自适应滤波算法是近几十年发展起来的一种信号处理方法，该方法无需输入信号的统计特性，具有较强的适应性，其中建立在梯度下降法基础上的最小均方（LMS）算法，因其结构简单、性能稳定等特点在工程实际中获得广泛应用。然而传统LMS算法收敛速度慢的缺陷，严重限制了它的进一步发展。分数阶微积分的引入，能使该问题得以改善，却可能导致其它问题的产生，如较大的稳态误差和计算复杂度，以及不期望的超调等。本项目拟从分数阶微积分非局部特性的角度，研究分数阶极值问题，系统地建立分数阶梯度下降法的理论框架；进而通过更新方向优化和更新阶次自适应的方式，完善分数阶LMS算法，使其在提高收敛速度的同时抑制更新方向的紊乱、减小稳态误差、降低计算复杂度等。希望通过本项目的研究，解决分数阶梯度下降法和分数阶LMS算法中的几个关键问题，并在理论研究的同时赋予其数值实现方案，为其在系统辨识和减振控制中的实际应用提供科学指导。

In the last decades, the adaptive filter algorithm has been developed and become a significant technique in the field of signal processing. It possessed adaptability strongly and the statistical property of input signal is not necessary any more. Among the great variety of adaptive filter algorithms, the least mean square (LMS) algorithm, which is based on the gradient descent method, has been widely used in the engineering application, due to its simplicity and stability. However, considering its slow convergence rate, the further development of traditional LMS algorithm is badly limited. This problem can be luckily solved by introducing the fractional order calculus, but at the same time, it would bring some other problems such as an increase of steady state error, computational complexity and unexpected overshoot. In this project, we will carry out our research from the perspective of the nonlocality involved in fractional order calculus. More specifically, we will firstly study the fractional order extreme value issue, and systematically build the framework for fractional order gradient descent method. Then, the fractional order LMS algorithm is expected to be improved by an optimized updating direction and a self-adapted updating order. Meanwhile, the proposed LMS algorithm can also speed up the convergence, suppress the disorder of update direction, and reduce the steady state error and computational complexity. It is expected that this project would be beneficial to the solutions of several key scientific problems existed in fractional order gradient descent method and fractional order LMS algorithm. Additionally, we will provide numerical implementation for our theoretical results, in order to apply them to the practical fields of system identification and vibration control.