Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

LI Tian-cheng; SU Jin-ya; LIU Wei; CORCHADO Juan M.

doi:10.1631/FITEE.1700379

Your Location：

Home >

Browse articles >

Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Invited Papers | Updated：2022-06-06

- Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond
- 近似高斯共轭：非线性、多模态、不确定以及约束下的参数递归滤波等
- Frontiers of Information Technology & Electronic Engineering Vol. 18, Issue 12, Pages: 1913-1939(2017)
- Affiliations：
  
  School of Sciences, University of Salamanca, Salamanca 37007, Spain
  Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, UK
  Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 4ET, UK
- Author bio：
  
  [ "", "Dr. Tian-cheng LI, first author of this invited review, received his Bachelor’s degree in Mechanical and Electrical Engineering, with a minor degree in Business Administration, from Harbin Engineering University (China), in 2008, his first PhD in Electrical and Electronic Engineering from London South Bank University (United Kingdom), in 2013, and his second doctoral degree in Mechatronics Engineering from Northwestern Polytechnical University (China), in 2015. He has been a Research Associate with the BISITE Group, University of Salamanca (Spain) since June 2014 and presently, he holds a Marie Skłodowska-Curie Individual Fellowship (H2020-MSCA-IF-2015) with the same Host and the Partner Vienna University of Technology (Austria).He has been on the Editorial Board of peer-reviewed journals: Advances in Distributed Computing and Artificial Intelligence Journal (2016–) and Frontiers of Information Technology & Electronic Engineering (2017–), and on the organizing and/or technical program committee of several international symposiums and conferences, including FUSION 2014–2017, DCAI 2015–2017, and ACM-SAC 2015–2017.His research interests lie in the general area of statistical signal processing and distributed information fusion, with particular interest in novel sensor data clustering, fitting and mining algorithms for advanced multi-sensor multi-object detection, tracking, and forecasting." ]
  [ "Jin-ya SU, E-mail: J.Su2@lboro.ac.uk" ]
  [ "Wei LIU, E-mail: w.liu@sheffield.ac.uk" ]
  [ "Juan M. CORCHADO, E-mail: corchado@usal.es" ]
- Funds：
- DOI：10.1631/FITEE.1700379
  CLC： TP391
- Received：11 June 2017，
  
  Revised：2017-11-22，
  
  Published：2017-12
- Accepted：
Scan QR Code
LI Tian-cheng, SU Jin-ya, LIU Wei, et al. Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(12): 1913-1939.
DOI：

LI Tian-cheng, SU Jin-ya, LIU Wei, et al. Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(12): 1913-1939. DOI： 10.1631/FITEE.1700379.

摘要

自上世纪60年代作为现代估计开山之作的卡尔曼滤波器（Kalman filter）的诞生，时间序列状态空间模型应用于各类动态估计问题吸引了大量的研究关注。特别是，寻求实现闭环马尔科夫?贝叶斯递归（比如，从一个高斯先验到一个高斯后验，本文称之为高斯共轭）的解析解成为一般时间序列滤波器设计的主流思路。其面临的主要挑战包括：系统的非线性、多模态（包括机动模型）、复杂不确定性（比如未知的系统输入，非高斯噪声等）和系统约束（包括循环随机变量）等。这些挑战不断触生新的理论、算法与滤波技术，以实现所期望的参数共轭递归。本文对最新研究进行分类、系统回顾，强调了一些容易被忽略的要点。着重介绍了高精观测非线性系统、高斯后验和机动多模态、以及复杂未知系统输入与约束，以弥补当前文献介绍的不足。同时，本文提出一些新的思考：一是一阶马尔科夫转移模型的替代模型，二是有关计算复杂度的滤波器评价。

Abstract

Since the landmark work of R. E. Kalman in the 1960s

considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular

parametric filters that seek analytical estimates based on a closed-form Markov-Bayes recursion

e.g.

recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed 'Gaussian conjugacy' in this paper)

form the backbone for a general time series filter design. Due to challenges arising from nonlinearity

multimodality (including target maneuver)

intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities)

etc.

new theories

algorithms

and technologies have been developed continuously to maintain such a conjugacy

or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper

we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular

specific attention is paid to nonlinear systems with an informative observation

multimodal systems including Gaussian mixture posterior and maneuvers

and intractable unknown inputs and constraints

to fill some gaps in existing reviews and surveys. In addition

we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity.

关键词

Keywords

references

N. Adurthi , P. Singla , T. Singh . Conjugate unscented transformation: applications to estimation and control . J. Dyn. Syst. Meas. Contr. , 2017 . 140 ( 3 ): 030907 DOI: 10.1115/1.4037783 http://doi.org/10.1115/1.4037783 .

H. Afshari , S. Gadsden , S. Habibi . Gaussian filters for parameter and state estimation: a general review of theory and recent trends . Signal Process. , 2017 . 135 218 - 238 . DOI: 10.1016/j.sigpro.2017.01.001 http://doi.org/10.1016/j.sigpro.2017.01.001 .

G. Agamennoni , E.M. Nebot . Robust estimation in non-linear state-space models with state-dependent noise . IEEE Trans. Signal Process. , 2014 . 62 ( 8 ): 2165 - 2175 . DOI: 10.1109/TSP.2014.2305636 http://doi.org/10.1109/TSP.2014.2305636 .

S.S. Ali-Loytty . Box Gaussian mixture filter . IEEE Trans. Autom. Contr. , 2010 . 55 ( 9 ): 2165 - 2169 . DOI: 10.1109/TAC.2010.2051486 http://doi.org/10.1109/TAC.2010.2051486 .

I. Arasaratnam , S. Haykin . Square-root quadrature Kalman filtering . IEEE Trans. Signal Process. , 2008 . 56 ( 6 ): 2589 - 2593 . DOI: 10.1109/TSP.2007.914964 http://doi.org/10.1109/TSP.2007.914964 .

I. Arasaratnam , S. Haykin . Cubature Kalman filters . IEEE Trans. Autom. Contr. , 2009 . 54 ( 6 ): 1254 - 1269 . DOI: 10.1109/TAC.2009.2019800 http://doi.org/10.1109/TAC.2009.2019800 .

A. Aravkin , J.V. Burke , G. Pillonetto . Robust and trend-following Kalman smoothers using Student's t . IFAC Proc. Vol. , 2012 . 45 ( 16 ): 1215 - 1220 . DOI: 10.3182/20120711-3-BE-2027.00283 http://doi.org/10.3182/20120711-3-BE-2027.00283 .

T. Ardeshiri , K. Granström , E. Ozkan , 等 . Greedy reduction algorithms for mixtures of exponential family . IEEE Signal Process. Lett. , 2015 . 22 ( 6 ): 676 - 680 . DOI: 10.1109/LSP.2014.2367154 http://doi.org/10.1109/LSP.2014.2367154 .

M.S. Arulampalam , S. Maskell , N. Gordon , 等 . A tutorial on particle filters for online nonlinear/nonGaussian Bayesian tracking . IEEE Trans. Signal Process. , 2002 . 50 ( 2 ): 174 - 188 . DOI: 10.1109/78.978374 http://doi.org/10.1109/78.978374 .

S.E. Azam , E. Chatzi , C. Papadimitriou . A dual Kalman filter approach for state estimation via outputonly acceleration measurements . Mech. Syst. Signal Process. , 2015 . 60-61 866 - 886 . DOI: 10.1016/j.ymssp.2015.02.001 http://doi.org/10.1016/j.ymssp.2015.02.001 .

V.A. Bavdekar , A.P. Deshpande , S.C. Patwardhan . Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter . J. Process Contr. , 2011 . 21 ( 4 ): 585 - 601 . DOI: 10.1016/j.jprocont.2011.01.001 http://doi.org/10.1016/j.jprocont.2011.01.001 .

B.M. Bell , F.W. Cathey . The iterated Kalman filter update as a Gauss-Newton method . IEEE Trans. Autom. Contr. , 1993 . 38 ( 2 ): 294 - 297 . DOI: 10.1109/9.250476 http://doi.org/10.1109/9.250476 .

T.R. Bielecki , J. Jakubowski , M. Niewegłowski . Conditional Markov chains: properties, construction and structured dependence . Stoch. Process. Their Appl. , 2017 . 127 ( 4 ): 1125 - 1170 . DOI: 10.1016/j.spa.2016.07.010 http://doi.org/10.1016/j.spa.2016.07.010 .

I. Bilik , J. Tabrikian . MMSE-based filtering in presence of non-Gaussian system and measurement noise . IEEE Trans. Aerosp. Electron. Syst. , 2010 . 46 ( 3 ): 1153 - 1170 . DOI: 10.1109/TAES.2010.5545180 http://doi.org/10.1109/TAES.2010.5545180 .

J.A. Bilmes . Buried Markov models for speech recognition . Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing , 1999 . p.713 - 716 . DOI: 10.1109/ICASSP.1999.759766 http://doi.org/10.1109/ICASSP.1999.759766 .

P.L. Bogler . Tracking a maneuvering target using input estimation . IEEE Trans. Aerosp. Electron. Syst. , 1987 . 23 ( 3 ): 298 - 310 . DOI: 10.1109/TAES.1987.310826 http://doi.org/10.1109/TAES.1987.310826 .

