Adaptive control based on hierarchical sliding mode for under-actuated systems

In this paper, an adaptive control law based on hierarchical sliding mode is addressed for under-actuated systems with 2 degrees of freedom. By capturing the physical nature, the class of systems can be considered to be made up of two subsystems so that the hierarchical structure of the sliding surfaces with double layers is constructed. From such kind of hierarchical structure, the adaptive law and the control law are deduced from Lyapunov direct method. The asymptotic stability of all the sliding surfaces is proven by Barbalat´s lemma and Lasalle´s invariance principle. Simulation results show the feasibility of this method.