Advertisement

Effective Computation for Nonlinear Systems

  • Pieter Collins
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

Abstract

Nonlinear dynamical and control systems are an important source of applications for theories of computation over the the real numbers, since these systems are usually to complicated to study analytically, but may be extremely sensitive to numerical error. Further, computer-assisted proofs and verification problems require a rigorous treatment of numerical errors. In this paper we will describe how to provide a semantics for effective computations on sets and maps and show how these operations have been implemented in the tool Ariadne for the analysis, design and verification of nonlinear and hybrid systems.

Keywords

computable analysis nonlinear systems Ariadne 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    van der Schaft, A., Schumacher, H.: An introduction to hybrid dynamical systems. Lecture notes in control and information sciences, vol. 251. Springer, London (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Weihrauch, K.: Computable analysis - An introduction. In: Texts in Theoretical Computer Science, Springer, Heidelberg (2000)Google Scholar
  3. 3.
    Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoretical Comp. Sci. 305, 43–76 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Stoltenberg-Hansen, V., Lindström, I., Griffor, E.R.: Mathematical Theory of Domains. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Collins, P.: Continuity and computability of reachable sets. Theor. Comput. Sci. 341, 162–195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs, N.J (1966)zbMATHGoogle Scholar
  7. 7.
    Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO-set oriented numerical methods for dynamical systems. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, pp. 145–174. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Mrozek, M., et al.: CAPD Library (2007), http://capd.wsb-nlu.edu.pl/
  9. 9.
    Nedialkov, N.S.: VNODE-LP: A validated solver for initial value problems in ordinary differential equations. Technical report, McMaster University, CAS-06-06-NN (2006)Google Scholar
  10. 10.
    Izaias Silva, B., Keith Richeson, B.K., Chutinan, A.: Modeling and verification of hybrid dynamical system using CheckMate. In: Proceedings of the International Conference on Automation of Mixed Processes. pp. 189–194 (2000)Google Scholar
  11. 11.
    Balluchi, A., Casagrande, A., Collins, P., Ferrari, A., Villa, T., Sangiovanni-Vincentelli, A.L.: Ariadne: a framework for reachability analysis of hybrid automata. In: Proceedings of the International Syposium on Mathematical Theory of Networks and Systems (2006)Google Scholar
  12. 12.
    Hanrot, G., et al.: The MPFR library (2000), http://www.mpfr.org/
  13. 13.
    Granlund, T., et al.: The GMP library (2005), http://swox.com/gmp/
  14. 14.
    Müller, N., et al.: iRRAM (2006), http://www.informatik.uni-trier.de/iRRAM/

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Pieter Collins
    • 1
  1. 1.Centrum voor Wiskunde en Informatica, Postbus 94079, 1090 GB AmsterdamThe Netherlands

Personalised recommendations