Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control

In this paper by using a combination of fuzzy identification and the sliding mode control a fuzzy adaptive sliding mode scheme is designed to stabilize the unstable periodic orbits of chaotic systems. The chaotic system is assumed to have an affine form x(n) = f(X) + g(X)u where f and g are unknown functions. Using only the input–output data obtained from the underlying dynamical system, two fuzzy systems are constructed for identification of f and g. Two distinct methods are utilized for fuzzy modeling, the least squares and the gradient descent techniques. Based on the estimated fuzzy models, an adaptive controller, which works through the sliding mode control, is designed to make the system track the desired unstable periodic orbits. The stability analysis of the overall closed loop system is presented in the paper and the effectiveness of the proposed adaptive scheme is numerically investigated. As a case of study, modified Duffing system is selected for applying the proposed method to stabilize its 2π and 4π periodic orbits. Simulation results show the high performance of the method for stabilizing the unstable periodic orbits of unknown chaotic systems.