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Lectures on Linear and Nonlinear Filtering

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Analysis and Estimation of Stochastic Mechanical Systems

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 303))

Abstract

Quite generally the filtering problem can be described as follows. Given a stochastic process x(t), t∈I ⊂ℝ, i.e. a sequence of random variables, and a (more or less related) second process y(t), t∈I, it is desired to find the best estimate of x at time t, i.e. the best estimate of x(t), given the (past observations) y(s), 0≤s≤t. Usually I = ℤ (discrete time) or I = ℝ (continuous time).

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References

  1. L. Arnold, Stochastische Differentialgleichungen, Oldenbourg Verlag, 1972. (Translation: Stochastic differential equations, Wiley, 1974 ).

    Google Scholar 

  2. J.S. Baras, Group invariance methods in nonlinear filtering of diffusion processes, In: [34], 565–572.

    Google Scholar 

  3. J.S. Baras, G.L. Blankenschip, W.E. Hopkins JR., Existence, uniqueness and asymptotic behaviour of solutions to a class of Zakai equations with unbounded coefficients, Ieee Trans. AC-28 (1983), 203–214.

    Google Scholar 

  4. V.E. Benes, Exact finite dimensional filters for certain diffusions with nonlinear drift, Stochastics 5 (1981), 65–92.

    Article  MATH  MathSciNet  Google Scholar 

  5. G.L. Blankenschip, C.-H. Liu, S.I. Marcus, Asymptotic expansions and Lie algebras for some nonlinear filter problems, Ieee Trans, AC-28 (1983), 787–797.

    Google Scholar 

  6. R.W. Brockett, Nonlinear systems and nonlinear extimation theory, In: [34], 441–478.

    Google Scholar 

  7. R.W. Brockett, Classification and equivalence in estimation theory, Proc, 1979 Ieee Conf. on decision and Control, Ft. Lauderdale, December 1979.

    Google Scholar 

  8. R.W. Brockett, Remarks on finite dimensional nonlinear estimation, In: C. Lobry (ed.), Analyse des systèmes, Astérisque 76 (1980), Soc. Math. de France.

    Google Scholar 

  9. R.W. Brockett, J.M.C. Clark, The geometry of the conditional density equation, In: O.L.R. Jacobs et al. (eds.), Analysis and optimization of stochastic systems, New York, 1980, 299–309.

    Google Scholar 

  10. R.S. Bucy, J.M.F. Moura (eds.), Nonlinear stochastic problems, Reidel, 1983.

    Google Scholar 

  11. R.S. Bucy, J. Pages, A priori error bounds for the cubic sensor problem, Ieee Trans. Automatic Control AC-24 (1979), 948–953.

    Google Scholar 

  12. M. Chaleyat-Maurel, D. Miche., Hypoellipticity theorems and conditional laws, Z. Wahrsch. and verw. Geb. 65 (1984), 573–597.

    Article  MATH  Google Scholar 

  13. S.D. Chikte, J.T.-H. Lo, Optimal filters for bilinear systems with nilpotent Lie algebras, Ieee Trans, Automatic Control AC-24 (1979), 948–953.

    Google Scholar 

  14. J.M.C. Clark, The design of robust approximations to the stochastic differential equations of nonlinear filtering, In: J.K. Skwirzynski (ed.), Communication systems and random process theory, Sijthoff & Noordhoff, 1978.

    Google Scholar 

  15. P.E. Crouch, Solvable approximations to control systems, Siam J. Control and Opt. 32 (1984), 40–54.

    Article  MathSciNet  Google Scholar 

  16. F.E. Daum, Exact nonlinear recursive filters and separation of variables, Proc. 25-th Cdc, Athens, Dec. 1986.

    Google Scholar 

  17. M.H.A. Davis, Exact finite dimensional nonlinear filters, Proc. 24-th Cdc, Las Vegas December 1985.

    Google Scholar 

  18. M.H.A. Davis, Pathwise nonlinear filtering, In: [34], 505–528.

    Google Scholar 

  19. M.H.A. Davis, S.I. Marcus, An introduction to nonlinear filtering, In: [34], 565–572.

    Google Scholar 

  20. T.E. Duncan, Probability densities for diffusion processes with applications to nonlinear filtering theory, Ph. D. thesis, Stanford, 1967.

    Google Scholar 

  21. M. Fujisaki, G. Kallianpur, H. Kunita, Stochastic differential equations for the nonlinear filtering problem, Osaka J. Math. 1 (1972), 19–40.

