“Mean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations”

Authors: Michael Basin, Juan J. Maldonado and Hamid Reza Karimi,
Affiliation: University of Nuevo Leon and University of Agder
Reference: 2011, Vol 32, No 2, pp. 47-55.

Keywords: Filter Design; Poisson Noises; Polynomial Observations

Abstract: In this paper, the mean-square filtering problem for polynomial system states confused with white Poisson noises over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third order sensor filtering problem for a quadratic state, assuming a conditionally Poisson initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.

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DOI forward links to this article:
[1] Michael Basin and Pablo Rodriguez-Ramirez (2013), doi:10.1080/00207721.2013.827265
[2] Zhankui Zeng, Shijie Zhang, Yanjun Xing and Xibin Cao (2014), doi:10.1155/2014/159149
[3] Michael Basin and Pablo Rodriguez-Ramirez (2013), doi:10.1109/ASCC.2013.6606188
[4] Garry A. Einicke (2014), doi:10.1109/SSP.2014.6884572
[5] Jun Hu, Zidong Wang, Dongyan Chen and Fuad E. Alsaadi (2016), doi:10.1016/j.inffus.2016.01.001
[6] M. V. Basin (2016), doi:10.1134/S000511791602003X
[7] Juan Jose Maldonado, Michael V. Basin and Miguel Hernandez-Gonzalez (2016), doi:10.1049/iet-cta.2015.1000
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[11] Qinyuan Liu, Zidong Wang and Xiao He (2019), doi:10.1007/978-3-030-00157-5_8
[12] Qinyuan Liu, Zidong Wang and Xiao He (2019), doi:10.1007/978-3-030-00157-5_7
[13] Wenjing Wang, Wei Wang, Yu Chen and Juan Li (2023), doi:10.23919/CCC58697.2023.10240008
References:
[1] Basin, M. (2008). New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems, Berlin: Springer.
[2] Basin, M., Alcorta-Garcia, M., Pena, D. (2007). Simulation and optimal filtering for linear systems with poisson white noises, Dynamics of Continuous, Discrete and Impulsive Systems. Series B, 14:221--231.
[3] Basin, M., Calderon-Alvarez, D., Skliar, M. (2008). Optimal filtering for incompletely measured polynomial states over linear observations, International J. Adaptive Control and Signal Processing, 22:482--494.
[4] Basin, M. Maldonado, J. (2011). Mean-square filter design for nonlinear polynomial systems with poisson noise, Proc. American Control Conference, page Ref. no. WeB02.1.
[5] Basin, M., Shi, P., Calderon-Alvarez, D. (2009). Optimal filtering for incompletely measured polynomial systems with multiplicative noises, Circuits, Systems and Signal Processing, 28:223--239 doi:10.1007/s00034-008-9083-2
[6] Basin, M., Shi, P., Calderon-Alvarez, D. (2010). Approximate finite-dimensional filtering for polynomial states over polynomial observations, International Journal of Control, 83:724--730.
[7] Benes, V. (1981). Exact finite-dimensional filters for certain diffusions with nonlinear drift, Stochastics, 5:65--92 doi:10.1080/17442508108833174
[8] Dupé, F., Fadili, M., Starck, J.-L. (2008). Image deconvolution under poisson noise using sparse representations and proximal thresholding iteration, Proc. IEEE ICASSP - Int. Conf. on Acoustics, Speech, and Signal Processing, pp. 761--764.
[9] Fleming, W. McEneaney, W. (2001). Robust limits of risk sensitive nonlinear filters, Mathematics of Control, Signals and Systems, 14:109--142 doi:10.1007/PL00009879
[10] Fridman, E. Shaked, U. (1997). On regional nonlinear H-Infinity filtering, Systems and Control Letters, 29:233--240 doi:10.1016/S0167-6911(96)00061-8
[11] Gao, H. Chen, T. (2007). H-Infinity estimation for uncertain systems with limited communication capacity, IEEE Transactions on Automatic Control, 52:2070--2084 doi:10.1109/TAC.2007.908316
[12] Gao, H., L., J.L., Xie, Wang, C. (2005). New approach to mixed H2/H-Infinity filtering for polytopic discrete-time systems, IEEE Transactions on Signal Processing, 53:3183--3192.
[13] Gao, H. Wang, C. (2004). A delay-dependent approach to robust H-Infinity filtering for uncertain discrete-time state-delayed systems, IEEE Transactions on Signal Processing, 52:1631--1640.
[14] Hannequin, P. Mas, J. (2002). Statistical and heuristic image noise extraction, shine: A new method for processing poisson noise in scintigraphic images. Phys. Med. Biol., 47:4329--4344 doi:10.1088/0031-9155/47/24/302
[15] Hazewinkel, M., Marcus, S., Sussmann, H. (1983). Nonexistence of exact finite-dimensional filters for conditional statistics of the cubic sensor problem, Systems and Control Letters, 5:331--340.
[16] Kalman, R. Bucy, R. (1961). New results in linear filtering and prediction theory, ASME Trans., Part D.J. of Basic Engineering, 83:95--108.
[17] Kolmanovsky, I. Maizenberg, T. (2002). Optimal containment control for a class of stochastic systems perturbed by poisson and wiener processes, Proc. American Control Conf., pp. 322--327.
