The basis of naming reaching laws for sliding surface in SMC approaches

A reaching law for non-linear sliding surface in sliding mode control (SMC) determines how states approach and converge onto the predefined sliding surface from various initial conditions or trajectories. It establishes dynamic equations governing state evolution towards this target region, enabling controlled motion along this manifold while ensuring stability properties. The term "non-linear sliding surface" in the context of SMC refers to the mathematical representation of the desired behavior that states or trajectories of the system should follow during system operation. The reaching law is an essential component within this non-linear sliding surface framework.A reaching law for a non-linear sliding surface depends on various issues such as “system dynamics”, “stability Analysis”, “sensitivity analysis”, “violent chattering phenomenon reduction”, “control-input constraints”, “Robustness against uncertainties and Bounded Unknown External disturbances (BUE-disturbances)”, “experimental validation”, “desired Performance Objectives”, “prior knowledge/experience”, and “computational complexity”. Therefore, each of these issues has a special impact on the design of a reaching law, and a designer must consider these issues in his design based on his request for the reaching law. A SMC approach has two parts: one is the reaching phase and the other is the sliding phase. The reaching law is the process of system’s states reaching a non-linear sliding surface. In this study, the basic methods of naming reaching laws were reported.