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The polynomial approach to H ∞-optimal regulation

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H∞-Control Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1496))

Abstract

After a review of the “polynomial approach” to the solution of the single-input-single-output minimum sensitivity and mixed sensitivity problems, a new solution is presented of the standard H ∞-optimal regulation problem based on polynomial J-spectral factorization. Besides a parametrization of all suboptimal compensators an explicit expression for all optimal compensators is obtained.

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References

  • P. Boekhoudt (1988), The H ∞ Control Design Method: A Polynomial Approach. Ph. D. Dissertation, University of Twente.

    Google Scholar 

  • F. M. Callier (1985), “On polynomial matrix spectral factorization by symmetric extraction.” IEEE Trans. Auto. Con., Vol. AC-30, pp. 453–464.

    Article  MathSciNet  Google Scholar 

  • F. M. Callier and C. A. Desoer (1982), Multivariable Feedback Systems. New York, etc.: Springer-Verlag.

    Book  Google Scholar 

  • J. C. Doyle (1979), “Robustness of multiloop linear feedback systems.” Proc. 17th IEEE Conf. Decision Control, pp. 12–18.

    Google Scholar 

  • J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis (1989), “State space solutions to standard H 2 and H ∞ control problems.” IEEE Trans. Aut. Cont., Vol. 34, pp. 831–847.

    Article  MATH  MathSciNet  Google Scholar 

  • B. A. Francis (1987), A Course in H ∞ Control Theory. Lecture Notes in Control and Information Sciences. Corrected first printing. Berlin etc.: Springer-Verlag.

    Book  MATH  Google Scholar 

  • B. A. Francis and J. C. Doyle (1987). “Linear control theory with an H ∞ optimality criterion.” SIAM J. Control Opt., Vol. 25, pp. 815–844.

    Article  MATH  MathSciNet  Google Scholar 

  • K. Glover and J. C. Doyle (1989), “A state space approach to H ∞ optimal control.” In: H. Nijmeijer and J. M. Schumacher (Eds.), Three Decades of Mathematical System Theory. Lecture Notes in Control and Information Sciences, Vol. 135. Berlin, etc.: Springer-Verlag.

    Chapter  Google Scholar 

  • M. Green (1989), “H ∞ controller synthesis by J-lossless coprime factorization.” Preprint, Department of Electrical Engineering, Imperial College, London.

    Google Scholar 

  • V. A. Jacubovič (1970), “Factorization of symmetric matrix polynomials.” Dokl. Akad. Nauk. SSSR, Tom 194, No. 3.

    Google Scholar 

  • T. Kailath (1980), Linear Systems. Englewood Cliffs, N. J.: Prentice-Hall.

    MATH  Google Scholar 

  • V. Kučera (1979), Discrete Linear Control: The Polynomial Equation Approach. Chichester, etc.: John Wiley.

    MATH  Google Scholar 

  • H. Kwakernaak (1983), “Robustness optimization of linear feedback systems.” Preprints, 22nd IEEE Conference on Decision and Control, San Antonio, Texas, U.S.A.

    Google Scholar 

  • H. Kwakernaak (1985), “Minimax frequency domain performance and robustness optimization of linear feedback systems.” IEEE Trans. Auto. Cont., Vol. AC-30, pp. 994–1004.

    Article  MathSciNet  Google Scholar 

  • H. Kwakernaak (1986), “A polynomial approach to minimax frequency domain optimization of multivariable systems.” Int. J. Control, Vol. 44, pp. 117–156.

    Article  MATH  MathSciNet  Google Scholar 

  • H. Kwakernaak (1987), “A polynomial approach to H ∞-optimization of control systems.” In: R. F. Curtain (Ed.),” Modelling, Robustness and Sensitivity Reduction in Control Systems. Berlin, etc.: Springer-Verlag.

    Google Scholar 

  • H. Kwakernaak (1990), “Progress in the polynomial solution of the standard H ∞ optimal control problem.” In: Preprints of the 1990 IFAC Congress, Tallinn, USSR.

    Google Scholar 

  • H. Kwakernaak and P.-O. Nyman (1989), “An equalizing approach to superoptimization.” Proc. 1989 IEEE Int. Conf. Control and Applications, Jerusalem.

    Google Scholar 

  • H. Kwakernaak and H. Westdijk (1985), “Regulability of a multiple inverted pendulum system.” Control-Theory and Advanced Technology, Vol. 1, pp. 1–9.

    Google Scholar 

  • D. J. N. Limebeer, G. D. Halikias, and K. Glover (1989), “State-space algorithms for the computation of superoptimal matrix interpolation functions.” Int. J. Cont., Vol. 50, pp. 2431–2466.

    Article  MATH  MathSciNet  Google Scholar 

  • I. Postlethwaite, M. C. Tsai, and D.-W. Gu (1989), “A state-space approach to discrete-time super-optimal H ∞ control problems.” Int. J. Cont., Vol. 49, pp 247–268.

    MATH  MathSciNet  Google Scholar 

  • M. C. Tsai, D.-W. Gu, and I. Postlethwaite (1988), “A state-space approach to super-optimal H ∞ control problems,” IEEE Trans. Aut. Cont., Vol. AC-33, pp. 833–843.

    Article  MathSciNet  Google Scholar 

  • M. Verma and E. Jonckheere (1984), “L ∞-compensation with mixed sensitity as a broadband matching problem.” Systems and Control Letters, Vol. 4, pp. 125–130.

    Article  MATH  MathSciNet  Google Scholar 

  • D. C. Youla, J. J. Bongiorno, and H. A. Jabr (1976), “Modern Wiener-Hopf Design of Optimal Controllers. Part I: The Single-Input-Output Case.” IEEE Trans. Auto. Cont., Vol. AC-21, pp. 3–13.

    Article  MathSciNet  Google Scholar 

  • N. J. Young (1986), “The Nevanlinna-Pick problem for matrix-valued functions.” J. Op. Th., Vol. 15, pp. 239–265.

    MATH  Google Scholar 

  • G. Zames (1981), “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and aproximate inverses.” IEEE Trans. Auto. Cont., Vol. AC-26, pp. 301–320.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Twente, P. O. Box 217, 7500 AE, Enschede, The Netherlands

    Huibert Kwakernaak

Authors
  1. Huibert Kwakernaak
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Editor information

Editors and Affiliations

  1. Dipartimento di Sistemi e Informatica, Università di Firenze, Via di Santa Marta 3, 1-50139, Firenze, Italy

    Edoardo Mosca

  2. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129, Torino, Italy

    Luciano Pandolfi

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© 1991 Springer-Verlag Berlin Heidelberg

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Kwakernaak, H. (1991). The polynomial approach to H ∞-optimal regulation. In: Mosca, E., Pandolfi, L. (eds) H∞-Control Theory. Lecture Notes in Mathematics, vol 1496. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084469

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  • DOI: https://doi.org/10.1007/BFb0084469

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54949-9

  • Online ISBN: 978-3-540-46604-8

  • eBook Packages: Springer Book Archive

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