Higher-Order Sliding-Mode Velocity Observer-Based Exponential Stabilization of the Second-Order Chained-Form System

This work addresses the problem of globally stabilizing underactuated systems with two inputs and three degrees of freedom. Through appropriate transformations, such dynamics can be expressed as second-order nonholonomic systems in chained form. A time-varying control framework is developed to overcome the absence of smooth static feedback laws and to drive the system trajectories to the origin. Explicit criteria for selecting feedback gains are derived, and velocity signals are reconstructed using a higher-order sliding-mode observer. A complete stability analysis is provided, ensuring exponential convergence of the closed-loop system under the proposed design. To illustrate its effectiveness, the method is applied to two benchmark problems: an underactuated hovercraft and a vertical takeoff and landing (VTOL) aircraft. Simulation results confirm the practicality and robustness of the proposed observer-based stabilization scheme for second-order chained systems.