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Learning HModel Sets from Data: The Set Membership Approach

  • Mario Milanese
  • Michele Taragna
Chapter
  • 306 Downloads
Part of the Trends in Mathematics book series (TM)

Abstract

The paper investigates the problem of identifying time-invariant, discrete-time, exponentially stable, possibly infinite dimensional, linear systems from noisy experimental data. The aim is to deliver not a single model, but a set of models whose size inH∞norm measures the uncertainty in the identification. The noise assumptions can account for information on its maximal magnitude and deterministic uncorrelation properties.

The paper overviews recent results of the authors, focusing on the optimality properties for finite data and on the tradeoff between optimality and complexity of the identified model set.A method is given for solving the consistency/prior validation problem, requiring the solution of one linear programming problem. An algorithm is presented for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency, i.e., the set of responses at that frequency of all systems not falsified by data. Such approximations provide a method for computing the identification error of any identified model set within any desired accuracy, while in the literature only bounds are available, which are often quite conservative. A method for evaluating an optimal model set at any given number of frequencies is presented. This optimal model set may prove to be too complex and simpler model sets may be looked for, at the expense of identification accuracy degradation. This is measured by the optimality level of the identified model set, i.e., the ratio between the achieved identification error and the minimal one. By suitably approximating the optimal model set, a ’-optimal algorithm is derived, called “nearly optimal”, thus improving over the 2-optimality of “almost optimal” algorithms available in the literature. Using these results, reduced-order model sets with nominal models inRH O are derived which tightly include the optimal model set, and their optimality levels are evaluated. Since the optimality level of reduced-order model sets can be made near to\by increasing their order, the model order selection can be performed by suitably trading off between model set complexity and optimality level degradation.

Keywords

Set Membership identification Identification for control Model sets Model quality Finite samples Complex systems 

