Using polynomial functions for a time varying sliding surface with angular information

Sliding surface translation that has been defined using angular information as a linear function of time provides an equally spaced rotation of the sliding surface and a smoother control under predefined error state constraints. In this study, second order single-input non-autonomous dynamic open-loop systems will be considered and a higher order polynomial will be used to define the time-varying structure of the angular information. Then, it will be shown via simulations that different phase plane trajectories could be obtained with the proposed method. For the simulations, the trajectory tracking problem will be considered. Then, the same phase plane characteristics obtained using a continuously time-varying linear sliding surface design method selected from the literature will be directly obtained by assigning the polynomial coefficients of the angular sliding surface.