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State space approach to the classical interpolation problem and its applications

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Three Decades of Mathematical System Theory

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 135))

Abstract

The Pick-Nevanlinna interpolation theory in classical analysis plays an important role in the recent progress of linear system theory in the frequency domain. In this paper, we shall show how the classical interpolation theory which relies heavily on function theoretic properties is described in the algebraic framework of the state space. The notion of conjugation, or more specifically, of J-lossless conjugation is shown to be the state-space representation of the classical interpolation, or its modern versions. Thus, the notion of J-lossless conjugation provides a unified treatment of the H ∞ control problems, as well as of robust stabilization and the model reduction.

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References

  1. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979.

    Google Scholar 

  2. J. A. Ball, "Nevanlinna-Pick interpolation: Generalizations and applications," Proc. of Special Year in Operator Theory, Indiana Universyity 1985–1986.

    Google Scholar 

  3. J. A. Ball and J. W. Helton, "A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory, "J. Operator Theory, vol.9, pp. 107–142, 1983

    Google Scholar 

  4. B. C. Chang and J. B. Pearson, "Optimal disturbance reduction in linear multivariable systems," IEEE Trans. Automat. Contr., vol. AC-29, pp. 880–887, 1984.

    Google Scholar 

  5. P. Delsarte, Y. Genin and Y. Kamp, "The Nevanlinna-Pick problem for matrix-valued functions," SIAM J. of Appl. Math., vol. 36, pp. 47–61, 1979.

    Article  Google Scholar 

  6. P. Delsarte, Y. Genin and Y. Kamp, "On the role of the Nevanlinna-Pick problem in circuit and system theory," Circuit Th. and Appl., vol.9, pp.177–187, 1981.

    Google Scholar 

  7. P. Dewilde and H. Dym, "Lossless chain scattering matrices and optimal linear prediction: The vector case," Circuit Theory Appl., vol. 9, pp. 135–175, 1981.

    Google Scholar 

  8. J. C. Doyle, "Synthesis of robust controllers and filters," Proc. IEEE Conf. Decision and Control, San Antonio, pp. 109–114, 1983.

    Google Scholar 

  9. H. Dym, "J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation," Monograph, Dept. of Theoretical Math., The Weizmann Inst. Science, 1988.

    Google Scholar 

  10. R. M. Fano, "Theoretical limitations on the broadband matching of arbitary impedances," J. Franklin Inst., vol. 249, pp. 57–83, 1960.

    Article  Google Scholar 

  11. I. P. Fedcina, "Solvability criteria of the Nevanlinna-Pick tangent problem," Mat. Issled. Kinshinev (in Russian) yp 4, pp. 213–227, 1972.

    Google Scholar 

  12. B. A. Francis and G. Zames, "On H ∞-optimal sensitivity theory for SISO feedback systems," IEEE Trans. Automat. Contr., vol. AC-29, pp. 880–887, 1984.

    Google Scholar 

  13. B. A. Francis, J. W. Helton and G. Zames, "H ∞ optimal feedback controllers for linear multivariable systems," IEEE Trans. Automat. Contr., vol. AC-29, pp. 888–900, 1984.

    Article  Google Scholar 

  14. B. A. Francis, A Course in tH ∞ Control Theory, Springer Verlag, New York, 1987.

    Google Scholar 

  15. J. S. Freudenberg and D. P. Looze, Frequency Domain Properties of Scalar and Multivariable Feedback Systems, Springer, New York 1988.

    Google Scholar 

  16. Y. Genin, P. van Dooren and T. Kailath, "On Σ-lossless transfer functions and related questions," Linear Algebra and Appl., vol. 50 pp. 251–275, 1983.

    Article  Google Scholar 

  17. T. T. Georgiou and P.P. Khargonekar, "Spectral factorization and Nevanlinna-Pick interpolation," SIAM J. of Control & Optimiz., vol.25, pp.754–766, 1987.

    Google Scholar 

  18. J. W. Helton, "Broadbanding: Gain equalization directly from data," IEEE Trans. Circuit and Systems, vol. CAS-28, pp. 1125–1137, 1981.

    Article  Google Scholar 

  19. J. W. Helton, "Non-Euclidean functional analysis and electronics," Bull. Amer. Math. Soc., vol. 7, pp. 1–64, 1982.

    Google Scholar 

  20. F. Itakura and S. Saito, "Digital filtering techniques for speech analysis and synthesis," in Proc. 7th Int. Cong., Acoust., Budapest, Paper 25-c-1, pp. 261–264, 1971.

    Google Scholar 

  21. V. E. Katsnel'son "Methods of J-theory in continuous interpolation problems of analysis," Harihov, translated by T. Ando, 1982.

    Google Scholar 

  22. R. Kawatani and H. Kimura, "Synthesis of reduced-order H ∞ controller," to appear in Int. J. Control.

    Google Scholar 

  23. P.P. Khargonekar and A. Tannenbaum, "Non-Euclidian metrics and the robust stabilization of systems with parameter uncertainty," IEEE Trans. Automat. Control, vol. AC-30, pp.1005–1013, 1985.

