Skip to main content
SpringerLink
Log in

Design of Causal Observers for Nonlinear Descriptor Systems

  • Conference paper
  • First Online:
Progress in Differential-Algebraic Equations

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

This contribution not only provides a new necessary and sufficient condition for causal observability of nonlinear descriptor systems but also a method to design the causal observer. The approach is based on the transformation of the descriptor system into a state-space form, the so called coupled state-space system. This description exists for all regular descriptor systems no matter if they are proper or not. If the new condition is satisfied, the coupled state-space system can be modified and can be used to design a state-space observer. It is shown, that this observer is also a causal observer for the original descriptor system. Two examples illustrate the capability of the new approach.

Mathematics Subject Classification (2010) 93B07 ⋅ 93B17

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

References

  1. Ailon, A.: On the reduced-order causal observer for generalized control systems. Int. J. Control 57(6), 1311–1323 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aliyu, M.D.S., Boukas, E.K.: Kalman filtering for affine nonlinear descriptor systems. Circuits Syst. Signal Process. 30, 125–142 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ascher, U.M., Chin, H., Reich, S.: Stabilization of DAEs and invariant manifolds. Numer. Math. 67, 131–149 (1994). http://dx.doi.org/10.1007/s002110050020

  4. Åslund, J., Frisk, E.: An observer for non-linear differential-algebraic systems. Automatica 42(6), 959–965 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972). http://www.sciencedirect.com/science/article/pii/0045782572900187

  6. Boutayeb, M., Darouach, M.: Observers design for nonlinear descriptor systems. In: Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, vol. 3, pp. 2369–2374 (1995)

    Google Scholar 

  7. Burgermeister, B., Arnold, M., Esterl, B.: DAE time integration for real-time applications in multi-body dynamics. ZAMM – Z. Angew. Math. Mech. 86(10), 759–771 (2006). http://dx.doi.org/10.1002/zamm.200610284

  8. Campbell, S.L., Delebecque, F., Nikoukhah, R.: Observer design for linear time varying descriptor systems. In: Proceedings Control of Industrial Systems, Belfort, vol. 1, pp. 507–512 (1997)

    Google Scholar 

  9. Dai, L.: Singular Control Systems. Springer, Berlin/Heidelberg (1989)

    Book  MATH  Google Scholar 

  10. Darouach, M., Boutat-Baddas, L.: Observers for a class of nonlinear singular systems. IEEE Trans. Autom. Control 53(11), 2627–2633 (2008)

    Article  MathSciNet  Google Scholar 

  11. Darouach, M., Boutat-Baddas, L.: Observers for Lipschitz nonlinear descriptor systems: application to unknown inputs systems. In: Proceedings of the 16th IEEE Mediterranean Conference on Control and Automation, Ajaccio, pp. 1369–1374 (2008)

    Google Scholar 

  12. Darouach, M., Boutayeb, M.: Design of observers for descriptor systems. IEEE Trans. Autom. Control 40(7), 1323–1327 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fahmy, M.M., O’Reilly, J.: Observers for descriptor systems. Int. J. Control 49(6), 2013–2028 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Frisk, E., Åslund, J.: An observer for semi-explicit differential-algebraic systems. In: Proceedings of the 16th IFAC World Congress (IFAC), Prague, pp. 671–676. Elsevier (2005)

    Google Scholar 

  15. Gao, Z., Ho, D.W.: State/noise estimator for descriptor systems with application to sensor fault diagnosis. IEEE Trans. Signal Process. 54(4), 1316–1326 (2006)

    Article  Google Scholar 

  16. Gear, C.W.: Differential-algebraic equation index transformations. SIAM J. Numer. Anal. 9, 39–47 (1988)

    MATH  MathSciNet  Google Scholar 

  17. Gerdin, M.: Local identifiability and observability of nonlinear differential-algebraic equations. In: Proceedings of the 14th IFAC Symposium on System Identification, Newcastle, vol. 14, pp. 802–807 (2006)

    Google Scholar 

  18. Hou, M., Müller, P.C.: Design of a class of Luenberger observers for descriptor systems. IEEE Trans. Autom. Control 40, 133–136 (1995)

    Article  MATH  Google Scholar 

  19. Hou, M., Müller, P.C.: Causal observability of descriptor systems. IEEE Trans. Autom. Control 44, 158–163 (1999)

    Article  MATH  Google Scholar 

  20. Hou, M., Müller, P.C.: Observer design for descriptor systems. IEEE Trans. Autom. Control 44, 164–168 (1999)

    Article  MATH  Google Scholar 

  21. Hou, M., Müller, P.C.: Observer design for a class of descriptor systems. In: Proceedings of the European Control Conference, Cambridge, 1–4 Sept 2003. CD–Rom

    Google Scholar 

  22. Kaprielian, S., Turi, J.: An observer for a nonlinear descriptor system. In: Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, vol. 1, pp. 975–976 (1992)

    Google Scholar 

  23. Koenig, D.: Observer design for unknown input nonlinear descriptor systems via convex optimization. IEEE Trans. Autom. Control 51(6), 1047–1052 (2006)

