Novel hybrid robust fractional interpolatory cubature Kalman filters. (English) Zbl 1429.93387
Summary: In this paper, a novel class of fractional interpolatory cubature Kalman filters (FICKFs) based on interpolatory cubature rule (ICR) is presented, which can achieve a custom degree of accuracy under the Bayesian filtering framework. The fractional cubature Kalman filter (FCKF) and the fractional unscented Kalman filter (FUKF) can be considered as particular types of the proposed FICKFs algorithm. In addition, a robust FICKF and also a hybrid robust FICKF are developed to estimate the states of a fractional-order nonlinear system, in the presence of uncertainty. This robust FICKF is formulated by combining a traditional FICKF and an uncertainty estimator. The hybrid robust FICKF is constructed by using a switching mechanism between a FICKF and a robust FICKF, such that it can provide accurate estimation in the presence or absence of uncertainty. Moreover, the algorithm performance is analyzed and evaluated by the state estimation of Malaria fractional nonlinear model with temporary immunity. Simulation results demonstrate that the FICKFs with suitable free parameters is better than the existing filters with the same degree. In especial case for uncertain systems, hybrid robust FICKF improves the convergence and accuracy of estimation.
MSC:
| 93E11 | Filtering in stochastic control theory |
| 93B35 | Sensitivity (robustness) |
| 26A33 | Fractional derivatives and integrals |
| 93C55 | Discrete-time control/observation systems |
| 93C10 | Nonlinear systems in control theory |
| 93C41 | Control/observation systems with incomplete information |
References:
| [1] | Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu-Batlle, V., Fractional-order Systems and Controls: Fundamentals and Applications (2010), Springer Science & Business Media · Zbl 1211.93002 |
| [2] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 198 (1998), Elsevier · Zbl 0922.45001 |
| [3] | Dalir, M.; Bashour, M., Applications of fractional calculus, Appl. Math. Sci., 4, 21, 1021-1032 (2010) · Zbl 1195.26011 |
| [4] | Daou, R. A.Z.; Moreau, X.; Francis, C., Study of the effects of structural uncertainties on a fractional system of the first kind-application in vibration isolation with the crone suspension, Signal Image Video Process., 6, 3, 463-478 (2012) |
| [5] | do Nascimento, J.; Damasceno, R.; de Oliveira, G.; Ramos, R., Quantum-chaotic key distribution in optical networks: from secrecy to implementation with logistic map, Quantum Inf. Process., 17, 12, 329 (2018) · Zbl 1402.81120 |
| [6] | Shao, S.; Chen, M.; Wu, Q., Tracking control for uncertain fractional-order chaotic systems based on disturbance observer and neural network, IMA J. Math. Control Inf., 34, 3, 1011-1030 (2016) · Zbl 1417.93179 |
| [7] | Ma, Y.-K.; Prakash, P.; Deiveegan, A., Optimization method for determining the source term in fractional diffusion equation, Math. Comput. Simul., 155, 168-176 (2019) · Zbl 1540.49042 |
| [8] | E. Okyere, F. Oduro, S. Amponsah, I. Dontwi, N. Frempong, Fractional order malaria model with temporary immunity, arXiv:1603.06416 (2016). |
| [9] | Djouambi, A.; Voda, A.; Charef, A., Recursive prediction error identification of fractional order models, Commun. Nonlinear Sci. Numer. Simul., 17, 6, 2517-2524 (2012) · Zbl 1252.35272 |
| [10] | N’Doye, I.; Darouach, M.; Voos, H.; Zasadzinski, M., On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters, IMA J. Math. Control Inf., 33, 4, 997-1014 (2015) · Zbl 1397.93186 |
| [11] | Jumarie, G., Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions, Appl. Math. Model., 32, 5, 836-859 (2008) · Zbl 1138.60324 |
| [12] | Ahmed, H. M., Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion, IMA J. Math. Control Inf., 32, 4, 781-794 (2015) · Zbl 1328.93056 |
| [13] | Åström, K. J., Introduction to Stochastic Control Theory (2012), Courier Corporation |
| [14] | Sierociuk, D.; Dzieliński, A., Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation, Int. J. Appl. Math. Comput. Sci., 16, 129-140 (2006) · Zbl 1334.93172 |
