Structural conditions for chattering avoidance in implicitly discretized sliding mode differentiators

The present paper considers the implicit discretization of Levant’s arbitrary order robust exact differentiator. It is shown that an improper implicit discretization may lead to an undesired bias in the differentiation error and, surprisingly, to discretization chattering despite the implicit discretization. Necessary and sufficient structural conditions for avoiding both of these problems are presented, which define a family of chattering-free discrete-time differentiators. A guideline for selecting a representative from this family is given. Numerical simulations illustrate the results.