Design of adaptive sliding mode controllers for systems with mismatched uncertainty to achieve asymptotic stability

Based on the Lyapunov stability theorem, an adaptive sliding mode control scheme is proposed in this paper for a class of mismatched, perturbed dynamic systems to solve regulation problems. This method is an extension of the existing methods where n ≤ 2 m has to be fulfilled (n is the dimension of the system, m is the number of inputs). It means that the proposed method in this paper can be used for the case n > 2 m. The sliding surface function is firstly designed by treating some state variables as pseudo controllers to stabilize the rest of state variables. The second step is to design the controllers so that the trajectories of the controlled systems are able to reach sliding surface in a finite time. The advantages of the existing methods, for instance, suppressing the perturbations without requiring their upper bounds when designing the sliding surface function and controllers, are still preserved. Once the controlled system enters the sliding mode, the property of asymptotic stability is achieved under certain conditions. A numerical example is given to demonstrate the feasibility of the proposed design technique.