This thesis proposes a design method for backpropagation (BP)- and linear matrix inequality (LMI)-based neuro-adaptive observer for uncertain nonlinear systems in discrete-time domain. The proposed scheme employs a neural network with a single hidden layer to approximate the unknown uncertainties. To avoid divergence risk associated with discretization, the observer is directly formulated and analyzed in the discrete-time domain. A Lyapunov function is constructed to guarantee the stability of both the linear observer and the weight updates of the neural network. The observer gain is determined by solving the LMI conditions, and the design is simplified by minimizing the number of tuning parameters, using a common gain structure for all vertices. Furthermore, designing an H_∞ observer can reduce the effect of neural network approximation error and the measurement noise. In conclusion, the proposed method minimizes the number of tuning parameters, accurately estimates the states and uncertainties, ensures LMI-based stability with backpropagation-based updates, suppresses disturbance effects through H_∞ design, and is directly applicable in the discrete-time domain. Simulation results indicate that the proposed method successfully tracks the actual states and the lumped nonlinear term and reduce the effects of neural network approximation error and the measurement noise with comparison of the root mean square error (RMSE) values.