Adaptive modeling of nonnegative environmental systems based on projectional Differential Neural Networks observer

A new design of a non-parametric adaptive approximate model based on Differential Neural Networks (DNNs) applied for a class of non-negative environmental systems with an uncertain mathematical model is the primary outcome of this study. The approximate model uses an extended state formulation that gathers the dynamics of the DNN and a state projector (pDNN). Implementing a non-differentiable projection operator ensures the positiveness of the identifier states. The extended form allows producing continuous dynamics for the projected model. The design of the learning laws for the weight adjustment of the continuous projected DNN considered the application of a controlled Lyapunov-like function. The stability analysis based on the proposed Lyapunov-like function leads to the characterization of the ultimate boundedness property for the identification error. Applying the Attractive Ellipsoid Method (AEM) yields to analyze the convergence quality of the designed approximate model. The solution to the specific optimization problem using the AEM with matrix inequalities constraints allows us to find the parameters of the considered DNN that minimizes the ultimate bound. The evaluation of two numerical examples confirmed the ability of the proposed pDNN to approximate the positive model in the presence of bounded noises and perturbations in the measured data. The first example corresponds to a catalytic ozonation system that can be used to decompose toxic and recalcitrant contaminants. The second one describes the bacteria growth in aerobic batch regime biodegrading simple organic matter mixture.