This paper studies a servo-valve controlled hydraulic cylinder system which is mostly used in industrial applications such as robotics, computer numerical control (CNC) machines and transportations. The system model consists of combination of two models: The first model involves nonlinear flow equations of the servo-valve, which are widely available in the literature. The second model employed in the system is a tailored asymmetric cylinder model. A fourth order nonlinear system model is then obtained by combining these two models. Two different neural network control algorithms are applied to the system. The first algorithm is "Neural Network Predictive Control (NNPC)," which employs identified neural network model to predict the future output of the system. The second algorithm is "Nonlinear Autoregressive Moving Average (NARMA-L2)" control, which transforms nonlinear system dynamics into linear system dynamics by eliminating the nonlinearities. On the simulation, NNPC and NARMA-L2 control are applied to the system model by using Matlab's Simulik simulation package and position control of the system is realized. A discussion regarding the advantages and disadvantages of the two control algorithms are also provided in the paper.