S. Bordonaro , P. Willett , Y. Bar-Shalom . Decorrelated unbiased converted measurement Kalman filter . IEEE Trans. Aerosp. Electron. Syst. , 2014 . 50 ( 2 ): 1431 - 1444 . DOI: 10.1109/TAES.2014.120563 http://doi.org/10.1109/TAES.2014.120563 .

S. Bordonaro , P. Willett , Y. Bar-Shalom . Consistent linear tracker with converted range, bearing and range rate measurements . IEEE Trans. Aerosp. Electron. Syst. , 2017 . 53 ( 6 ): 3135 - 3149 . DOI: 10.1109/TAES.2017.2730980 http://doi.org/10.1109/TAES.2017.2730980 .

M.F. Bugallo , V. Elvira , L. Martino , 等 . Adaptive importance sampling: the past, the present, and the future . IEEE Signal Process. Mag. , 2017 . 34 ( 4 ): 60 - 79 . DOI: 10.1109/MSP.2017.2699226 http://doi.org/10.1109/MSP.2017.2699226 .

O. Cappé , S.J. Godsill , E. Moulines . An overview of existing methods and recent advances in sequential Monte Carlo . Proc. IEEE , 2007 . 95 ( 5 ): 899 - 924 . DOI: 10.1109/JPROC.2007.893250 http://doi.org/10.1109/JPROC.2007.893250 .

A. Carrassi , M. Bocquet , L. Bertino , 等 . Data assimilation in the geosciences-an overview on methods, issues and perspectives . arXiv:1709.02798. , 2017 . http://arxiv.org/abs/1709.02798 http://arxiv.org/abs/1709.02798 , .

L. Chang , B. Hu , A. Li , 等 . Transformed unscented Kalman filter . IEEE Trans. Autom. Contr. , 2013 . 58 ( 1 ): 252 - 257 . DOI: 10.1109/TAC.2012.2204830 http://doi.org/10.1109/TAC.2012.2204830 .

B. Chen , J.C. Principe . Maximum correntropy estimation is a smoothed MAP estimation . IEEE Signal Process. Lett. , 2012 . 19 ( 8 ): 491 - 494 . DOI: 10.1109/LSP.2012.2204435 http://doi.org/10.1109/LSP.2012.2204435 .

B. Chen , X. Liu , H. Zhao , 等 . Maximum correntropy Kalman filter . Automatica , 2017 . 76 70 - 77 . DOI: 10.1016/j.automatica.2016.10.004 http://doi.org/10.1016/j.automatica.2016.10.004 .

F.C. Chen , C.S. Hsieh . Optimal multistage Kalman estimators . IEEE Trans. Autom. Contr. , 2000 . 45 ( 11 ): 2182 - 2188 . DOI: 10.1109/9.887678 http://doi.org/10.1109/9.887678 .

H.D. Chen , K.C. Chang , C. Smith . Constraint optimized weight adaptation for Gaussian mixture reduction . SPIE , 2010 . 7697 76970N DOI: 10.1117/12.851993 http://doi.org/10.1117/12.851993 .

R. Chen , J.S. Liu . Mixture Kalman filters . J. R. Stat. Soc. Ser. B , 2000 . 62 ( 3 ): 493 - 508 . DOI: 10.1111/1467-9868.00246 http://doi.org/10.1111/1467-9868.00246 .

Y. Cheng , H. Ye , Y. Wang , 等 . Unbiased minimum-variance state estimation for linear systems with unknown input . Automatica , 2009 . 45 ( 2 ): 485 - 491 . DOI: 10.1016/j.automatica.2008.08.009 http://doi.org/10.1016/j.automatica.2008.08.009 .

J.M.C. Clark , R.B. Vinter , M.M. Yaqoob . Shifted Rayleigh filter: a new algorithm for bearings-only tracking . IEEE Trans. Aerosp. Electron. Syst. , 2007 . 43 ( 4 ): 1373 - 1384 . DOI: 10.1109/TAES.2007.4441745 http://doi.org/10.1109/TAES.2007.4441745 .

J.L. Crassidis , F.L. Markley , Y. Cheng . Survey of nonlinear attitude estimation methods . J. Guid. Contr. Dyn. , 2007 . 30 ( 1 ): 12 - 28 . DOI: 10.2514/1.22452 http://doi.org/10.2514/1.22452 .

D.F. Crouse , P. Willett , K. Pattipati , 等 . A look at Gaussian mixture reduction algorithms . 14th Int. Conf. on Information Fusion , 2011 . p.1 - 8 . .

M. Darouach , M. Zasadzinski . Unbiased minimum variance estimation for systems with unknown exogenous inputs . Automatica , 1997 . 33 ( 4 ): 717 - 719 . DOI: 10.1016/S0005-1098(96)00217-8 http://doi.org/10.1016/S0005-1098(96)00217-8 .

F. Daum , J. Huang . Generalized particle flow for nonlinear filters . SPIE , 2010 . 7698 76980I DOI: 10.1117/12.839421 http://doi.org/10.1117/12.839421 .

M.P. Deisenroth , R.D. Turner , M.F. Huber , 等 . Robust filtering and smoothing with Gaussian processes . IEEE Trans. Autom. Contr. , 2012 . 57 ( 7 ): 1865 - 1871 . DOI: 10.1109/TAC.2011.2179426 http://doi.org/10.1109/TAC.2011.2179426 .

P. del Moral , D. Arnaud . Particle methods: an introduction with applications . Proc. ESAIM , 2014 . 44 1 - 46 . DOI: 10.1051/proc/201444001 http://doi.org/10.1051/proc/201444001 .

K.J. DeMars , R.H. Bishop , M.K. Jah . Entropybased approach for uncertainty propagation of nonlinear dynamical systems . J. Guid. Contr. Dyn. , 2013 . 36 ( 4 ): 1047 - 1057 . DOI: 10.2514/1.58987 http://doi.org/10.2514/1.58987 .

P.M. Djurić , J. Miguez . Sequential particle filtering in the presence of additive Gaussian noise with unknown parameters . Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing , 2002 . p.1621 - 1624 . DOI: 10.1109/ICASSP.2002.5744928 http://doi.org/10.1109/ICASSP.2002.5744928 .

H. Dong , Z. Wang , H. Gao . Robust H ∞ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts . IEEE Trans. Signal Process. , 2010 . 58 ( 4 ): 1957 - 1966 . DOI: 10.1109/TSP.2009.2038965 http://doi.org/10.1109/TSP.2009.2038965 .

Z. Duan , X.R. Li . The role of pseudo measurements in equality-constrained state estimation . IEEE Trans. Aerosp. Electron. Syst. , 2013 . 49 ( 3 ): 1654 - 1666 . DOI: 10.1109/TAES.2013.6558010 http://doi.org/10.1109/TAES.2013.6558010 .

Z. Duan , X.R. Li . Analysis, design, and estimation of linear equality-constrained dynamic systems . IEEE Trans. Aerosp. Electron. Syst. , 2015 . 51 ( 4 ): 2732 - 2746 . DOI: 10.1109/TAES.2015.140441 http://doi.org/10.1109/TAES.2015.140441 .

J. Duník , M. Šimandl , O. Straka . Multiple-model filtering with multiple constraints . Proc. American Control Conf. , 2010 . p.6858 - 6863 . DOI: 10.1109/ACC.2010.5531573 http://doi.org/10.1109/ACC.2010.5531573 .

J. Duník , O. Straka , M. Šimandl . Stochastic integration filter . IEEE Trans. Autom. Contr. , 2013 . 58 ( 6 ): 1561 - 1566 . DOI: 10.1109/TAC.2013.2258494 http://doi.org/10.1109/TAC.2013.2258494 .

J. Duník , O. Straka , M. Šimandl , 等 . Randompoint-based filters: analysis and comparison in target tracking . IEEE Trans. Aerosp. Electron. Syst. , 2015 . 51 ( 2 ): 1403 - 1421 . DOI: 10.1109/TAES.2014.130136 http://doi.org/10.1109/TAES.2014.130136 .

J. Duník , O. Straka , M. Mallick , 等 . Survey of nonlinearity and non-Gaussianity measures for state estimation . 19th Int. Conf. on Information Fusion , 2016 . p.1845 - 1852 . .

J. Duník , O. Straka , J. Ajgl , 等 . From competitive to cooperative filter design . Proc. 20th Int. Conf. on Information Fusion , 2017a . p.235 - 243 . DOI: 10.23919/ICIF.2017.8009652 http://doi.org/10.23919/ICIF.2017.8009652 .

J. Duník , O. Straka , O. Kost , 等 . Noise covariance matrices in state-space models: a survey and comparison of estimation methods-part I . Int. J. Adapt. Contr. Signal Process. , 2017b . 31 ( 11 ): 1505 - 1543 . DOI: 10.1002/acs.2783 http://doi.org/10.1002/acs.2783 .

Y.C. Eldar . Rethinking biased estimation: improving maximum likelihood and the Cramér-Rao bound . Found. Trends Signal Process. , 2008 . 1 ( 4 ): 305 - 449 . DOI: 10.1561/2000000008 http://doi.org/10.1561/2000000008 .