    Article  Google Scholar 

  24. H. Kimura, "Robust stabilizability for a class of transfer functions," IEEE Trans. Automat. Contr., vol. AC-29, pp. 788–793, 1984.

    Article  Google Scholar 

  25. H. Kimura, "Directional interpolation approach to H∞-optimization and robust stabilization,", vol. AC-32, pp. 1085–1093, 1987.

    Google Scholar 

  26. H. Kimura, "Directional interpolation in the state space," Systems and Control Letters, vol. 10, pp. 317–324, 1988.

    Article  Google Scholar 

  27. H. Kimura, "Conjugation, interpolation and model-matching in H∞," Int. J. Control, vol. 49, pp. 269–307, 1989.

    Google Scholar 

  28. H. Kimura and R. Kawatani, "Synthesis of H∞ controller based on conjugation," Proc. IEEE Conf. on Decision and Control, Austin, pp. 7–13, 1988.

    Google Scholar 

  29. H. Kimura, "Conjugation of Hamilton systems and model-matching in H∞," under preparation.

    Google Scholar 

  30. D. J. Limebeer and B. D. O. Anderson, "An interpolation theory approach to H∞ controller degree bounds," Linear Algebra and its Appl., vol. 98, pp.347–386, 1988.

    Article  Google Scholar 

  31. K. Z. Liu and T. Mita, "Conjugation and H∞ control of discrete-time systems," to appear in Int. J. Control.

    Google Scholar 

  32. A. G. J. MacFarlane (ed.), Frequency-Response Methods in Control Systems, IEEE Press, NY., 1979.

    Google Scholar 

  33. D. Q. Mayne, "The design of linear multivariable systems," Automatica, vol. 9, pp. 201–207, 1973.

    Article  Google Scholar 

  34. R. Nevanlinna, "Über beschränkte Funktionen die in gegebenen Punkten vorgeschreibene Funktionswerte bewirkt werden," Ann. Acad. Sci. Fenn., Ser A, vol. 13, pp. 1–71, 1919.

    Google Scholar 

  35. G. Pick, "Über die beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden," Math. Ann.,vol. 77, pp. 7–23, 1916.

    Article  Google Scholar 

  36. V. P. Potapov, "The multiplicative structure of J-contractive matrix functions," Amer. Math. Soc. Transl., vol. 15, pp. 131–243, 1960.

    Google Scholar 

  37. A. C. M. Ran, "State space formulas for a model matching problem," Systems and Control Letters, vol. 12, pp. 17–21, 1989.

    Article  Google Scholar 

  38. H. H. Rosenbrock, Computer-Aided Control System Design, Academic Press, 1974.

    Google Scholar 

  39. D. Sarason, "Generalized interpolation in H∞," Trans. Amer. Math. Soc., vol. 127, pp. 180–203, 1967.

    Google Scholar 

  40. I. Schur, "Uber die Potenzreihen, die im Inneren des Einheitskreises beschrankt sind," 1; 2, J. Reine Angew. Math., vol. 147, pp. 205–232, 1917; vol. 148, pp. 122–145, 1918.

    Google Scholar 

  41. A. Tannenbaum, "Modified Nevanlinna-Pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor," Int. J. Control, vol. 36, pp. 331–336, 1982.

    Google Scholar 

  42. M. Vidyasagar and H. Kimura, "Robust controllers for uncertain linear multivariable systems," Automatica, vol. 22, pp. 85–94, 1986.

    Article  Google Scholar 

  43. D. C. Youla, "A new theory of broadband matching," IEEE Trans. on Circuit Th., vol. CT-11, pp. 30–50, 1964.

    Google Scholar 

  44. D. C. Youla and M. Saito, "Interpolation with positive real functions," J of Franklin Inst., vol. 284, pp.77–108, 1967.

    Article  Google Scholar 

  45. G. Zames, "Feedback and optimal sensitivity; model reference transformations, multiplicative seminorms, and approximate inverses," IEEE Trans. Automat. Control, vol. AC-23, pp. 301–320, 1981.

    Article  Google Scholar 

  46. G. Zames and B. A. Francis, "Feedback, minimax sensitivity and optimal robustness," IEEE Trans. Automat. Contr., vol. AC-28, pp.585–601, 1983.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering for Computer-Controlled Machinery, Osaka University, 2-1, Yamada-oka, 565, Suita, Japan

    H. Kimura

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  1. H. Kimura
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Hendrik Nijmeijer Johannes M. Schumacher

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© 1989 Springer-Verlag

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Kimura, H. (1989). State space approach to the classical interpolation problem and its applications. In: Nijmeijer, H., Schumacher, J.M. (eds) Three Decades of Mathematical System Theory. Lecture Notes in Control and Information Sciences, vol 135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0008465

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  • DOI: https://doi.org/10.1007/BFb0008465

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51605-7

  • Online ISBN: 978-3-540-46709-0

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