Advertisement

Mathematics of Control, Signals and Systems

, Volume 9, Issue 4, pp 370–385 | Cite as

Finite-dimensional filters with nonlinear drift, VI: Linear structure of Ω

  • Jie Chen
  • Stephen S. -T. Yau
Article

Abstract

Ever since the concept of estimation algebra was first introduced by Brockett and Mitter independently, it has been playing a crucial role in the investigation of finite-dimensional nonlinear filters. Researchers have classified all finite-dimensional estimation algebras of maximal rank with state space less than or equal to three. In this paper we study the structure of quadratic forms in a finite-dimensional estimation algebra. In particular, we prove that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂f j /∂x i −∂f i /∂x j )matrix, wheref denotes the drift term, is a linear matrix in the sense that all the entries in Ω are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank.

Key words

Finite-dimensional filters Nonlinear drift Estimation algebra of maximal rank 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    R. W. Brokett, Nonlinear systems and nonlinear estimation theory, inThe Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems, eds., Reidel, Dordrecht, 1981.Google Scholar
  2. [BC]
    R. W. Brockett and J. M. C. Clark, The geometry of the conditional density functions, inAnalysis and Optimization of Stochastic Systems, O. L. R. Jacobset al, eds., Academic Press, New York, 1980, pp. 299–309.Google Scholar
  3. [CM]
    M. Chaleyat-Maurel and D. Michel, Des resultants de non-existence de filter de dimension finie,Stochastics 13 (1984), 83–102.zbMATHMathSciNetGoogle Scholar
  4. [CLY]
    J. Chen, C. W. Leung, and S. S.-T. Yau, Finite-dimensional filters with nonlinear drift, IV: Classification of finite-dimensional estimation algebra of maximal rank with state space dimension 3,SIAM J. Control Optim. 34(1) (1996), 179–198.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [CY]
    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) (1994), 297–310.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [C]
    M. Cohen De Lara, Contribution des methods geometriques au filtrage de dimension finie, Ph.D. thesis, Ecole des Mines de Paris, 1991.Google Scholar
  7. [D]
    M. H. A. Davis, On a multiplicative functional transformation arising in nonlinear filtering theory,Z. Wahrsch. Verw. Gebiete 54 (1980), 125–139.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [DM]
    M. H. A. Davis and S. I. Marcus, An introduction to nonlinear filtering, inThe Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems, eds., Reidel, Dordrecht, 1981, pp. 53–75.Google Scholar
  9. [DTWY]
    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) (1991), 866–877.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [DWY]
    R. T. Dong, W. S. Wong, and S. S.-T. Yau, Filtering systems with finite dimensional estimation algebras, Preprint, 1995.Google Scholar
  11. [FKK]
    M. Fujisaki, G. Kallianpur, and H. Kunita, Stochastic differential equations for the non-linear filtering problem,Osaka J. Math. 1 (1972), 19–40.MathSciNetGoogle Scholar
  12. [HP]
    U. G. Haussman and E. Pardoux, A conditionally almost linear filtering problem with non-Gaussian initial condition,Stochastics 23 (1988), 241–275.MathSciNetGoogle Scholar
  13. [H]
    H. Hazewinkel, Lecture on linear and nonlinear filtering, inAnalysis and Estimation of Stochastic Mechanical Systems, CISM Courses and Lectures, vol. 303, W. Shiehlen and W. Wedig, eds., Springer-Verlag, Vienna, 1988, pp. 103–135.Google Scholar
  14. [HMS]
    M. Hazewinkel, S. I. Marcus, and H. J. Sussmann, Nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem,Systems Control Lett. 3 (1983), 331–340.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [M1]
    A. Makowski, Filtering formula for partially observed linear systems with non-Gaussian initial conditions,Stochastics 16 (1986), 1–24.zbMATHMathSciNetGoogle Scholar
  16. [M2]
    S. I. Marcus, Algebraic and geometric methods in nonlinear filtering,SIAM J. Control Optim. 22 (1984), 817–844.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [M3]
    S. K. Mitter, On the analogy between mathematical problems of non-linear filtering and quantum physics,Ricerche Automat. 10 (1979), 163–216.MathSciNetGoogle Scholar
  18. [O]
    D. Ocone, Finite dimensional estimation algebras in nonlinear filtering, inThe Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems, eds., Reidel, Dordrecht, 1981, pp. 629–636.Google Scholar
  19. [R]
    B. L. Rozovsky, Stochastic partial differential equations arising in nonlinear filtering problem (in Russian),Uspekhi Mat. Nauk. 27 (1972), 213–214.Google Scholar
  20. [TWY]
    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 (1990), 173–185.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [W]
    W. S. Wong, On a new class of finite dimensional estimation algebras,Systems Control Lett. 9 (1987), 79–83.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [Y]
    S. S.-T. Yau, Finite-dimensional filters with nonlinear drift, I: A class of filters including both Kalman-Bucy filters and Benes filters,J. Math. Systems Estim. Control 4(2) (1994), 181–203.zbMATHMathSciNetGoogle Scholar
  23. [YL]
    S. S.-T. Yau and C. W. Leung, Recent result on classification of finite dimensional maximal rank estimation algebras with state space dimension 3,Proceedings of the 31st Conference on Decision and Control, Tucson, Arizona, Dec. 1992, pp. 2247–2250.Google Scholar

Copyright information

© Springer-Verlag London Limited 1996

Authors and Affiliations

  • Jie Chen
    • 1
  • Stephen S. -T. Yau
    • 1
  1. 1.Control and Information Laboratory, MSCS, M/C 249University of Illinois at ChicagoChicagoUSA

Personalised recommendations