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Anisotropy-based suboptimal filtering for the linear discrete time-invariant systems

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Abstract

Consideration was given to the problem of robust filtering for the finite-dimensional linear discrete time-invariant system with measured and estimated outputs. The system is exposed to a random disturbance with the imprecisely known probability distribution. In the information-theoretical terms, the stochastic uncertainty of the input disturbance is defined by the functional of mean anisotropy. The error of estimation was quantified by the anisotropy norm. A sufficient condition for an estimator to exist and ensure that the error is less than the given threshold value was derived in the form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities.

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References

  1. Doyle, J.C., Glover, K., Khargonekar, P.P., et al., State-space Solutions to Standard H 2 and H ∞ Control Problems, IEEE Trans. Automat. Control, 1989, vol. 34, no. 8, pp. 831–848.

    Article  MathSciNet  MATH  Google Scholar 

  2. Francis, B.A., A Course in H ∞ Control Theory, Lecture Notes in Control and Information Sciences, New York: Springer, 1987.

    Book  MATH  Google Scholar 

  3. Gu, D.-W., Tsai, M.C., O’Young, S.D., et al., State-space Formulae for Discrete-time H ∞ Optimization, Int. J. Control, 1989, vol. 49, no. 1, pp. 1683–1723.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Green, M. and Limebeer, D.J.N., Linear Robust Control, Englewood Cliffs: Prentice Hall, 1995.

    MATH  Google Scholar 

  7. 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 

  8. 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 

  9. Bernstein, D.S. and Haddad, W.M., Steady-state Kalman Filtering with an H ∞ Error Bound, Syst. Control Lett., 1989, vol. 12, pp. 9–16.

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. 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 

  12. Zhou, K., Glover, K., Bodenheimer, B., et al., Mixed H 2 and H ∞ Performance Objectives I: Robust Performance Analysis, IEEE Trans. Automat. Control, 1994, vol. 39, no. 8, pp. 1564–1574.

    Article  MathSciNet  MATH  Google Scholar 

  13. Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  16. Vladimirov, I.G., Kurdyukov, A.P., and Semenov, A.V., Anisotropy of Signals and Entropy of the Linear Time-invariant Systems, Dokl. Ross. Akad. Nauk, 1995, no. 3, pp. 583–585.

    Google Scholar 

  17. Vladimirov, I.G., Kurdjukov, A.P., and Semyonov, A.V., On Computing the Anisotropic Norm of Linear Discrete-Time-Invariant Systems, in Proc. 13th IFAC World Congr., San-Francisco, 1996, pp. 179–184.

    Google Scholar 

  18. Vladimirov, I.G., Kurdyukov, A.P., and Semenov, A.V., Stochastic Problem of H ∞-optimization, Dokl. Ross. Akad. Nauk, 1995, vol. 343, no. 5, pp. 607–609.

    MathSciNet  Google Scholar 

  19. Vladimirov, I.G., Kurdjukov, A.P., and Semyonov, A.V., State-Space Solution to Anisotropy-Based Stochastic H ∞-Optimization Problem, in Proc. 13th IFAC World Congr., San-Francisco, 1996, pp. 427–432.

    Google Scholar 

  20. Vladimirov, I., Anisotropy-based Optimal Filtering in Linear Discrete Time Invariant Systems, CADSMAP Report 01-03, Centre for Applied Dynamical Systems, Centre for Applied Dynamical Systems, Mathematical Analysis and Probability, The University of Queensland, 2001, http://smp.uq.edu.au/node/1278.

    Google Scholar 

  21. Mariton, M. and Bertrand, R., A Homotopy Algorithm for Solving Coupled Riccati Equations, Optim. Control Appl. Meth., 1985, vol. 6, pp. 351–357.

    Article  MathSciNet  Google Scholar 

  22. Chaikovskii, M.M. and Kurdyukov, A.P., Criterion for Strict Boundedness of the Anisotropy-based Norm by the Given Valie in Terms of the Matrix Inequalities, Dokl. Ross. Akad. Nauk, 2011, vol. 441, no. 3, pp. 318–321.

    MathSciNet  Google Scholar 

  23. Tchaikovsky, M.M., Kurdyukov, A.P., and Timin, V.N., Synthesis of Anisotropic Suboptimal Controllers by Convex Optimization, Submitted Eur. J. Control, 2011 (preprint available from http://arxiv.org/abs/1108.4982).

    Google Scholar 

  24. Timin, V.N., Chaikovskii, M.M., and Kurdyukov, A.P., Solution of the Problem of Anisotropy-based Suboptimal Filtering by the Method of Convex Optimization, Dokl. Ross. Akad. Nauk, 2012, vol. 444, no. 6, pp. 612–615.

    MathSciNet  Google Scholar 

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

    Google Scholar 

  26. 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 

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Authors and Affiliations

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

    V. N. Timin

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  1. V. N. Timin
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Original Russian Text © V.N. Timin, 2013, published in Avtomatika i Telemekhanika, 2013, No. 11, pp. 3–19.

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Timin, V.N. Anisotropy-based suboptimal filtering for the linear discrete time-invariant systems. Autom Remote Control 74, 1773–1785 (2013). https://doi.org/10.1134/S0005117913110015

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