Adaptive Fuzzy Hierarchical Sliding-Mode Control for the Trajectory Tracking of Uncertain Underactuated Nonlinear Dynamic Systems

The trajectory tracking of uncertain underactuated nonlinear dynamic systems is tackled by an adaptive fuzzy hierarchical sliding-mode control (AFHSMC). First, one of the subsystems is assigned as the first layer sliding surface. Next, a second layer sliding surface from the first layer sliding surface and the sliding surface of another subsystem is constructed. In this paper, the nth layer is supposed to be the top layer (or hierarchical layer) for including the sliding surfaces of all subsystems. Because two nonlinear system functions and the time-varying external disturbance of each subsystem are supposed to be unknown, different online fuzzy models are employed to approximate these nonlinear system functions and the upper bounded functions of external disturbances. Moreover, the upper bound of uncertainties caused by these fuzzy modeling errors is estimated online. Based on these learning fuzzy models and the estimated upper bound of these modeling errors, an AFHSMC is developed. The stability analysis and tracking performance of the closed-loop system are verified by Lyapunov stability theory. Finally, two simulation examples including different amplitudes of external disturbance and comparison with hierarchical sliding-mode control confirm the effectiveness and robustness of the proposed control.