Hybrid extended‐cubature Kalman filters for non‐linear continuous‐time fractional‐order systems involving uncorrelated and correlated noises using fractional‐order average derivative

Abstract

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 nonlinear 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.

A Review of Recent Advances in Fractional-Order Sensing and Filtering Techniques

“…These aspects have been discussed by [91], another recent paper which clearly highlights the advantages of using a hybrid robust fractional interpolatory cubature Kalman filter instead of a traditional one. Extended Kalman filters for nonlinear fractional-order systems perturbed by colored noises 2020 [11] Hybrid extended-cubature Kalman filters for nonlinear continuous-time fractional-order systems involving uncorrelated and correlated noises using fractional-order average derivative 2020 [24] Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises 2020 [97] Novel hybrid robust fractional interpolatory cubature Kalman filters 2020 [91] Adaptive fractional-order Kalman filters for continuous-time nonlinear fractional-order systems with unknown parameters and fractional orders 2021 [96] As indicated above, the methods reviewed in the current manuscript are applied to handle similar problems as in [154][155][156][157][158][159]. The methods reviewed in this paper could be applied to the systems in [154][155][156][157][158][159], but the systems need to be altered and generalized to a fractional-order representation.…”

“…Using this hybrid extended-unscented Kalman filter, the accuracy of state estimation is improved since this allows for a third-order approximations for the nonlinear functions. A similar approach is taken for the design of the hybrid extended-cubature Kalman filter in [ 24 ]. The fractional-order average derivative method is used instead of the Grünwald–Letnikov difference method.…”

“…The applicability of digital fractional-order filters spans on a manifold of domains from data transmission and networking applications [ 23 , 24 , 25 ], electrical vehicle manufacturing (through the determination of state of charge in lithium-ion batteries [ 26 ], aerial vehicle orientation using fractional-order filtering of yaw, pitch and roll signals [ 27 ], air-quality assessments through pollution and humidity factors [ 28 ], civil engineering targeting the measurement and data processing of various characteristics of buildings such as stiffness and damping [ 29 ], different biomedical processes, image processing and many more. Most of the existing implementations of fractional-order filters are in the fields of data transmission and battery estimation, as will be shown in a dedicated section that highlights the benefits of fractional-order filters in real-life applications.…”

Full adaptive Kalman filters for nonlinear fractional-order systems containing unknown parameters and fractional-orders

“…In Liu et al (2019), the FOKFs were studied to solve the problems on the uncorrelated and correlated noises involved in linear FOSs. Combining the extended Kalman filter and the cubature Kalman filter, the hybrid extended-cubature Kalman filters were investigated in Yang et al (2020), which were used to improve the accuracy of state estimation for continuous-time nonlinear FOSs with uncorrelated and correlated noises. Meanwhile, a fractional-order distributed Kalman filter algorithm was presented in Firouzabadi et al (2020) to estimate the state information in sensor networks and ensure the accuracy of sensor measurements.…”

Study on initial value problem for fractional-order cubature Kalman filters of nonlinear continuous-time fractional-order systems

“…In [32], the CKF method was applied to obtain the higher estimation accuracy compared with EKF method for nonlinear continuous-time FOSs contain white Gaussian noise. Meanwhile, hybrid FOKFs for nonlinear FOSs were also offered to enhance the estimation accuracy in [33] and [34].…”