Projectional differential neural network observer with stable adaptation weights

A class of dynamic neural network (DNN) observers involving a projection operator inside is considered. Such observers seem to be useful when an uncertain nonlinear system, affected by external perturbations, keeps its states in an a priori known compact set, defined by the given state constraints independently of the measurement noise effects. Since the projection method introduces discontinuities into the trajectory dynamics, the standard Lyapunov method is not applicable to describe the convergence property of this class of observers. This problem is suggested to be resolved using a Lyapunov-Krasovski functional including both the estimation error and the weights involved in the DNN description. The stable adaptive laws for the DNN-weights adjustment are derived. The upper bound for the estimation error is obtained based on linear matrix inequality (LMI) technique implementation. An illustrative example clearly shows the effectiveness of the suggested approach. It deals with an environment control problem, related to the soil contaminants degradation by ozonation.