Contribution in Information Signal Processing for Solving State Space Nonlinear Estimation Problems

Hamza Benzerrouk, Alexander Nebylov, Hassen Salhi
DOI: 10.4236/jsip.2013.44048                

Abstract

In this paper, comprehensive methods to apply several formulations of nonlinear estimators to integrated navigation problems are considered and developed. The problem of linear and nonlinear filters such as Kalman Filter (KF) and Extended Kalman Filter (EKF) is stated. Analog solution which is based on fisher information matrix propagation for linear and nonlinear filtering is also developed. Additionally, the idea of iterations is included through the update step both for Kalman filters and Information filters in order to improve accuracy. Through this development, two new formulations of High order Kalman filters and High order Information filters are presented. Finally, in order to compare these different nonlinear filters, special applications are analyzed by using the proposed techniques to estimate two well-known mathematical state space models, which are based on nonlinear time series used to apply these estimation algorithms. A criterion used for comparison is the root mean square error RMSE and several simulations under specific conditions are illustrated.

Keywords

Kalman Filter; Information Filter; Extended Kalman Filter; Extended Information Filter; 2nd Order Kalman Filter; 2nd Order Information Filter

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. E. Kalman and R. S. Bucy “A New Approach to Linear Filtering and Prediction Problems,” Journal of Basic Engineering, Vol. 82, No. 1, 1960, pp. 35-45. http://dx.doi.org/10.1115/1.3662552
[2]	R. E. Kalman and R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Journal of Basic Engineering, Vol. 83, No. 1, 1961, pp. 95-108. http://dx.doi.org/10.1115/1.3658902
[3]	J. Kim, “Autonomous Navigation for Airborne Applications,” Department of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, 2004.
[4]	A. V. Nebylov, “Ensuring Control Accuracy,” Springer Verlag, Heidelberg, 2004. 244 p.
[5]	T. Lefebvre, et al. “Kalman Filters for Nonlinear Systems,” Nonlinear Kalman Filtering, Vol. 19, 2005, pp. 51-76.
[6]	S. Thrun, D. Koller, Z. Ghahramani, H. Durrant-Whyte and Y. Ng Andrew, “Simultaneous Mapping and Localization with Sparse Extended Information Filters: Theory and Initial Results,” University of Sydney, Sydney, 2002.
[7]	Y. Liu and S. Thrun, “Results for Outdoor-SLAM Using Sparse Extended Information Filters,” Proceedings of ICRA, 2003.
[8]	M. Simandl, “Lectures Notes on State Estimation of Non-Linear Non-Gaussian Stochastic Systems,” Department of Cybernetics, Faculty of Applied Sciences, University of West Bohemia, Pilsen, 2006.
[9]	A. Mutambara, “Decentralised Estimation and Control for Multisensor Systems,” CRC Press, LLC, Boca Raton, 1998.
[10]	J. Manyika and H. Durrant-Whyte, “Data Fusion and Sensor Management: A Decentralized Information-Theoretic Approach,” Prentice Hall, Upper Saddle River, 1994.
[11]	A. Gasparri, F. Pascucci and G. Ulivi, “A Distributed Extended Information Filter for Self-Localization in Sensor Networks,” Personal, Indoor and Mobile Radio Communications, 2008.
[12]	M. Walter, F. Hover and J. Leonard, “SLAM for Ship Hull Inspection Using Exactly Sparse Extended Information Filters,” Massachusetts Institute of Technology, 2008.
[13]	G. Borisov, A. S. Ermilov, T. V. Ermilova and V. M. Sukhanov, “Control of the Angular Motion of a Semiactive Bundle of Bodies Relying on the Estimates of Nonmeasurable Coordinated Obtained by Kalman Filtration Methods,” Institute of Control Sciences, Russian Academy of Sciences, Moscow, 2004.
[14]	N. V. Medvedeva and G. A. Timofeeva, “Comparison of Linear and Nonlinear Methods of Confidence Estimation for Statistically Uncertain Systems,” Ural State Academy of Railway Transport, Yekaterinburg, 2006.
[15]	H. Benzerrouk and A. Nebylov, “Robust Integated Navigation System Based on Joint Application of Linear and Non-Linear Filters,” IEEE Aerospace Conference, Big Sky, 2011.
[16]	H. Benzerrouk and A. Nebylov, “Experimental Navigation System Based on Robust Adaptive Linear and Non-Linear Filters,” 19th International Integrated Navigation System Conference, Elektropribor-Saint Petersburg, 2011.
[17]	H. Benzerrouk and A. Nebylov, “Robust Non-Linear Filtering Applied to Integrated Navigation System INS/ GNSS under Non-Gaussian Noise Effect, Embedded Guidance, Navigation and Control in Aerospace (EGNCA),” 2012.
[18]	T. Vercauteren and X. Wang, “Decentralized Sigma-Point Information Filters for Target Tracking in Collaborative Sensor Networks,” IEEE Transactions on Signal Processing, 2005.
[19]	G. J. Bierman, “Square-Root Information Filtering and Smoothing for Precision Orbit Determination,” Factorized Estimation Applications, Inc., Canoga Park, 1980.
[20]	M. V. Kulikova and I. V. Semoushin, “Score Evaluation within the Extended Square-Root Information Filter,” In: V. N. Alexandrov, et al., Eds., Springer-Verlag, Berlin, 2006, pp. 473-481.
[21]	G. J. Bierman, “The Treatment of Bias in the Square-Root Information Filter/Smoother,” Journal of Optimization Theory and Applications, Vol. 16, No. 1-2, 1975, pp. 165-178.
[22]	C. Lanquillon, “Evaluating Performance Indicators for Adaptive Information Filtering,” Daimler Chrysler Research and Technology, Germany.
[23]	V. Yu. Tertychnyi-Dauri, “Adaptive Optimal Nonlinear Filtering and Some Adjacent Questions,” State Institute of Fine Mechanics and Optics, St. Petersburg, 2000.
[24]	O. M. Kurkin, “Guaranteed Estimation Algorithms for Prediction and Interpolation of Random Processes,” Scientic Research Institute of Radio Engineering, Moscow, 1999.
[25]	A. G. Chentsov, “Construction of Limiting Process Operations Using Ultrafilters of Measurable Spaces,” Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 2006.
[26]	A. V. Borisov, “Backward Representation of Markov Jump Processes and Related Problems. II. Optimal Non-linear Estimation,” Institute of Informatics Problems, Russian Academy of Sciences, Moscow, 2006.
[27]	A. A. Pervozvansky, “Learning Control and Its Applications. Part 1: Elements of General Theory,” Avtomatika i Telemekhanika, No.11, 1995.
[28]	A. A. Pervozvansky, “Learning Control and Its Applications. Part 2: Frobenious Systems and Learning Control for Robot Manipulators,” Avtomatika i Telemekhanika, No.12, 1995.
[29]	V. I. Kulakova and A. V. Nebylov, “Guaranteed Estimation of Signals with Bounded Variances of Derivatives,” Automation and Remote Control, Vol. 69, No. 1, 2008, pp. 76-88.
[30]	Maybeck. “Stochastic Models, Estimations and Control,” Vol. 1-2, Academic Press, 1982.