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Characterization of a subclass of finite-dimensional estimation algebras with maximal rank application to filtering

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Abstract

Finite-dimensional estimation Lie algebras play a crucial role in the study of finite-dimensional filters for partially observed stochastic process. When the dynamics noise is Gaussian we can characterize the so-called estimation Lie algebras with maximal rank in terms of the observation functions (necessarily affine) and the drift (necessarily a sum of a skew-symmetric linear term and a gradient vector field, with a functional relationship), under the assumption that the estimation algebra has one and only one operator of order greater or equal to two in any of its basis.

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References

  1. V.E. Beneš. Exact finite-dimensional filters for certain diffusions with nonlinear drift.Stochastics, 5:65–92, 1982.

    Google Scholar 

  2. R.W. Brockett. Nonlinear systems and nonlinear estimation theory. In M. Hazewinkel and J.C. Willems, editors,Stochastic Systems: The Mathematics of Filtering and Identification and Applications, pages 442–477. Reidel, Dordrecht, 1981.

    Google Scholar 

  3. R.W. Brockett and J.M.C. Clark. The geometry of the conditional density equation. In O.L.R. Jacobset al., editors,Analysis and Optimization of Stochastic Systems, pages 299–310. Academic Press, New York, 1980.

    Google Scholar 

  4. W.-L. Chiou and S.S.-T. Yau. Finite-dimensional filters with nonlinear drift, II: Brockett's problem on classification of finite-dimensional estimation algebras.SIAM J. Control Optim., 32(1):297–310, 1994.

    Google Scholar 

  5. M. Cohen de Lara. Application of symmetry semi-groups to discrete and continuous time filtering problems. In B. Bonnard, B. Bride, J.P. Gauthier, and I. Kupka, editors,Analysis of Controlled Dynamical Systems, pages 146–155. Birkhäuser, Boston, 1991.

    Google Scholar 

  6. M. Cohen de Lara. A note on the symmetry group and perturbation algebra of a parabolic partial differential operator.J. Math. Phys., 32(6):1444–1449, 1991.

    Google Scholar 

  7. M. Cohen de Lara. Geometric and symmetry properties of a nondegenerate diffusion process.Ann. Probab., 23(4):1557–1604, October 1995.

    Google Scholar 

  8. M. Cohen de Lara. Finite dimensional filters. Part I: the Wei-Norman technique.SIAM J. Control Optim., 35(3):980–1001, 1997.

    Google Scholar 

  9. M. Cohen de Lara. Finite dimensional filters. Part II: invariance group techniques.SIAM J. Control Optim., 35(3):1002–1029, 1997.

    Google Scholar 

  10. M.H.A. Davis and S.I. Marcus. An introduction to nonlinear filtering. In M. Hazewinkel and J.C. Willems, editors,Stochastic Systems: The Mathematics of Filtering and Identification and Applications, pages 53–75. Reidel, Dordrecht, 1981.

    Google Scholar 

  11. R.T. Dong, L.F. Tam, W.S. Wong, and S.S.T. Yau. Structure and classification theorems of finite-dimensional exact estimation algebras.SIAM J. Control Optim., 29(4):866–877, 1991.

    Google Scholar 

  12. J. Lévine. Finite dimensional filters for a class of nonlinear systems and immersion in a linear system.SIAM J. Control Optim., 25(6):1430–1439, 1987.

    Google Scholar 

  13. D.L. Ocone. Topics in nonlinear filtering theory. Ph.D. thesis, Massachussets Institute of Technology, Cambridge, MA, 1980.

    Google Scholar 

  14. D.L. Ocone. Finite dimensional Lie algebras in nonlinear filtering. In M. Hazewinkel and J.C. Willems, editors,Stochastic Systems: The Mathematics of Filtering and Identification and Applications, pages 629–636. Reidel, Dordrecht, 1981.

    Google Scholar 

  15. E. Pardoux. Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. École d'été de Probabilités de Saint-Flour, 1989.

  16. L.F. Tam, W.S. Wong, and S.S.T. Yau. On a necessary and sufficient condition for finite dimensionality of estimation algebras.SIAM J. Control Optim., 28(1):173–185, 1990.

    Google Scholar 

  17. E. Wong.Stochatic Processes in Information and Dynamical Systems. McGraw-Hill, New York, 1971.

    Google Scholar 

  18. W.S. Wong. New classes of finite dimensional filters.Systems Control Lett., 3:155–164, 1983.

    Google Scholar 

  19. S.S.-T. Yau. Finite dimensional filters with nonlinear drift I: A class of filters including both Kalman-Bucy filters and Beneš filters.J. Math. Systems Estim. Control, 4(2):181–203, 1994.

    Google Scholar 

  20. M. Zakai. On the optimal filtering of diffusion processes.Z. Wahrsch. Verw. Gebiete, 11:230–243, 1969.

    Google Scholar 

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  1. Cergrene, École nationale des ponts et chaussées, 6 et 8 avenue Biaise Pascal, Cité Descartes, 77455, Marne la Vallée Cedex 2, France

    M. Cohen de Lara

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de Lara, M.C. Characterization of a subclass of finite-dimensional estimation algebras with maximal rank application to filtering. Math. Control Signal Systems 10, 237–246 (1997). https://doi.org/10.1007/BF01211505

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  • DOI: https://doi.org/10.1007/BF01211505

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