Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems
Abstract
Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.
References
[1] , “A New Approach to Linear Filtering and Prediction Problems,” Transactions of the ASME–Journal of Basic Engineering, Vol. 82, Series D, March 1960, pp. 35–45. 0021-9223
[2] , “Application of Statistical Filter Theory to the Optimal Estimation of Position and Velocity on Board a Circumlunar Vehicle,” NASA TR-R-135, Jan. 1962.
[3] , “Suboptimal State Estimation for Continuous-Time Nonlinear Systems from Discrete Noisy Measurements,” IEEE Transactions on Automatic Control, Vol. 13, No. 5, Oct. 1968, pp. 504–514. doi:https://doi.org/10.1109/TAC.1968.1098986 IETAA9 0018-9286
[4] , “Navigation Algorithms and Observability Analysis for Formation Flying Missions,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 4, 2009, pp. 1218–1231. doi:https://doi.org/10.2514/1.41288 JGCDDT 0162-3192
[5] , “Underweighting Nonlinear Measurements,” Journal of Guidance, Control, and Dynamics, Vol. 33, No. 5, 2010, pp. 1670–1675. doi:https://doi.org/10.2514/1.50596 JGCDDT 0162-3192
[6] , “Detection of Non-Linearity Effects During Orbit Estimation,” 20th AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 10-239, Feb. 2010.
[7] , “A New Approach for Filtering Nonlinear Systems,” Proceedings of the American Control Conference, Vol. 3, June 1995, pp. 1628–1632. doi:https://doi.org/10.1109/ACC.1995.529783
[8] , “Unscented Filtering and Nonlinear Estimation,” Proceedings of the IEEE, Vol. 92, No. 3, 2004, pp. 401–422. doi:https://doi.org/10.1109/JPROC.2003.823141 IEEPAD 0018-9219
[9] , “Gaussian Filters for Nonlinear Filtering Problems,” IEEE Transactions on Automatic Control, Vol. 45, No. 5, May 2000, pp. 910–927. doi:https://doi.org/10.1109/9.855552 IETAA9 0018-9286
[10] , Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State-Space Models, Ph.D. Thesis, Oregon Health and Science Univ., Portland, Oregon, 2004.
[11] , Simultaneous Map Building and Localization for Real Time Applications, Ph.D. Thesis, Univ. of Oxford, Oxford, England, U.K., 1994.
[12] , “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
[13] , “Nonlinear Bayesian Estimation Using Gaussian Sum Approximations,” IEEE Transactions in Automatic Control, Vol. AC-17, No. 4, Aug. 1972, pp. 439–448. doi:https://doi.org/10.1109/TAC.1972.1100034 0018-9286
[14] , “Gaussian Sum Filters for Space Surveillance: Theory and Simulations,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, 2011, pp. 1839–1851. doi:https://doi.org/10.2514/1.53793 JGCDDT 0162-3192
[15] , “Uncertainty Propagation for Nonlinear Dynamic Systems Using Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 6, 2008, pp. 1623–1633. doi:https://doi.org/10.2514/1.36247 JGCDDT 0162-3192
[16] , “Orbit Determination Performance Improvements for High Area-to-Mass Ratio Space Object Tracking Using an Adaptive Gaussian Mixtures Estimation Algorithm,” 21st International Symposium on Space Flight Dynamics,
Toulouse, France , Sept. 2009.[17] , “An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009-6082, Aug. 2009.
[18] , Stochastic Processes and Filtering Theory, Dover, New York, 1998, pp. 126–139.
[19] , “A Mathematical Theory of Communication,” The Bell System Technical Journal, Vol. 27, July–Oct. 1948, pp. 379–423.
[20] , Elements of Information Theory, Wiley, New York, 1991, pp. 224–225.
[21] , “On Measures of Entropy and Information,” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, edited by Neyman J., Vol. 1 of
Contributions to the Theory of Statistics , Univ. of California Press, Berkeley, CA, June–July 1961, pp. 547–561.[22] , “Expressions for Rényi and Shannon Entropies for Multivariate Distributions,” Statistics & Probability Letters, Vol. 71, No. 1, Jan. 2005, pp. 71–84. doi:https://doi.org/10.1016/j.spl.2004.10.023
[23] , Kalman Filtering: Theory and Practice, Prentice–Hall, Upper Saddle River, NJ, 1993, pp. 165–171.
[24] , “Information Entropy and State Observation of a Dynamical System,” Uncertainty in Knowledge-Based Systems, edited by Bouchon B. and Yager R., Vol. 286 of
Lecture Notes in Computer Science , Springer–Verlag, Berlin, 1987, pp. 403–405.[25] , “Robust and Efficient Estimation by Minimising a Density Power Divergence,” Biometrika, Vol. 85, No. 3, 1998, pp. 549–559. doi:https://doi.org/10.1093/biomet/85.3.549 BIOKAX 0006-3444
[26] , “Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities,” Entropy, Vol. 12, No. 6, 2010, pp. 1532–1568. doi:https://doi.org/10.3390/e12061532 ENTRFG 1099-4300
[27] , “On Entropy Approximation for Gaussian Mixture Random Vectors,” IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems, Paper TA2-3, 2008, pp. 181–188.
[28] , “Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 2, 2011, pp. 367–379. doi:https://doi.org/10.2514/1.52016 JGCDDT 0162-3192
[29] , Nonlinear Orbit Uncertainty Prediction and Rectification for Space Situational Awareness, Ph.D. Thesis, Univ. of Texas at Austin, Austin, TX, 2010.
