This paper presents a solution to the filtering problem for a class of stochastic discrete-time nonlinear polynomial systems with switching in the state equation over linear observations and its application to a mechatronic system. The switching in the state equation is performed between two different nonlinear functions according to a sequence of independent Bernoulli random variables that take the quantities of zero and one. The mean-square filtering solution is obtained for a general nonlinear discrete-time polynomial system and a finite-dimensional system of filtering equations is then obtained for a second degree polynomial system as a particular case. The mean-square estimates of polynomial state terms are expressed as functions of the estimate and covariance matrix. Finally, some numerical simulations are carried out to reconstruct the variable states given a vector output measurement for a linear system, a second degree polynomial system, and an induction motor model to show effectiveness of the proposed algorithm. The proposed method is compared with an extended Kalman filter-based algorithm for discrete-time switched nonlinear systems.
AfshariHHGadsdenSAHabibiS (2017) Gaussian filters for parameter and state estimation: A general review of theory and recent trends. Signal Processing135(6): 218–238.
2.
AlessandriABagliettoMBattistelliG (2005) Receding–horizon estimation for switched discrete–time linear systems. IEEE Transactions on Automatic Control50(11): 1736–1748.
EvansJSEvansRJ (1997) State estimation for Markov switching systems with modal observations. In: Proceedings of the 36th Conference on Decision and Control, IEEE, San Diego, CA, USA, pp. 1688–1693.
13.
FisherJRStearEB (1967a) Optimal nonlinear filtering for independent increment processes–Part I. IEEE Transactions on Information Theory3(4): 558–568.
14.
FisherJRStearEB (1967b) Optimal nonlinear filtering for independent increment processes–Part II. IEEE Transactions on Information Theory3(4): 568–578.
15.
Hernandez–GonzalezMBasinMV (2014) Discrete–time filtering for nonlinear polynomial systems over linear observations. International Journal of Systems Science45(7): 1461–1472.
16.
Hernandez–GonzalezMBasinMV (2017) State estimation for stochastic polynomial systems with switching in the state equation. Transactions of the Institute of Measurement and Control40(9): 1–8.
17.
Hernandez-GonzalezMBasinMVStepanovO (2018) Discrete–time state estimation for stochastic polynomial systems over polynomial observation. International Journal of General Systems47(5): 460–476.
18.
HoYCLeeCK (1964) A Bayesian approach to problems in stochastic estimation and control. IEEE Transactions on Automatic Control9(4): 333–339.
19.
JafarzadehSLascuCFadaliMS (2011) State estimation of induction motor drives using unscented Kalman filter. IEEE Transactions on Industrial Electronics59(11): 4207–4216.
20.
JulierSJUhlmannJK (2004) Unscented filtering and nonlinear estimation. Proceedings of the IEEE92(3): 401–422.
21.
KalmanRE (1960) A new approach to linear filtering and prediction problems. Transactions of the ASME-Journal of Basic Engineering82 (Series D): 35–45.
22.
KalmanREBucyRS (1961) New results in linear filtering and prediction theory. Transactions of the ASME. Series D, Journal of the Basic Engineering83(1): 95–107.
23.
KaliYAyalaMRodasJ, et al. (2020) Time delay estimation based discrete–time supertwisting current control for a six phase induction motor. IEEE Transactions on Power Electronics35(1): 12570–12580.
24.
KhalilHK (2002) Nonlinear Systems. New Jersey. Prentice Hall.
25.
KushnerHJ (1967) Approximations to optimal nonlinear filters. IEEE Transactions on Automatic Control12(5): 546–556.
26.
LinHAntsaklisPJ (2009) Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control54(2): 308–322.
27.
LinLLinawati JasaLAmbikairajahE (2002) A hybrid state estimation scheme for power systems. In: Asia–Pacific Conference on Circuits and Systems, IEEE, Bali, Indonesia, 28–31 October, pp. 555–558.
28.
LiuWHwangI (2014) On hybrid state estimation for stochastic hybrid systems. IEEE Transactions on Automatic Control59(10): 2615–2628.
29.
LiuDLiuYAlsaadiFE (2018) Recursive state estimation based-on the outputs of partial nodes for discrete-time stochastic complex networks with switched topology, Journal of the Franklin Institute355(11): 4686–4707.
30.
LiuLZhaoYZhangZMaB (2018) State estimation for nonlinear switched systems under constrained switching. In: 2018 33rd Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, 18–20 May, Nanjing, China, pp. 946–951.
31.
MarinoRTomeiPVerrelliCM (2010) Induction Motor Control Design. London: Springer.
32.
McGonigalDIonescuD (1995) An outline for a Kalman filter and recursive parameter estimation approach applied to stock market forecasting. In: Proceedings 1995 Canadian Conference on Electrical and Computer Engineering, IEEE, Montreal, Quebec, Canada, 5–8 September, pp. 1148–1151.
33.
NaYAhmadM (2019) A fault detection scheme for switched systems with noise under asynchronous switching. In: 2019 9th International Conference on Information Science and Technology (ICIST), IEEE, 2–5 August, Hulunbuir, China, pp. 258–262.
34.
