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2009 | 19 | 3 | 485-499
Tytuł artykułu

Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Interval arithmetic techniques such as VALENCIA-IVP allow calculating guaranteed enclosures of all reachable states of continuous-time dynamical systems with bounded uncertainties of both initial conditions and system parameters. Considering the fact that, in naive implementations of interval algorithms, overestimation might lead to unnecessarily conservative results, suitable consistency tests are essential to obtain the tightest possible enclosures. In this contribution, a general framework for the use of constraints based on physically motivated conservation properties is presented. The use of these constraints in verified simulations of dynamical systems provides a computationally efficient procedure which restricts the state enclosures to regions that are physically meaningful. A branch and prune algorithm is modified to a consistency test, which is based on these constraints. Two application scenarios are studied in detail. First, the total energy is employed as a conservation property for the analysis of mechanical systems. It is shown that conservation properties, such as the energy, are applicable to any Hamiltonian system. The second scenario is based on constraints that are derived from decoupling properties, which are considered for a high-dimensional compartment model of granulopoiesis in human blood cell dynamics.
Rocznik
Tom
19
Numer
3
Strony
485-499
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-09-22
poprawiono
2008-12-15
Twórcy
  • 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
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  • de Figueiredo, L. H., van Iwaarden, R. and Stolfi, J. (1997). Fast interval branch-and-bound methods for unconstrained global optimization with affine arithmetic, Technical Report IC-97-08, Institute of Computing, University of Campinas, Campinas, Brazil.
  • Hofer, E. P., Fan, Y. and Tibken, B. (1991a). Extraction of rules for model based estimation of granulocytopoiesis, in M. Frik (Ed.), 5th German-Japanese Seminar Nonlinear Problems in Dynamical Systems-Theory and Applications, Daun, Vulkaneifel, pp. 58-68.
  • Hofer, E. P., Tibken, B. and Fliedner, T. M. (1991b). Modern control theory as a tool to describe the biomathematical model of granulocytopoiesis, in D. Möller and O. Richter (Eds.), Analyse dynamischer Systeme in Medizin, Biologie, Ökologie, Vol. 275, Springer-Verlag, Berlin, pp. 33-39.
  • Kearfott, R. B. (1992). An interval branch and bound algorithm for bound constrained optimization problems, Journal of Global Optimization 2(3): 259-280.
  • Keil, C. (2007). PROFIL/BIAS, Version 2.0.4, Available at: http://www.ti3.tu-harburg.de/keil/profil/.
  • Maschke, B. M. and van der Schaft, A. J. (2000). Portcontrolled Hamiltonian representation of distributed parameter systems, in N. E. Leonard and R. Ortega (Eds.), Proceedings of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, NJ, USA, pp. 28-38.
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  • Nedialkov, N. S. (2007). Interval tools for ODEs and DAEs, 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.
  • Pfeiffer, F. and Reithmeier, E. (1987). Roboterdynamik, Teubner, Stuttgart, (in German).
  • Rauh, A. (2008). Theorie und Anwendung von Intervallmethoden für Analyse und Entwurf robuster und optimaler Regelungen dynamischer Systeme, FortschrittBerichte VDI, Reihe 8, Nr. 1148, PhD thesis, University of Ulm, Ulm, (in German).
  • Rauh, A., Auer, E. and Hofer, E. P. (2007). 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., Brill, M. and Günther, C. (2009). A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP, International Journal of Applied Mathematics and Computer Science 19(3): 381-397.
  • Singer, A. B. and Barton, P. I. (2006). Bounding the solutions of parameter dependent nonlinear ordinary differential equations, SIAM Journal on Scientific Computing 27(6): 2167-2182.
  • The American Heritage Medical Dictionary (2007). Houghton Mifflin Company, Boston, MA.
  • van der Schaft, A. J. (2005). Network modeling and control of physical systems, DISC theory of port-Hamiltonian systems, Available at: http://www.vf.utwente.nl/~schaftaj/downloads-diversen/DISCportbased1.pdf.
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv19i3p485bwm
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