On initial value problem for fractional-order Kalman filters of linear continuous-time fractional-order systems

2021 ◽  
Author(s):  
Chuang Yang ◽  
Zhe Gao ◽  
Xiaoou Ma ◽  
Yue Miao
Keyword(s):  
Fractional Order ◽  
Initial Value Problem ◽  
Continuous Time ◽  
Kalman Filters ◽  
Fractional Order Systems ◽  
Initial Value

Fractional-order Kalman filters for continuous-time fractional-order systems involving correlated and uncorrelated process and measurement noises

2018 ◽  
Vol 41 (7) ◽  
pp. 1933-1947 ◽  
Author(s):  
Fanghui Liu ◽  
Zhe Gao ◽  
Chao Yang ◽  
Ruicheng Ma
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Kalman Filters ◽  
Equation Model ◽  
Order System ◽  
Fractional Order Systems ◽  
Derivative Method ◽  
Measurement Noises ◽  
Computation Load ◽  
Average Derivative

This paper presents fractional-order Kalman filters using the fractional-order average derivative method for linear fractional-order systems involving process and measurement noises. By using the fractional-order average derivative method, a difference equation model is obtained by discretizing the investigated continuous-time fractional-order system, and the accuracy of state estimation is improved. Meanwhile, compared with the Tustin generating function, the fractional-order average derivative method proposed in this paper can reduce computation load and save calculation time. Two kinds of fractional-order Kalman filters are given, for the correlated and uncorrelated cases, in terms of the process and measurement noises for linear fractional-order systems, respectively. Finally, simulation results are illustrated to verify the effectiveness of the proposed Kalman filters using the fractional-order average derivative method.

scholarly journals Solution of linear fractional order systems with variable coefficients

2020 ◽  
Vol 23 (3) ◽  
pp. 753-763
Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Transition Matrix ◽  
State Transition ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Theoretical Results

AbstractThe paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.

scholarly journals Study on the Initial Value Problem for Fractional-order Cubature Kalman Filters of Nonlinear Continuous-time Fractional-order Systems

2021 ◽  
Author(s):  
Chuang Yang ◽  
Zhe Gao ◽  
Yue Miao ◽  
Tao Kan
Keyword(s):  
Kalman Filter ◽  
State Estimation ◽  
Fractional Order ◽  
Continuous Time ◽  
Kalman Filters ◽  
The State ◽  
Order System ◽  
Initial Value ◽  
Cubature Kalman Filter ◽  
Augmented State

Abstract To realize the state estimation of a nonlinear continuous-time fractional-order system, two types of fractional-order cubature Kalman filters (FOCKFs) designed to solve problem on the initial value influence. For the first type of cubature Kalman filter (CKF), the initial value of the estimated system are also regarded as the augmented state, the augmented state equation is constructed to obtain the CKF based on Grünwald-Letnikov difference. For the second type of CKF, the fractional-order hybrid extended-cubature Kalman filter (HECKF) is proposed to weaken the influence of initial value by the first-order Taylor expansion and the third-order spherical-radial rule. These two methods can effectively reduce the influence of initial value on the state estimation. Finally, the effectiveness of the proposed CKFs is verified by two simulation examples.

Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises

2020 ◽  
Vol 42 (9) ◽  
pp. 1618-1631
Author(s):  
Xiaojiao Chen ◽  
Zhe Gao ◽  
Ruicheng Ma ◽  
Xiaomin Huang
Keyword(s):  
Kalman Filter ◽  
Fractional Order ◽  
Continuous Time ◽  
Kalman Filters ◽  
Unscented Kalman Filter ◽  
Nonlinear Function ◽  
Fractional Order Systems ◽  
Error Matrix ◽  
Current Time ◽  
Measurement Noises

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.

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