Practical stability analysis for DNN observation

The most important fact for differential neural networks dynamics is related to its weights time evolution. This is a consequence for the higher nonlinear structure describing the matrix differential equations, which are associated with the adaptive capability for this kind of neural networks. However, as we know, there is no any analytical demonstration of the weights stability. In fact, this is the main inconvenient to design real applications of differential neural network observers, especially for control uncertain nonlinear systems. This paper deals with the stability proof for the weights dynamics using an adaptive procedure to adjust the weights ODE. A new dynamic neuro-observer, using the classical Luenberger structure based on practical stability theory, is suggested. This methodology aviods the averaged convergence for the state estimation and provides an upper bound for the weights trajectories. A numerical example dealing with the ozonization process state estimation is presented to illustrate the effectiveness of the suggested approach.