[18] Kolmanovsky, I. Maizenberg, T. (2002). Optimal containment control for a class of stochastic systems perturbed by poisson and wiener processes, IEEE Trans. on Automatic Control, 47:1641--1645.
[19] Kushner, H. (1964). On differential equations satisfied by conditional probability densities of markov processes, SIAM J. Control, 12:106--119.
[20] Lu, H., Liang, Z., Chen, D. (2001). A combined transformation of ordering spect sinograms for signal extraction from measurements with poisson noise, SPIE Proc., 4322:1431--1438.
[21] Mahmoud, M. Shi, P. (2003). Robust kalman filtering for continuous time-lag systems with markovian jump parameters, Automatica, 50:98--105.
[22] Nguang, S. Fu, M. (1996). Robust nonlinear H-Infinity filtering, International Journal of Robust and Nonlinear Control, 32:1195--1199 doi:10.1016/0005-1098(96)00067-2
[23] Oksendal, B. (2006). Stochastic Differential Equations, Berlin: Springer.
[24] Pugachev, V. (1984). Probability Theory and Mathematical Statistics for Engineers, London, Pergamon.
[25] Pugachev, V. Sinitsyn, I. (2001). Stochastic Systems: Theory and Applications, Singapore: World Scientific.
[26] Shen, B., Wang, Z., Shu, H., Wei, G. (2009). H-Infinity filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays, Automatica, 45:1032--1037 doi:10.1016/j.automatica.2008.11.009
[27] Shi, P. (1998). Filtering on sampled-data systems with parametric uncertainty, IEEE Transactions on Automatic Control. 43:1022--1027.
[28] Tucker, H. (1967). A Graduate Course in Probability, New York: Academic Press.
[29] Wang, Z., Ho, D., Liu, Y., Liu, X. (2009). Robust H-Infinity infinity control for a class of nonlinear discrete time-delay stochastic systems with missing measurements, Automatica, 45:684--691 doi:10.1016/j.automatica.2008.10.025
[30] Wang, Z., Lam, J., Liu, X. (2003). Nonlinear filtering for state delayed systems with markovian switching, IEEE Transactions on Signal Processing, 51:2321--2328 doi:10.1109/TSP.2003.815373
[31] Wang, Z., Liu, Y., Liu, X. (2008). H-Infinity filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44:1268--1277 doi:10.1016/j.automatica.2007.09.016
[32] Wei, G., Wang, Z., Shu, H. (2009). Robust filtering with stochastic nonlinearities and multiple missing measurements, Automatica, 45:836--841 doi:10.1016/j.automatica.2008.10.028
[33] Wonham, W. (1965). Some applications of stochastic differential equations to nonlinear filtering, SIAM J. Control, 2:347--369.
[34] Xie, L., Souza, C.D., Wang, Y. (1996). Robust filtering for a class of discrete-time uncertain nonlinear systems, International Journal of Robust and Nonlinear Control, 6:297--312 doi:10.1002/(SICI)1099-1239(199605)6:4andlt;297::AID-RNC234andgt;3.0.CO;2-V
[35] Xu, S. Chen, T. (2003). Robust H-Infinity filtering for uncertain impulsive stochastic systems under sampled measurements, Automatica, 39:509--516 doi:10.1016/S0005-1098(02)00248-0
[36] Xu, S. van Dooren, P. (2002). Robust H-Infinity filtering for a class of nonlinear systems with state delay and parameter uncertainty, Int. J. Control, 75:766--774 doi:10.1080/00207170210141815
[37] Xu, S., Lam, J., Gao, H., Zhou, Y. (2005). Robust H-Infinity filtering for uncertain discrete stochastic systems with time delays, Circuits, Systems and Signal Processing, 24:753--770 doi:10.1007/s00034-005-0921-1
[38] Yau, S. S.-T. (1994). Finite-dimensional filters with nonlinear drift i: a class of filters including both kalman-bucy and benes filters, J. Math. Systems, Estimation, and Control, 4:181--203.
[39] Yaz, E. Yaz, Y. (2001). State estimation of uncertain nonlinear systems with general criteria, Applied Mathematics Letters, 14:605--610 doi:10.1016/S0893-9659(00)00201-9
[40] Zhang, B., Fadili, M., Starck, J., Dige, S. (2008). Fast poisson noise removal by biorthogonal haar domain hypothesis testing, Statistical Methodology, 5:387--396 doi:10.1016/j.stamet.2008.02.004
[41] Zhang, B., Fadili, M., Starck, J.-L. (2008). Wavelets, ridgelets, and curvelets for poisson noise removal, IEEE Trans. on Image Processing, 17:1093--1108 doi:10.1109/TIP.2008.924386
[42] Zhang, H., Basin, M., Skliar, M. (2007). Ito-volterra optimal state estimation with continuous, multirate, randomly sampled, and delayed measurements, IEEE Transactions on Automatic Control, 52:401--416 doi:10.1109/TAC.2007.892383
[43] Zhang, W., Chen, B., Tseng, C. (2005). Robust H-Infinity filtering for nonlinear stochastic systems, IEEE Transactions on Signal Processing, 53:589--598 doi:10.1109/TSP.2004.840724


BibTeX:
@article{MIC-2011-2-1,
  title={{Mean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations}},
  author={Basin, Michael and Maldonado, Juan J. and Karimi, Hamid Reza},
  journal={Modeling, Identification and Control},
  volume={32},
  number={2},
  pages={47--55},
  year={2011},
  doi={10.4173/mic.2011.2.1},
  publisher={Norwegian Society of Automatic Control}
};