Circuits, Systems & Signal Processing

, Volume 28, Issue 2, pp 223–239 | Cite as

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

  • Michael Basin
  • Dario Calderon-Alvarez
New Trends in Optimum and Robust Filtering


In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman–Bucy filter.


Filtering Stochastic system Nonlinear polynomial system Bilinear system 


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  1. 1.
    L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974) zbMATHGoogle Scholar
  2. 2.
    K.J. Åström, Introduction to Stochastic Control Theory (Academic Press, New York, 1970) zbMATHGoogle Scholar
  3. 3.
    M.V. Basin, On optimal filtering for polynomial system states. ASME Trans. J. Dyn. Syst. Meas. Control 125, 123–125 (2003) CrossRefGoogle Scholar
  4. 4.
    M.V. Basin, M.A. Alcorta-Garcia, Optimal filtering and control for third degree polynomial systems. Dyn. Contin. Discrete Impuls. Syst. B 10, 663–680 (2003) zbMATHMathSciNetGoogle Scholar
  5. 5.
    M.V. Basin, J. Perez, D. Calderon-Alvarez, Optimal filtering for linear systems over polynomial observations. Int. J. Innov. Comput. Inf. Control 4(2), 313–320 (2008) Google Scholar
  6. 6.
    M.V. Basin, J. Perez, M. Skliar, Optimal filtering for polynomial systems with partially measured states and multiplicative noises. In: Proc. 45th IEEE Conference on Decision and Control, pp. 4169–4174, San Diego, CA, USA, 2006 Google Scholar
  7. 7.
    M.V. Basin, J. Perez, M. Skliar, Optimal filtering for polynomial system states with polynomial multiplicative noise. Int. J. Robust Nonlinear Control 16, 287–298 (2006) CrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Basin, E. Sanchez, R. Martinez-Zuniga, Optimal linear filtering for systems with multiple state and observation delays. Int. J. Innov. Comput. Inf. Control 3(5), 1309–1320 (2007) Google Scholar
  9. 9.
    V.E. Benes, Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics 5, 65–92 (1981) zbMATHMathSciNetGoogle Scholar
  10. 10.
    H. Gao, J. Lam, L. Xie, C. Wang, New approach to mixed H 2/H -filtering for polytopic discrete-time systems. IEEE Trans. Signal Process. 53, 3183–3192 (2005) CrossRefMathSciNetGoogle Scholar
  11. 11.
    H. Gao, C. Wang, A delay-dependent approach to robust H filtering for uncertain discrete-time state-delayed systems. IEEE Trans. Signal Process. 52, 1631–1640 (2004) CrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Gao, C. Wang, Delay-dependent robust H and L 2-L filtering for a class of uncertain nonlinear time-delay systems. IEEE Trans. Autom. Control 48, 1661–1666 (2003) CrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Hazewinkel, S.I. Marcus, H.J. Sussmann, Nonexistence of exact finite-dimensional filters for conditional statistics of the cubic sensor problem. Syst. Control Lett. 5, 331–340 (1983) CrossRefMathSciNetGoogle Scholar
  14. 14.
    R.E. Kalman, R.S. Bucy, New results in linear filtering and prediction theory. ASME Trans. Part D: J. Basic Eng. 83, 95–108 (1961) MathSciNetGoogle Scholar
  15. 15.
    H.J. Kushner, On differential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control 12, 106–119 (1964) MathSciNetGoogle Scholar
  16. 16.
    M. Mahmoud, P. Shi, Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters. IEEE Trans. Circuits Syst. 50, 98–105 (2003) CrossRefMathSciNetGoogle Scholar
  17. 17.
    X. Mao, Stochastic Differential Equations and Their Applications (Horwood, Chichester, 1997) zbMATHGoogle Scholar
  18. 18.
    V.S. Pugachev, I.N. Sinitsyn, Stochastic Systems: Theory and Applications (World Scientific, Singapore, 2001) zbMATHGoogle Scholar
  19. 19.
    J. Sheng, T. Chen, S.L. Shah, Optimal filtering for multirate systems. IEEE Trans. Circuits Syst. 52, 228–232 (2005) CrossRefGoogle Scholar
  20. 20.
    P. Shi, Filtering on sampled-data systems with parametric uncertainty. IEEE Trans. Autom. Control 43, 1022–1027 (1998) zbMATHCrossRefGoogle Scholar
  21. 21.
    A. Tanikawa, On new smoothing algorithms for discrete-time linear stochastic systems with unknown disturbances. Int. J. Innov. Comput. Inf. Control 4, 1 (2008) Google Scholar
  22. 22.
    W.M. Wonham, Some applications of stochastic differential equations to nonlinear filtering. SIAM J. Control 2, 347–369 (1965) zbMATHMathSciNetGoogle Scholar
  23. 23.
    S. Xu, T. Chen, H–infinity model reduction in the stochastic framework. SIAM J. Control Optim. 42, 1293–1309 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    S. Xu, T. Chen, Robust H–infinity control for uncertain stochastic systems with state delay. IEEE Trans. Autom. Control 47, 2089–2094 (2002) CrossRefMathSciNetGoogle Scholar
  25. 25.
    S. Xu, P.V. van Dooren, Robust H –filtering for a class of nonlinear systems with state delay and parameter uncertainty. Int. J. Control 75, 766–774 (2002) zbMATHCrossRefGoogle Scholar
  26. 26.
    S.S.T. Yau, Finite-dimensional filters with nonlinear drift I: a class of filters including both Kalman–Bucy and Benes filters. J. Math. Syst. Estim. Control 4, 181–203 (1994) Google Scholar
  27. 27.
    X. Zhong, X.H. Xing, K. Fujimoto, Sliding mode variable structure control for uncertain stochastic systems. Int. J. Innov. Comput. Inf. Control 3(2), 397–406 (2007) Google Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Department of Physical and Mathematical SciencesAutonomous University of Nuevo LeonSan Nicolas de los GarzaMexico

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