In this paper, two types of adaptive Kalman filters are proposed by using the Grünwald-Letnikov (G-L) difference method to estimate the state information of continuous-time nonlinear fractional-order systems with unknown parameters and fractional-orders. An adaptive extended Kalman filter is designed by using the first-order Taylor expansion to deal with the nonlinear function in a nonlinear fractional-order system with unknown parameters and fractional-order. Based on the third-degree spherical-radial rule, an adaptive cubature Kalman filter as another adaptive fractional-order Kalman filter discussed in this paper is provided by the cubature points to deal with the nonlinear function. The augmented vector consisting of the unknown state vectors, parameters and fractional-order is constructed, and the corresponding augmented equation is established to solve the estimation problem with unknown parameters and fractional-order. The state estimations of nonlinear fractional-order systems with unknown parameters and fractional-orders are carried out by the augmented vector method. Finally, four examples are given to verify the effectiveness of the proposed adaptive Kalman filters with unknown parameters and fractional-orders in this paper.

本文提出了两种自适应卡尔曼滤波器，利用Grünwald-Letnikov(GL)差分法估计未知参数和分数阶的连续时间非线性分数阶系统的状态信息。利用一阶泰勒展开设计了一种自适应扩展卡尔曼滤波器来处理未知参数和分数阶的非线性分数阶系统中的非线性函数。基于三次球半径法则，自适应体积卡尔曼滤波器作为本文讨论的另一种自适应分数阶卡尔曼滤波器，由体积点提供来处理非线性函数。构造由未知状态向量、参数和分数阶组成的增广向量，并建立相应的增广方程来解决未知参数和分数阶的估计问题。具有未知参数和分数阶的非线性分数阶系统的状态估计是通过增广向量方法进行的。最后，给出了四个例子来验证本文提出的未知参数和分数阶自适应卡尔曼滤波器的有效性。