G. Evensen . The ensemble Kalman filter: theoretical formulation and practical implementation . Ocean Dyn. , 2003 . 53 ( 4 ): 343 - 367 . DOI: 10.1007/s10236-003-0036-9 http://doi.org/10.1007/s10236-003-0036-9 .

H. Fan , Y. Zhu , Q. Fu . Impact of mode decision delay on estimation error for maneuvering target interception . IEEE Trans. Aerosp. Electron. Syst. , 2011 . 47 ( 1 ): 702 - 711 . DOI: 10.1109/TAES.2011.5705700 http://doi.org/10.1109/TAES.2011.5705700 .

H. Fang , R.A. de Callafon . On the asymptotic stability of minimum-variance unbiased input and state estimation . Automatica , 2012 . 48 ( 12 ): 3183 - 3186 . DOI: 10.1016/j.automatica.2012.08.039 http://doi.org/10.1016/j.automatica.2012.08.039 .

F. Faubel , J. McDonough , D. Klakow . The split and merge unscented Gaussian mixture filter . IEEE Signal Process. Lett. , 2009 . 16 ( 9 ): 786 - 789 . DOI: 10.1109/LSP.2009.2024859 http://doi.org/10.1109/LSP.2009.2024859 .

B. Friedland . Treatment of bias in recursive filtering . IEEE Trans. Autom. Contr. , 1969 . 14 ( 4 ): 359 - 367 . DOI: 10.1109/TAC.1969.1099223 http://doi.org/10.1109/TAC.1969.1099223 .

R. Frigola-Alcade . Bayesian Time Series Learning with Gaussian Pocesses . PhD Thesis, University of Cambridge, Cambridge, UK , 2015 .

C. Fritsche , U. Orguner , F. Gustafsson . On parametric lower bounds for discrete-time filtering . Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing , 2016 . p.4338 - 4342 . DOI: 10.1109/ICASSP.2016.7472496 http://doi.org/10.1109/ICASSP.2016.7472496 .

A. F. García-Fernández , L. Svensson . Gaussian map filtering using Kalman optimization . IEEE Trans. Autom. Contr. , 2015 . 60 ( 5 ): 1336 - 1349 . DOI: 10.1109/TAC.2014.2372909 http://doi.org/10.1109/TAC.2014.2372909 .

A. F. García-Fernández , M.R. Morelande , J. Grajal , 等 . Adaptive unscented Gaussian likelihood approximation filter . Automatica , 2015a . 54 166 - 175 . DOI: 10.1016/j.automatica.2015.02.005 http://doi.org/10.1016/j.automatica.2015.02.005 .

A. F. García-Fernández , L. Svensson , M.R. Morelande , 等 . Posterior linearization filter: principles and implementation using sigma points . IEEE Trans. Signal Process. , 2015b . 63 ( 20 ): 5561 - 5573 . DOI: 10.1109/TSP.2015.2454485 http://doi.org/10.1109/TSP.2015.2454485 .

J.D. Geeter , H.V. Brussel , J.D. Schutter , 等 . A smoothly constrained Kalman filter . IEEE Trans. Patt. Anal. Mach. Intell. , 1997 . 19 ( 10 ): 1171 - 1177 . DOI: 10.1109/34.625129 http://doi.org/10.1109/34.625129 .

T. Gerstner , M. Griebel . Numerical integration using sparse grids . Numer. Algor. , 1998 . 18 ( 3 ): 209 - 232 . DOI: 10.1023/A:1019129717644 http://doi.org/10.1023/A:1019129717644 .

E. Ghahremani , I. Kamwa . Dynamic state estimation in power system by applying the extended Kalman filter with unknown inputs to phasor measurements . IEEE Trans. Power Syst. , 2011 . 26 ( 4 ): 2556 - 2566 . DOI: 10.1109/TPWRS.2011.2145396 http://doi.org/10.1109/TPWRS.2011.2145396 .

A. Ghoreyshi , T.D. Sanger . A nonlinear stochastic filter for continuous-time state estimation . IEEE Trans. Autom. Contr. , 2015 . 60 ( 8 ): 2161 - 2165 . DOI: 10.1109/TAC.2015.2409910 http://doi.org/10.1109/TAC.2015.2409910 .

G. Gigerenzer , H. Brighton . Homo heuristicus: why biased minds make better inferences . Top. Cogn. Sci. , 2009 . 1 ( 1 ): 107 - 143 . DOI: 10.1111/j.1756-8765.2008.01006.x http://doi.org/10.1111/j.1756-8765.2008.01006.x .

S. Gillijns , B.D. Moor . Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough . Automatica , 2007 . 43 ( 5 ): 934 - 937 . DOI: 10.1016/j.automatica.2006.11.016 http://doi.org/10.1016/j.automatica.2006.11.016 .

F. J. Girón , J.C. Rojano . Bayesian Kalman filtering with elliptically contoured errors . Biometrika , 1994 . 81 ( 2 ): 390 - 395 . .

S. Godsill , T. Clapp . Improvement strategies for Monte Carlo particle filters . In : Doucet A., de Freitas N., Gordon, N. (Eds.), Sequential Monte Carlo Methods in Practice. Springer, New York, USA , 2001 . DOI: 10.1007/978-1-4757-3437-9_7 http://doi.org/10.1007/978-1-4757-3437-9_7 .

N. Gordon , J. Percival , M. Robinson . The KalmanLévy filter and heavy-tailed models for tracking manoeuvring targets . Proc. 6th Int. Conf. on Information Fusion , 2003 . p.1024 - 1031 . DOI: 10.1109/ICIF.2003.177351 http://doi.org/10.1109/ICIF.2003.177351 .

J.D. Gorman , A.O. Hero . Lower bounds for parametric estimation with constraints . IEEE Trans. Inform. Theory , 1990 . 36 ( 6 ): 1285 - 1301 . DOI: 10.1109/18.59929 http://doi.org/10.1109/18.59929 .

K. Granström , P. Willett , Y. Bar-Shalom . Systematic approach to IMM mixing for unequal dimension states . IEEE Trans. Aerosp. Electron. Syst. , 2015 . 51 ( 4 ): 2975 - 2986 . DOI: 10.1109/TAES.2015.150015 http://doi.org/10.1109/TAES.2015.150015 .

M.S. Grewal , A.P. Andrews . Kalman Filtering: Theory and Practice with MATLAB , : New York, USA Wiley-IEEE Press , 2014 .

Y. Guo , K. Fan , D. Peng , 等 . A modified variable rate particle filter for maneuvering target tracking . Front. Inform. Technol. Electron. Eng. , 2015 . 16 ( 11 ): 985 - 994 . DOI: 10.1631/FITEE.1500149 http://doi.org/10.1631/FITEE.1500149 .

Y. Guo , R. Tharmarasa , S. Rajan , 等 . Passive tracking in heavy clutter with sensor location uncertainty . IEEE Trans. Aerosp. Electron. Syst. , 2016 . 52 ( 4 ): 1536 - 1554 . DOI: 10.1109/TAES.2016.140820 http://doi.org/10.1109/TAES.2016.140820 .

U.D. Hanebeck , K. Briechle , A. Rauh . Progressive Bayes: a new framework for nonlinear state estimation . SPIE , 2003 . 5099 256 - 267 . DOI: 10.1117/12.487806 http://doi.org/10.1117/12.487806 .

G. Hendeby . Performance and Implementation Aspects of Nonlinear Filtering . PhD Thesis, Linköping University, Linköping, Sweden , 2008 .

R.J. Hewett , M.T. Heath , M.D. Butala , 等 . A robust null space method for linear equality constrained state estimation . IEEE Trans. Signal Process. , 2010 . 58 ( 8 ): 3961 - 3971 . DOI: 10.1109/TSP.2010.2048901 http://doi.org/10.1109/TSP.2010.2048901 .

Y. Ho , R. Lee . A Bayesian approach to problems in stochastic estimation and control . IEEE Trans. Autom. Contr. , 1964 . 9 ( 4 ): 333 - 339 . DOI: 10.1109/TAC.1964.1105763 http://doi.org/10.1109/TAC.1964.1105763 .

C.S. Hsieh . Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs . Automatica , 2009 . 45 ( 9 ): 2149 - 2153 . DOI: 10.1016/j.automatica.2009.05.004 http://doi.org/10.1016/j.automatica.2009.05.004 .

C.S. Hsieh . Robust two-stage Kalman filters for systems with unknown inputs . IEEE Trans. Autom. Contr. , 2000 . 45 ( 12 ): 2374 - 2378 . DOI: 10.1109/9.895577 http://doi.org/10.1109/9.895577 .

X. Hu , M. Bao , X.P. Zhang , 等 . Generalized iterated Kalman filter and its performance evaluation . IEEE Trans. Signal Process. , 2015 . 63 ( 12 ): 3204 - 3217 . DOI: 10.1109/TSP.2015.2423266 http://doi.org/10.1109/TSP.2015.2423266 .

Y. Huang , Y. Zhang , X. Wang , 等 . Gaussian filter for nonlinear systems with correlated noises at the same epoch . Automatica , 2015 . 60 122 - 126 . DOI: 10.1016/j.automatica.2015.06.035 http://doi.org/10.1016/j.automatica.2015.06.035 .

