Analysis of linear dynamic systems of low rank

Abstract

The objective of this paper is to show how the procedures of traditional chemometrics like stepwise evaluation of the model, graphic analysis of the latent structure, etc., can be applied to common modeling methods in chemical engineering like for instance Kalman filtering. Procedures of how to find stable solutions for linear dynamic systems are presented. Different types of models are considered. The procedures are based on the H-principle of mathematical modeling. The basic idea is to approximate the solution by rank 1 parts. Each of them is found by optimizing the estimation and prediction part of the model. The approximations stop, when the prediction ability of the model cannot be improved for the present data. Therefore, the present methods give better prediction results than traditional methods that are based on exact solutions. The vectors used in the approximations can be used to carry out graphic analysis of the dynamic systems. It is shown how score vectors can display the low dimensional variation in data, the loading vectors display the correlation structure, and the transformation vectors how the variables generate the resulting variation in data; these graphic analysis have proven their importance in traditional chemometric methods. These graphics methods are important in supervising and controlling the process in light of the variation in data. The algorithms can provide with solutions of models having hundreds or thousands of variables. It is shown here how these algorithms can provide with solutions involving NIR data for process control having over 1000 variables.