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Suboptimal anisotropic filtering in a finite horizon

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Abstract

Consideration was given to the problem of robust stochastic filtering in a finite horizon for the linear discrete time-varying system. A random disturbance with inaccurately known probabilistic distribution is fed to the system input. Uncertainty of the input disturbance is defined in the information-theoretical terms by the anisotropy functional of a random vector. The sufficient condition for strict boundedness of the anisotropic norm of linear discrete timevarying system assigned by the threshold value (lemma of real boundedness) was proved in terms of the matrix inequalities. Sufficient conditions for boundedness of the anisotropic norm of two limiting cases of the anisotropy levels of the input disturbance (a = 0 and a → ∞) were established. A sufficient existence condition for the estimator guaranteeing boundedness of the anisotropic norm of the estimation error operator by the given threshold value was formulated and proved. Sufficient existence conditions for the estimators of two limiting cases of the anisotropy levels of input disturbance were obtained. The estimation algorithm relies on the recurrent solution of a system of matrix inequalities.

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References

  1. Kalman, R., A New Approach to Linear Filtering and Prediction Problems, Trans. ASME. J. Basic Eng., 1960, vol. 82D, pp. 34–45.

    Google Scholar 

  2. Anderson, B.D.O. and Moore, J.B., Optimal Filtering, Englewood Cliffs: Prentice Hall, 1979.

    MATH  Google Scholar 

  3. Hassibi, B., Sayed, A., and Kailath, T., Indefinite Quadratic Estimation and Control: A Unified Approach to H 2 and H ∞ Theories, Philadelphia: SIAM, 1999.

    Book  Google Scholar 

  4. Doyle, J.C., Guaranteed Margins for LQG Regulators, IEEE Trans. Automat. Control, 1978, vol. AC-23, no. 4, pp. 756–757.

    Article  Google Scholar 

  5. Simon, D., Optimal State Estimation: Kalman, Infinity, and Nonlinear Approaches, New Jersey: Wiley, 2006.

    Book  Google Scholar 

  6. Nagpal, K.M. and Khargonekar, P.P., Filtering and Smoothing in an H ∞-setting, IEEE Trans. Automat. Control, 1991, vol. 36, no. 2, pp. 152–166.

    Article  MathSciNet  MATH  Google Scholar 

  7. Grimble, M.J. and Elsayed, A., Solution of the H ∞ Optimal Linear Filtering Problem for Discrete-time Systems, IEEE Trans. Acoust. Speech Signal Proc., 1990, vol. 38, pp. 1092–1104.

    Article  MathSciNet  MATH  Google Scholar 

  8. Shaked, U. and Theodor, Y., H ∞-optimal Estimation: A Tutorial, in Proc. 31 IEEE CDC, 1992, pp. 2278–2286.

    Google Scholar 

  9. Theodor, Y., Shaked, U., and de Souza, C.E., A Game Theory Approach to Robust Discrete-time H ∞ Estimation, IEEE Trans. Signal Proc., 1994, vol. 42, pp. 1486–1495.

    Article  Google Scholar 

  10. Zhou, K., Glover, K., Bodenheimer, B., and Doyle, J., Mixed H 2 and H ∞ Performance Objectives I: Robust Performance Analysis, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1564–1574.

    Article  MathSciNet  MATH  Google Scholar 

  11. Doyle, J., Zhou, K., Gover, K., and Bodenheimer, B., Mixed H 2 and H ∞ Performance Objectives II: Optimal Control, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1575–1587.

    Article  MathSciNet  MATH  Google Scholar 

  12. Limebeer, D.J.N., Anderson, B.D.O., and Hendel, B., Mixed H 2/H ∞ Filtering by the Theory of Nash Games, in Proc. Workshop on Robust Control, Tokyo, Japan, 1992, pp. 9–15.

    Chapter  Google Scholar 

  13. Haddad, W.M., Bernstein, D.S., and Mustafa, D., Mixed-norm H 2/H ∞ Regulation and Estimation: The Discrete-time Case, Syst. Control Lett., 1991, vol. 16, no. 4, pp. 235–247.

    Article  MathSciNet  MATH  Google Scholar 

  14. Khargonekar, P.P., Rotea, M.A., and Baeyens, E., Mixed H 2/H ∞ Filtering, Int. J. Robust Nonlin. Control, 1996, vol. 6, pp. 313–330.

    Article  MathSciNet  MATH  Google Scholar 

  15. Huaizhong, Li and Minyue, Fu, A Linear Matrix Inequality Approach to Robust H ∞ Filtering, IEEE Trans. Signal Proc., 1997, vol. 45, no. 9, pp. 2338–2350.

    Article  Google Scholar 

  16. Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    Book  MATH  Google Scholar 

  17. Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Synthesis of the Control Laws on the Basis of Matrix Linear Inequalities), Moscow: Fizmatlit, 2007.

