PD Controller Based on Second Order Sliding Mode Differentiation

The Proportional Derivative controller (PD) has been successfully implemented in many real-time applications. It is well-known, that the PD is composed by a proportional and a derivative terms of the signal error. However, the main problem in the implementation of this controller is related with the error signal derivative. Most of the existing results obtain the derivative based on first order filters. This approach is not useful if the signal is noisy and uncertain. In the present paper, the so-called Super-Twisting Algorithm (STA), that is a second order sliding mode approach, is implemented as a robust exact differentiator because it can reach the derivative of a signal in finite time. The closed loop stability of the proposed controller is analyzed in terms of a non-smooth Lyapunov function. Finite time convergence of the tracking error into a boundary layer is obtained. With a slightly modification in the control law with the addition of a discontinuous term, finite time convergence of the error to zero is obtained. Numerical results are given to show the difference between the classical PD and the proposed PD with the STA differentiator.