How to implement Super-Twisting Controller based on sliding mode observer?

Implementation of the Super-Twisting Control (STC) requires the first time derivative of the sliding surface must be Lipschitz in time. It is shown that if we will use STC based on the absolutely continuous estimation of the surface, controller can not be implemented. In this paper two methodologies are proposed to avoid the above problem. For simplicity we are considering here only perturbed double integrator which can be generalized for an arbitrary order. Numerical simulations are also presented to show the effectiveness of the proposed method.