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Single-phase inverters are widely used in critical applications such as dynamic voltage restorer (DVR) and uninterrupted power supply (UPS) system. In such applications, the inverter must be capable of operating with low total harmonic distortion (THD) and quick transient response. This paper and its companion paper present two promising control methods based on neural networks. This paper presents a novel neural network-based adaptive controller for high performance single-phase inverters capable of maintaining a high quality output voltage with very low THD of less than 2% in the presence of unknown loads, either linear or nonlinear. In the proposed scheme, a voltage source inverter with its corresponding LC load-filter is controlled by an adaptive linear neuron (ADALINE) controller with only one sensor. The on-line adaptive learning algorithm developed in this paper guarantees steady-state controller stability. The principle, stability, robustness, and the special features of the control scheme are discussed. The scheme is simple to implement and has superior performance compared to other popular control strategies, such as the dead-beat controller. An important merit of the proposed control scheme is that it can be designed and implemented without knowing the exact parameters of the PWM inverter system. Simulation and experimental results are presented to verify the feasibility and performance of proposed approach.