Skip to main content
SpringerLink
Log in

Global Bilinearization and Reachability Analysis of Control-Affine Nonlinear Systems

  • Chapter
  • First Online:
The Koopman Operator in Systems and Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 484))

Abstract

Nonlinear systems are ubiquitous in real-world applications, but the control design for them is not an easy task. Hence, methods are sought to transform a nonlinear system into linear or bilinear forms to alleviate the problem of nonlinear controllability and control design. While there are linearization techniques like Carleman linearization for embedding a finite-dimensional nonlinear system into an infinite-dimensional space, they depend on the analytic property of the vector fields and work only on polynomial space. The Koopman-based approach described here utilizes the Koopman canonical transform (KCT) to transform the dynamics and ensures bilinearity from the projection of the Koopman operator associated with the control vector fields on the eigenspace of the drift Koopman operator. The resulting bilinear system is then subjected to controllability analysis using the Myhill semigroup method and Lie algebraic structures.

© [2017] IEEE. Sections 4.1.1, 4.2, and 4.4 are reprinted, with permission, from [6].

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
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

References

  1. Brockett, R.W.: Finite Dimensional Linear Systems, pp. 67–122. Wiley, New York (1970)

    Google Scholar 

  2. Brockett, R.W.: System theory on group manifolds and coset spaces. SIAM J. Control 10(2), 265–284 (1972)

    Article  MathSciNet  Google Scholar 

  3. Brockett, R.W.: Lie Algebras and Lie Groups in Control Theory, pp. 43–82. Springer, Dordrecht (1973)

    Chapter  Google Scholar 

  4. Budisić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos: Interdiscip. J. Nonlinear Sci. 22(4) (2012)

    Article  MathSciNet  Google Scholar 

  5. Gantmacher, R.F.: Theory of Matrices. Chelsea, New York (1959)

    MATH  Google Scholar 

  6. Goswami, D., Paley, D.A.: Global bilinearization and controllability of control-affine nonlinear systems: a Koopman spectral approach. In: 2017 IEEE 56th Annual Conference on Decision and Control, pp. 6107–6112 (2017)

    Google Scholar 

  7. Jurdjevic, V., Sussmann, H.J.: Control systems on Lie groups. J. Differ. Equ. 12(2), 313–329 (1972)

    Article  MathSciNet  Google Scholar 

  8. Kowalski, K., Steeb, W.H.: Nonlinear Dynamical Systems and Carleman Linearization, pp. 73–102. World Scientific, Singapore (2011)

    Google Scholar 

  9. Lan, Y., Mezić, I.: Linearization in the large of nonlinear systems and Koopman operator spectrum. Phys. D: Nonlinear Phenom. 242(1), 42–53 (2013)

    Article  MathSciNet  Google Scholar 

  10. Martin, J.: Some results on matrices which commute with their derivatives. SIAM J. Appl. Math. 15(5), 1171–1183 (1967)

    Article  MathSciNet  Google Scholar 

  11. Mauroy, A., Mezić, I.: Global stability analysis using the eigenfunctions of the Koopman operator. IEEE Trans. Autom. Control 61(11), 3356–3369 (2016)

    Article  MathSciNet  Google Scholar 

  12. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309–325 (2005)

    Article  MathSciNet  Google Scholar 

  13. Mohr, R., Mezić, I.: Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator (2014). arxiv.org/abs/1403.6559

  14. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  Google Scholar 

  15. Surana, A.: Koopman operator based observer synthesis for control-affine nonlinear systems. In: 55th IEEE Conference on Decision and Control, pp. 6492–6499 (2016)

    Google Scholar 

  16. Surana, A., Banaszuk, A.: Linear observer synthesis for nonlinear systems using Koopman operator framework. IFAC-PapersOnLine 49(18), 716–723 (2016)

    Article  Google Scholar 

  17. Wei, J., Norman, E.: On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Am. Math. Soc. 15(2), 327–334 (1964)

    Article  MathSciNet  Google Scholar 

  18. Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Maryland, College Park, MD, 20740, USA

    Debdipta Goswami & Derek A. Paley

Authors
  1. Debdipta Goswami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Derek A. Paley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debdipta Goswami .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Namur, Namur, Belgium

    Alexandre Mauroy

  2. Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA, USA

    Igor Mezić

  3. Department of Electrical and Information Systems, Osaka Prefecture University, Sakai, Japan

    Yoshihiko Susuki

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Goswami, D., Paley, D.A. (2020). Global Bilinearization and Reachability Analysis of Control-Affine Nonlinear Systems. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation