Robust Stabilizing Control of Underactuated Cart-Pendulum System Using Fixed-Time Hierarchical Sliding Mode Method

The cart-pendulum system has two degrees of freedom (DOF) and one control input. It is a typical 2-DOF underactuated mechanical system. This paper discusses the stabilizing control problem for this mechanical system with external disturbances. A fixed-time control method is developed based on hierarchical sliding mode idea. First, the system's Euler-Lagrange motion equations are established. And they are divided into two subsystems. Second, a first-order sliding surface is constructed for each subsystem. Subsequently, a second-level sliding mode surface is constructed by linearly combining two first-order sliding surfaces. After that, a robust fixed-time sliding mode control law is designed, which makes the cart-pendulum system reach the second-level surface within a fixed-time. This enables the system's robust stabilizing control objective to be quickly realized. Finally, the effectiveness of our presented method is verified by numerical simulation examples.