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References

  1. 1.
    L. Andersson, A. Rantzer, and C. Beck, “Model comparison and simplification,”International Journal of Robust and Nonlinear Controlvol. 9, pp. 157–181, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    C. L. Beck, J. Doyle, and K. Glover, “Model reduction of multidimensional and uncertain systems,”IEEE Transactions on Automatic Controlvol. AC-41, no. 10, pp. 1466–1477, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J. Chen and G. GuControl-Oriented System Identification: An H Approach.New York: John Wiley & Sons, Inc., 2000.Google Scholar
  4. 4.
    J. Chen and C. N. Nett, “The Carathéodory—Fejér problem and H/t1 identification: A time domain approach,”IEEE Transactions on Automatic Controlvol. AC-40, no. 4, pp. 729–735, 1995.MathSciNetGoogle Scholar
  5. 5.
    J. Chen, C. N. Nett, and M. K. H. Fan, “Worst case system identification in H∞: Validation ofa prioriinformation, essentially optimal algorithms, and error bounds,”IEEE Transactions on Automatic Controlvol. AC-40, no. 7, pp. 1260–1265, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. Garulli, A. Tesi, and A. Vicino, Eds.Robustness in Identification and Control.Lecture Notes in Control and Information Sciences, vol. 245, Godalming, UK: Springer-Verlag, 1999.Google Scholar
  7. 7.
    L. Giarré and M. Milanese, “SM identification of approximating models for Hrobust control,”International Journal of Robust and Nonlinear Controlvol. 9, pp. 319–332, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. Giarré, M. Milanese, and M. TaragnaHidentification and model quality evaluation,”IEEE Transactions on Automatic Controlvol. AC-42, no. 2, pp. 188–199, 1997.zbMATHCrossRefGoogle Scholar
  9. 9.
    G. Gu and J. Chen, “A nearly interpolatory algorithm for Hidentification with mixed time and frequency response data,”IEEE Transactions on Automatic Controlvol. AC-46, no. 3, pp. 464–469, 2001.zbMATHCrossRefGoogle Scholar
  10. 10.
    G. Gu and P. P. Khargonekar, “A class of algorithms for identification inH “ Automaticavol. 28, no. 2, pp. 299–312, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    G. Gu, D. Xiong, and K. Zhou, "Identification inH using Pick“s interpolation,”Systems Control Lettersvol. 20, pp. 263–272, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    A. J. Helmicki, C. A. Jacobson, and C. N. Nett, “Control oriented system identification: A worst-case/deterministic approach inH “IEEE Transactions on Automatic Control, vol. AC-36, no. 10, pp. 1163–1176, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    I. Jikuya and H. Kimura, “Representation and reduction of model sets,” inProc. of the 38th IEEE Conference on Decision and ControlPhoenix, AZ, 1999, pp. 1482–1487.Google Scholar
  14. 14.
    M. A. Kon and R. Tempo, “On linearity of spline algorithms,”Journal of Complexityvol. 5, no. 2, pp. 251–259, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    L. LjungSystem Identification: Theory for the User2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999.Google Scholar
  16. 16.
    L. Ljung and L. Guo, “The role of model validation for assessing the size of the unmodeled dynamics,”IEEE Transactions on Automatic Controlvol. AC-42, no. 9, pp. 1230–1239, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P.M.MäkiläJ.R.PartingtonT.K.Gustafsson“Worst-case control-relevant identification”Automatica vol. 31, no. 12, pp. 1799–1819, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M. Milanese, J. Norton, H. Piet-Lahanier, and é. Walter, Eds.Bounding Approaches to System Identification.New York: Plenum Press, 1996.zbMATHGoogle Scholar
  19. 19.
    M. Milanese and M. Taragna, “Inputs for convergent SM identification with approximated models,” inProc. of the 37th IEEE Conference on Decision and ControlTampa, FL, 1998, pp. 4458–4463.Google Scholar
  20. 20.
    M. Milanese and M. Taragna, “Suboptimality evaluation of approximated models inH midentification,“ inProc. of the 38th IEEE Conference on Decision and ControlPhoenix, AZ, 1999, pp. 1494–1499.Google Scholar
  21. 21.
    M. Milanese and M. Taragna, “Set Membership identification forH mrobust control design,“ inProc. of 12th IFAC Symposium on System Identification SYSID 2000Santa Barbara, CA, 2000.Google Scholar
  22. 22.
    M. Milanese and M. Taragna, “Nearly optimal model sets inH midentification,“ inProc. of the European Control Conference 2001Porto, Portugal, 2001, pp. 1704–1709.Google Scholar
  23. 23.
    M. Milanese and M. Taragna, “Optimality, approximation, and complexity in Set MembershipH midentification,“IEEE Transactions on Automatic Controlvol. AC-47, no. 10, pp. 1682–1690, 2002.MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Milanese and R. Tempo, “Optimal algorithms theory for estimation and prediction,”IEEE Transactions on Automatic Controlvol. AC-30, pp. 730–738, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M. Milanese, R. Tempo, and A. Vicino, Eds.Robustness in Identification and Control.New York: Plenum Press, 1989.zbMATHCrossRefGoogle Scholar
  26. 26.
    M. Milanese and A. Vicino, “Optimal estimation theory for dynamic systems with set membership uncertainty: An overview,”Automaticavol. 27, no. 6, pp. 997–1009, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Milanese and A. Vicino, “Information-based complexity and non-parametric worst-case system identification,”Journal of Complexityvol. 9, pp. 427–446, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    B. Ninness and G. C. Goodwin, “Estimation of model quality,”Automaticavol. 31, no. 12, pp. 1771–1797, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    F. Paganini, “A set-based approach for white noise modeling,”IEEE Transactions on Automatic Controlvol. AC-41, no. 10, pp. 1453–1465, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    P. A. Parrilo, M. Sznaier, and R. S. Sánchez Peña, “Mixed time/ frequency-domain based robust identification,”Automaticavol. 34, pp. 1375–1389, 1998.zbMATHCrossRefGoogle Scholar
  31. 31.
    J. R. Partington, “Robust identification and interpolation in H∞ International Journal of Controlvol. 54, no. 5, pp. 1281–1290, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    J. R. PartingtonInterpolation Identification and Sampling.London Mathematical Society Monographs New Series, vol. 17, New York: Clarendon Press - Oxford, 1997.Google Scholar
  33. 33.
    K. R. PopperConjectures and Refutations: The Growth of Scientific Knowledge.London, UK: Rontedge and Kegan Paul, 1969.Google Scholar
  34. 34.
    R. E. Scheid and D. S. Bayard, “A globally optimal minimax solution for spectral overbounding and factorization,”IEEE Transactions on Automatic Controlvol. AC-40, no. 4, pp. 712–716, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    R. S. Smith and M. Dahleh, Eds.The Modeling of Uncertainty in Control Systems.Lecture Notes in Control and Information Sciences, vol. 192, London, UK: Springer-Verlag, 1994.Google Scholar
  36. 36.
    M. Taragna, “Uncertainty model identification forH robust control,“ inProc. of the 37th IEEE Conference on Decision and ControlTampa, FL, 1998, pp. 3403–3405.Google Scholar
  37. 37.
    J. F. Traub, G. W. Wasilkowski, and H. WozniakowskiInformation-Based Complexity.New York: Academic Press, 1988.zbMATHGoogle Scholar
  38. 38.
    S. R. Venkatesh and M. A. Dahleh, “Identification in the presence of classes of unmodeled dynamics and noise,”IEEE Transactions on Automatic Controlvol. AC-42, no. 12, pp. 1620–1635, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    M. VidyasagarA Theory of Learning and Generalization with Application to Neural Networks and Control Systems.Springer-Verlag, 1996.Google Scholar
  40. 40.
    E. Weyer, “Finite sample properties of system identification of ARX models under mixing conditions,”Automaticavol. 36, no. 9, pp. 1291–1299, 2000.MathSciNetCrossRefGoogle Scholar
  41. 41.
    T. Zhou and H. Kimura, “Structure of model uncertainty for a weakly corrupted plant,”IEEE Transactions on Automatic Controlvol. AC-40, no. 4, pp. 639–655, 1995.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Mario Milanese
    • 1
  • Michele Taragna
    • 1
  1. 1.Dipartimento di Automatica e InformaticaPolitecnico di TorinoTorinoItaly

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