Skip to main content
SpringerLink
Log in

Two Approaches for Guaranteed State Estimation of Nonlinear Continuous-Time Models

  • Chapter
  • First Online:
Modeling, Design, and Simulation of Systems with Uncertainties

Part of the book series: Mathematical Engineering ((MATHENGIN,volume 3))

Abstract

This paper deals with the estimation of the state vector of a nonlinear continuous-time state-space model, such as those frequently encountered in the context of knowledge-based modeling. Unknown and possibly time-varying parameters may be included in an extended state vector to deal with the simultaneous estimation of state and parameters. Observations depending on the (possibly extended) state are assumed to take place at discrete measurement times. Given bounds on the size of the additive measurement errors, guaranteed estimation should then provide bounds on the possible values of the state at any given time. Two recently developed approaches are presented and their performance is compared on a simple test case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable Computing 4, 361–369 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chernousko, F.L.: Optimal guaranteed estimates of indeterminacies with the aid of ellipsoids. Engrg. Cybernetics 18, 1–9 (1980)

    Google Scholar 

  3. Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton, FL (1994)

    Google Scholar 

  4. Gennat, M., Tibken, B.: Simulation of uncertain systems with guaranteed bounds. In: 11th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics. Fukuoka, Japan (2004)

    Google Scholar 

  5. Gouz´e, J.L., Rapaport, A., Hadj-Sadok, Z.M.: Interval observers for uncertain biological systems. Journal of Ecological Modelling (133), 45–56 (2000)

    Google Scholar 

  6. Hoefkens, J., Berz, M., Makino, K.: Efficient high-order methods for ODEs and DAEs. In: G. Corliss, C. Faure, A. Griewank (eds.) Automatic Differentiation : From Simulation to Optimization, pp. 341–351. Springer-Verlag, New-York, NY (2001)

    Google Scholar 

  7. Jaulin, L.: Nonlinear bounded-error state estimation of continuous-time systems. Automatica 38, 1079–1082 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer-Verlag, London (2001)

    MATH  Google Scholar 

  9. Jaulin, L., Walter, E.: Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica 29(4), 1053–1064 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kieffer, M.,Walter, E.: Guaranteed parameter estimation for cooperative systems. In: L. Benvenuti, A. De Santis, L. Farina (eds.) Positive Systems – LNCIS 294, pp. 103–110. Springer, Berlin (2003)

    Google Scholar 

  11. Kieffer, M., Walter, E.: Interval analysis for guaranteed nonlinear parameter and state estimation. Mathematical and Computer Modelling of Dynamic Systems 11(2), 171–181 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kieffer, M., Walter, E.: Guaranteed nonlinear state estimation for continuous–time dynamical models from discrete–time measurements. In: Preprints of the 5th IFAC Symposium on Robust Control Design (2006)

    Google Scholar 

  13. Kieffer, M., Walter, E.: Guaranteed estimation of the parameters of nonlinear continuous–time models: contributions of interval analysis. Int. J. Adap. Contr. Sig. Proc. (2010). Doi : 10.1002/acs.1194

  14. Kletting, M.: Verified Methods for State and Parameter Estimators for Nonlinear Uncertain Systems with Applications in Engineering. Ph.D. thesis, Institute of Measurement, Control, and Microtechnology,University of Ulm, Germany (2009)

    Google Scholar 

  15. K¨uhn, W.: Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61(1), 47–67 (1998)

    Google Scholar 

  16. Kurzhanski, A., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. Birkh¨auser, Boston, MA (1997)

    Google Scholar 

  17. Lin, Y., Stadtherr, M.A.: Validated solution of ODEs with parametric uncertainties. In: W. Marquardt, C. Pantelides (eds.) Proc. 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, Computer Aided Chemical Engineering, vol. 21, pp. 167 – 172. Elsevier (2006). DOI 10. 1016/S1570-7946(06)80041-6. URL http://www.sciencedirect.com/science/article/B8G5G-4P37F2K-S/2/a22680956d2786784b23e88e9b272db6

  18. Lohner, R.: Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value-problem. In: J.R. Cash, I. Gladwell (eds.) Computational Ordinary Differential Equations, pp. 425–435. Clarendon Press, Oxford (1992)

