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.
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References
Ackermann, J.: Robust Control. Communication and Control Engineering (1993)
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)
Braun, M.: Differential equations and their applications. In: Applied Mathematical Sciences, 3rd edn., Springer, Heidelberg (1983)
Brown, C.W.: Improved projection for cylindrical algebraic decomposition. J. Symb. Computation 32(5), 447–465 (2001)
Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5(1–2), 29–35 (1988)
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)
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)
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)
Dorato, P., Yang, W., Abdallah, C.: Robust multi-object feedback design by quantifier elimination. J. Symb. Computation 24(2), 153–159 (1997)
Hong, R., Liska, H., Steinberg, S.: Testing stability by quantifier elimination. J. Symb. Comp. 24(2), 161–187 (1997)
Liska, R., Steinberg, S.: Applying quantifier elimination to stability analysis of difference schemes. The Computer Journal 36, 497–509 (1993)
Seidl, A.: Cylindrical Decomposition under Application-Oriented Paradigms. PhD thesis, FMI, Univ. Passau (2006)
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)
Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1–2), 3–27 (1988)
Weispfenning, V.: Parametric linear and quadratic optimization by elimination. Technical Report MIP-9404, FMI, Universität Passau, D-94030 Passau, Germany (April 1994)
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)
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Weispfenning, V. (2007). Robust Stability for Parametric Linear ODEs. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_32
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DOI: https://doi.org/10.1007/978-3-540-75187-8_32
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