Skip to main content
SpringerLink
Log in

Algorithms for robust identification in ℋ∞ with nonuniformly spaced frequency response data

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

In this paper, first a two-stage robustly covergent identification algorithm in ℋ∞ for nonuniformly spaced data is proposed. The worst-case error of the algorithm converges to zero faster than polynomial rates in the noise-free case when the identified system is an exponentially stable discrete-time system. The algorithm is characterized by a rational interpolation step with fixed poles at zero and infinity. Next, a minimax algorithm with better convergence properties is introduced. Sensitivity of the algorithms to small variations in the frequency values is also studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [AGK1] H. Akçay, G. Gu, and P. P. Khargonekar. A class of algorithms for identification in ℋ∞: continuous-time case.IEEE Trans. Automat. Control, 38:289–294, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  • [AGK2] H. Akçay, G. Gu, and P. P. Khargonekar. Identification in ℋ∞ with nonuniformly spaced frequency response measurements.Internat. J. Robust Nonlinear Control, 4:613–629, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • [BP] I. Barrodale and C. Phillips. Algorithm 495, solution of an overdetermined system of linear equations in the Chebyshey norm [f4].ACM Trans. Math. Software, 1:264–270, 1975.

    Article  MATH  Google Scholar 

  • [CN] J. Chen and C. N. Nett. The Carethéodory-Fejér problem and ℋ∞ identification: a time domain approach.IEEE Trans. Automat. Control, 40:729–735, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • [CNF1] J. Chen, C. N. Nett, and M. K. H. Fan. Optimal nonparametric identification from arbitrary corrupt finite time series.IEEE Trans. Automat. Control., 40:769–776, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • [CNF2] J. Chen, C. N. Nett, and M. K. H. Fan. Worst case system identification in ℋ∞: validation ofa priori information, essentially optimal algorithms, and error bounds.IEEE Trans. Automat. Control, 40:1260–1265, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • [FK] J. H. Friedman and P. P. Khargonekar. Application of identification in ℋ∞ to lightly damped systems: two case studies.IEEE Trans. Control Systems Technol., 3(3):279–289, 1995.

    Article  Google Scholar 

  • [G] G. Gu. Suboptimal algorithms for worst-case identification in ℋ∞ and model validation.IEEE Trans. Automat. Control, 39:1657–1661, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • [GK1] G. Gu and P. P. Khargonekar. A class of algorithms for identification in ℋ∞.Automatica, 28:229–312, 1992.

    Article  MathSciNet  Google Scholar 

  • [GK2] G. Gu and P. P. Khargonekar. Linear and nonlinear algorithms for identification in ℋ∞ with error bounds.IEEE Trans. Automat. Control, 37:953–963, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  • [GKL] G. Gu, P. P. Khargonekar, and Y. Li. Robust convergence of two-stage nonlinear algorithms for identification in ℋ∞.Systems Control Lett., 18:253–263, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  • [GXZ] G. Gu, D. Xiong, and K. Zhou. Identification in ℋ∞ using Pick’s interpolation.Systems Control Lett., 20:263–272, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  • [HJN1] A. J. Helmicki, C. A. Jacobson, and C. N. Nett. Control-oriented system identification: a worst-case/deterministic approach in ℋ∞.IEEE Trans. Automat. Control, 36:1163–1176, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • [HJN2] A. J. Helmicki, C. A. Jacobson, and C. N. Nett. Worst-case/deterministic identification in ℋ∞: the continuous-time case.IEEE Trans. Automat. Control, 37:604–610, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  • [HJ] R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, 1985.

  • [K] S. Y. Kung. A new identification and model reduction algorithm via singular value decomposition. InProc. 12th Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, CA, pages 705–714, 1978.

  • [M] P. M. Mäkilä. Robust identification and Galois sequences.Internat. J. Control, 54:1189–1200, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • [MP] P. M. Mäkilä and J. R. Partington. Robust identification of strongly stabilizable systems.IEEE Trans. Automat. Control, 37:1709–1716, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  • [P1] J. R. Partington. Robust identification and interpolation in ℋ∞.Internat. J. Control, 54: 1281–1290, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • [P2] J. R. Partington. Robust identification in ℋ∞.J. Math. Anal. Appl., 166:428–441, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  • [P3] J. R. Partington. Algorithms for identification in ℋ∞ with unequally spaced function measurements.Internat. J. Control., 58:21–31, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  • [P4] J. R. Partington. Interpolation in normed spaces from the values of linear functionals.Bull. London Math. Soc., 26:165–170, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • [P5] J. R. Partington. Recovery of functions by interpolation and sampling.J. Math. Anal. Appl., 198:301–309, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  • [P6] A. Pinkus.n-Widths in Approximation Theory. Springer-Verlag, Berlin, 1985.

    MATH  Google Scholar 

  • [RAG] L. Reichel, G. S. Ammar, and W. B. Gragg. Discrete least squares approximation by trigonometric polynomials.Math. Comp., 57:273–289, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • [S1] L. L. Schumaker.Spline Functions. Wiley, New York, 1981.

    MATH  Google Scholar 

  • [S2] G. Somorjai. On discrete linear operators in the function spaceA. InProc. Constructive Function Theory, Blagoevgrad, 1977, pages 489–500. H. Bulgarian Academy of Science, Sofia, 1980.

    Google Scholar 

  • [Z] A. Zygmund.Trigonometric Series. Cambridge University Press. Cambridge, 1959.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Mathematics, Tübitak, Marmara Research Center, P. O. Box 21, Gebze-Kocaeli, Turkey

    Hüseyin Akçay

Authors
  1. Hüseyin Akçay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hüseyin Akçay.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akçay, H. Algorithms for robust identification in ℋ∞ with nonuniformly spaced frequency response data. Math. Control Signal Systems 11, 161–181 (1998). https://doi.org/10.1007/BF02741889

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02741889

Key words

Search

Navigation