Y. Huang , Y. Zhang , N. Li , 等 . Design of Gaussian approximate filter and smoother for nonlinear systems with correlated noises at one epoch apart . Circ. Syst. Signal Process. , 2016a . 35 ( 11 ): 3981 - 4008 . DOI: 10.1007/S00034-016-0256-0 http://doi.org/10.1007/S00034-016-0256-0 .

Y. Huang , Y. Zhang , N. Li , 等 . Design of Sigma-point Kalman filter with recursive updated measurement . Circ. Syst. Signal Process. , 2016b . 35 ( 5 ): 1767 - 1782 . DOI: 10.1007/s00034-015-0137-y http://doi.org/10.1007/s00034-015-0137-y .

Y. Huang , Y. Zhang , N. Li , 等 . A novel robust Student's t-based Kalman filter . IEEE Trans. Aerosp. Electron. Syst. , 2017 . 53 ( 3 ): 1545 - 1554 . DOI: 10.1109/TAES.2017.2651684 http://doi.org/10.1109/TAES.2017.2651684 .

M.F. Huber . Nonlinear Gaussian Filtering: Theory, Algorithms, and Applications . KIT Scientific Publishing, Karlsruhe, Germany , 2015 .

M.F. Huber , U.D. Hanebeck . Progressive Gaussian mixture reduction . 11th Int. Conf. on Information Fusion , 2008 . p.1 - 8 . .

S. Ishihara , M. Yamakita . Constrained state estimation for nonlinear systems with non-Gaussian noise . 48th IEEE Conf. on Decision Control , 2009 . p.1279 - 1284 . DOI: 10.1109/CDC.2009.5399627 http://doi.org/10.1109/CDC.2009.5399627 .

K. Ito , K. Xiong . Gaussian filters for nonlinear filtering problems . IEEE Trans. Autom. Contr. , 2000 . 45 ( 5 ): 910 - 927 . DOI: 10.1109/9.855552 http://doi.org/10.1109/9.855552 .

A.H. Jazwinski . Stochastic Processes and Filtering Theory . Academic Press, New York, USA , 1970 . p.349 - 351 . .

B. Jia , M. Xin , Y. Cheng . Sparse-grid quadrature nonlinear filtering . Automatica , 2012 . 48 ( 2 ): 327 - 341 . DOI: 10.1016/j.automatica.2011.08.057 http://doi.org/10.1016/j.automatica.2011.08.057 .

B. Jia , M. Xin , Y. Cheng . High-degree cubature Kalman filter . Automatica , 2013 . 49 ( 2 ): 510 - 518 . DOI: 10.1016/j.automatica.2012.11.014 http://doi.org/10.1016/j.automatica.2012.11.014 .

K. Judd . Tracking an object with unknown accelerations using a shadowing filter . arXiv:1502.07743 , 2015 . http://arxiv.org/abs/1502.07743 http://arxiv.org/abs/1502.07743 , .

K. Judd , T. Stemler . Failures of sequential Bayesian filters and the successes of shadowing filters in tracking of nonlinear deterministic and stochastic systems . Phys. Rev. E , 2009 . 79 ( 6 ): 066206 DOI: 10.1103/PhysRevE.79.066206 http://doi.org/10.1103/PhysRevE.79.066206 .

S.J. Julier , J.J. LaViola . On Kalman filtering with nonlinear equality constraints . IEEE Trans. Signal Process. , 2007 . 55 ( 6 ): 2774 - 2784 . DOI: 10.1109/TSP.2007.893949 http://doi.org/10.1109/TSP.2007.893949 .

S.J. Julier , J.K. Uhlmann . Unscented filtering and nonlinear estimation . Proc. IEEE , 2004 . 92 ( 3 ): 401 - 422 . DOI: 10.1109/JPROC.2003.823141 http://doi.org/10.1109/JPROC.2003.823141 .

R. Kalman . A new approach to linear filtering and prediction problems . J. Basic Eng. , 1960 . 82 ( 1 ): 35 - 45 . DOI: 10.1115/1.3662552 http://doi.org/10.1115/1.3662552 .

D.S. Kalogerias , A.P. Petropulu . Grid based nonlinear filtering revisited: recursive estimation asymptotic optimality . IEEE Trans. Signal Process. , 2016 . 64 ( 16 ): 4244 - 4259 . DOI: 10.1109/TSP.2016.2557311 http://doi.org/10.1109/TSP.2016.2557311 .

R. Kandepu , B. Foss , L. Imsland . Applying the unscented Kalman filter for nonlinear state estimation . J. Process Contr. , 2008 . 18 ( 7-8 ): 753 - 768 . DOI: 10.1016/j.jprocont.2007.11.004 http://doi.org/10.1016/j.jprocont.2007.11.004 .

K.S. Kim , K.H. Rew . Reduced order disturbance observer for discrete-time linear systems . Automatica , 2013 . 49 ( 4 ): 968 - 975 . DOI: 10.1016/j.automatica.2013.01.014 http://doi.org/10.1016/j.automatica.2013.01.014 .

K. Kim , G. Shevlyakov . Why Gaussianity? . IEEE Signal Process. Mag. , 2008 . 25 ( 2 ): 102 - 113 . DOI: 10.1109/MSP.2007.913700 http://doi.org/10.1109/MSP.2007.913700 .

P.K. Kitanidis . Unbiased minimum-variance linear state estimation . Automatica , 1987 . 23 ( 6 ): 775 - 778 . DOI: 10.1016/0005-1098(87)90037-9 http://doi.org/10.1016/0005-1098(87)90037-9 .

J. Ko , D. Fox . GP-Bayes filters: Bayesian filtering using Gaussian process prediction and observation models . Auton. Robots , 2009 . 27 ( 1 ): 75 - 90 . DOI: 10.1007/s10514-009-9119-x http://doi.org/10.1007/s10514-009-9119-x .

S. Ko , R.R. Bitmead . State estimation for linear systems with state equality constraints . Automatica , 2007 . 43 ( 8 ): 1363 - 1368 . DOI: 10.1016/j.automatica.2007.01.017 http://doi.org/10.1016/j.automatica.2007.01.017 .

K.R. Koch , Y. Yang . Robust Kalman filter for rank deficient observation models . J. Geod. , 1998 . 72 ( 7-8 ): 436 - 441 . DOI: 10.1007/s001900050183 http://doi.org/10.1007/s001900050183 .

E. Kostelich , T. Schreiber . Noise-reduction in chaotic time-series data: a survey of common methods . Phys. Rev. E , 1993 . 48 ( 3 ): 1752 - 1763 . DOI: 10.1103/PhysRevE.48.1752 http://doi.org/10.1103/PhysRevE.48.1752 .

J.H. Kotecha , P. M. Djurić . Gaussian particle filtering . IEEE Trans. Signal Process. , 2003a . 51 ( 10 ): 2592 - 2601 . DOI: 10.1109/TSP.2003.816758 http://doi.org/10.1109/TSP.2003.816758 .

J.H. Kotecha , P. M. Djurić . Gaussian sum particle filtering . IEEE Trans. Signal Process. , 2003b . 51 ( 10 ): 2602 - 2612 . DOI: 10.1109/TSP.2003.816754 http://doi.org/10.1109/TSP.2003.816754 .

G. Kurz , I. Gilitschenski , U.D. Hanebeck . Recursive Bayesian filtering in circular state spaces . IEEE Aerosp. Electron. Syst. Mag. , 2016 . 31 ( 3 ): 70 - 87 . DOI: 10.1109/MAES.2016.150083 http://doi.org/10.1109/MAES.2016.150083 .

W.H. Kwon , P.S. Kim , P. Park . A receding horizon Kalman FIR filter for discrete time-invariant systems . IEEE Trans. Autom. Contr. , 1999 . 44 ( 9 ): 1787 - 1791 . DOI: 10.1109/9.788554 http://doi.org/10.1109/9.788554 .

H. Lan , Y. Liang , F. Yang , 等 . Joint estimation and identification for stochastic systems with unknown inputs . IET Contr. Theory Appl. , 2013 . 7 ( 10 ): 1377 - 1386 . DOI: 10.1049/iet-cta.2013.0996 http://doi.org/10.1049/iet-cta.2013.0996 .

J. Lan , X.R. Li . Nonlinear estimation by LMMSEbased estimation with optimized uncorrelated augmentation . IEEE Trans. Signal Process. , 2015 . 63 ( 16 ): 4270 - 4283 . DOI: 10.1109/TSP.2015.2437834 http://doi.org/10.1109/TSP.2015.2437834 .

J. Lan , X.R. Li . Multiple conversions of measurements for nonlinear estimation . IEEE Trans. Signal Process. , 2017 . 65 ( 18 ): 4956 - 4970 . DOI: 10.1109/TSP.2017.2716901 http://doi.org/10.1109/TSP.2017.2716901 .

J. Lan , X.R. Li , V.P. Jilkov , 等 . Second-order Markov chain based multiple-model algorithm for maneuvering target tracking . IEEE Trans. Aerosp. Electron. Syst. , 2013 . 49 ( 1 ): 3 - 19 . DOI: 10.1109/TAES.2013.6404088 http://doi.org/10.1109/TAES.2013.6404088 .

