基于变结构奇异值分解的信号处理理论及其在故障诊断中的应用

The idea of variable structure of matrix is introduced into singular value decomposition (SVD) and the concept of variable structure SVD is put forward. It is proposed that the adjustment of matrix structure and recursive decomposition be combined together in this variable structure SVD, and the characteristic that the components of signal are sensitive to the structure of matrix in the SVD based signal processing is utilized, so the adaptation for the different components of the complicated signal can be formed via the variable structure of matrix, and the defect of fixed structure of matrix in the usual SVD based signal processing is overcome, and a multi-gradation adaptive decomposition of the complicated signal can be implemented with SVD. By this adaptive decomposition way the different components can be isolated gradually from strong to weak from the original signal, a series of principal components, which reflect the main energy of the original signal, and a series of subsidiary components, which only occupy the small energy proportion of the original signal, will be got. In total three variable structure ways of matrix are proposed in this project, and decomposition and reconstruction algorithms of the corresponding variable structure SVD will be solved theoretically. The signal decomposition essence and performance of the variable structure SVD will be ascertained by the study on the relationship and property of the principal and subsidiary vector spaces of the different gradations. Moreover, it is proposed that subsidiary components be further decomposed so that the weak fault feature information can be extracted from the original complicated signal. Finally the variable structure SVD is applied to the engineering practice, and its separation effect for the different components of the complicated signal, extraction effect for the faint signal, noise reduction effect and its application to fault feature extraction and diagnosis of the weak inchoate fault of mechanical devices will be studied systematically. The object of this project is to establish a complete theory and application system of the variable structure SVD, and a new research and application field of SVD can be developed.

将矩阵变结构思想引入到奇异值分解(SVD)中来，提出变结构SVD的概念，在SVD中将矩阵结构可调性和递推分解相结合，利用信号成分对矩阵结构敏感，通过变化的矩阵结构形成对信号不同成分的适应性，克服以往SVD中矩阵结构固定不变这一不足，利用SVD实现对复杂信号不同成分由强到弱的多层次适应性分解，获得一系列反映原信号主要能量构成的主分量和占小能量比例的副分量。提出三种具体的矩阵变结构方式，研究解决相应变结构SVD的分解和重构算法，并通过对各层主、副向量空间关系和性质的研究，明确变结构SVD的信号分解特性。提出对副分量做进一步细分，以实现对复杂背景信号中弱故障特征信息的提取。最后将变结构SVD应用到工程实践，研究它对复杂信号不同成分的分离效果、对弱信号的提取效果、消噪效果以及它在特征提取和设备早期微弱故障诊断中的应用，形成一种较为完备的变结构SVD理论和应用体系，为SVD开拓一个新的应用研究领域。

奇异值分解(SVD)的信号分离效果和矩阵的结构有紧密联系，目前SVD中都采用固定的矩阵结构，这难以适应信号中复杂多变的频率成分。本项目将矩阵变结构思想引入到SVD中，提出变结构SVD的概念，将矩阵结构可调性和递推分解相结合，通过变化的矩阵结构形成对信号不同成分的适应性，克服以往SVD中矩阵结构固定不变这一不足，利用SVD实现了对信号不同成分由强到弱的多层次适应性分解。提出了具体的矩阵变结构方式，推导了变结构SVD的分解和重构算法，研究了变结构SVD的信号分解特性和分解本质，利用向量空间的基变换理论，证明了变结构SVD的三个性质：正交性、加性分解性和单调性。研究了变结构SVD的信号分离机理，证明了两个相应的定理，揭示了变结构矩阵构造时各行向量的正、反相关系与行向量之间的线性相关性，从理论上对变结构SVD的信号分离机理给出了解释。研究了变结构SVD副分量信号的进一步分解，证明了其对弱信号的提取效果。将变结构SVD应用于工程实际，用于提取铣削力、转子振动和轴承振动信号的特征信息，验证了此方法对不同特性信号的特征提取效果，结果优于多分辨SVD和小波变换。此外，推导了Hankel矩阵下非零奇异值和频率的定量关系，发现了SVD的频率分离条件，进而提出了一种频率分离算法，并提出了非零奇异值的三条选择原则，实现了对单个频率的分离。研究了非零奇异值和小波变换各尺度信号的关系，明确了信号和噪声的连续小波变换结果的奇异值的分布特点，分析了矩阵结构变化对这两种奇异值分布特性的影响，实现了对小波变换结果中噪声和特征信息的分离。提出了一种奇异值能量谱方法，克服了传统Shannon熵优化方法的不足，得到了最优的小波尺度参数。研究了SVD对非行满秩矩阵冗余数据的压缩，通过SVD得到了非行满秩矩阵的最大线性无关向量组，实现了对矩阵数据的压缩和提纯。发现了正弦信号非零奇异值的波动特性，从理论上对这种波动性进行了解释。项目的研究结果对SVD理论是一个全新的发展，对信号处理和故障诊断具有积极意义。