An Adaptive fuzzy sliding mode controller applied to a chatoic pendulum
Chaotic response is related to a dense set of unstable periodic orbits (UPOs) and the system often visits the neighborhood of each one of them. Moreover, chaos has sensitive dependence to initial conditions, which implies that the system evolution may be altered by small perturbations. Chaos control is based on the richness of chaotic behavior and may be understood as the use of tiny perturbations for the stabilization of an UPO embedded in a chaotic attractor. It makes this kind of behavior to be desirable in a variety of applications, since one of these UPO can provide better performance than others in a particular situation. This contribution proposes a robust controller that can be applied to stabilize UPOs of chaotic attractors. The adopted approach is based on the sliding mode control strategy and enhanced by a stable adaptive fuzzy inference system to cope with modeling inaccuracies and external disturbances that can arise. The boundedness of all closed-loop signals and the convergence properties of the tracking error are analytically proven using Lyapunov's direct method and Barbalat's lemma. The general procedure is applied to a nonlinear pendulum that presents chaotic response. Numerical simulations are carried out showing the stabilization of some UPOs of the chaotic attractor showing an efective response, demonstrating the controller performance.