Proportional derivative fuzzy control supplied with second order sliding mode differentiation

The fuzzy logic controller (FLC) has the ability of handling parametric uncertainties and external perturbations for unknown systems. A regular structure for a FLC is the proportional derivative (PD) form. The proportional derivative fuzzy controller (PDF) could be seen as a variable gain PD controller. Despite this characteristic, the most common drawback for any PD controller, with unknown dynamics or even with unmodeled dynamics is the error signal differentiation. In this manuscript this disadvantage was overtaken implementing the super-twisting algorithm (STA) as a robust exact differentiator (RED). The information provided by the STA was injected into the PDF to enhace its performance. In this study, the stability of the nonlinear system under the fuzzy super twisting PD controller (FSTPD) in closed loop was analyzed using the concept of the second Lyapunov's method. Numerical simulations were designed to show the effectiveness and advantages of the proposed FSTPD over the classical PD structure supplied with the STA and a PDF with the derivative part obtained by a linear filter. A first example to stabilize a simple pendulum was developed applying the FSTPD. A second example for solving a tracking control problem was designed for a robot manipulator with six degrees of freedom. In both cases, the FSTPD showed better performance and a significant reduction of the control energy.