Power system stabilizers are widely used to damp out the low frequency oscillations in power systems. In power system control literature, there is a lack of stability analysis for proposed controller designs. This paper proposes a Neural Network (NN) based stabilizing controller design based on a sixth order single machine infinite bus power system model. The NN is used to compensate the complex nonlinear dynamics of power system. To speed up the learning process, an adaptive signal is introduced to the NN's weights updating rule. The NN can be directly used online without offline training process. Magnitude constraint of the activators is modeled as saturation nonlinearities and is included in the stability analysis. The proposed controller design is compared with Conventional Power System Stabilizers whose parameters are optimized by Particle Swarm Optimization. Simulation results demonstrate the effectiveness of the proposed controller design.