Skip to main content
Search
Search
Quick Search anywhere
Quick Search fdjslkfh
Advanced search
Cart
Join AIAA Institution Login
Skip main navigation
Journal of Guidance, Control, and Dynamics logo
Skip to article control options
No AccessFull-Length Paper

Approximate Estimators for Linear Systems with Additive Cauchy Noises

Published Online:4 Jul 2017https://doi.org/10.2514/1.G002775

Abstract

The recently published optimal Cauchy estimator poses practical implementation challenges due to its time-growing complexity. Alternatively, addressing impulsive measurement and process noises, while using common estimation approaches, requires heuristic schemes. Approximate methods, such as particle and Gaussian-sum filters, were suggested to tackle the estimation problem in a heavy-tailed-noise environment when constraining the computational load. In this paper, the performances of a particle filter and a Gaussian-sum filter, designed for a linear system with specified Cauchy-noise parameters, are compared numerically to a Cauchy filter-based approximation showing the advantages of the latter.

References

  • [1] Taleb N. N., The Black Swan: The Impact of the Highly Improbable, Random House, New York, 2007. Google Scholar

  • [2] Carpenter J. R. and Mashiku A. K., “Cauchy Drag Estimation for Low Earth Orbiters,” AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, 2015, pp. 2731–2746. Google Scholar

  • [3] Speyer J. L. and Chung W. H., Stochastic Processes, Estimation, and Control, Soc. for Industrial and Applied Mathematics, Philadelphia, PA, 2008, pp. 22–145. CrossrefGoogle Scholar

  • [4] Kalman R. E., “A New Approach to Linear Filtering and Prediction Problems,” Journal of Basic Engineering, Vol. 82, No. 1, 1960, pp. 35–45. doi:https://doi.org/10.1115/1.3662552 JBAEAI 0021-9223 CrossrefGoogle Scholar

  • [5] Schick I. C. and Mitter S. K., “Robust Recursive Estimation in the Presence of Heavy-Tailed Observation Noise,” Annals of Statistics, Vol. 22, No. 2, 1994, pp. 1045–1080. doi:https://doi.org/10.1214/aos/1176325511 CrossrefGoogle Scholar

  • [6] Idan M. and Speyer J. L., “Cauchy Estimation for Linear Scalar Systems,” IEEE Transactions on Automatic Control, Vol. 55, No. 6, 2010, pp. 1329–1342. doi:https://doi.org/10.1109/TAC.2010.2042009 IETAA9 0018-9286 CrossrefGoogle Scholar

  • [7] Idan M. and Speyer J. L., “State Estimation for Linear Scalar Dynamic Systems with Additive Cauchy Noises: Characteristic Function Approach,” SIAM Journal on Control and Optimization, Vol. 50, No. 4, 2012, pp. 1971–1994. doi:https://doi.org/10.1137/110831362 SJCODC 0363-0129 CrossrefGoogle Scholar

  • [8] Idan M. and Speyer J. L., “Multivariate Cauchy Estimator with Scalar Measurement and Process Noises,” SIAM Journal on Control and Optimization, Vol. 52, No. 2, 2014, pp. 1108–1141. doi:https://doi.org/10.1137/120891897 SJCODC 0363-0129 CrossrefGoogle Scholar

  • [9] Fernandez J. H., “Methods for Estimation and Control of Linear Systems Driven by Cauchy Noises,” Ph.D. Thesis, Univ. of California, Los Angeles, 2013. Google Scholar

  • [10] Fernandez J. H., Speyer J. L. and Idan M., “Stochastic Estimation for Two-State Linear Dynamic Systems with Additive Cauchy Noises,” IEEE Transactions on Automatic Control, Vol. 60, No. 12, 2015, pp. 3367–3372. doi:https://doi.org/10.1109/TAC.2015.2422478 IETAA9 0018-9286 CrossrefGoogle Scholar

  • [11] Gordon N. J., Salmond D. J. and Smith A. F. M., “Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation,” IEE Proceedings-F: Radar and Signal Processing, Vol. 140, No. 2, 1993, pp. 107–113. doi:https://doi.org/10.1049/ip-f-2.1993.0015 CrossrefGoogle Scholar