    Google Scholar 

  19. Makino, K., Berz, M.: Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning. International Journal of Differential Equations and Applications 10(4), 353–384 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Makino, K., Berz, M.: Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by shrink wrapping. International Journal of Differential Equations and Applications 10(4), 385–403 (2005)

    MathSciNet  Google Scholar 

  21. Meslem, N., Ramdani, N., Candau, Y.: Guaranteed state bounding estimation for uncertain non linear continuous systems using hybrid automata. In: Proc. IFAC International Conference on Informatics in Control, Automation Robotics. Funchal, Madeira (2008)

    Google Scholar 

  22. Meslem, N., Ramdani, N., Candau, Y.: Interval observers for uncertain nonlinear systems. application to bioreactors. In: Proc. IFAC World Congress, pp. 9667–9672. Seoul, Korea (2008)

    Google Scholar 

  23. Moisan, M., Bernard, O., Gouz´e, J.L.: Near optimal interval observers bundle for uncertain bioreactors. Automatica 45(1), 291–295 (2009)

    Google Scholar 

  24. Mu¨ller, M.: U¨ ber das Fundamentaltheorem in der Theorie der gewo¨hnlichen Differentialgleichungen. Math. Z. 26, 619–645 (1926)

    Google Scholar 

  25. Nedialkov, N.S., Jackson, K.R.: Methods for initial value problems for ordinary differential equations. In: U. Kulisch, R. Lohner, A. Facius (eds.) Perspectives on Enclosure Methods, pp. 219–264. Springer-Verlag, Vienna (2001)

    Google Scholar 

  26. Raissi, T., Ramdani, N., Candau, Y.: Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica 40(10), 1771–1777 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Raissi, T., Videau, G., Zolghadri, A.: Interval observer design for consistency checks of nonlinear continuous-time systems. Automatica 46(3), 518–527 (2010)

    Article  MATH  Google Scholar 

  28. Rauh, A., Auer, E., Minisini, J., Hofer, E.P.: Extensions of ValEncIA-IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization. Proc. Appl. Math. Mech. 7(1), 1023,0011023,002 (2007). DOI 10.1002/pamm.200700022

  29. Rauh, A., Hofer, E.P., Auer, E.: VALENCIA-IVP: A comparison with other initial value problem solvers. In: Proc. 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), p. 36. IEEE Computer Society, Duisburg, Germany (2006). DOI 10.1109/SCAN.2006.47

  30. Veres, S.M., Mayne, D.Q.: Geometric bounding toolbox. Tech. rep. URL http://www.sysbrain.com/gbt/

  31. Walter, E., Kieffer, M.: Interval analysis for guaranteed nonlinear estimation. In: Proceedings of the 13th IFAC Symposium on System Identification (SYSID), pp. 259–270 (2003)

    Google Scholar 

  32. Walter, E., Kieffer, M.: Guaranteed nonlinear parameter estimation in knowledge–based models. Journal of Computational and Applied Mathematics 199(2), 277–285 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Multi-Function Airborne Radars (OPES22), Cassidian Electronics, Woerthstr. 85, 89077, Ulm, Germany

    Marco Kletting

  2. Laboratoire des Signaux et Systèmes - CNRS - SUPELEC, Univ Paris-Sud, 3 rue Joliot-Curie, F-91192, Gif-sur-Yvette cedex, France

    Michel Kieffer & Eric Walter

  3. LTCI - CNRS - Telecom ParisTech, 46 rue Barault, F-75013, Paris, France

    Michel Kieffer

Authors
  1. Marco Kletting
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Kieffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Kieffer .

Editor information

Editors and Affiliations

  1. , Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Str. 6, Rostock, 18059, Germany

    Andreas Rauh

  2. , Faculty of Engineering, INKO, University of Duisburg-Essen, Duisburg, 47048, Germany

    Ekaterina Auer

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kletting, M., Kieffer, M., Walter, E. (2011). Two Approaches for Guaranteed State Estimation of Nonlinear Continuous-Time Models. In: Rauh, A., Auer, E. (eds) Modeling, Design, and Simulation of Systems with Uncertainties. Mathematical Engineering, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15956-5_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15956-5_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15955-8

  • Online ISBN: 978-3-642-15956-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation