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Robust Stability for Parametric Linear ODEs

  • Volker Weispfenning
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4770)

Abstract

The study of linear ordinary differential equations (ODEs) with parametric coefficients is an important topic in robust control theory. A central problem is to determine parameter ranges that guarantee certain stability properties of the solution functions. We present a logical framework for the formulation and solution of problems of this type in great generality. The function domain for both parametric functions and solutions is the differential ring D of complex exponential polynomials. The main result is a quantifier elimination algorithm for the first-order theory T of D in a language suitable for global and local stability questions, and a resulting decision procedure for T. For existential formulas the algorithm yields also parametric sample solution functions. Examples illustrate the expressive power and algorithmic strengh of this approach concerning parametric stability problems. A contrasting negative theorem on undecidability shows the boundaries of extensions of the method.

Keywords

Normal Form Function Variable Robust Stability Solution Function Atomic Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ackermann, J.: Robust Control. Communication and Control Engineering (1993)Google Scholar
  2. 2.
    Anai, H., Hara, S.: Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In: Proceedings of ACC 2000 (to appear)Google Scholar
  3. 3.
    Braun, M.: Differential equations and their applications. In: Applied Mathematical Sciences, 3rd edn., Springer, Heidelberg (1983)Google Scholar
  4. 4.
    Brown, C.W.: Improved projection for cylindrical algebraic decomposition. J. Symb. Computation 32(5), 447–465 (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5(1–2), 29–35 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dolzmann, A., Gilch, L.A.: Generic hermitian quantifier elimination. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 80–92. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Dolzmann, A., Seidl, A.: Redlog – first-order logic for the masses. Journal of Japan Society for Symbolic and Algebraic Computation 10(1), 23–33 (2003)Google Scholar
  8. 8.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.-M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Berlin (1998)Google Scholar
  9. 9.
    Dorato, P., Yang, W., Abdallah, C.: Robust multi-object feedback design by quantifier elimination. J. Symb. Computation 24(2), 153–159 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hong, R., Liska, H., Steinberg, S.: Testing stability by quantifier elimination. J. Symb. Comp. 24(2), 161–187 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Liska, R., Steinberg, S.: Applying quantifier elimination to stability analysis of difference schemes. The Computer Journal 36, 497–509 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Seidl, A.: Cylindrical Decomposition under Application-Oriented Paradigms. PhD thesis, FMI, Univ. Passau (2006)Google Scholar
  13. 13.
    Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In: Sendra, R. (ed.) Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC 2003), Philadelphia, Pennsylvania, pp. 240–247. ACM Press, New York (2003)CrossRefGoogle Scholar
  14. 14.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1–2), 3–27 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Weispfenning, V.: Parametric linear and quadratic optimization by elimination. Technical Report MIP-9404, FMI, Universität Passau, D-94030 Passau, Germany (April 1994)Google Scholar
  16. 16.
    Weispfenning, V.: Solving linear differential problems with parameters. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 469–488. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Volker Weispfenning
    • 1
  1. 1.University of Passau, D-94030 PassauGermany

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