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Adaptive algorithms with filtered regressor and filtered error

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Abstract

This paper presents a unified framework for the analysis of several discrete time adaptive parameter estimation algorithms, including RML with nonvanishing stepsize, several ARMAX identifiers, the Landau-style output error algorithms, and certain others for which no stability proof has yet appeared. A general algorithmic form is defined, incorporating a linear time-varying regressor filter and a linear time-varying error filter. Local convergence of the parameters in nonideal (or noisy) environments is shown via averaging theory under suitable assumptions of persistence of excitation, small stepsize, and passivity. The excitation conditions can often be transferred to conditions on external signals, and a small stepsize is appropriate in a wide range of applications. The required passivity is demonstrated for several special cases of the general algorithm.

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References

  1. B. D. O. Anderson, R. R. Bitmead, C. R. Johnson, Jr., P. V. Kokotovic, R. L. Kosut, I. M. Y. Mareels, L. Praly, and B. D. Riedle,Stability of Adaptive Systems: Passivity and Averaging Analysis, MIT Press, Cambridge, MA, 1986.

    Google Scholar 

  2. B. D. O. Anderson and M. Gevers, Identifiability of linear stochastic systems operating under linear feedback,Automatica,18 (1982), 195–213.

    Article  MathSciNet  MATH  Google Scholar 

  3. B. D. O. Anderson and C. R. Johnson, Jr., Exponential convergence of adaptive identification and control algorithms,Automatica,18 (1982), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. R. Bitmead, Persistence of excitation conditions and the convergence of adaptive schemes,IEEE Trans. Inform. Theory,30 (1984), 183–191.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. R. Bitmead and B. D. O. Anderson, Performance of adaptive estimation algorithms in dependent random environments,IEEE Trans. Automat. Control,25 (1980), 788–794.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. R. Bitmead and C. R. Johnson, Jr., Discrete averaging principles and robust adaptive identification, inControl and Dynamic Systems: Advances in Theory and Applications (C. T. Leondes, ed.), Vol. 25, pp. 237–271, Academic Press, Orlando, FL, 1987.

    Google Scholar 

  7. S. Boyd and S. S. Sastry, On parameter convergence in adaptive control,Systems Control Lett.,3 (1983), 311–319.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Dasgupta and A. S. Bhagwat, Conditions for designing strictly positive real transfer functions for adaptive output error identification.IEEE Trans. Circuits and Systems,34 (1987), 731–736.

    Article  MathSciNet  Google Scholar 

  9. C. A. Desoer, Slowly varying discrete systemx i+1 =A i x i ,Electrons. Lett.,6 (1970), 339–340.

    Article  Google Scholar 

  10. L. Dugard and G. C. Goodwin, Global convergence of Landau’s “Output error with adjustable compensator” adaptive algorithm,IEEE Trans. Automat. Control,30 (1985), 593–595.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Faurre, M. Clerget, and F. Germain,Operateurs Rationnels Positifs, Dunod, Paris, (1979).

    MATH  Google Scholar 

  12. B. Friedlander, System identification techniques for adaptive signal processing,IEEE Trans. Acoust. Speech Signal Process.,30 (1982), 240–246.

    Article  Google Scholar 

  13. W. Hahn,Stability of Motion, Springer-Verlag, Berlin, 1967.

    MATH  Google Scholar 

  14. C. R. Johnson, Jr., A convergence proof for a hyperstable adaptive recursive filter,IEEE Trans. Inform. Theory,25 (1979), 745–749.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. R. Johnson, Jr., Adaptive IIR filtering: current results and open issues,IEEE Trans. Inform. Theory,30 (1984), 237–250.

    Article  MathSciNet  Google Scholar 

  16. C. R. Johnson, Jr., and T. Taylor, Failure of a parallel adaptive identifier with adaptive error filtering,IEEE Trans. Automat. Control,25 (1980), 1248–1250.

