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 or , 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.

中文翻译：

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具有外部扰动和未知控制方向的分数阶非线性系统的模糊自适应控制

本文研究了带有外部扰动和未知控制方向的未知非线性分数阶系统的模糊自适应控制问题。我们将控制增益矩阵分解成对称的正定矩阵，对角矩阵或对角矩阵，和一个统一的上三角矩阵 模糊逻辑系统用于估计未知的非线性函数。基于分数阶Lyapunov直接法和提出的引理，设计了一种新型的模糊自适应控制器。所提出的方法可以保证闭环系统中的所有信号保持有界，并且跟踪误差收敛到原点的任意小区域。另外，为了更新模糊系统的参数，提出了分数阶自适应定律。最后，给出一个说明性的例子来证明所提出的结果的有效性。