Sliding mode control design based on nonsmooth linear Lipschitz continuous surface

By utilizing the notion of Filippov solution, Clarke generalized gradient and nonsmooth Lyapunov stability theory, a further discuss on sliding mode control is presented for second-order systems with a nonsmooth linear Lipschitz continuous sliding surface. Firstly, a discontinuous control law is proposed, which drives the trajectories of the closed-loop system reaching the designed surface in finite time. Furthermore, in the framework of Filippov, the asymptotical stability of the trajectories on the linear Lipschitz continuous surface is proved. The sliding mode control is constructed for more general sense, which improves the performance of the closed-loop system. Finally, simulation result shows the effectiveness and correction of the designs.