A second order sliding mode differentiator with a variable exponent

In this article, a new second order sliding mode differentiator with a variable exponent is proposed. Inspired by the classical super twisting differentiator, the dedicated differentiator allows to give a solution for reducing the effect of variable noise of sensor output measurement. To achieve this objective, the parameter α that is fixed in a super twisting differentiator is made variable in the proposed differentiator. First of all, the proposed differentiator is presented in free noise case, after that the extension to the case with output noise is given in details. In both cases the practical convergences of the observation error are guaranteed. In the first the radius of the practical stability is depending on the considered unknown input while in the second case this radius depends also on the noise. Finally some simulation results are given in order to show the performances and the effectiveness of the proposed differentiator compared to existing one.