Cart Pendulum Control: A Fas Approach

This paper presents a switching control strategy for the cart-pendulum system subject to symmetric constraints on the pendulum angle. Based on the fully actuated system (FAS) approach, the proposed framework addresses the underactuated dynamics and input coupling through a three-stage design. First, a sub-FAS controller is developed for the pendulum subsystem to ensure stabilization for large-angle deviations and high-velocity conditions. By analyzing the system dynamics in reverse time, we analytically characterize the region of exponential attraction (RoEA) boundaries for underdamped, critically damped, and overdamped damping regimes, providing an explicit description of the RoEA. Second, to achieve stabilization of both the cart position and pendulum angle near the upright equilibrium, a local linearized FAS model is derived. Through a state transformation involving the cart and pendulum coordinates, the dynamics are converted into a fourth-order linear fully actuated form, enabling exponential stabilization via parametric design. Furthermore, a smooth switching mechanism based on statedependent sigmoid functions is introduced to seamlessly blend the sub-FAS controller with the local linear controller, mitigating control discontinuities. Numerical simulations demonstrate the effectiveness of the proposed approach in achieving stabilization while strictly satisfying state constraints.