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In this study, full adaptive Kalman filters are designed for continuous-time nonlinear fractional-order systems (FOSs) containing unknown parameters and fractional-orders. First, the estimated FOS is discretized using the Grünwald–Letnikov difference method to transform the fractional-order differential equation into a difference equation. Then, in terms of the nonlinear function contained in the investigated system, the Taylor expansion formula is adopted to linearize the discretized equation. Based on the method of augmented vector, an augmented state equation is established by state equations, unknown parameter equations, and fractional-order equation to achieve the state estimation. Besides, the sigmoid function is brought to ensure that the estimation of the fractional-order is performed in a suitable range. Because the covariance matrices of noises are difficult to be measured in physical systems, we also concern the problems on the state estimation, parameter estimation, and fractional-order estimation under the cases that the covariance matrix of process noise is unknown or the covariance matrix of measurement noise is unknown. Considering that the initial value can produce an error of state estimation and parameter identification for the FOS defined under the Caputo sense, the augmented vector method is used to achieve the initial value compensation. Finally, the effectiveness of the proposed algorithms is validated by four examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, hybrid extended‐cubature Kalman filters (HECKFs) for non‐linear continuous‐time fractional‐order systems with uncorrelated and correlated noises are investigated. A non‐linear continuous‐time fractional‐order system using the fractional‐order average derivative is discretised to gain a difference equation. The fractional‐order average derivative method can achieve more accurate state estimation compared with the Grünwald–Letnikov difference method and the non‐linear functions in the system description are dealt with the extended Kalman filter (EKF) and cubature Kalman filter (CKF). The first‐order Taylor expansion used in the EKF method is implemented for a non‐linear function at the current time. Meanwhile, by using the third‐degree spherical‐radical rule, the functions in the state equation and output equation are performed by the cubature points. Based on these cubature points, the CKFs for uncorrelated and correlated noises are presented to achieve the effective state estimation. Besides, the simulation results are offered to validate the effectiveness of the HECKF proposed in this study. [ABSTRACT FROM AUTHOR]

Hybrid extended-unscented Kalman filters (HEUKFs) for continuous-time nonlinear fractional-order systems with process and measurement noises are investigated in this paper. The Grünwald-Letnikov difference and the fractional-order average derivative (FOAD) method are adopted to discretize the investigated nonlinear fractional-order system, and the nonlinear functions in the system description are coped with the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). The first-order Taylor expansion used in the EKF method is performed for the nonlinear function at the current time. Meanwhile, the unscented transformation used in the UKF is also concerned for the nonlinear function at the previous time. By using the HEUKF designed in this paper, the third-order approximations for the nonlinear function can be achieved to enhance the accuracy of state estimation and estimation error matrix. Finally, numerical examples are provided to illustrate the effectiveness of the proposed HEUKF for nonlinear fractional-order systems. [ABSTRACT FROM AUTHOR]

Summary: By using the Grünwald‐Letnikov (G‐L) difference method and the Tustin generating function method, this study presents extended Kalman filters to achieve satisfactory state estimation for fractional‐order nonlinear continuous‐time systems that containing some unknown parameters with the correlated fractional‐order colored noises. Based on the G‐L difference method and the Tustin generating function method, the difference equations corresponding to fractional‐order nonlinear continuous‐time systems are constructed respectively. The first‐order Taylor expansion is used to linearize the nonlinear functions in the estimated system, which provides the system model for extended Kalman filters. Using the augmented vector method, the unknown parameters are regarded as new state vectors, and the augmented difference equation is constructed. Based on the augmented difference equation, extended Kalman filters are designed to estimate the state of fractional‐order nonlinear systems with process noise as fractional‐order colored noise or measurement noise as fractional‐order colored noise. Meanwhile, the extended Kalman filters proposed in this paper can also estimate the unknown parameters effectively. Finally, the effectiveness of the proposed extended Kalman filters is validated in simulation with two examples. [ABSTRACT FROM AUTHOR]