Generalization of Gao's Reaching Law for Higher Relative Degree Sliding Variables

In this paper, a generalized discrete time reaching law for sliding variables with arbitrary relative degree is proposed. The reaching law is based on Gao's strategy and ensures higher order switching-type sliding motion, which is a novel concept in the field of discrete time plants. The controller synthesis process is simplified by performing a specific system state transformation. It is demonstrated that the width of the layer to which the representative point of the system is driven decreases as the relative degree of the sliding variable gets bigger. It has further been proven that achieving a narrower quasi-sliding mode band width is reflected in reduced state error. Finally, the reaching phase elimination for variables with arbitrary relative degree is considered. This is obtained by implementing a time-varying sliding surface in a finite number of initial time instants. Uniform ultimate boundedness of all state variables is guaranteed.