    Article  MATH  Google Scholar 

  17. R. L. Kosut, B. D. O. Anderson, and I. M. Y. Mareels, Stability theory for adaptive systems: methods of averaging and persistence of excitation,Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL, 1985, pp. 478–483.

  18. I. D. Landau, Unbiased recursive identification using model reference adaptive techniques,IEEE Trans. Automat. Control,21 (1976), 194–202.

    Article  MATH  Google Scholar 

  19. I. D. Landau, Elimination of the real positivity condition in the design of parallel MRAS,IEEE Trans. Automat. Control,23 (1978), 1015–1020.

    Article  MATH  Google Scholar 

  20. I. D. LandauAdaptive Control: The Model Reference Approach, Marcel Dekker, New York, 1979.

    MATH  Google Scholar 

  21. M. G. Larimore, J. R. Treichler, and C. R. Johnson, Jr., SHARF: An algorithm for adapting IIR digital filters,IEEE Trans. Acoust. Speech Signal Process.,28 (1980), 428–440.

    Article  Google Scholar 

  22. D. A. Lawrence, Adaptive System Stability Analysis via Energy Exchange, Ph.D. Thesis, Cornell University, June 1985.

  23. D. A. Lawrence and C. R. Johnson Jr., Recursive parameter identification algorithm stability analysis via π-sharing,IEEE Trans. Automat. Control,31 (1986), 16–25.

    Article  MATH  Google Scholar 

  24. L. Ljung, On positive real transfer functions and the convergence of some recursive schemes,IEEE Trans. Automat. Control,22 (1977), 539–551.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Ljung and T. Soderstrom,Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983.

    MATH  Google Scholar 

  26. D. G. Luenberger,Optimization by Vector Space Methods, Wiley, New York, 1968.

    Google Scholar 

  27. J. M. Mendel,Discrete Techniques of Parameters Estimation: The Equation Error Formulation, Marcel Dekker, New York, 1973.

    Google Scholar 

  28. B. Riedle, L. Praly, and P. V. Kokotovics, Examination of the SPR condition in output error parameter estimation,Automatica,22, (1986), 495–498.

    Article  MATH  Google Scholar 

  29. J. A. Sanders and F. Verhulst,Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  30. V. Solo, The convergence of AML,IEEE Trans. Automat. Control 24 (1979), 958–962.

    Article  MATH  Google Scholar 

  31. J. R. Treichler, C. R. Johnson, Jr., and M. G. Larimore,Theory and Design of Adaptive Filters, Wiley-Interscience, New York, 1987.

    MATH  Google Scholar 

  32. M. Vidyasagar,Nonlinear Systems Analysis, Prentice Hall, Englewood Cliffs, NJ, 1978.

    Google Scholar 

  33. B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson, Jr., Stationary and nonstationary learning characteristics of the LMS adaptive filter,Proc. IEEE,64 (1976), 1151–1162.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Wisconsin, 53706, Madison, Wisconsin, USA

    W. A. Sethares

  2. Department of Systems Engineering, Australian National University, 2601, Canberra, ACT, Australia

    B. D. O. Anderson

  3. School of Electrical Engineering, Cornell University, 14853, Ithaca, New York, USA

    C. R. Johnson Jr.

Authors
  1. W. A. Sethares
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  2. B. D. O. Anderson
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  3. C. R. Johnson Jr.
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Additional information

The first and third authors were supported by NSF Grants ECS-8506149, INT-8513400, and MIP-8608787.

Research done while at the School of Electrical Engineering, Cornell University, Ithaca, New York 14853, U.S.A.

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Sethares, W.A., Anderson, B.D.O. & Johnson, C.R. Adaptive algorithms with filtered regressor and filtered error. Math. Control Signal Systems 2, 381–403 (1989). https://doi.org/10.1007/BF02551278

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  • DOI: https://doi.org/10.1007/BF02551278

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