In this paper, the power system with an excitation controller is represented as a class of large-scale, uncertain, interconnected nonlinear continuous-time system in strict-feedback form. Subsequently, dynamic surface control (DSC)-based adaptive neural network (NN) controller is designed to overcome the repeated differentiation of the control input that is observed in the conventional backstepping approach. The NNs are utilized to approximate the unknown subsystem and the interconnection dynamics. By using novel online NN weight update laws with quadratic error terms, the closed-loop signals are shown to be locally asymptotically stable via Lyapunov stability analysis, even in the presence of NN approximation errors in contrast with other NN techniques where a bounded stability is normally assured. Simulation results on the IEEE 14-bus power system with generator excitation control are provided to show the effectiveness of the approach in damping oscillations that occur after disturbances are removed. The end result is a nonlinear decentralized adaptive state-feedback excitation controller for damping power systems oscillations in the presence of uncertain interconnection terms.