Skip to main content

Parameter Identification in a Respiratory Control System Model with Delay

  • 2177 Accesses

Part of the Lecture Notes in Mathematics book series (LNMBIOS,volume 2064)

Abstract

In this paper we study parameter identification issues by computational means for a set of nonlinear delay equations which have been proposed to model the dynamics of a simplified version of the respiratory control system. We design specific inputs for our system to produce “information rich” output data needed to determine values of unknown parameters. We also consider the effects of noisy measurements in the identification process. Several case studies are included.

Keywords

  • Respiratory Control System
  • Nonlinear Delay Equations
  • Piecewise Constant Argument
  • Parameter Estimation Process
  • Transport Delay

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions
Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7
Fig. 6.8

References

  1. Banks, H.T., Burns, J.A., Cliff, E.M.: Parameter estimation and identification for systems with delays. SIAM J. Contr. Optim. 19(6), 791–828 (1981)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Banks, H.T., Daniel Lamm, P.K.: Estimation of delays and other parameters in nonlinear functional differential equations. SIAM J. Contr. Optim. 21(6), 895–915 (1983)

    Google Scholar 

  3. Batzel, J.J., Tran, H.T.: Stability of the human respiratory control system: I. Analysis of a two-dimensional delay state-space model. J. Math. Biol. 41, 45–79 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Batzel, J.J., Tran, H.T.: Stability of the human respiratory control system: II. Analysis of a three-dimensional delay state-space model. J. Math. Biol. 41, 80–102 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Cooke, K.L., Turi, J.: Stability, instability in delay equations modeling human respiration. J. Math. Biol. 32, 535–543 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Cooke, K.L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265–297 (1984)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Cooke, K.L., Wiener, J.: Stability regions for linear equations with piecewise continuous delays. Comput. Math. Appl. 12A, 695–701 (1986)

    CrossRef  MathSciNet  Google Scholar 

  8. Cooke, K.L., Wiener, J.: A survey of differential equations with piecewise continuous arguments. In: Busenberg, S., Martelli, M. (eds.) Delay Differential Equations and Dynamical Systems. Lecture Notes in Mathematics, vol. 1475, pp. 1–15. Springer, Berlin (1991)

    CrossRef  Google Scholar 

  9. Dennis, J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs, NJ (1983)

    MATH  Google Scholar 

  10. Győri, I.: On approximation of the solutions of delay differential equations by using piecewise constant arguments. Int. J. Math. Math. Sci. 14(1), 111–126 (1991)

    CrossRef  Google Scholar 

  11. Győri, I., Hartung, F., Turi, J.: Numerical approximations for a class of differential equations with time- and state-dependent delays. Appl. Math. Lett. 8(6), 19–24 (1995)

    CrossRef  MathSciNet  Google Scholar 

  12. Hartung, F., Herdman, T.L., Turi, J.: On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays. Appl. Numer. Math. 24, 393–409 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Hartung, F., Herdman, T.L., Turi, J.: Parameter identification in classes of hereditary systems of neutral type. Appl. Math. Comput. 89(1-3), 147–160 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Hartung, F., Herdman, T.L., Turi, J.: Parameter identifications in classes of neutral differential equations with state-dependent delays. Nonlinear Anal. 39(3), 305–325 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Hartung, F., Krisztin, T., Walther, H.O., Wu, J.: Functional differential equations with state-dependent delays: Theory and applications. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, vol. 3, pp. 435–545. Elsevier, Amsterdam (2006)

    Google Scholar 

  16. Hartung, F., Turi, J.: Identification of parameters in delay equations with state-dependent delays. Nonlinear Anal. 29(11), 1303–1318 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Hartung, F., Turi, J.: Linearized stability in functional differential equations with state-dependent delays. Discrete and Continuous Dynamical Systems (Added Volume), 416–425 (2001). Dynamical Systems and Delay Differential Equations (Kennesaw, GA, 2000)

    Google Scholar 

  18. Lunel, S.M.V.: Parameter identifiability of differential delay equations. Int. J. Adapt. Contr. Signal Process. 15, 655–678 (2001)

    CrossRef  MATH  Google Scholar 

  19. Nakagiri, S., Yamamoto, M.: Identifiability of linear retarded systems in Banach spaces. Funkcial. Ekvac. 31, 315–329 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)

    MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the National Science Foundation under grant DMS-0705247 (FH and JT) and by Hungarian NFSR Grant No. K101217 (FH).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Janos Turi .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hartung, F., Turi, J. (2013). Parameter Identification in a Respiratory Control System Model with Delay. In: Batzel, J., Bachar, M., Kappel, F. (eds) Mathematical Modeling and Validation in Physiology. Lecture Notes in Mathematics(), vol 2064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32882-4_6

Download citation