Key Words: Empirical MIMO model; biological wastewater treatment; Wiener model structure; Principal Component Analysis; polynomial approximation; neural networks.
Abstract. The aim of the present work is to develop data based MIMO mathematical model for biological wastewater treatment, designed for real-time work, and a procedure for creating mathematical models of this class. An analysis of the processes of biological wastewater treatment for the purposes of their mathematical modelling is made. The study includes variables that are known to have sensors worldwide or to have software sensors developed. In conducting the research published in the present work, a combination of real and synthetic data is used. The constructive parameters of the considered installation correspond to settlements with an average number of equivalent inhabitants for the country. To develop a MIMO nonlinear dynamic mathematical model, the structure of Wiener model was chosen – series-connected linear dynamic and nonlinear static parts. The procedure for creating the mathematical model includes: processing of incoming data by the principal components method (PCA); to form the nonlinear static part of the model and to compare the predictive abilities, polynomial dependences for each of the intermediate and target variables are derived as a function of the normalized values of the three principal components and two types of neural networks for each variable are trained. In one case the independent variables are the normalized values of the principal components and in the other – the natural values of principal components. In some cases, higher accuracy of approximation is obtained in polynomial dependencies, in others in neural networks. In neural networks, the same approximation accuracy with polynomial models is obtained with a larger number of parameters. Based on simulation studies, the dynamic characteristics of an installation for biological wastewater treatment are derived. A block diagram of the mathematical model for is presented. The created mathematical model can be used on a modular basis with respect to the target variables of interest, regardless of the other target variables.