  • [12] Doucet A., Godsill S. and Andrieu C., “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering,” Statistics and Computing, Vol. 10, No. 3, 2000, pp. 197–208. doi:https://doi.org/10.1023/A:1008935410038 STACE3 0960-3174 CrossrefGoogle Scholar

  • [13] Sorenson H. W. and Alspach D. L., “Recursive Bayesian Estimation Using Gaussian Sums,” Automatica, Vol. 7, No. 4, 1971, pp. 465–479. doi:https://doi.org/10.1016/0005-1098(71)90097-5 ATCAA9 0005-1098 CrossrefGoogle Scholar

  • [14] Alspach D. L. and Sorenson H. W., “Nonlinear Bayesian Estimation Using Gaussian Sum Approximations,” IEEE Transactions on Automatic Control, Vol. 17, No. 4, 1972, pp. 439–448. doi:https://doi.org/10.1109/TAC.1972.1100034 IETAA9 0018-9286 CrossrefGoogle Scholar

  • [15] Fonod R., Idan M. and Speyer J. L., “State Estimation for Linear Systems with Additive Cauchy Noises: Optimal and Suboptimal Approaches,” Proceedings of European Control Conference, IEEE Publ., Piscataway, NJ, 2016, pp. 1434–1439. doi:https://doi.org/10.1109/ECC.2016.7810491 Google Scholar

  • [16] Doucet A., de Freitas N. and Gordon N., Sequential Monte-Carlo Methods in Practice, Springer–Verlag, New York, 2001. CrossrefGoogle Scholar

  • [17] Arulampalam M. S., Maskell S., Gordon N. and Clapp T., “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” IEEE Transactions on Signal Processing, Vol. 50, No. 2, 2002, pp. 174–188. doi:https://doi.org/10.1109/78.978374 ITPRED 1053-587X CrossrefGoogle Scholar

  • [18] Li T., Bolic M. and Djuric P. M., “Resampling Methods for Particle Filtering: Classification, Implementation, and Strategies,” IEEE Signal Processing Magazine, Vol. 32, No. 3, 2015, pp. 70–86. doi:https://doi.org/10.1109/MSP.2014.2330626 ISPRE6 1053-5888 CrossrefGoogle Scholar

  • [19] Kitagawa G., “Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models,” Journal of Computational and Graphical Statistics, Vol. 5, No. 1, 1996, pp. 1–25. doi:https://doi.org/10.2307/1390750 1061-8600 Google Scholar

  • [20] Achieser N. I., Theory of Approximation, Dover Publ., New York, 1992, Chap. 6. Google Scholar

  • [21] Psiaki M. L., Schoenberg J. R. and Miller I. T., “Gaussian Sum Reapproximation for Use in a Nonlinear Filter,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 2, 2015, pp. 292–303. doi:https://doi.org/10.2514/1.G000541 JGCODS 0731-5090 LinkGoogle Scholar

  • [22] Kitagawa G., “The Two-Filter Formula for Smoothing and an Implementation of the Gaussian-Sum Smoother,” Annals of the Institute of Statistical Mathematics, Vol. 46, No. 4, 1994, pp. 605–623. doi:https://doi.org/10.1007/BF00773470 AISXAD 0020-3157 CrossrefGoogle Scholar

  • [23] Kitagawa G., “Non-Gaussian Seasonal Adjustment,” Computers & Mathematics with Applications, Vol. 18, No. 6, 1989, pp. 503–514. doi:https://doi.org/10.1016/0898-1221(89)90103-X CrossrefGoogle Scholar

  • [24] Williams J. L. and Maybeck P. S., “Cost-Function-Based Gaussian Mixture Reduction for Target Tracking,” Proceedings of the Sixth International Conference of Information Fusion, Vol. 2, IEEE Publ., Piscataway, NJ, 2003, pp. 1047–1054. doi:https://doi.org/10.1109/ICIF.2003.177354 Google Scholar

  • [25] Venkataraman P., Applied Optimization with MATLAB Programming, Wiley, New York, 2002, pp. 265–316. Google Scholar

Previous article
Next article