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A stable neural network (NN)-based observer for general multivariable nonlinear systems is presented in this paper. Unlike most previous neural network observers, the proposed observer uses a nonlinear-in-parameters neural network (NLPNN). Therefore, it can be applied to systems with higher degrees of nonlinearity without any a priori knowledge about system dynamics. The learning rule for the neural network is a novel approach based on the modified backpropagation (BP) algorithm. An e-modification term is added to guarantee robustness of the observer. No strictly positive real (SPR) or any other strong assumption is imposed on the proposed approach. The stability of the recurrent neural network observer is shown by Lyapunov's direct method. Simulation results for a flexible-joint manipulator are presented to demonstrate the enhanced performance achieved by utilizing the proposed neural network observer.