  18. Balandin, D.V. and Kogan, M.M., Generalized H ∞-optimal Filtering under External and Initial Disturbances, Differ. Uravn., 2012, vol. 48, no. 11, pp. 1507–1514.

    MathSciNet  Google Scholar 

  19. Kogan, M.M., LMI Based Minimax Estimation and Filtering under Unknown Covariances, Int. J. Control, 2014, vol. 87, pp. 1216–1226.

    Article  MATH  Google Scholar 

  20. Gershon, E., Shaked, U., and Yaesh, I., H ∞ Control and Estimation of State-Multiplicative Linear Systems, London: Springer, 2005.

    Google Scholar 

  21. Diamond, P., Vladimirov, I., Kurdjukov, A., and Semyonov, A., Anisotropy-based Perfomance Analysis of Linear Discrete Time Invariant Control Systems, Int. J. Control, 2001, vol. 74, no. 1, pp. 28–42.

    Article  MathSciNet  MATH  Google Scholar 

  22. Vladimirov, I., Anisotropy-based Optimal Filtering in Linear Discrete Time Invariant Systems, 2012 http://arxiv.org/abs/1412.3010.

    Google Scholar 

  23. Timin, V.N., Tchaikovsky, M.M., and Kurdyukov, A.P., A Solution to Anisotropic Suboptimal Filtering Problem by Convex Optimization, Dokl. Math., 2012, vol. 85, pp. 443–445.

    Article  MathSciNet  MATH  Google Scholar 

  24. Timin, V.N., Anisotropy-based Suboptimal Filtering for the Linear Discrete Time-invariant Systems, Autom. Remote Control, 2013, vol. 74, no. 11, pp. 1773–1785.

    Article  MathSciNet  MATH  Google Scholar 

  25. Vladimirov, I.G., Diamond, P., and Kloeden, P., Anisotropy-based Robust Performance Analysis of Finite Horizon Linear Discrete Time Varying Systems, Autom. Remote Control, 2006, vol. 67, no. 8, pp. 1265–1282.

    Article  MathSciNet  MATH  Google Scholar 

  26. Vladimirov, I. and Diamond, P., Robust Filtering in Finite Horizon Linear Discrete Time Varying Systems by Minimum Anisotropic Norm Criterion, CADSMAP Research Report 01-05, Center for Applied Dynamical Systems/Center for Applied Dynamical Systems, Mathematical Analysis and Probability, Univ. of Queensland, Brisbane, Australia, 2001.

    Google Scholar 

  27. Maximov, E.A., Kurdyukov, A.P., and Vladimirov, I.G., Anisotropic Norm Bounded Real Lemma for Linear Time-varying System, in Preprint 18 IFAC World Congr., Milano, 2011, pp. 4701–4706.

    Google Scholar 

  28. Yaesh, I. and Stoica, A.-M., Linear Time-varying Anisotropic Filtering its Application to Nonlinear Systems State Estimation, in Proc. Eur. Control Conf., Strasbourg, 2014, June 24–27, pp. 975–980.

    Google Scholar 

  29. Cover, T.M. and Thomas, J.A., Elements of Information Theory, New York: Wiley, 1991.

    Book  MATH  Google Scholar 

  30. Ben-Tal, A. and Nemirovski, A., Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization, Philadelphia: SIAM, 2001.

    Book  Google Scholar 

  31. Yaesh, I. and Shaked, U., Game Theory Approach to Optimal Linear Estimation in Minimum H ∞-norm Sense, in Proc. 28 Conf. Decision Control, Tampa, Florida, December 1989, pp. 421–425.

    Google Scholar 

  32. Lofberg, J., YALMIP: A Toolbox for Modeling and Optimization in MATLAB, in Proc. CACSD Conf., Taipei, Taiwan, 2004, http://users.isy.liu.se/johanl/yalmip/.

    Google Scholar 

  33. Sturm, J.F., Using SeDuMi 1.02, a MATLAB Toolbox for Optimization over Symmetric Cones, Optim. Methods Software, 1999, vol. 11, pp. 625–653.

    Article  MathSciNet  Google Scholar 

  34. Freiling, G. and Ionescu, V., Time-varing Discrete Riccati Equation: Some Monotonicity Resuits, Linear Algebra Appl., 1999, vol. 286, pp. 135–148.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. N. Timin & A. P. Kurdyukov

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  1. V. N. Timin
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  2. A. P. Kurdyukov
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Correspondence to V. N. Timin.

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Original Russian Text © V.N. Timin, A.P. Kurdyukov, 2016, published in Avtomatika i Telemekhanika, 2016, No. 1, pp. 5–29.

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Timin, V.N., Kurdyukov, A.P. Suboptimal anisotropic filtering in a finite horizon. Autom Remote Control 77, 1–20 (2016). https://doi.org/10.1134/S000511791601001X

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