Modelling of Wastewater Treatment Plant for Monitoring and Control Purposes by State – Space Wavelet Networks

Abstract

Most of industrial processes are nonlinear, not stationary, and dynamical with at least few different time scales in their internal dynamics and hardly measured states. A biological wastewater treatment plant falls into this category. The paper considers modelling such processes for monitorning and control purposes by using State -Space Wavelet Neural Networks (SSWN). The modelling method is illustrated based on bioreactors of the wastewater treatment plant. The learning algorithms and basis function (multidimensional wavelets) are also proposed. The simulation results based on real data record are presented.

Adaptive Prediction of Stock Exchange Indices by State Space Wavelet Networks

“…Stability conditions expressed in terms of network parameters were derived for neural networks in, e.g., (Sanchez and Perez, 1999;Kulawski and Brdyś, 2000;Nguyen and Brdyś, 2006). This is still under research for our SSWN, for which stability was demonstrated by simulation for a wide range of the parameter values in (Borowa et al, 2007). Let us denote the initial state discharge time by J.…”

“…This is done by using historical data and searching for parameters values such that the corresponding prediction error is minimal. The procedure is called network training (Zhang et al, 2007;Borowa et al, 2007). The SSWN structure and its optimised parameters for one session ahead prediction are illustrated in Fig.…”

“…Hence, we shall separate determining the number of states M and the number of wavelons L, which are the integer valued SSWN structure parameters, from determining the continuously valued parameters of the SSWN, which are W, d and t. The former will be presented later, and for the time being M and L are assumed known. The latter will be determined by applying a global stochastic optimiser being a further development of an algorithm presented in (Borowa et al, 2007). When preparing the training data, special attention needs to be paid to the impact of the network initial state on its output.…”

“…The solver turned out to be efficient in training SSWN for modelling a wastewater treatment plant (Borowa et al, 2007). Each iteration of the solver starts from discharging the network state initial condition for the actual parameter values.…”

“…The SSWN was initially proposed for modelling nonlinear and nonstationary processes with multiple time scales in internal dynamic and hardly measured states under uncertainty in the inputs and dynamic models. It was successfully applied to the input-output modelling in a state-space form of a wastewater treatment plant (Borowa et al, 2007). The wastewater treatment…”