D. Lerro , Y. Bar-Shalom . Tracking with debiased consistent converted measurements versus EKF . IEEE Trans. Aerosp. Electron. Syst. , 1993 . 29 ( 3 ): 1015 - 1022 . DOI: 10.1109/7.220948 http://doi.org/10.1109/7.220948 .

B. Li . State estimation with partially observed inputs: a unified Kalman filtering approach . Automatica , 2013 . 49 ( 3 ): 816 - 820 . DOI: 10.1016/j.automatica.2012.12.007 http://doi.org/10.1016/j.automatica.2012.12.007 .

T. Li , M. Bolić , P.M. Djurić . Resampling methods for particle filtering: classification, implementation, and strategies . IEEE Signal Process. Mag. , 2015a . 32 ( 3 ): 70 - 86 . DOI: 10.1109/MSP.2014.2330626 http://doi.org/10.1109/MSP.2014.2330626 .

T. Li , G. Villarrubia , S. Sun , 等 . Resampling methods for particle filtering: identical distribution, a new method, and comparable study . Front. Inform. Technol. Electron. Eng. , 2015b . 16 ( 11 ): 969 - 984 . DOI: 10.1631/FITEE.1500199 http://doi.org/10.1631/FITEE.1500199 .

T. Li , J.M. Corchado , J. Bajo , 等 . Effectiveness of Bayesian filters: an information fusion perspective . Inform. Sci. , 2016a . 329 670 - 689 . DOI: 10.1016/j.ins.2015.09.041 http://doi.org/10.1016/j.ins.2015.09.041 .

T. Li , J. Prieto , J.M. Corchado . Fitting for smoothing: a methodology for continuous-time target track estimation . Int. Conf. on Indoor Positioning and Indoor Navigation , 2016b . p.1 - 8 . DOI: 10.1109/IPIN.2016.7743582 http://doi.org/10.1109/IPIN.2016.7743582 .

T. Li , J.M. Corchado , S. Sun , 等 . Clustering for filtering: multi-object detection and estimation using multiple/massive sensors . Inform. Sci. , 2017a . 388-389 172 - 190 . DOI: 10.1016/j.ins.2017.01.028 http://doi.org/10.1016/j.ins.2017.01.028 .

T. Li , J. Corchado , J. Prieto . Convergence of distributed flooding and its application for distributed Bayesian filtering . IEEE Trans. Signal Inform. Process. Netw. , 2017b . 3 ( 3 ): 580 - 591 . DOI: 10.1109/TSIPN.2016.2631944 http://doi.org/10.1109/TSIPN.2016.2631944 .

T. Li , H. Chen , S. Sun , 等 . Joint smoothing, tracking, and forecasting based on continuous-time target trajectory fitting . arXiv:1708.02196 , 2017c . http://arxiv.org/abs/1708.02196 http://arxiv.org/abs/1708.02196 , .

T. Li , J. Corchado , H. Chen , 等 . Track a smoothly maneuvering target based on trajectory estimation . Proc. 20th Int. Conf. on Information Fusion , 2017d . p.800 - 807 . DOI: 10.23919/ICIF.2017.8009731 http://doi.org/10.23919/ICIF.2017.8009731 .

T. Li , F.D. la Prieta Pintado , J.M. Corchado , 等 . Multi-source homogeneous data clustering for multitarget detection from cluttered background with misdetection . Appl. Soft Comput. , 2018a . 60 436 - 446 . DOI: 10.1016/j.asoc.2017.07.012 http://doi.org/10.1016/j.asoc.2017.07.012 .

T. Li , J. Corchado , S. Sun , 等 . Partial consensus and conservative fusion of Gaussian mixtures for distributed PHD fusion . arXiv:1711.10783 , 2018b . http://arxiv.org/abs/1711.10783 http://arxiv.org/abs/1711.10783 , .

X.R. Li , Y. Bar-Shalom . Multiple-model estimation with variable structure . IEEE Trans. Autom. Contr. , 1996 . 41 ( 4 ): 478 - 493 . DOI: 10.1109/9.489270 http://doi.org/10.1109/9.489270 .

X.R. Li , V.P. Jilkov . Survey of maneuvering target tracking: decision-based methods . SPIE , 2002 . 4728 511 - 534 . DOI: 10.1117/12.478535 http://doi.org/10.1117/12.478535 .

X.R. Li , V.P. Jilkov . Survey of maneuvering target tracking. Part V. Multiple-model methods . IEEE Trans. Aerosp. Electron. Syst. , 2005 . 41 ( 4 ): 1255 - 1321 . DOI: 10.1109/TAES.2005.1561886 http://doi.org/10.1109/TAES.2005.1561886 .

X.R. Li , V.P. Jilkov . A survey of maneuvering target tracking, Part VIc: approximate nonlinear density filtering in discrete time . SPIE , 2012 . 8393 83930V DOI: 10.1117/12.921508 http://doi.org/10.1117/12.921508 .

Y. Liang , D.X. An , D.H. Zhou , 等 . A finitehorizon adaptive Kalman filter for linear systems with unknown disturbances . Signal Process. , 2004 . 84 ( 11 ): 2175 - 2194 . DOI: 10.1016/j.sigpro.2004.06.021 http://doi.org/10.1016/j.sigpro.2004.06.021 .

Y. Liang , D.H. Zhou , L. Zhang , 等 . Adaptive filtering for stochastic systems with generalized disturbance inputs . IEEE Signal Process. Lett. , 2008 . 15 645 - 648 . DOI: 10.1109/LSP.2008.2002707 http://doi.org/10.1109/LSP.2008.2002707 .

D.V. Lindley , A.F.M. Smith . Bayes estimates for the linear model . J. R. Stat. Soc. Ser. B , 1972 . 34 ( 1 ): 1 - 41 . .

W. Liu . Optimal estimation for discrete-time linear systems in the presence of multiplicative and timecorrelated additive measurement noises . IEEE Trans. Signal Process. , 2015 . 63 ( 17 ): 4583 - 4593 . DOI: 10.1109/TSP.2015.2447491 http://doi.org/10.1109/TSP.2015.2447491 .

W. Liu , P.P. Pokharel , J.C. Principe . Correntropy: properties and applications in non-Gaussian signal processing . IEEE Trans. Signal Process. , 2007 . 55 ( 11 ): 5286 - 5298 . DOI: 10.1109/TSP.2007.896065 http://doi.org/10.1109/TSP.2007.896065 .

Y. Liu , X.R. Li . Measure of nonlinearity for estimation . IEEE Trans. Signal Process. , 2015 . 63 ( 9 ): 2377 - 2388 . DOI: 10.1109/TSP.2015.2405495 http://doi.org/10.1109/TSP.2015.2405495 .

Y. Liu , X.R. Li , H. Chen . Generalized linear minimum mean-square error estimation with application to space-object tracking . Asilomar Conf. on Signals, Systems, and Computers , 2013 . p.2133 - 2137 . DOI: 10.1109/ACSSC.2013.6810685 http://doi.org/10.1109/ACSSC.2013.6810685 .

J. Loxam , T. Drummond . Student-t mixture filter for robust, real-time visual tracking . European Conf. on Computer Vision , 2008 . p.372 - 385 . DOI: 10.1007/978-3-540-88690-7_28 http://doi.org/10.1007/978-3-540-88690-7_28 .

R. Ma , T.P. Coleman . Generalizing the posterior matching scheme to higher dimensions via optimal transportation . 49th Annual Allerton Conf. on Communication, Control, and Computing , 2011 . p.96 - 102 . DOI: 10.1109/Allerton.2011.6120155 http://doi.org/10.1109/Allerton.2011.6120155 .

R. Mahler . Advances in Statistical MultisourceMultitarget Information Fusion . Artech House, Norwood, USA , 2014 .

L. Martino , J. Read , V. Elvira , 等 . Cooperative parallel particle filters for online model selection and applications to urban mobility . Dig. Signal Process. , 2017 . 60 172 - 185 . DOI: 10.1016/j.dsp.2016.09.011 http://doi.org/10.1016/j.dsp.2016.09.011 .

D.Q. Mayne . Optimal non-stationary estimation of the parameters of a linear system with Gaussian inputs . J. Electron. Contr. , 1963 . 14 ( 1 ): 101 - 112 . DOI: 10.1080/00207216308937480 http://doi.org/10.1080/00207216308937480 .

H. Michalska , D.Q. Mayne . Moving horizon observers and observer-based control . IEEE Trans. Autom. Contr. , 1995 . 40 ( 6 ): 995 - 1006 . DOI: 10.1109/9.388677 http://doi.org/10.1109/9.388677 .

S.K. Mitter , N.J. Newton . A variational approach to nonlinear estimation . SIAM J. Contr. Optim. , 2003 . 42 ( 5 ): 1813 - 1833 . DOI: 10.1137/S0363012901393894 http://doi.org/10.1137/S0363012901393894 .

G. Mohammaddadi , N. Pariz , A. Karimpour . Modal Kalman filter . Asian J. Contr. , 2017 . 19 ( 2 ): 728 - 738 . DOI: 10.1002/asjc.1425 http://doi.org/10.1002/asjc.1425 .