| [15] | Sierociuk, D.; Tejado, I.; Vinagre, B. M., Improved fractional Kalman filter and its application to estimation over lossy networks, Signal Process., 91, 3, 542-552 (2011) · Zbl 1203.94055 |
| [16] | Torabi, H.; Pariz, N.; Karimpour, A., A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems, J. Vib. Control, 24, 24, 5880-5897 (2018) |
| [17] | Caballero-Aguila, R.; Hermoso-Carazo, A.; Linares-Pérez, J., Extended and unscented filtering algorithms in nonlinear fractional order systems with uncertain observations, Appl. Math. Sci., 6, 29-32, 1471-1486 (2012) · Zbl 1253.62067 |
| [18] | Sadeghian, H.; Salarieh, H.; Alasty, A.; Meghdari, A., On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment, Appl. Math. Model., 38, 3, 961-973 (2014) · Zbl 1449.37071 |
| [19] | Wang, X.; Liang, Y.; Pan, Q.; Huang, H., Gaussian/Gaussian-mixture filters for non-linear stochastic systems with delayed states, IET Control Theory Appl., 8, 11, 996-1008 (2014) |
| [20] | Wang, X.; Liang, Y.; Pan, Q.; Zhao, C., Gaussian filter for nonlinear systems with one-step randomly delayed measurements, Automatica, 49, 4, 976-986 (2013) · Zbl 1284.93236 |
| [21] | Ramezani, A.; Safarinejadian, B., A modified fractional-order unscented Kalman filter for nonlinear fractional-order systems, Circuits Syst. Signal Process., 37, 9, 3756-3784 (2018) · Zbl 1427.93249 |
| [22] | Huang, Y.; Zhang, Y.; Li, N.; Zhao, L., Gaussian approximate filter with progressive measurement update, Proceedings of the 54th IEEE Conference on Decision and Control (CDC), 4344-4349 (2015), IEEE |
| [23] | Huang, Y.; Zhang, Y.; Li, N.; Zhao, L., Design of sigma-point Kalman filter with recursive updated measurement, Circuits Syst. Signal Process., 35, 5, 1767-1782 (2016) · Zbl 1346.93378 |
| [24] | Zhang, Y.; Huang, Y.; Li, N.; Zhao, L., Embedded cubature Kalman filter with adaptive setting of free parameter, Signal Process., 114, 112-116 (2015) |
| [25] | Huang, Y.; Zhang, Y.; Li, N.; Chambers, J., Robust studentst based nonlinear filter and smoother, IEEE Trans. Aerosp. Electron. Syst., 52, 5, 2586-2596 (2016) |
| [26] | Huang, Y.; Zhang, Y.; Li, N.; Wu, Z.; Chambers, J. A., A novel robust student’st-based Kalman filter, IEEE Trans. Aerosp. Electr. Syst., 53, 3, 1545-1554 (2017) |
| [27] | Huang, Y.; Zhang, Y.; Xu, B.; Wu, Z.; Chambers, J., A new outlier-robust student’st based gaussian approximate filter for cooperative localization, IEEE/ASME Trans. Mech., 22, 5, 2380-2386 (2017) |
| [28] | Huang, Y.; Zhang, Y.; Wu, Z.; Li, N.; Chambers, J., A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices, IEEE Trans. Autom. Control, 63, 2, 594-601 (2017) · Zbl 1390.93790 |
| [29] | Huang, Y.; Zhang, Y.; Xu, B.; Wu, Z.; Chambers, J. A., A new adaptive extended Kalman filter for cooperative localization, IEEE Trans. Aerosp. Electr. Syst., 54, 1, 353-368 (2017) |
| [30] | Chang, L.; Hu, B.; Chang, G.; Li, A., Huber-based novel robust unscented Kalman filter, IET Sci. Measur. Technol., 6, 6, 502-509 (2012) |
| [31] | Zhang, L.; Li, S.; Zhang, E.; Chen, Q., Robust measure of non-linearity-based cubature Kalman filter, IET Sci. Measur. Technol., 11, 7, 929-938 (2017) |
| [32] | Sun, Y.; Wu, X.; Cao, J.; Wei, Z.; Sun, G., Fractional extended Kalman filtering for non-linear fractional system with Lévy noises, IET Control Theory Appl., 11, 3, 349-358 (2016) |
| [33] | Jia, B.; Xin, M.; Cheng, Y., High-degree cubature Kalman filter, Automatica, 49, 2, 510-518 (2013) · Zbl 1259.93118 |
| [34] | Wang, S.; Feng, J.; Chi, K. T., Spherical simplex-radial cubature Kalman filter, IEEE Signal Process. Lett., 21, 1, 43-46 (2013) |
| [35] | Zhang, Y.; Huang, Y.; Li, N.; Zhao, L., Interpolatory cubature Kalman filters, IET Control Theory Appl., 9, 11, 1731-1739 (2015) |
| [36] | Tumwiine, J.; Mugisha, J.; Luboobi, L. S., A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. Math. Comput., 189, 2, 1953-1965 (2007) · Zbl 1117.92039 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
