Derivation of function space analysis based PCA control charts for batch process monitoring

A technique of the multivariate statistical process control for the analysis and monitoring of batch processes is developed. This technique, called FSPCA, combines the function space analysis and the principal component analysis method (PCA). The function space analysis is based on the concept of the orthonormal function approximation. The trajectories of process measurements in the batches are mapped onto the new feature parameters in the function space. Then the concept of the multivariate statistical process control can be applied for this type of new parameters to extract the correlated features. Like the philosophy of statistical process control in the traditional PCA, FSPCA can generate simple monitoring charts, easy tracking of the progress in each batch run and monitoring the occurrence of observable upsets. The proposed technique are that not only the process variables are significantly reduced but also the problem of the varying time in the batch runs is eliminated. Furthermore, the proposed technique can extract the nonlinear feature without heavy computation load. Two major contributions of this paper are made. First, FSPCA is systematically derived. It is proved that the statistic properties of coefficient matrix derived from the orthogonal function follow Gaussian distribution. PCA, thus, can be properly applied. Second, FSPCA is a methodology of general purposes since both fixed operating time and varying operating time are considered. The control chart's performance, design and usage are also included. By making comparison with the other methods, the effectiveness of the proposed method is shown through two detailed simulation studies to demonstrate the potential applications of FSPCA.