M.R. Morelande , A. F. García-Fernández . Analysis of Kalman filter approximations for nonlinear measurements . IEEE Trans. Signal Process. , 2013 . 61 ( 22 ): 5477 - 5484 . DOI: 10.1109/TSP.2013.2279367 http://doi.org/10.1109/TSP.2013.2279367 .

N. Morrison . Tracking Filter Engineering: the Gauss-Newton and Polynomial Filters . IET, London, UK , 2012 . DOI: 10.1049/PBRA023E http://doi.org/10.1049/PBRA023E .

K.P. Murphy . Conjugate Bayesian Analysis of the Gaussian Distribution . Technical Report, University of British Columbia, Vancouver, Canada , 2007 .

R. Nadjiasngar , M. Inggs . Gauss-Newton filtering incorporating Levenberg-Marquardt methods for tracking . Dig. Signal Process. , 2013 . 23 ( 5 ): 1662 - 1667 . DOI: 10.1016/j.dsp.2012.12.005 http://doi.org/10.1016/j.dsp.2012.12.005 .

M. Nørgaard , N.K. Poulsen , O. Ravn . New developments in state estimation for nonlinear systems . Automatica , 2000 . 36 ( 11 ): 1627 - 1638 . DOI: 10.1016/S0005-1098(00)00089-3 http://doi.org/10.1016/S0005-1098(00)00089-3 .

H. Nurminen , T. Ardeshiri , R. Piché , 等 . Robust inference for state-space models with skewed measurement noise . IEEE Signal Process. Lett. , 2015 . 22 ( 11 ): 1898 - 1902 . DOI: 10.1109/LSP.2015.2437456 http://doi.org/10.1109/LSP.2015.2437456 .

H. Nurminen , R. Piché , S. Godsill . Gaussian flow sigma point filter for nonlinear Gaussian state-space models . Proc. 20th Int. Conf. on Information Fusion , 2017 . p.1 - 8 . DOI: 10.23919/ICIF.2017.8009682 http://doi.org/10.23919/ICIF.2017.8009682 .

M. Ostendorf , V.V. Digalakis , O.A. Kimball . From HMM's to segment models: a unified view of stochastic modeling for speech recognition . IEEE Trans. Speech Audio Process. , 1996 . 4 ( 5 ): 360 - 378 . DOI: 10.1109/89.536930 http://doi.org/10.1109/89.536930 .

N. Oudjane , C. Musso . Progressive correction for regularized particle filters . 3rd Int. Conf. on Information Fusion: THB2/10 , 2000 . DOI: 10.1109/IFIC.2000.859873 http://doi.org/10.1109/IFIC.2000.859873 .

S. Park , E. Serpedin , K. Qaraqe . Gaussian assumption: the least favorable but the most useful[lecture notes] . IEEE Signal Process. Mag. , 2013 . 30 ( 3 ): 183 - 186 . DOI: 10.1109/MSP.2013.2238691 http://doi.org/10.1109/MSP.2013.2238691 .

S.C. Patwardhan , S. Narasimhan , P. Jagadeesan , 等 . Nonlinear Bayesian state estimation: a review of recent developments . Contr. Eng. Pract. , 2012 . 20 ( 10 ): 933 - 953 . DOI: 10.1016/j.conengprac.2012.04.003 http://doi.org/10.1016/j.conengprac.2012.04.003 .

D.J. Petrucci . Gaussian Mixture Reduction for Bayesian Target Tracking in Clutter . BiblioScholar, Sydney, Australia , 2005 .

R. Piché . Cramér-Rao lower bound for linear filtering with t-distributed measurement noise . 19th Int. Conf. on Information Fusion , 2016 . p.536 - 540 . .

R. Piché , S. Särkkä , J. Hartikainen . Recursive outlier-robust filtering and smoothing for nonlinear systems using the multivariate student-t distribution . IEEE Int. Workshop on Machine Learning for Signal Processing , 2012 . p.1 - 6 . DOI: 10.1109/MLSP.2012.6349794 http://doi.org/10.1109/MLSP.2012.6349794 .

L. Pishdad , F. Labeau . Analytic MMSE bounds in linear dynamic systems with Gaussian mixture noise statistics . arXiv:1505.01765 , 2015 . http://arxiv.org/abs/1506.07603 http://arxiv.org/abs/1506.07603 , .

W. Qi , P. Zhang , Z. Deng . Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances . Signal Process. , 2014 . 99 185 - 200 . DOI: 10.1016/j.sigpro.2013.12.013 http://doi.org/10.1016/j.sigpro.2013.12.013 .

M. Raitoharju , L. Svensson , á. F. García-Fernández , 等 . Damped posterior linearization filter . arXiv:1704.01113 , 2017 . http://arxiv.org/abs/1704.01113 http://arxiv.org/abs/1704.01113 , .

C.E. Rasmussen , C.K.I. Williams . Gaussian Processes for Machine Learning , : Cambridge, USA MIT Press , 2005 .

S. Reece , S. Roberts . Generalised covariance union: a unified approach to hypothesis merging in tracking . IEEE Trans. Aerosp. Electron. Syst. , 2010 . 46 ( 1 ): 207 - 221 . DOI: 10.1109/TAES.2010.5417157 http://doi.org/10.1109/TAES.2010.5417157 .

B. Ristic , X. Wang , S. Arulampalam . Target motion analysis with unknown measurement noise variance . Proc. 20th Int. Conf. on Information Fusion , 2017 . p.1663 - 1670 . DOI: 10.23919/ICIF.2017.8009853 http://doi.org/10.23919/ICIF.2017.8009853 .

A.V.I. Rosti , M.J.F. Gales . Switching linear dynamical systems for speech recognition . UK Speech Meeting , 2003 .

M. Roth , E. Özkan , F. Gustafsson . A Student's t filter for heavy tailed process and measurement noise . Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing , 2013 . p.5770 - 5774 . DOI: 10.1109/ICASSP.2013.6638770 http://doi.org/10.1109/ICASSP.2013.6638770 .

M. Roth , G. Hendeby , F. Gustafsson . Nonlinear Kalman filters explained: a tutorial on moment computations and sigma point methods . J. Adv. Inform. Fus. , 2016 . 11 ( 1 ): 47 - 70 . .

M. Roth , T. Ardeshiri , E. Özkan , 等 . Robust Bayesian filtering and smoothing using student's t distribution . arXiv:1703.02428 , 2017a . http://arxiv.org/abs/1703.02428 http://arxiv.org/abs/1703.02428 , .

M. Roth , G. Hendeby , C. Fritsche , 等 . The ensemble Kalman filter: a signal processing perspective . arXiv:1702.08061 , 2017b . http://arxiv.org/abs/1702.08061 http://arxiv.org/abs/1702.08061 , .

J. Ru , V.P. Jilkov , X.R. Li , 等 . Detection of target maneuver onset . IEEE Trans. Aerosp. Electron. Syst. , 2009 . 45 ( 2 ): 536 - 554 . DOI: 10.1109/TAES.2009.5089540 http://doi.org/10.1109/TAES.2009.5089540 .

A.R. Runnalls . Kullback-Leibler approach to Gaussian mixture reduction . IEEE Trans. Aerosp. Electron. Syst. , 2007 . 43 ( 3 ): 989 - 999 . DOI: 10.1109/TAES.2007.4383588 http://doi.org/10.1109/TAES.2007.4383588 .

D.J. Salmond . Mixture reduction algorithms for target tracking in clutter . SPIE , 1990 . 1305 434 - 445 . DOI: 10.1117/12.21610 http://doi.org/10.1117/12.21610 .

S. Särkkä , J. Hartikainen , L. Svensson , 等 . On the relation between Gaussian process quadratures and sigma-point methods . J. Adv. Inform. Fus. , 2016 . 11 ( 1 ): 31 - 46 . DOI: 10.1016/j.automatica.2014.08.030 http://doi.org/10.1016/j.automatica.2014.08.030 .

J. Sarmavuori , S. Särkkä . Fourier-Hermite Kalman filter . IEEE Trans. Autom. Contr. , 2012 . 57 ( 6 ): 1511 - 1515 . DOI: 10.1109/TAC.2011.2174667 http://doi.org/10.1109/TAC.2011.2174667 .

L.K. Saul , M.I. Jordan . Mixed memory Markov models: decomposing complex stochastic processes as mixtures of simpler ones . Mach. Learn. , 1999 . 37 ( 1 ): 75 - 87 . DOI: 10.1023/A:1007649326333 http://doi.org/10.1023/A:1007649326333 .

L.A. Scardua , J.J. da Cruz . Complete offline tuning of the unscented Kalman filter . Automatica , 2017 . 80 54 - 61 . DOI: 10.1016/j.automatica.2017.01.008 http://doi.org/10.1016/j.automatica.2017.01.008 .

D. Schieferdecker , M.F. Huber . Gaussian mixture reduction via clustering . 12th Int. Conf. on Information Fusion , 2009 . p.1536 - 1543 . .

