Design of adaptive sliding mode controllers for mismatched perturbed systems with application to underactuated systems

A methodology of designing adaptive sliding mode controllers for a class of nonlinear systems with matched and mismatched perturbations is proposed in this paper. Firstly, A specific designed sliding surface function is presented, and the controller with adaptive mechanisms embedded is also designed in accordance with the designated sliding surface function. Secondly, the coefficients of designated sliding surface function are determined by using Lyapunov stability theorem and linear matrix inequality (LMI) optimization technique. Not only the proposed control scheme can drive the trajectories of the controlled systems into the designed sliding surface within a finite time, but also it is able to suppress the mismatched perturbations when the controlled systems are in the sliding mode, and guarantee asymptotic stability. Furthermore, the proposed control scheme can be directly applied to a class of underactuated systems. A numerical example is given for demonstrating the feasibility of the proposed control scheme.