Adaptive fractional-order Kalman filters for continuous-time nonlinear fractional-order systems with unknown parameters and fractional-orders

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.