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.
AbdeljawadT (2011) On Riemann and Caputo fractional differences. Computers & Mathematics with Applications62(3): 1602–1611.
3.
AzimiVMuntherDFakoorianSAet al. (2017) Hybrid extended Kalman filtering and noise statistics optimization for produce wash state estimation. Journal of Food Engineering212: 136–145.
4.
BettayebMMansouriRAl-SaggafUet al. (2017) Smith predictor based fractional-order-filter PID controllers design for long time delay systems. Asian Journal of Control19(2): 587–598.
5.
BrzezinskiDWOstaiczykP (2016) About accuracy increase of fractional order derivative and integral computations by applying the Grünwald–Letnikov formula. Communications in Nonlinear Science and Numerical Simulation40: 151–162.
6.
ChoiEJYoonJCLeeBSet al. (2010) Onboard orbit determination using GPS observations based on the unscented Kalman filter. Advances in Space Research46(11): 1440–1450.
7.
ChoiHDAhnCKLimMT (2013) Time-domain filtering for estimation of linear systems with colored noises using recent finite measurements. Measurement46(8): 2792–2797.
8.
GaoZ (2017) Fractional-order Kalman filters for continuous time linear and nonlinear fractional-order systems using Tustin generating function. International Journal Control. Epub ahead of print 20 September 2017. DOI: 10.1080/00207179.2017.1378438.
9.
GarrappaR (2016) Grünwald–Letnikov operators for fractional relaxation in Havriliak–Negami models. Communications in Nonlinear Science and Numerical Simulation38: 178–191.
10.
GengHWangZChengYet al. (2017) Tobit Kalman filter with fading measurements. Signal Processing140: 60–68.
11.
GuoFWangXYZhuCYet al. (2017) Stochastic multiresonance for a fractional linear oscillator with time-delayed kernel and quadratic noise. Physica A: Statistical Mechanics and its Applications487: 205–214.
12.
GyörgyKKelemenADávidL (2014) Unscented Kalman filters and particle filter methods for nonlinear state estimation. Procedia Technology12: 65–74.
13.
LambaRSingalSKSondhiS (2017) Fractional order PID controller for power control in perturbed pressurized heavy water reactor. Nuclear Engineering and Design323: 84–94.
14.
LiaoRNingLChenZet al. (2017) Performance of unscented Kalman filter tractography in edema: analysis of the two-tensor model. NeuroImage: Clinical15: 819–831.
15.
LiuDXuYLiJ (2017) Randomly-disordered-periodic-induced chaos in a piezoelectric vibration energy harvester system with fractional-order physical properties. Journal of Sound and Vibration399(7) 182–196.
16.
LüLLiCLiGet al (2017) Projective synchronization for uncertain network based on modified sliding mode control technique. International Journal of Adaptive Control and Signal Processing31(3): 429–440.
17.
LuoR (2015) The robust adaptive control of chaotic systems with unknown parameters and external disturbance via a scalar input. International Journal of Adaptive Control and Signal Processing29(10): 1296–1307.
18.
MacDonaldCLBhattacharyaNSprouseBPet al. (2015) Efficient computation of the Grünwald–Letnikov fractional diffusion derivative using adaptive time step memory. Journal of Computational Physics297: 221–236.
19.
MandicPDSekaraTBLazarevicMPet al. (2017) Dominant pole placement with fractional order PID controllers: D-decomposition approach. ISA Transactions67: 76–86.
20.
NosratiKVolosCAzemiA (2017) Cubature Kalman filter-based chaotic synchronization and image encryption. Signal Processing58: 35–48.
21.
PanFLiuLXueD (2016) Optimal PID controller design with Kalman filter for Qball-X4 quad-rotor unmanned aerial vehicle. Transactions of the Institute of Measurement and Control39(12): 1785–1797.
22.
PanHLüZLinWet al. (2017) State of charge estimation of lithium-ion batteries using a grey extended Kalman filter and a novel open-circuit voltage model. Energy138: 764–775.
23.
