On non-stationarity of dynamic systems

Past years have shown great advances in measurement instrumentation. Many chemical companies are using NIR (Near Infra-Red) instruments for process control, because it can offer various advantages compared to traditional on-line analysers. These new instruments use often, e.g. PLS models for predicting the results, and their success depends mostly on the quality of the spectra and the models. However, there is a need for new methods that can handle data from these modern instruments. Typically, a large amount of data is received and needs to be processed. This data usually show very low rank. Covariance structure of dynamic systems tends to vary over time. Here some procedures to find stable solutions to linear dynamic systems with low rank are presented. Subsets of variables and samples to be included in a model are considered. The procedures are based on the H-principle of mathematical modelling. The basic idea is to approximate the solution by rank one parts. Each of them is found by optimising the estimation and prediction part of the model. The aim is to balance improvement in fit and precision. Therefore, the present methods give better prediction results than traditional methods that are based on exact solutions. With in few seconds the algorithms can provide with solutions of models having hundreds or thousands of variables. The procedure is described mathematically and demonstrated for a dynamic industrial case. It is shown how the algorithms can provide solutions involving NIR data for process control. The method is simple to apply and the motivation of the procedure is obvious for industrial applications. It can be used, e.g., when modelling on-line systems.