M. Šimandl , J. Duník . Derivative-free estimation methods: new results and performance analysis . Automatica , 2009 . 45 ( 7 ): 1749 - 1757 . DOI: 10.1016/j.automatica.2009.03.008 http://doi.org/10.1016/j.automatica.2009.03.008 .

M. Šimandl , J. Královec , T. Söderström . Advanced point-mass method for nonlinear state estimation . Automatica , 2006 . 42 ( 7 ): 1133 - 1145 . DOI: 10.1016/j.automatica.2006.03.010 http://doi.org/10.1016/j.automatica.2006.03.010 .

D. Simon . Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches , : New York, USA John Wiley &Sons , 2006 .

D. Simon . Kalman filtering with state constraints: a survey of linear and nonlinear algorithms . IET Contr. Theory Appl. , 2010 . 4 ( 8 ): 1303 - 1318 . DOI: 10.1049/iet-cta.2009.0032 http://doi.org/10.1049/iet-cta.2009.0032 .

N.D. Singpurwalla , N.G. Polson , R. Soyer . From least squares to signal processing and particle filtering . Technometrics , 2017 . 2017 1 - 15 . DOI: 10.1080/00401706.2017.1341341 http://doi.org/10.1080/00401706.2017.1341341 .

L.A. Smith , M.C. Cuellar , H. Du , 等 . Exploiting dynamical coherence: a geometric approach to parameter estimation in nonlinear models . Phys. Lett. A , 2010 . 374 ( 26 ): 2618 - 2623 . DOI: 10.1016/j.physleta.2010.04.032 http://doi.org/10.1016/j.physleta.2010.04.032 .

L. Snidaro , J. García , J. Llinas . Context-based information fusion: a survey and discussion . Inform. Fus. , 2015 . 25 ( Supplement C ): 16 - 31 . DOI: 10.1016/j.inffus.2015.01.002 http://doi.org/10.1016/j.inffus.2015.01.002 .

P. Song . Monte Carlo Kalman filter and smoothing for multivariate discrete state space models . Can. J. Statist. , 2000 . 28 ( 3 ): 641 - 652 . DOI: 10.2307/3315971 http://doi.org/10.2307/3315971 .

H.W. Sorenson . Least-squares estimation: from Gauss to Kalman . IEEE Spectr. , 1970 . 7 ( 7 ): 63 - 68 . DOI: 10.1109/MSPEC.1970.5213471 http://doi.org/10.1109/MSPEC.1970.5213471 .

H. Sorenson , D. Alspach . Recursive Bayesian estimation using Gaussian sums . Automatica , 1971 . 7 ( 4 ): 465 - 479 . DOI: 10.1016/0005-1098(71)90097-5 http://doi.org/10.1016/0005-1098(71)90097-5 .

D. Sornette , K. Ide . The Kalman-Lévy filter . Phys. D , 2001 . 151 ( 2-4 ): 142174 DOI: 10.1016/S0167-2789(01)00228-7 http://doi.org/10.1016/S0167-2789(01)00228-7 .

D. Spinello , D.J. Stilwell . Nonlinear estimation with state-dependent Gaussian observation noise . IEEE Trans. Autom. Contr. , 2010 . 55 ( 6 ): 1358 - 1366 . DOI: 10.1109/TAC.2010.2042006 http://doi.org/10.1109/TAC.2010.2042006 .

P. Stano , Z. Lendek , J. Braaksma , 等 . Parametric Bayesian filters for nonlinear stochastic dynamical systems: a survey . IEEE Trans. Cybern. , 2013 . 43 ( 6 ): 1607 - 1624 . DOI: 10.1109/TSMCC.2012.2230254 http://doi.org/10.1109/TSMCC.2012.2230254 .

J. Steinbring , U.D. Hanebeck . Progressive Gaussian filtering using explicit likelihoods . 17th Int. Conf. on Information Fusion , 2014 . p.1 - 8 . .

P. Stoica , P. Babu . The Gaussian data assumption leads to the largest Cramér-Rao bound[lecture notes] . IEEE Signal Process. Mag. , 2011 . 28 ( 3 ): 132 - 133 . DOI: 10.1109/MSP.2011.940411 http://doi.org/10.1109/MSP.2011.940411 .

P. Stoica , R.L. Moses . On biased estimators and the unbiased Cramér-Rao lower bound . Signal Process. , 1990 . 21 ( 4 ): 349350 DOI: 10.1016/0165-1684(90)90104-7 http://doi.org/10.1016/0165-1684(90)90104-7 .

O. Straka , J. Duník , M. Šimandl . Truncation nonlinear filters for state estimation with nonlinear inequality constraints . Automatica , 2012 . 48 ( 2 ): 273 - 286 . DOI: 10.1016/j.automatica.2011.11.002 http://doi.org/10.1016/j.automatica.2011.11.002 .

O. Straka , J. Duník , M. Šimandl . Unscented Kalman filter with advanced adaptation of scaling parameter . Automatica , 2014 . 50 ( 10 ): 2657 - 2664 . DOI: 10.1016/j.automatica.2014.08.030 http://doi.org/10.1016/j.automatica.2014.08.030 .

J. Su , W.H. Chen . Model-based fault diagnosis system verification using reachability analysis . IEEE Trans. Syst. Man Cybern. Syst. , 2017 . 99 1 - 10 . DOI: 10.1109/TSMC.2017.2710132 http://doi.org/10.1109/TSMC.2017.2710132 .

J. Su , B. Li , W.H. Chen . On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs . Automatica , 2015a . 53 149 - 154 . DOI: 10.1016/j.automatica.2014.12.044 http://doi.org/10.1016/j.automatica.2014.12.044 .

J. Su , B. Li , W.H. Chen . Simultaneous state and input estimation with partial information on the inputs . Syst. Sci. Contr. Eng. , 2015b . 3 ( 1 ): 445 - 452 . DOI: 10.1080/21642583.2015.1082512 http://doi.org/10.1080/21642583.2015.1082512 .

J. Su , W.H. Chen , J. Yang . On relationship between time-domain and frequency-domain disturbance observers and its applications . J. Dyn. Syst. Meas. Contr. , 2016 . 138 ( 9 ): 091013 DOI: 10.1115/1.4033631 http://doi.org/10.1115/1.4033631 .

A. Svensson , T. B. Schön , F. Lindsten . Learning of state-space models with highly informative observations: a tempered sequential Monte Carlo solution . arXiv:1702.01618 , 2017 . http://arxiv.org/abs/1702.01618 http://arxiv.org/abs/1702.01618 , .

M. Tahk , J.L. Speyer . Target tracking problems subject to kinematic constraints . IEEE Trans. Autom. Contr. , 1990 . 35 ( 3 ): 324 - 326 . DOI: 10.1109/9.50348 http://doi.org/10.1109/9.50348 .

B.O. Teixeira , L. A. Tôrres , L.A. Aguirre , 等 . On unscented Kalman filtering with state interval constraints . J. Process Contr. , 2010 . 20 ( 1 ): 45 - 57 . DOI: 10.1016/j.jprocont.2009.10.007 http://doi.org/10.1016/j.jprocont.2009.10.007 .

G. Terejanu , P. Singla , T. Singh , 等 . Adaptive Gaussian sum filter for nonlinear Bayesian estimation . IEEE Trans. Autom. Contr. , 2011 . 56 ( 9 ): 2151 - 2156 . DOI: 10.1109/TAC.2011.2141550 http://doi.org/10.1109/TAC.2011.2141550 .

P. Tichavsky , C.H. Muravchik , A. Nehorai . Posterior Cramér-Rao bounds for discrete-time nonlinear filtering . IEEE Trans. Signal Process. , 1998 . 46 ( 5 ): 1386 - 1396 . DOI: 10.1109/78.668800 http://doi.org/10.1109/78.668800 .

M.E. Tipping , N.D. Lawrence . Variational inference for Student-t models: robust Bayesian interpolation and generalised component analysis . Neurocomputing , 2005 . 69 ( 1-3 ): 123 - 141 . DOI: 10.1016/j.neucom.2005.02.016 http://doi.org/10.1016/j.neucom.2005.02.016 .

R. van der Merwe , A. Doucet , N. de Freitas , 等 . The unscented particle filter . Proc. NIPS , 2000 . p.563 - 569 . .

H.L. van Trees . Detection, Estimation and Modulation Theory , : New York, USA Wiley , 1968 .

H.L. van Trees , K.L. Bell . Bayesian bounds for parameter estimation and nonlinear filtering/tracking . IET Radar Sonar Navig. , 2007 . 3 ( 3 ): 285 - 286 . DOI: 10.1049/iet-rsn:20099030 http://doi.org/10.1049/iet-rsn:20099030 .

B.N. Vo , W.K. Ma . The Gaussian mixture probability hypothesis density filter . IEEE Trans. Signal Process. , 2006 . 54 ( 11 ): 4091 - 4104 . DOI: 10.1109/TSP.2006.881190 http://doi.org/10.1109/TSP.2006.881190 .

J.M. Wang , D.J. Fleet , A. Hertzmann . Gaussian process dynamical models for human motion . IEEE Trans. Patt. Anal. Mach. Intell. , 2008 . 30 ( 2 ): 283 - 298 . DOI: 10.1109/TPAMI.2007.1167 http://doi.org/10.1109/TPAMI.2007.1167 .

