Quantifier elimination approach to frequency domain design

  • Peter Dorato
  • Wei Yang
  • Chaouki T. Abdallah
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 227)


The design, in the frequency domain, of robust feedback systems with bounded control effort can be reduced to quantifier elimination problem. However due to computational complexities, only problems of modest size can be solved. Nevertheless it may be possible to solve some practical problems where no analytic design procedures exist.


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  1. 1.
    C. Abdallah, P. Dorato, W. Yang, R. Liska, and S. Steinberg, Applications of quantifier elimination theory to control system design, 4th IEEE Mediterranean Symposium on Control & Automation, Chania, Crete, Greece, June 10–14, 1996.Google Scholar
  2. 2.
    B.D.O. Anderson, N.K. Bose, and E.I. Jury, Output feedback and related problems-Solution via Decision methods, IEEE Trans. on Automatic Control, AC-20, pp.53–65, 1975.Google Scholar
  3. 3.
    S. Basu, R. Pollack, and M.F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, Proc. 35th Symposium on Foundations of Computer Science, Santa Fe, NM, pp. 632–641, 1994.Google Scholar
  4. 4.
    G. E. Collins, Quantifier Elimination in the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition, Lecture Notes in Computer Science, Spring Verlag, Berlin, Vol. 33, pp. 134–183, 1975.Google Scholar
  5. 5.
    G.E. Collins and H. Hong, Partial cylindrical algebraic decomposition for quantifier elimination, J. Symbolic Computation, 12, pp. 299–328, 1991.Google Scholar
  6. 6.
    P. Dorato, W. Yang, and C. Abdallah, Robust multi-objective feedback design by quantifier elimination, submitted to J. Symbolic Computation, 1996.Google Scholar
  7. 7.
    G.Fiorio, S. Malan, M.Milanese, and M. Taragna, Robust Performance Design of Fixed Structure Controller with Uncertain Parameters, Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX, pp. 3029–3031.Google Scholar
  8. 8.
    H. Hong, Improvements in CAD-based Quantifier Elimination, Ph.D. Thesis, The Ohio State University, 1990.Google Scholar
  9. 9.
    H. Hong, Simple Solution Formula Construction in Cylindrical Algebraic Decomposition based Quantifier Elimination, ISSAC'92, International Symposium on Symbolic and Algebraic Computation, July 27–29, Berkeley, California (Editor P.S. Wang), ACM Press, New York, pp. 177–188, 1992.Google Scholar
  10. 10.
    R. Liska and S. Steinberg, Applying Quantifier Elimination to Stability Analysis of Difference Schemes, The Computer Journal, vol. 36, No. 5, pp. 497–503, 1993.CrossRefGoogle Scholar
  11. 11.
    A. Seidenberg, A New Decision Method for Elementary Algebra, Annals of Math., 60, pp. 365–374, 1954.Google Scholar
  12. 12.
    M. Jirstrand, Algebraic methods for modeling and design in control, Linköping Studies in Science and Technology, Thesis no. 540, Linköping University, 1996.Google Scholar
  13. 13.
    V.L. Syrmos, C.T. Abdallah, P.Dorato, and K. Grigoriadis, Static Output Feedback: A Survey, Scheduled for publication in Automatica, Feb., 1997.Google Scholar
  14. 14.
    A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd Ed., Berkeley, University of California Press, 1951.Google Scholar

Copyright information

© Springer-Verlag London Limited 1997

Authors and Affiliations

  • Peter Dorato
    • 1
  • Wei Yang
    • 1
  • Chaouki T. Abdallah
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of New MexicoAlbuquerqueUSA

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