Advertisement

Automation and Remote Control

, Volume 73, Issue 6, pp 962–975 | Cite as

On efficient parametric identification methods for linear discrete stochastic systems

  • Yu. V. Tsyganova
  • M. V. Kulikova
Stochastic Systems, Queueing Systems

Abstract

We construct a numerically stable algorithm (with respect to machine rounding errors) of adaptive Kalman filtering in order to solve the parametric identification problem for linear stationary stochastic discrete systems. We solve the problem in the state space. The proposed algorithm is formulated in terms of an orthogonal square-root covariance filter which lets us avoid a standard implementation of the Kalman filter.

Keywords

Remote Control Cholesky Decomposition Sensitivity Equation Standard Implementation Newton Type Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ljung, L., System Identification: Theory for the User, Englewood Cliffs: Prentice Hall, 1987. Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Moscow: Nauka, 1991.zbMATHGoogle Scholar
  2. 2.
    Aström, K.J., Maximum Likelihood and Prediction Error Methods, Automatica, 1980, vol. 16, pp. 551–574.zbMATHCrossRefGoogle Scholar
  3. 3.
    Gupta, N.K. and Mehra, R.K., Computational Aspects of Maximum Likelihood Estimation and Reduction in Sensitivity Function Calculations, IEEE Trans. Automat. Control, 1994, vol. AC-19, no. 6, pp. 774–783.MathSciNetGoogle Scholar
  4. 4.
    Bierman, G.J., Belzer, M.R., Vandergraft, J.S., and Porter, D.W., Maximum Likelihood Estimation Using Square Root Information Filters, IEEE Trans. Automat. Control, 1990, vol. 35, no. 12, pp. 1293–1299.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ogarkov, M.A., Metody statisticheskogo otsenivaniya parametrov sluchainykh protsessov (Statistical Estimation Methods for Parameters of Random Processes), Moscow: Energoatomizdat, 1990.Google Scholar
  6. 6.
    Verhaegen, M. and Van Dooren, P., Numerical Aspects of Different Kalman Filter Implementations, IEEE Trans. Automat. Control, 1986, vol. AC-31, no. 10, pp. 907–917.CrossRefGoogle Scholar
  7. 7.
    Kailath, T., Sayed, A.H., and Hassibi, B., Linear Estimation, New Jersey: Prentice Hall, 2000.Google Scholar
  8. 8.
    Potter, J.E. and Stern, R.G., Statistical Filtering of Space Navigation Measurements, Proc. 1963 AIAA Guidance and Control Conf., 1963.Google Scholar
  9. 9.
    Dyer, P. and McReynolds, S., Extension of Square-Root Filtering to Include Process Noise, J. Optim. Theory Appl., 1969, no. 3, pp. 444–459.Google Scholar
  10. 10.
    Kaminski, P.G., Bryson, A.E., and Schmidt, S.F., Discrete Square-Root Filtering: A Survey of Current Techniques, IEEE Trans. Automat. Control, 1971, vol. AC-16, no. 6, pp. 727–735.CrossRefGoogle Scholar
  11. 11.
    Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, New York: Academic, 1977.zbMATHGoogle Scholar
  12. 12.
    Kailath, T., Some New Algorithms for Recursive Estimation in Constant Linear Systems, IEEE Trans. Inf. Theory, 1973, vol. IT-19, no. 11, pp. 750–760.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Morf, M., Sidhu, G.S., and Kailath, T., Some New Algorithms for Recursive Estimation in Constant, Linear Discrete-Time Systems, IEEE Trans. Automat. Control, 1974, vol. AC-19, no. 4, pp. 315–323.CrossRefGoogle Scholar
  14. 14.
    Morf, M. and Kailath, T., Square-Root Algorithms for Least-Squares Estimation, IEEE Trans. Automat. Control, 1975, vol. AC-20, no. 4, pp. 487–497.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sayed, A.H. and Kailath, T., Extended Chandrasekhar Recursions, IEEE Trans. Automat. Control, 1994, vol. 39, no. 3, pp. 619–622.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Park, P. and Kailath, T., New Square-Root Algorithms for Kalman Filtering, IEEE Trans. Automat. Control, 1995, vol. 40, no. 5, pp. 895–899.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Zhdanov, A.I., Direct Recursive Algorithms for Solving Linear Problems in Least Squares Estimation, Zh. Vychisl. Mat. Mat. Fiz., 1994, vol. 34, no. 6, pp. 805–814.MathSciNetGoogle Scholar
  18. 18.
    Lee, E.K.B. and Haykin, S., Parallel Implementation of the Eextended Square-Root Covariance Filter for Tracking Applications, IEEE Trans. Parallel Distr. Syst., 1993, vol. 4, no. 4, pp. 446–457.CrossRefGoogle Scholar
  19. 19.
    Hassibi, B. and Sayed, A.H., Array Algorithms for H Estimation, IEEE Trans. Automat. Control, 2000, vol. 45, no. 4, pp. 702–706.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Rodriguez, A. and Ruiz, E., Bootstrap Prediction Mean Squared Errors of Unobserved States Based on the Kalman Filter with Estimated Parameters, Computat. Statist. Data Anal., 2012, vol. 56, pp. 62–74.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fomin, V.N., Rekurrentnoe otsenivanie i adaptivnaya fil’tratsiya (Recursive Estimation and Adaptive Filtering), Moscow: Nauka, 1984.Google Scholar
  22. 22.
    Golub, G. and Van Loan, Ch., Matrix Computations, Baltimor: John Hopkins Univ. Press, 1989, 2nd ed. Translated under the title Matrichnye vychisleniya, Moscow: Mir, 1999.zbMATHGoogle Scholar
  23. 23.
    Kulikova, M.V., Likelihood Gradient Evaluation Using Square-Root Covariance Filters, IEEE Trans. Automat. Control, 2009, vol. 54, no. 3, pp. 646–651.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tsyganova, Yu.V., Computing the Gradient of the Auxiliary Quality Functional in the Parametric Identification Problem for Stochastic Systems, Autom. Remote Control, 2011, vol. 72, no. 9, pp. 1925–1940.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Verdult, V. and Verhaegen, M., Subspace Identification of Multivariable Linear Parameter-Varying Systems, Automatica, 2002, no. 38, pp. 805–814.Google Scholar
  26. 26.
    Venttsel’, E.S., Teoriya veroyatnostei (Probability Theory), Moscow: Vysshaya Shkola, 1999.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • Yu. V. Tsyganova
    • 1
  • M. V. Kulikova
    • 2
  1. 1.Ul’yanovsk State UniversityUl’yanovskRussia
  2. 2.Lisbon Technical UniversityLisbonPortugal

Personalised recommendations