    Article  MathSciNet  Google Scholar 

  24. Labisch, D.: Verkopplungsbasierte Methoden zum Regler- und Beobachterentwurf für nichtlineare Deskriptorsysteme. Fortschritt-Berichte VDI, Reihe 8, Nr. 1227, VDI-Verlag, Düsseldorf (2013)

    Google Scholar 

  25. Labisch, D., Konigorski, U.: Entwurf kausaler Beobachter für lineare Deskriptorsysteme. In: Adamy, J., Sawodny, O. (eds.) GMA Fachausschuss 1.40 “Theoretische Verfahren der Regelungstechnik”, Workshop, 16–19 Sept 2011, Salzburg, pp. 289–301 (2012)

    Google Scholar 

  26. Labisch, D., Manderla, M., Konigorski, U.: Control, simulation and stability analysis of nonlinear regular proper DAEs. In: Proceedings of the 20th Mediterranean Conference on Control and Automation, Barcelona, pp. 392–397 (2012)

    Google Scholar 

  27. Lu, G., Ho, D.W.: Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach. IEEE Trans. Circuits Syst. II: Express Briefs 53(7), 563–567 (2006)

    Article  Google Scholar 

  28. Manderla, M.: Eine Methodik zum Regler- und Beobachterentwurf für Mehrgrößensysteme in Deskriptorform. VDI-Verlag, Düsseldorf (2011)

    Google Scholar 

  29. Manderla, M., Labisch, D., Konigorski, U.: Dualität von Regler- und Beobachterentwurf für reguläre Deskriptorsysteme. In: Adamy, J., Sawodny, O. (eds.) GMA Fachausschuss 1.40 “Theoretische Verfahren der Regelungstechnik”, Workshop, 18–21 Sept 2011, Salzburg, pp. 320–331 (2011)

    Google Scholar 

  30. Mehrmann, V., Stykel, T.: Descriptor systems: a general mathematical framework for modelling, simulation and control. At – Automatisierungstechnik 54, 405–415 (2006)

    Google Scholar 

  31. Müller, P.C., Hou, M.: On the observer design for descriptor systems. IEEE Trans. Autom. Control 38, 1666–1671 (1993)

    Article  MATH  Google Scholar 

  32. Nijmeijer, H., van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, New York (1990)

    Book  MATH  Google Scholar 

  33. Ploix, S., Maquin, D., Ragot, J.: Asymptotic observer for a non linear descriptor system. In: IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS’97), pp. 654–658. Hull, Royaume-Uni (1997)

    Google Scholar 

  34. Respondek, W.: Right and left invertibility of nonlinear control systems. In: Sussmann, H.J. (ed.) Nonlinear Control and Optimal Control, pp. 133–176. Marcel Dekker, New York (1990)

    Google Scholar 

  35. Shields, D.N.: Observer design for discrete nonlinear descriptor systems. In: UKACC International Conference on Control, Exeter, vol. 2, pp. 831–836 (1996)

    Google Scholar 

  36. Shields, D.N.: Observer design and detection for nonlinear descriptor systems. Int. J. Control 67(2), 153–168 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Terrell, W.J.: Local observability of nonlinear differential-algebraic equations (DAEs) from the linearization along a trajectory. IEEE Trans. Autom. Control 46(12), 1947–1950 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yang, C., Zhang, Q., Chai, T.: Observer design for a class of nonlinear descriptor systems. In: Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, pp. 8232–8237 (2009)

    Google Scholar 

  39. Yang, C., Zhang, Q., Zhou, L.: Stability Analysis and Design for Nonlinear Singular Systems, vol. 435. Springer, Berlin/Heidelberg (2013)

    MATH  Google Scholar 

  40. Zimmer, G., Meier, J.: On observing nonlinear descriptor systems. Syst. Control Lett. 32(1), 43–48 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Siemens AG, Industry Sector, Industry Automation Division, Karlsruhe, Germany

    Daniel Labisch

  2. Department of Control Engineering and Mechatronics, Technische Universität Darmstadt, Darmstadt, Germany

    Ulrich Konigorski

Authors
  1. Daniel Labisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ulrich Konigorski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Labisch .

Editor information

Editors and Affiliations

  1. Technische Universität Darmstadt, Graduate School CE, Darmstadt, Germany

    Sebastian Schöps

  2. Bergische Universität Wuppertal, Applied Math. and Numerical Analysis, Wuppertal, Germany

    Andreas Bartel

  3. Bergische Univeirsität Wuppertal, Applied Math. and Numerical Analysis, Wuppertal, Germany

    Michael Günther

  4. Eindhoven University of Technology, Dept. Mathematics & Comp. Sci., Eindhoven, The Netherlands

    E. Jan W. ter Maten

  5. Bergische Universität Wuppertal, Dept. Safety Control Engineering, Wuppertal, Germany

    Peter C Müller

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Labisch, D., Konigorski, U. (2014). Design of Causal Observers for Nonlinear Descriptor Systems. In: Schöps, S., Bartel, A., Günther, M., ter Maten, E., Müller, P. (eds) Progress in Differential-Algebraic Equations. Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44926-4_3

Download citation

Publish with us

Policies and ethics

Search

Navigation