    MathSciNet  Google Scholar 

  22. R.W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lnm 562, Springer, 1976.

    Google Scholar 

  23. M. Hazewinkel, On deformations, approximations and nonlinear filtering, Systems Control Leu. 1 (1982), 29–62.

    Google Scholar 

  24. M. Hazewinkel, The linear systems Lie algebra, the Segal-Shale-Weil representation and all Kalman-Bucy filters, J. Syst. Sci. & Math. Sci. 5 (1985), 94–106.

    MATH  MathSciNet  Google Scholar 

  25. M. Hazewinkel, S.I. Marcus, On Lie algebras and finite dimensional filtering, Stochastics 7 (1982), 29–62.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Hazewinkel, S.I. Marcus, H.J. Sussmann, Nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem, Systems Control Len. 3 (1983), 331–340.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Hazewinkel, A tutorial introduction to differentiable manifolds and calculus on differentiable manifolds, This volume.

    Google Scholar 

  28. M. Hazewinkel, A tutorial on Lie algebras, This volume.

    Google Scholar 

  29. M. Hazewinkel, Lie algebraic methods in filtering and identification, In: M. Mebkhout, R. Sénéor (eds.), Proc. Viiith Int. Cong. Math. Physics, Luming 1986, World Scientific, 1987, 120–137 (also to appear in Proc. 1-st World Cong. Bernouilli Society, Tashkent, 1986, Vnu, 1988 ).

    Google Scholar 

  30. M. Hazewinkel, Identification, filtering, the symplectic group, and Wei-Norman theory. To appear in Proc. 2-nd workshop on the Road-vehicle system and related mathematics, Torino, 1987, Reidel-Teubner, 1988.

    Google Scholar 

  31. M. Hazewinkel, On filtering and polynomials in n independent Brownian motions, Acta Scientifica,Caracas, to appear.

    Google Scholar 

  32. M. Hazewinkel, Moduli and canonical forms for linear dynamical systems II: the topological case, Math. Systems Th. 10 (1977), 363–385.

    Article  MATH  MathSciNet  Google Scholar 

  33. M. Hazewinkel, (Fine) moduli (spaces) for linear systems: what are they and what are they good for, In: C.I. Byrnes, C.F. Martin (eds.), Geometric methods for linear systems theory (Harvard, June 1979), Reidel, 1980, 125–193.

    Chapter  Google Scholar 

  34. M. Hazewinkel, J.C. Willems, (eds.), Stochastic systems: the mathematics of filtering and identification and applications, Reidel, 1981.

    Google Scholar 

  35. M. Hazewinkel, An introduction to nilpotent approximation filtering. To appear in Proc. 2-nd European Symp. on Math. in Industry (Ecmi 2), Oberwolfach, 1987, Reidel-Teubner, 1988.

    Google Scholar 

  36. R.C. Hermann, Lie algebras and quantum mechanics, Benjamin, 1970.

    Google Scholar 

  37. R.C. Hermann, Infinite dimensional Lie algebras and current algebras, In: Lect. Notes in Physics 6, Springer, 1969.

    Google Scholar 

  38. R.C. Hermann, Current algebras, the Sugawara model and differential geometry, J. Math. Phys. 11: 6, (1970), 1825–1829.

    Article  MATH  MathSciNet  Google Scholar 

  39. O.B. Himn, Finite dimensional causal functionals of brownian motion, In [6], 425–436.

    Google Scholar 

  40. A.A. Kirilov, Local Lie algebras, Usp. Mat. Mauk. 81, 57–76.

    Google Scholar 

  41. A.J. Krener, On the equivalence of control systems and the linearization of nonlinear systems, Siam J. Control 11, (1973), 670–676.

    Article  MATH  MathSciNet  Google Scholar 

  42. P.S. Krishnaprasad, S.I. Marcus, M. Hazewinkel, Current algebras and the identification problem, Stochastics 11 (1983), 65–101.

    Article  MATH  MathSciNet  Google Scholar 

  43. TH.G. Kurtz, D. Ocone, A martingale problem for conditional distributions and uniqueness for the nonlinear filtering equations, Lect. Notes Control and Inf. Sci. 69 (1985), 224–235.

    Article  MathSciNet  Google Scholar 

  44. H.J. Kushner, On the dynamical equations of conditional probability functions with application to optimal stochastic control theory, J. Math. Anal. Appl. 8 (1964), 332–344.

