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

Liu, Fanghui; Gao, Zhe; Yang, Chao; Ma, Ruicheng

doi:10.1177/0142331218790786

Cited by 8 publications

(8 citation statements)

References 44 publications

(42 reference statements)

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“…To simplify the calculation and analysis, the variables u = aω α /c and v = bω β /c are selected to replace the previous variables. Then (27) can be written as…”

Section: B Equal Order Fractional Butterworth Filter Designmentioning

confidence: 99%

“…Systematic design procedures are widely available for IO filters to determine these coefficients, but the procedures for determining the coefficients of FO filters are not common. Only a few studies focused on the calculation of the coefficients, where FO filters in the type of Chebyshev filter [26], Kalman filter [27], Butterworth (BW) filter [28]- [33] are proposed for an ideal configuration.…”

Section: Introductionmentioning

confidence: 99%

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Internal Model Control Design Based on Equal Order Fractional Butterworth Filter for Multivariable Systems

Liu

Chen

2020

IEEE Access

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An internal model control with inverted decoupling (ID-IMC) controller design method based on equal fractional Butterworth (EFBW) filter is proposed for Multiple Input-Multiple Output (MIMO) systems with multiple time delays and Right Half Plane (RHP) zeros. There has been finite memory and limited flexibility for multivariable processes developed using a direct ID-IMC method. This paper presents a novel procedure to approximate Butterworth (BW) filters using fractional-order (FO) theories, so that the degree-of-freedom for tunable parameters is increased. The proposed ID-IMC controller cascaded with EFBW filter combines the computational simplicity of the ID-IMC structure with the greater flexibility of the EFBW filter, in conjunction with a better set-point tracking and disturbance rejection performance. Further, the stability analysis of the designed controller is given to ensure the stability of the closed-loop system. Dynamic performance indicators and sensitivity functions are carried out for the time domain and robustness analysis. Two illustrative examples are presented to show the merits of the proposed method.

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“…To simplify the calculation and analysis, the variables u = aω α /c and v = bω β /c are selected to replace the previous variables. Then (27) can be written as…”

Section: B Equal Order Fractional Butterworth Filter Designmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Internal Model Control Design Based on Equal Order Fractional Butterworth Filter for Multivariable Systems

Liu

Chen

2020

IEEE Access

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“…For the problem on state estimation, the reduced-order FOKFs were presented in Gao (2018) for continuous-time linear FOSs. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Huang

Gao

2022

Transactions of the Institute of Measurement and Control

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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.

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“…The FOKFs can obtain more satisfactory state estimation than integer-order KFs. The fractionalorder EKFs were proposed for linear FOSs with white Gaussian noise and coloured noise respectively in [28], [29]. In [30], KFs for linear and nonlinear discrete-time FOSs were extended to estimate the parameters or order of FOSs.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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To realize the state estimation of a nonlinear continuous-time fractionalorder 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.

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

Cited by 8 publications

References 44 publications

Internal Model Control Design Based on Equal Order Fractional Butterworth Filter for Multivariable Systems

Internal Model Control Design Based on Equal Order Fractional Butterworth Filter for Multivariable Systems

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

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

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