PetriIVukosaviS (2020) High-performance speed estimation of induction motor machines based on adaptive filtering, IET Electric Power Applications14(4): 695–704.
35.
PletschenNDiepoldKJ (2017) Nonlinear state estimation for suspension control applications: A Takagi–Sugeno Kalman filtering approach. Control Engineering Practice61(4): 292–306.
36.
RossS (2010) A First Course in Probability. New Jersey: Pearson Education, Inc.
37.
SavkinAVEvansRJ (2002) Hybrid Dynamical Systems. Controller and Sensor Switching Problems. Birkhäuser Basel, New York.
38.
SellamiLAbderrahimK (2015) State estimation for switched linear systems. In: 2015 12th International Multi–Conference on Systems, Signals & Devices (SSD15), IEEE, Mahdia, Tunisia, 16–19 March, pp. 1–5.
39.
ShiSFeiZShiPAhnCK (2019) Asynchronous filtering for discrete-time switched T-S fuzzy systems. IEEE Transactions on Fuzzy Systems28(8): 1531–1541.
40.
SimonD (2010) Kalman filtering with state constraints: A survey of linear and nonlinear algorithms, IET Control Theory and Applications4(8): 1303–1318.
41.
SunZGeSS (2005) Switched Linear Systems. Control and Design. London: Springer–Verlag.
42.
SunX-LTianY-ZWangW-B (2008) The application of Kalman filtering to corporate bankruptcy prediction. In: 2008 International Conference on Management Science & Engineering, IEEE, Long Beach, USA, 10–12 September, pp. 670–676.
43.
TangWWangZShenY (2019) Interval estimation for discrete-time linear systems: A two-step method. Systems & Control Letters123(1): 69–74.
44.
TayamaTKuroseYMarinhoMM, et al. (2018) Autonomous positioning of eye surgical robot using the tool shadow and Kalman filtering. In: 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), IEEE, Honolulu, HI, USA, 18–21 July, pp. 1723–1726.
45.
VargasANCostaEFAchoLdo ValJBR (2018) Switching stochastic nonlinear systems with application to an automotive throttle. IEEE Transactions on Automatic Control63(9): 3098–3104.
46.
WangYKi AhnCYanHXieS (2020) Fuzzy control and filtering for nonlinear singularly perturbed Markov jump Systems. IEEE Transactions on Cybernetics (Early Access): 1–12.
47.
WangWMaSWangJSongS (2012) Robust stability and stabilization for nonlinear uncertain discrete–time descriptor switched systems. In: 2012 24th Chinese Control and Decision Conference (CCDC), IEEE, Taiyuan, China, 23–25 May, pp. 4025–4030.
48.
Wayne BeatyHKirtleyJLJr (1998) Electric Motor Handbook. McGraw-Hill. New York, NY.
49.
XiaoXParkJHZhouL (2018) Event-triggered H∞ filtering of discrete-time switched linear systems. ISA Transactions77(6): 112–121.
50.
YanHZhangHYangF, et al. (2017) Distributed H∞ filtering for switched repeated scalar nonlinear systems with randomly occurred sensor nonlinearities and asynchronous switching. IEEE Transactions on Systems, Man, and Cybernetics: Systems48(12): 2263–2270.
51.
YangJWuCCuiLChenY (2018) Finite time state estimation and fault detection for linear switched systems with unknown inputs. In: 2018 IEEE 7th Data Driven Control and Learning Systems Conference, IEEE, pp. 1079–1084, 25–27 May, Hubei Province, China.
52.
ZhangZBoTSunC, et al. (2020) State estimation for induction motor speed–sensorless control based on strong tracking. In: 2020 38th Chinese Control Conference (CCC), IEEE, 27–29 July, Shenyang, China.
53.
ZhangDYuLWangQ-GOngC-J (2012) Estimator design for discrete–time switched neural networks with asynchronous switching and time–varying delay. IEEE Transactions on Neural Networks on Learning Systems23(5): 827–834.
54.
ZhangHMaoX (2009) Fuzzy-based Kalman filter for ship navigation. In: Proceedings of the 2009 IEEE International Conference on Mechatronics and Automation, IEEE, Changchun, China, 9–12 August, pp. 2388–2392.
55.
ZhangJZhangTJiangXWangS (2012) Tightly coupled GPS/INS integrated navigation algorithm based on Kalman filter. In: 2012 Second International Conference on Business Computing and Global Information, IEEE, Shanghai, China, 12–14 October, pp. 588–591.
56.
ZhaoSHuangBLiuF (2017a) Linear optimal unbiased filter for time-variant systems without apriori information in initial conditions. IEEE Transactions on Automatic Control62(2): 882–887.
57.
ZhaoSHuangBShmaliyYS (2017b) Bayesian state estimation on finite horizons: The case of linear state-space model. Automatica85: 91–99.
58.
ZhouJXuYLiJ-YLiY (2018) Robust state estimation for discrete-time systems with colored noises and communication constraints. Journal of the Franklin Institute355(13): 5790–7810.
59.
ZhuZMengZZhangZ, et al. (2017) Robust particle filter for state estimation using measurements with different types of gross errors. ISA Transactions69(7): 281–295.