PashaeiSBadamchizadehM (2016) A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances. ISA Transactions63: 39–48.
24.
PourdehiSAzamiAShabaniniaF (2015) Fuzzy Kalman-type filter for interval fractional-order systems with finite-step auto-correlated process noises. Neurocomputing159(2): 44–49.
25.
RaitoharjuMGarcía-FernándezÁFPichéR (2017) Kullback–Leibler divergence approach to partitioned update Kalman filter. Signal Processing130: 289–298.
26.
SadeghianHSalariehHAlastyAet al. (2014) On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment. Applied Mathematical Modelling38(3): 961–973.
27.
SchererRKallaSLTangYet al. (2011) The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications62(3): 902–917.
28.
SedraouiMAmieurTBouiadjraBRet al. (2016) Robustified fractional-order controller based on adjustable fractional weights for a doubly fed induction generator. Transactions of the Institute of Measurement and Control39(5): 660–674.
29.
SierociukD (2013) Fractional Kalman filter algorithms for correlated system and measurement noises. Control & Cybernetics42(2): 471–490.
30.
SierociukDDzielińskiA (2006) Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. International Journal of Applied Mathematics & Computer Science16(1): 129–140.
31.
SircoulombVHoblosGChafoukH (2010) A gain modification form for Kalman filtering under state inequality constraints. International Journal of Adaptive Control and Signal Processing24(12): 1051–1069.
32.
StanislawskiRRydelMLatawiecKJ (2017) Modeling of discrete-time fractional-order state space systems using the balanced truncation method. Journal of the Franklin Institute354(7): 3008–3020.
33.
SunYWangYWuXet al. (2018) Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements. Journal of the Franklin Institute355: 361–380.
34.
TarasovVE (2014) Lattice model of fractional gradient and integral elasticity: long-range interaction of Grünwald–Letnikov–Riesz type. Mechanics of Materials70: 106–114.
35.
UllahNAsgharMKhattakAet al. (2017) Comparison of integer and fractional order robust controllers for DC/DC converter feeding constant power load in a DC microgrid. Sustainable Energy, Grids and Networks12: 1–9.
36.
WangTQiuJFuSet al. (2017a) Distributed fuzzy H∞ filtering for nonlinear multirate networked double-layer industrial processes. IEEE Transactions on Industrial Electronics64(6): 5203–5211.
37.
WangTQiuJGaoHet al. (2017b) Network-based fuzzy control for nonlinear industrial processes with predictive compensation strategy. IEEE Transactions on Systems, Man, and Cybernetics: Systems47(8): 2137–2147.
38.
WangTQiuJGaoH (2017c) Adaptive neural control of stochastic nonlinear time-delay systems with multiple constraints. IEEE Transactions on Systems, Man, and Cybernetics: Systems47(8): 1875–1883.
39.
WangJYeYPanXet al. (2015) Parallel-type fractional zero-phase filtering for ECG signal denoising. Biomedical Signal Processing and Control18: 36–41.
40.
WeiYTsePWYaoZet al. (2017) The output feedback control synthesis for a class of singular fractional order systems. ISA Transactions69: 1–9.
41.
WuXSunYLuZet al. (2015) A modified Kalman filter algorithm for fractional system under Lévy noises. Journal of the Franklin Institute352(5): 1963–1978.
42.
ZhangYPuYZhangHet al. (2014) An extended fractional Kalman filter for inferring gene regulatory networks using time-series data. Chemometrics and Intelligent Laboratory Systems138(15): 57–63.
43.
ZhongFLiHZhongS (2016) State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems. Signal Processing127: 168–184.
44.
ZhongQZhongFChengJet al. (2017) State of charge estimation of lithium-ion batteries using fractional order sliding mode observer. ISA Transactions66: 448–459.
45.
ZhouLLiXShanLet al. (2017) Hierarchical recursive least squares parameter estimation of non-uniformly sampled Hammerstein nonlinear systems based on Kalman filter. Journal of the Franklin Institute354(10): 4231–4246.
