Sliding mode control design with a class of piecewise smooth switching surfaces

In order to achieve more flexibility, a new design method is proposed to construct the switching surface of sliding mode control. We first prove that the available switching surface is not unique for the fixed reaching phase design. A new switching surface is developed by the combination of these available switching surfaces. Unlike typical sliding mode control design, the new switching surface is continuous, in other words, piecewise smooth. Furthermore, a method is provided to obtain the sliding motion on the non-smooth region of the switching surface, where a high-dimensional cross product is introduced to calculate the direction of the trajectories on the smooth half surfaces with respect to the non-smooth regions. Based on this method, we show that the sliding motion is still smooth on the non-smooth switching surface.