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The 2-block superoptimalAAK problem

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Abstract

This paper presents a solution to a superoptimal version of the 2-blockAAK problem: Given a rational and antistable matrix functionR(s)=[R 11(s)R 12(s)] and a nonnegative integerk, find all superoptimal approximationsQ(s), with no more thank poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error functionE(s)=[R 11 (s) R 12 (s)+Q(s)], with respect to lexicographic ordering. Conditions are given for which the superoptimal approximation is unique. In addition, ana priori upper bound on the MacMillan degree of the approximation is provided. The algorithm may be stopped after minimizing a given number of the singular values. This premature termination carries with it a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations, and is therefore easily implemented.

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References

  • [AAK] V. M. Adamjan, D. Z. Arov and M. G. Krein, Infinite block Hankel matrices and related extension problems.AMS Trans.,111 (1978), 133–156.

    Google Scholar 

  • [AV] B. D. O. Anderson and S. Vongpanitlerd,Network Analysis and Synthesis. A Modern Systems Approach, Prentice-Hall, Englewood-Cliffs, NJ, 1973.

    Google Scholar 

  • [G1] K. Glover, All optimal Hankel norm approximations of linear multivariable systems and their ℒ∞ error bounds,Internat. J. Control,39 (1984), 1115–1193.

    Article  MATH  MathSciNet  Google Scholar 

  • [G2] K. Glover, A tutorial on Hankel-norm approximation, inFrom Data to Model (ed. J. C. Willems), Springer-Verlag, New York, 1989, pp. 26–48.

    Google Scholar 

  • [GLD+] K. Glover, D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally and M. G. Safonov, A characterization of all solutions to the four-block general-distance problem.SIAM J. Control Optim.,29 (1991), 283–324.

    Article  MATH  MathSciNet  Google Scholar 

  • [GLH] K. Glover, D. J. N. Limebeer and Y. S. Hung, A structured approximation problem with applications to frequency-weighted model reduction,IEEE Trans. Automat Control,37 (1992), 447–465.

    Article  MATH  MathSciNet  Google Scholar 

  • [GTP] D. W. Gu, M. C. Tsai and I. Postlethwaite, An algorithm for superoptimal ℋ∞ design: the 2-block case.Automatica,25 (1989), 437–440.

    Article  MathSciNet  Google Scholar 

  • [H] G. D. Halikias, State-space analysis for a class of ℋ∞ optimal control problems, Ph.D. Thesis, Imperial College, University of London, 1990.

  • [HJ] G. D. Halikias, and I. M. Jaimoukha, Hierarchical optimization in, ℋ∞,Proc. 34th IEEE Conf. on Decision and Control, pp. 3664–3669, 1995.

  • [HLG] G. D. Halikias, D. J. N. Limebeer and K. Glover, State-space algorithm for the superoptimal Hankel-norma approximation problem,SIAM J. Control Optim.,31 (1993), 960–982.

    Article  MATH  MathSciNet  Google Scholar 

  • [J] I. M. Jaimoukha, The two-block super-optimal distance problem, Ph.D. Thesis, Imperial College, University of London, 1990.

  • [JL] I. M. Jaimoukha and D. J. N. Limebeer, A state-space algorithm for the solution of the 2-block superoptimal distance problem,SIAM J. Control Optim.,31 (1993), 1115–1134.

    Article  MATH  MathSciNet  Google Scholar 

  • [LCL+] N. A. Lehtomaki, D. A. Castanon, B. C. Levy, G. Stein, N. R. Sandell and M. Athans, Robustness and modeling error characterization.IEEE Trans. Automat. Control,29 (1984), 212–220.

    Article  MATH  Google Scholar 

  • [LHG] D. J. N. Limebeer, G. D. Halikias and K. Glover, State-space algorithm for the computation of super-optimal matrix interpolating functions,Internat. J. Control,50 (1989), 2431–2466.

    Article  MATH  MathSciNet  Google Scholar 

  • [PY] V. V. Peller and N. J. Young, Superoptimal approximation by meromorphic functions.Math. Proc. Cambridge Philos. Soc.,119 (1996), 497–511.

    Article  MATH  MathSciNet  Google Scholar 

  • [T] S. Treil, On superoptimal approximation by analytic and meromorphic matrix valued functions.J. Funct. Anal.,131 (1995), 386–414.

    Article  MATH  MathSciNet  Google Scholar 

  • [TGP] M. C. Tsai, D. W. Gu and I. Postlethwaite, A state space approach to superoptimal ℋ∞ control problems,IEEE Trans. Automat. Control,33 (1988), 833–843.

    Article  MATH  MathSciNet  Google Scholar 

  • [Y] N. J. Young, The Nevanlinna-Pick problem for matrix valued functions,J. Operator Theory,15 (1986), 239–265.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electronic and Electrical Engineering, University of Leeds, LS2 9JT, Leeds, England

    George D. Halikias

  2. Department of Electrical and Electronic Engineering and Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, SW7 2BT, London, England

    Imad M. Jaimoukha

Authors
  1. George D. Halikias
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  2. Imad M. Jaimoukha
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Correspondence to George D. Halikias.

Additional information

This work has been supported by Engineering and Physical Sciences Research Council Grant GR/J42533.

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Halikias, G.D., Jaimoukha, I.M. The 2-block superoptimalAAK problem. Math. Control Signal Systems 11, 244–264 (1998). https://doi.org/10.1007/BF02741893

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  • DOI: https://doi.org/10.1007/BF02741893

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