    Article  MATH  MathSciNet  Google Scholar 

  45. H.J. Kushner, Dynamical equations for optimal nonlinear filtering, J. Diff. Equations 3 (1967), 179–190.

    Article  MATH  MathSciNet  Google Scholar 

  46. R.S. Liptser, A.N. Shiryayev, Statistics of Random Processes I. New York: Springer-Verlag, 1977.

    Book  MATH  Google Scholar 

  47. C.-H. Liu, S.I. Marcus, The Lie algebraic structure of a class of finite dimensional nonlinear filters, In: C.I. Byrnes, C.F. Martin (eds.), Algebraic and geometric methods in linear systems theory, Amer. Math. Soc., (1980), 277–279.

    Google Scholar 

  48. S.I. Marcus, Algebraic and geometric methods in nonlinear filtering, Siam J. Control Opt. 22 (1984), 817–844.

    Article  MATH  Google Scholar 

  49. S.I. Marcus, A.S. Willsky, Algebraic structure and finite dimensional nonlinear estimation, Siam J. Math. Anal. 9 (1978), 312–327.

    Article  MATH  MathSciNet  Google Scholar 

  50. S.I. Marcus, Optimal nonlinear estimation for a class of discrete-time stochastic systems, Ieee Trans. Automatic Control AC-24 (1979), 297–302.

    Google Scholar 

  51. S.I. Marcus, S.K. Mitter, D. Ocone, Finite dimensional nonlinear estimation for a class of systems in continuous and discrete time, In: Analysis and Optimization of Stochastic Systems, O.L.R. Jacobs et al. Eds, Academic Press, 1980, 387–406.

    Google Scholar 

  52. D. Michel, Régularité des lois conditionelles en théorie du filtrage non-linéaire et calcul des variations stochastiques, J. Funct. Anal. 14 (1981), 8–36.

    Article  Google Scholar 

  53. S.K. Mitter, On the analogy between mathematical problems of non-linear filtering and quantum physics, Richerche di Automatica 10 (1980), 163–216.

    MathSciNet  Google Scholar 

  54. S.K. Mitter, Nonlinear filtering and stochastic machanics, In: [34], 479–504.

    Google Scholar 

  55. R.E. Mortensen, Optimal control of continuous time stochastic systems, Ph. D thesis, Berkeley, 1966.

    Google Scholar 

  56. D. Ocone, Topics in non-linear filtering, Thesis, M.I.T., June 1980.

    Google Scholar 

  57. E. Pardoux, D. Talay, Discretization and simulation of stochstic differential equations, Acta Appl. Math. 3 (1982), 182–203.

    MathSciNet  Google Scholar 

  58. K.R. Parthasarathy, K. Schmidt, A new method for constructing factorizable representations for current group and current algebras, Comm. Math. Phys. 50 (1976), 167–175.

    Article  MATH  MathSciNet  Google Scholar 

  59. L.E. Pursell, M.E. Shanks, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc. 5 (1954), 468–472.

    Article  MathSciNet  Google Scholar 

  60. CH. Rockland, Intrinsic nilpotent approximation, Acta Appl. Math. 8 (1987), 213–270.

    Article  MATH  MathSciNet  Google Scholar 

  61. L..P. Rothschild, E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320.

    Article  MathSciNet  Google Scholar 

  62. S. Steinberg, Application of the Lie algebraic formulas of Baker, Campbell, Hausdorff and Zassenhaus to the calculation of explicit solutions of partial differential equations, J. DifJ. Equations 26 (1977), 404–434.

    Article  MATH  Google Scholar 

  63. H.J. Sussmann, Existence and uniqueness of minimal realisations of nonlinear systems, Math. Syst. Theory 10 (1977), 349–356.

    MathSciNet  Google Scholar 

  64. J. Wei, E. Norman, On the global representation of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc. 15 (1964), 327–334.

    Article  MATH  MathSciNet  Google Scholar 

  65. M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrsch. verw. Geb. 11 (1969), 230–243.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CWI, Amsterdam, The Netherlands

    M. Hazewinkel

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  1. M. Hazewinkel
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Editors and Affiliations

  1. University of Stuttgart, Stuttgart, Germany

    Werner Schiehlen

  2. University of Karlsruhe, Karlsruhe, Germany

    Walter Wedig

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© 1988 Springer-Verlag Wien

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Hazewinkel, M. (1988). Lectures on Linear and Nonlinear Filtering. In: Schiehlen, W., Wedig, W. (eds) Analysis and Estimation of Stochastic Mechanical Systems. International Centre for Mechanical Sciences, vol 303. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2820-6_3

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  • DOI: https://doi.org/10.1007/978-3-7091-2820-6_3

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82058-2

  • Online ISBN: 978-3-7091-2820-6

  • eBook Packages: Springer Book Archive

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