New Results on the Generalized Discrete Reaching Law with Positive or Negative Decay Factors

Dynamic characteristics of sliding variables for the generalized discrete reaching law are investigated, such as phenomena of reaching condition and ultimate boundedness of sliding variables. The discussed reaching law is relatively more general than some others in the literature, because it contains a wider range of decay factors, which could be positive or negative. First, a sufficient and necessary condition is obtained ensuring that the generalized reaching law satisfies the reaching condition. Then, the dependence between the ultimate bounding of sliding variables (i.e., the range of steady-state chattering) and parameters in the generalized reaching law is discussed. The discussions are stated in three different cases, i.e., negative decay factors, adjustable negative decay factors, and adjustable positive decay factors. The obtained results of the generalized reaching law are further applied to the discrete control system with unmatched uncertainties to demonstrate the usefulness of this generalized discrete reaching law in sliding-mode control theory. Finally, two simulation examples are given to illustrate the validity of our findings.