Adaptive fuzzy sliding-mode control of under-actuated systems with unknown input gain function

This research addresses the design of an Adaptive Fuzzy Sliding-Mode Control (AFSMC) for a group of nonlinear under-actuated systems with unknown input gain functions. In the proposed approach, to provide a structure for controlling the under-actuated subsystem, the sliding manifold corresponding to the whole dynamic system is derived based on the backstepping technique. In addition, two separate fuzzy inference systems are utilized for approximating the system's internal dynamics and the unknown input gain function. In order to derive the robust part of the controller, a novel method is proposed to define the upper bound of the uncertain term in the form of a state-dependent second-degree polynomial. It should be mentioned that the output vectors of the fuzzy systems and the coefficients of the robust part of the controller are obtained by the designed adaptation laws. The asymptotic stability analysis for the proposed AFSMC structure is provided by the second theorem of Lyapunov and Barbalat's lemma. In the end, the AFSMC effectuality is confirmed by the depth control of a REMUS Autonomous Underwater Vehicle (AUV).