X. Wang , M. Fu , H. Zhang . Target tracking in wireless sensor networks based on the combination of KF and MLE using distance measurements . IEEE Trans. Mob. Comput. , 2012 . 11 ( 4 ): 567 - 576 . DOI: 10.1109/TMC.2011.59 http://doi.org/10.1109/TMC.2011.59 .

X. Wang , Y. Liang , Q. Pan , 等 . Design and implementation of Gaussian filter for nonlinear system with randomly delayed measurements and correlated noises . Appl. Math. Comput. , 2014 . 232 1011 - 1024 . DOI: 10.1016/j.amc.2013.12.168 http://doi.org/10.1016/j.amc.2013.12.168 .

X. Wang , Y. Liang , Q. Pan , 等 . Nonlinear Gaussian smoothers with colored measurement noise . IEEE Trans. Autom. Contr. , 2015 . 60 ( 3 ): 870 - 876 . DOI: 10.1109/TAC.2014.2337991 http://doi.org/10.1109/TAC.2014.2337991 .

X. Wang , B. Song , Y. Liang , 等 . EM-based adaptive divided difference filter for nonlinear system with multiplicative parameter . Int. J. Robust Nonl. Contr. , 2017 . 27 ( 13 ): 2167 - 2197 . DOI: 10.1002/rnc.3674 http://doi.org/10.1002/rnc.3674 .

W. Wen , H.F. Durrant-Whyte . Model-based multisensor data fusion . Proc. IEEE Int. Conf. on Robotics and Automation , 1992 . p.1720 - 1726 . DOI: 10.1109/ROBOT.1992.220130 http://doi.org/10.1109/ROBOT.1992.220130 .

J.L. Williams , P.S. Maybeck . Cost-function-based hypothesis control techniques for multiple hypothesis tracking . Math. Comput. Model. , 2006 . 43 ( 9-10 ): 976 - 989 . DOI: 10.1016/j.mcm.2005.05.022 http://doi.org/10.1016/j.mcm.2005.05.022 .

Y. Wu , D. Hu , M. Wu , 等 . A numerical-integration perspective on Gaussian filters . IEEE Trans. Signal Process. , 2006 . 54 ( 8 ): 2910 - 2921 . DOI: 10.1109/TSP.2006.875389 http://doi.org/10.1109/TSP.2006.875389 .

Z. Wu , J. Shi , X. Zhang , 等 . Kernel recursive maximum correntropy . Signal Process. , 2015 . 117 11 - 16 . DOI: 10.1016/j.sigpro.2015.04.024 http://doi.org/10.1016/j.sigpro.2015.04.024 .

L. Xu , X.R. Li , Z. Duan , 等 . Modeling and state estimation for dynamic systems with linear equality constraints . IEEE Trans. Signal Process. , 2013 . 61 ( 11 ): 2927 - 2939 . DOI: 10.1109/TSP.2013.2255045 http://doi.org/10.1109/TSP.2013.2255045 .

L. Xu , X.R. Li , Z. Duan . Hybrid grid multiple-model estimation with application to maneuvering target tracking . IEEE Trans. Aerosp. Electron. Syst. , 2016 . 52 ( 1 ): 122 - 136 . DOI: 10.1109/TAES.2015.140423 http://doi.org/10.1109/TAES.2015.140423 .

C. Yang , E. Blasch . Kalman filtering with nonlinear state constraints . IEEE Trans. Aerosp. Electron. Syst. , 2009 . 45 ( 1 ): 70 - 84 . DOI: 10.1109/TAES.2009.4805264 http://doi.org/10.1109/TAES.2009.4805264 .

T. Yang , R.S. Laugesen , P.G. Mehta , 等 . Multivariable feedback particle filter . Automatica , 2016 . 71 10 - 23 . DOI: 10.1016/j.automatica.2016.04.019 http://doi.org/10.1016/j.automatica.2016.04.019 .

Y. Yang , H. He , G. Xu . Adaptively robust filtering for kinematic geodetic positioning . J. Geod. , 2001 . 75 ( 2-3 ): 109 - 116 . DOI: 10.1007/s001900000157 http://doi.org/10.1007/s001900000157 .

D. Yi , J. Su , C. Liu , 等 . Data-driven situation awareness algorithm for vehicle lane change . 19th IEEE Int. Conf. on Intelligent Transportation Systems , 2016 . p.998 - 1003 . DOI: 10.1109/ITSC.2016.7795677 http://doi.org/10.1109/ITSC.2016.7795677 .

S.Z. Yong , M. Zhu , E. Frazzoli . A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems . Automatica , 2016 . 63 321 - 329 . DOI: 10.1016/j.automatica.2015.10.040 http://doi.org/10.1016/j.automatica.2015.10.040 .

R. Zanetti . Recursive update filtering for nonlinear estimation . IEEE Trans. Autom. Contr. , 2012 . 57 ( 6 ): 1481 - 1490 . DOI: 10.1109/TAC.2011.2178334 http://doi.org/10.1109/TAC.2011.2178334 .

H. Zen , K. Tokuda , T. Kitamura . Reformulating the HMM as a trajectory model by imposing explicit relationships between static and dynamic feature vector sequences . Comput. Speech Lang. , 2007 . 21 ( 1 ): 153 - 173 . DOI: 10.1016/J.CSL.2006.01.002 http://doi.org/10.1016/J.CSL.2006.01.002 .

R. Zhan , J. Wan . Iterated unscented Kalman filter for passive target tracking . IEEE Trans. Aerosp. Electron. Syst. , 2007 . 43 ( 3 ): 1155 - 1163 . DOI: 10.1109/TAES.2007.4383605 http://doi.org/10.1109/TAES.2007.4383605 .

C. Zhang , R. Zhi , T. Li , 等 . Adaptive Mestimation for robust cubature Kalman filtering . Sensor Signal Processing for Defence , 2016 . p.114 - 118 . DOI: 10.1109/SSPD.2016.7590586 http://doi.org/10.1109/SSPD.2016.7590586 .

Y. Zhang , Y. Huang , N. Li , 等 . Embedded cubature Kalman filter with adaptive setting of free parameter . Signal Process. , 2015 . 114 112 - 116 . DOI: 10.1016/j.sigpro.2015.02.022 http://doi.org/10.1016/j.sigpro.2015.02.022 .

S. Zhao , Y.S. Shmaliy , F. Liu . Fast Kalman-like optimal unbiased FIR filtering with applications . IEEE Trans. Signal Process. , 2016a . 64 ( 9 ): 2284 - 2297 . DOI: 10.1109/TSP.2016.2516960 http://doi.org/10.1109/TSP.2016.2516960 .

S. Zhao , Y.S. Shmaliy , F. Liu , 等 . Unbiased, optimal, and in-betweens: the trade-off in discrete finite impulse response filtering . IET Signal Process. , 2016b . 10 ( 4 ): 325 - 334 . DOI: 10.1049/iet-spr.2015.0360 http://doi.org/10.1049/iet-spr.2015.0360 .

Y. Zheng , O. Ozdemir , R. Niu , 等 . New conditional posterior Cramér-Rao lower bounds for nonlinear sequential Bayesian estimation . IEEE Trans. Signal Process. , 2012 . 60 ( 10 ): 5549 - 5556 . DOI: 10.1109/TSP.2012.2205686 http://doi.org/10.1109/TSP.2012.2205686 .

D.H. Zhou , P.M. Frank . Strong tracking filtering of nonlinear time-varying stochastic systems with coloured noise: application to parameter estimation and empirical robustness analysis . Int. J. Contr. , 1996 . 65 ( 2 ): 295 - 307 . DOI: 10.1080/00207179608921698 http://doi.org/10.1080/00207179608921698 .

L. Zuo , R. Niu , P.K. Varshney . Conditional posterior Cramér-Rao lower bounds for nonlinear sequential Bayesian estimation . IEEE Trans. Signal Process. , 2011 . 59 ( 1 ): 1 - 14 . DOI: 10.1109/TSP.2010.2080268 http://doi.org/10.1109/TSP.2010.2080268 .

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

Recursive filtering of multi-rate cyber-physical systems with unknown inputs under adaptive event-triggered mechanisms

A hybrid visual servo control method for simultaneously controlling a nonholonomic mobile and a manipulator

Related Author

Ying SUN

Miaomiao FU

Jingyang MAO

Guoliang WEI

Wei LI

Rong XIONG

Related Institution

Business School, University of Shanghai for Science and Technology

Shanghai Caoyang Vocational School

School of Electrical and Electronic Engineering, Shanghai Institute of Technology

School of Intelligent Emergency Management, University of Shanghai for Science and Technology

State Key Laboratory of Industrial Control Technology, Zhejiang University

Chat

Address：Zhejiang University Press, 148 Tianmushan Road, Hangzhou, China Postal code：310028
Tel：+86-571-88273162 Email：fitee@zju.edu.cn
It is recommended to read the content of this site in Chrome&IE9+. Please switch to extreme mode in browser 360.
Cookies We use cookies to help provide and enhance our service and tailor content. By continuing, you agree to the use of cookies.

⁰