Skip to main content
SpringerLink
Log in

Nonlinear Observers and Separation Principle

  • Chapter
  • First Online:
Lectures in Feedback Design for Multivariable Systems

Part of the book series: Advanced Textbooks in Control and Signal Processing ((C&SP))

  • 2719 Accesses

Abstract

This chapter considers the design of asymptotic state observers for a single-output nonlinear system. A fundamental property that makes the design of such observers possible is the existence of change of coordinates by means of which the system is brought to a special form in which a property of observability, uniform with respect to the input, is highlighted. For such systems, it is possible to design global asymptotic state observers. Then, a nonlinear equivalent of the so-called separation principle of linear system theory is developed. It is shown how to combine a state feedback stabilizer with a nonlinear observer, so as to obtain a dynamic output feedback by means of which asymptotic stability with guaranteed domain of attraction is obtained.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Notes

  1. 1.

    High-gain observers have been considered by various authors in the literature. Here we closely follow the approach of Gauthier and Kupca, who have thoroughly investigated the design of high-gain observers in [1].

  2. 2.

    See [1].

  3. 3.

    Condition (i) is a “regularity” condition. Condition expresses the independence of \({\mathscr {K}}_i(x,u)\) on the parameter u.

  4. 4.

    See [1], Chap. 3, Theorem 2.1.

  5. 5.

    For convenience, we drop the “tilde” above \(h(\cdot )\) and the \(f_i(\cdot )\)’s.

  6. 6.

    See also [1] and reference to Dayawansa therein.

  7. 7.

    The \(g_i(t)\)’s are continuous functions that never vanish. Thus, each of them has a well-defined sign.

  8. 8.

    In this section, we continue to essentially follow the approach of [1]. A slightly alternative approach and additional relevant results can be found in [2, 3].

  9. 9.

    This phenomenon is sometimes referred to as “peaking”.

  10. 10.

    See Example B.3 in Appendix B in this respect.

  11. 11.

    The idea of saturating the control, outside a fixed region on which the trajectories are expected to range, so as to prevent finite escape times, has been originally suggested by Khalil in [4, 5], see also [6].

  12. 12.

    See, in this respect, Sect. B.1 of Appendix B.

  13. 13.

    Form more details, see [1].

References

  1. J.P. Gauthier, I. Kupka, Deterministic Observation Theory and Applications (Cambridge University Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  2. A.R. Teel, L. Praly, Global stabilizability and observability imply semi-global stabilizability by output feedback. Syst. Control Lett. 22, 313–325 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. A.R. Teel, L. Praly, Tools for semi-global stabilization by partial state and output feedback. SIAM J. Control Optim. 33, 1443–1488 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Esfandiari, H.K. Khalil, Output feedback stabilization of fully linearizable systems. Int. J. Control 56, 1007–1037 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. H.K. Khalil, F. Esfandiari, Semiglobal stabilization of a class of nonlinear systems using output feedback. IEEE Trans. Autom. Control 38, 1412–1415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. A.N. Atassi, H.K. Khalil, A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Autom. Control 44, 1672–1687 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Università degli Studi di Roma “La Sapienza”, Rome, Italy

    Alberto Isidori

Authors
  1. Alberto Isidori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Isidori .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Isidori, A. (2017). Nonlinear Observers and Separation Principle. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-42031-8_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42030-1

  • Online ISBN: 978-3-319-42031-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation