Fuzzy Adaptive Control for Fractional Nonlinear Systems with External Disturbances and Unknown Control Directions

In this paper, the problem of fuzzy adaptive control of unknown nonlinear fractional-order systems with external disturbances and unknown control directions is studied. We exploit a decomposition of the control gain matrix into a symmetric positive-definite matrix, a diagonal matrix with diagonal entries + 1 or 1 , and a unity upper triangular matrix. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed lemmas, a novel fuzzy adaptive controller is designed. The proposed method can guarantee that all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. In addition, for updating the parameters of the fuzzy system, fractional-order adaptations laws are proposed. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.