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Exponential Enclosure Techniques for Initial Value Problems with Multiple Conjugate Complex Eigenvalues

  • Andreas RauhEmail author
  • Ramona Westphal
  • Harald Aschemann
  • Ekaterina Auer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9553)

Abstract

The computation of guaranteed state enclosures has a large variety of applications in engineering if initial value problems for sets of ordinary differential equations are concerned. One possible scenario is the use of such state enclosures in the design and verification of linear and nonlinear feedback controllers as well as in predictive control procedures. In many of these applications, system models are characterized by a dominant linear part (commonly after a suitable coordinate transformation) and by a not fully negligible nonlinear part. To compute guaranteed state enclosures for such systems, general purpose approaches relying on a Taylor series expansion of the solution can be employed. However, they do not exploit knowledge about the specific system structure. The exponential state enclosure technique makes use of this structure, allowing users to compute tight enclosures that contract over time for asymptotically stable dynamics. This paper firstly gives an overview of exponential enclosure techniques, implemented in ValEncIA-IVP, and secondly focuses on extensions to dynamic systems with single and multiple conjugate complex eigenvalues.

Keywords

Ordinary differential equations Initial value problems Complex interval arithmetic ValEncIA-IVP 

References

  1. 1.
    Auer, E., Rauh, A., Hofer, E.P., Luther, W.: Validated modeling of mechanical systems with SmartMOBILE: improvement of performance by ValEncIA-IVP. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds.) Real Number Algorithms. LNCS, vol. 5045, pp. 1–27. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Deville, Y., Janssen, M., van Hentenryck, P.: Consistency techniques for ordinary differential equations. Constraint 7(3–4), 289–315 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin Heidelberg (2000)zbMATHGoogle Scholar
  4. 4.
    Jordan, C.: Traité des substitutions et des équations algébriques. Gauthier-Villars, Paris (1870). in FrenchzbMATHGoogle Scholar
  5. 5.
    Lin, Y., Stadtherr, M.A.: Validated solution of initial value problems for odes with interval parameters. In: NSF Workshop Proceeding on Reliable Engineering Computing. Savannah GA, February 22–24 2006Google Scholar
  6. 6.
    Nedialkov, N.S.: Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. Ph.D. thesis, Graduate Department of Computer Science, University of Toronto (1999)Google Scholar
  7. 7.
    Nedialkov, N.S.: Implementing a rigorous ODE solver through literate programming. In: Rauh, A., Auer, E. (eds.) Modeling, Design, and Simulation of Systems with Uncertainties. Mathematical Engineering, pp. 3–19. Springer, Heidenberg (2011)CrossRefGoogle Scholar
  8. 8.
    Petković, M., Petković, L.: Complex Interval Arithmetic and Its Applications. Wiley-VCH Verlag GmbH, Berlin (1998)zbMATHGoogle Scholar
  9. 9.
    Rauh, A., Auer, E., Hofer, E.P.: ValEncIA-IVP: a comparison with other initial value problem solvers. In: CD-Proceedings of 12th GAMM-IMACS Intenational Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006. IEEE Computer Society, Duisburg, Germany (2007)Google Scholar
  10. 10.
    Rauh, A., Kersten, J., Auer, E., Aschemann, H.: Sensitivity-based feedforward and feedback control for uncertain systems. Computing 2–4, 357–367 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rauh, A., Westphal, R., Aschemann, H.: Verified simulation of control systems with interval parameters using an exponential state enclosure technique. In: CD-Proceedings of IEEE International Conference on Methods and Models in Automation and Robotics MMAR. Miedzyzdroje, Poland (2013)Google Scholar
  12. 12.
    Rauh, A., Westphal, R., Auer, E., Aschemann, H.: Exponential enclosure techniques for the computation of guaranteed state enclosures in ValEncIA-IVP. In: Proceedings of 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2012, vol. 19(1), pp. 66–90. Novosibirsk, Russia, Special Issue of Reliable Computing (2013)Google Scholar
  13. 13.
    Rump, S.M.: IntLab - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluver Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

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Authors and Affiliations

  • Andreas Rauh
    • 1
    Email author
  • Ramona Westphal
    • 1
  • Harald Aschemann
    • 1
  • Ekaterina Auer
    • 2
  1. 1.Chair of MechatronicsUniversity of RostockRostockGermany
  2. 2.University of Applied Sciences Wismar, Faculty of EngineeringWismarGermany

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