基于秩-1矩阵摄动的递归主元分析算法

Recursive PCA algorithm based on rank-one matrix perturbation

针对传统主元分析(PCA)算法仅适用于定常系统监测的不足,提出了一种基于秩-1矩阵摄动的递归主元分析(RPCA)算法以适应实际工业过程的时变特性.RPCA算法首先对初始化样本协方差矩阵进行特征值分解,得到特征向量矩阵与特征值矩阵;然后在各时刻采用秩-1矩阵摄动算法对这两个矩阵递归更新并对其各向量与各元素排序,同时以累计方差百分比(CPV)为标准选取主元数目,从而显著降低了运算复杂度,节省了存储量.青霉素间歇发酵过程在线监测的仿真结果表明,RPCA算法大大降低了系统的误警率,并及时监测出过程中存在的故障.

As traditional PCA-based methods are limited to the application in time-invariant systems, a recursive PCA （RPCA） algorithm based on rank-one matrix perturbation was proposed to fit with the timevariant characteristics of practical industrial processes. Firstly the covariance matrix of initial samples was decomposed to an eigenvector matrix and a diagonal eigenvalue matrix. Then the eigenvector and eigenvalue matrices were updated with every new data sample, and the number of the selected components was determined according to the cumulative percent variance （CPV） criterion simultaneously. Thus the computational complexity was greatly reduced and the memory saved. The proposed method was applied to on-line monitoring a fed-batch penicillin fermentation process and compared with the conventional PCA monitoring methods. The results clearly illustrated the superiority of the proposed method, with fewer false alarms within normal batch processes and small fault detection delay when faults existed.