The underactuated, fast-responding and nonlinear system is a typical hard-to-control plant in real applications, such as multi-stage inverted pendulum. This paper considers modeling and stabilization control of a Linear Two-Stage Inverted Pendulum（LTSIP）. To avoid the potential problems resulted using first principle model-based control to the plant, such as that some model parameters are unknown or inaccurate, this paper uses a data-driven modeling approach. A State-Dependent Auto Regressive eX ogenous（SD-ARX） model without offset term, whose state-dependent functional coefficients are approximated by Gaussian radial basis function（RBF） neural networks, is built to describe the dynamic behavior of the LTSIP system. Based on the identification model, an infinite horizon model predictive control（MPC） strategy is proposed to implement stabilization control of the LTSIP plant,which is designed by using the locally linearized state-space model and obtaining the locally optimal state feedback control low via solving an algebra Riccati equation online. The real-time control experiment results of the proposed approach and the comparisons with traditional Linear Quadratic Regulator（LQR） method to the plant demonstrate that the modeling and control method proposed in this paper are very effective and superior in modeling and controlling the underactuated, fast-responding and nonlinear system.