Nonlinear Projection Filter Based on Galerkin Approximation

Journal of Guidance, Control, and Dynamics

Volumn 22, Issue 2, 1999, Pages 258-266

  Beard, Randal ^a   Kenney, John ^a   Gunther, Jacob ^a,b   Lawton, Jonathan ^a   Stirling, Wynn ^a  

^a BRIGHAM YOUNG UNIVERSITY   (United States)

^b Merisoft   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; COSINE TRANSFORMS; GALERKIN METHODS; PROBABILITY DENSITY FUNCTION; STATE ESTIMATION; STOCHASTIC CONTROL SYSTEMS;

KOLMOGOROV'S FORWARD EQUATION; NONLINEAR PROJECTION FILTERS;

NONLINEAR FILTERING;

EID: 0033098658     PISSN: 07315090     EISSN: 15333884     Source Type: Journal    
DOI: 10.2514/2.4403     Document Type: Article

References (33)

1. Kalman, R. E., "A New Approach to Linear Filtering and Prediction Problems," Transactions of the ASME, Journal of Basic Engineering, Vol. 82, No. 1, 1960, pp. 34, 35.
2. Kalman R. E., and Bucy, R. S., "New Results in Linear Filtering and Prediction Theory," Transaction of the ASME, Journal of Basic Engineering, Vol. 83, 1961, 95-108.
3. Jazwinski, A. H., Stochastic Processes and Filtering Theory, Vol. 64, Mathematics in Science and Engineering, Academic Press, New York, 1970.
4. Kushner, H. J., "Dynamical Equations for Optimal Nonlinear Filter," Journal of Differential Equations, Vol. 3, No. 2, 1967, pp. 179-190.
5. Kushner, H. J., "Nonlinear Filtering: The Exact Dynamical Equations Satisfied by the Conditional Mode," IEEE Transactions on Automatic Control, Vol. 12, 1967, pp. 262-267.
6. Stratonovich, R. L., "Conditional Markov Processes," Theory of Probability and Its Applications, Vol. 5, 1960, pp. 156-178.
7. Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Hemisphere, New York, 1975.
8. Lewis, F. L., Optimal Estimation: With an Introduction to Stochastic Control Theory, Wiley, New York, 1986.
9. Ohap, R. F., and Stubberud, A. R., "A Technique for Estimating the State of a Nonlinear System," IEEE Transactions on Automatic Control, Vol. 10, No. 4, 1965, pp. 150-155.
10. Gelb, A. (ed.), Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974.
11. Ljung, L., "Asymptotic Behavior of the Extended Kalman Filter Using Filtered State Estimates," IEEE Transactions on Automatic Control, Vol. 24, No. 1, 1979, pp. 36-50.
12. Ursin, B., "Asymptotic Convergence Properties of the Extended Kalman Filter Using Filtered State Estimates," IEEE Transactions on Automatic Control, Vol. 25, No. 12, 1980, pp. 1207-1211.
13. Galkowski, P. J., and Islam, M. A., "An Alternative Derivation of the Modified Gain Function of Song and Speyer," IEEE Transactions on Automatic Control, Vol. 36, No. 11, 1991, pp. 1323-1326.
14. Bell, B. M., and Cathey, F. W., "The Iterated Kalman Filter Update as a Gauss-Newton Method," IEEE Transactions on Automatic Control, Vol. 38, No. 2, 1993, pp. 294-297.
15. Bell, B. M., "The Iterated Kalman Smoother as a Gauss-Newton Method," SIAM Journal of Optimization, Vol. 4, No. 8, 1994, pp. 626-636.
16. Mikhlin, S. G., Variational Methods in Mathematical Physics, MacMillan, New York, 1964.
17. Kantorovich, L. V., and Krylov, V. I., Approximate Methods of Higher Analysis, Interscience Publishers, New York, 1958.
18. Risken, H., The Folker-Planck Equation: Methods of Solution and Application, Vol. 18, Springer Series in Synergetics, Springer-Verlag, Berlin, 1989.
19. Ling, F. H., and Wu, X. X., "Fast Galerkin Method and Its Application to Determine Periodic Solutions of Non-Linear Oscillators," International Journal of Non-Linear Mechanics, Vol. 22, No. 2, 1987, pp. 89-98.
20. Bouc, R., and Defilippi, M., "A Galerkin Multiharmonic Procedure for Nonlinear Multidimentional Random Vibration," International Journal of Engineering Science, Vol. 25, No. 6, 1987, pp. 723-733.
21. Weber, H., "Nonlinear Solution of a Class of Nonlinear Boundary Value Problems for Analytic Functions," Journal of Applied Mathematics and Physics, Vol. 33, May 1982, pp. 301-314.
22. Beard, R., "Improving the Closed-Loop Performance of Nonlinear Systems," Ph.D. Thesis, Dept. of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Inst., Troy, NY, 1995.
23. Beard, R., Saridis, G., and Wen, J., "Improving the Performance of Stabilizing Control for Nonlinear Systems," Control Systems Magazine, Vol. 16, No. 10, 1996, pp. 27-35.
24. Beard, R., Saridis, G., and Wen, J., "Galerkin Approximation of the Generalized Hamilton-Jacobi-Bellman Equation," Automatica, Vol. 33, No. 12, 1997, pp. 2159-2177.
25. Beard, R., Saridis, G., and Wen, J., "Approximate Solution to the Time-Invariant Hamilton-Jacobi-Bellman Equation," Journal of Optimization Theory and Application, Vol. 96, March 1998, pp. 589-626.
26. Baras, J. S., "Real Time Architectures for the Zakai Equation and Application," Stochastic Analysis: Liber Amicorum for Moshe Zakai, edited by E. Mayer-Wolf, E. Merzbach, and A. Shwartz, Academic Press, Boston, MA, 1991, pp. 15-38.
27. Zakai, M., "On the Optimal Filtering of Diffusion Processes," Zeitschrift für Wahrscheinlichkeitstheorie und verw. Geb., Vol. 2, 1969, pp. 230-243.
28. Daum, F. E., "Solution of the Zakai, Equation by Separation of Variables," IEEE Transactions on Automatic Control, Vol. 32, No. 10, 1987, pp. 941-943.
29. Daum, F. E., "Exact Finite-Dimensional Nonlinear Filters," IEEE Transactions on Automatic Control, Vol. 31, No. 7, 1986, pp. 616-622.
30. Brigo, D., Hanzon, B., and LeGlanc, F., "A Differential Geometric Approach to Nonlinear Filtering: The Projection Filter," IEEE Transactions on Automatic Control, Vol. 43, No. 2, 1998, pp. 247-252.
31. Zeidler, E., Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators, Springer-Verlag, Berlin, 1990.
32. H. K., Khalil, Nonlinear Systems, Macmillan, New York, 1992.
33. Jain, A. K., Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.