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2009 | 19 | 3 | 381-397
Tytuł artykułu

A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, VALENCIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of VALENCIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.
Rocznik
Tom
19
Numer
3
Strony
381-397
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-09-22
poprawiono
2008-12-15
Twórcy
autor
  • Chair of Mechatronics, University of Rostock, D-18059 Rostock, Germany
  • Institute of Measurement, Control, and Microtechnology, University of Ulm, D-89069 Ulm, Germany
  • Institute of Measurement, Control, and Microtechnology, University of Ulm, D-89069 Ulm, Germany
Bibliografia
  • Auer, E., Rauh, A., Hofer, E. P. and Luther, W. (2008). Validated modeling of mechanical systems with S MART MOBILE: Improvement of Performance by VALENCIA-IVP, Proceedings of the Dagstuhl Seminar 06021-Reliable Implementation of Real Number Algorithms: Theory and Practice, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5045, Springer-Verlag, Berlin/Heidelberg, pp. 1-27.
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  • Lin, Y. and Stadtherr, M. A. (2007). Deterministic global optimization for dynamic systems using interval analysis, CD-Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA.
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  • Nedialkov, N. S. and Pryce, J. D. (2008). DAETS-DifferentialAlgebraic Equations by Taylor Series, Available at: http://www.cas.mcmaster.ca/~nedialk/daets/.
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  • Rauh, A. (2008). Theorie und Anwendung von Intervallmethoden für Analyse und Entwurf robuster und optimaler Regelungen dynamischer Systeme, FortschrittBerichte VDI, Reihe 8, Nr. 1148, Ph.D. thesis, University of Ulm, Ulm, (in German).
  • Rauh, A. and Auer, E. (2008). Verified simulation of ODEs and DAEs in VALENCIA-IVP, Proceedings of the 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2008, El Paso, TX, USA, (under review).
  • Rauh, A., Auer, E., Freihold, M., Hofer, E. P. and Aschemann, H. (2008). Detection and reduction of overestimation in guaranteed simulations of hamiltonian systems, Proceedings of the 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2008, El Paso, TX, USA, (under review).
  • Rauh, A., Auer, E. and Hofer, E. P. (2007a). VALENCIA-IVP: A comparison with other initial value problem solvers, CD-Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA.
  • Rauh, A., Auer, E., Minisini, J. and Hofer, E. P. (2007b). Extensions of VALENCIA-IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization, Proceedings of the 6th International Congress on Industrial and Applied Mathematics, Minisymposium on Taylor Model Methods and Interval Methods-Applications, PAMM, Zurich, Switzerland, Vol. 7(1), pp. 1023001-1023002.
  • Rauh, A. and Hofer, E. P. (2009). Interval methods for optimal control, in A. Frediani and G. Buttazzo (Eds.), Proceedings of the 47th Workshop on Variational Analysis and Aerospace Engineering 2007, Erice, Italy, Springer-Verlag, New York, NY, pp. 397-418.
  • Rauh, A., Minisini, J. and Hofer, E. P. (2009). Towards the development of an interval arithmetic environment for validated computer-aided design and verification of systems in control engineering, Proceedings of the Dagstuhl Seminar 08021: Numerical Validation in Current Hardware Architectures, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5492, Springer-Verlag, Berlin/Heidelberg, pp. 175-188.
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Typ dokumentu
Bibliografia
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