Skip to main content
Springer Nature Link
Log in

On the robustness of feedback linearization

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

Feedback linearization is possibly the most widely used technique for controlling nonlinear systems without resorting to local linearization. In this algorithm the system nonlinearities are canceled by the control input, but accurate cancelation requires a precise modeling of the system. The possibility that an exact cancelation may not occur as a consequence of modeling errors has raised justifiable concerns about the robustness of this methodology and feedback linearization is widely cited in the literature as a non-robust design technique. This paper examines the robustness—or lack thereof—of the feedback linearization methodology and introduces two methodologies that provide clear guidelines on how to design a feedback linearized controller that is stable in the presence of Lipschitz bounded uncertainties. It is shown that a feedback linearized controller can be designed to be robust in face of parametric uncertainties, by simply pushing the closed-loop eigenvalues far enough to the left in the complex plane if the eigenvalues are sufficiently far apart. For structural uncertainties, such a strong result could not be obtained in general. Examples of stable feedback linearization of uncertain nonlinear dynamics are presented at the end.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Availability of data and material

Availability of data and materials Data used in this work is available in the public domain.

Abbreviations

x :

State vector

u :

Input

\(\Phi\) :

Known portion of the nonlinear dynamics

\(\hat{\Phi }\) :

Unknown portion of the dynamics

A :

Invariant matrix in the controllable form

\(\Lambda\) :

Diagonal matrix with the eigenvalues of A

a i :

Coefficient of the characteristic polynomial of A

\(\phi\) :

Lipschitz constant

\(\lambda\) :

Eigenvalue of A

T :

Eigenvectors matrix of A

K T :

Condition number of the T

F :

State and time dependent matrix representation of \(\hat{\Phi }\)

f ij :

Entry of \(F\)

\(\overline{f}_{ij} , \underline {f}_{ij}\) :

Max, min of fi.j

\(\overline{F},\; \underline {F}\) :

Invariant matrices composed of \(\overline{f}_{ij} , \underline {f}_{ij}\)

V :

Lyapunov function

P, Q :

Solution pair of a Lyapunov equation

p ij, q ij :

Entry of P, Q

P, \(\tilde{Q}\) :

Solution of a Lyapunov equation considering only the linear dynamics, \(\tilde{Q} \) diagonal

\(\tilde{q}_{i}\) :

Entry of \(\tilde{Q}\)

\(\alpha\) :

Parameter used to solve the desired \(\lambda\)

References

  1. Krener AJ (1973) On the equivalence of control systems and the linearization of nonlinear systems. SIAM J Control 11(4):670–676

    Article  MathSciNet  Google Scholar 

  2. Isidori A (1985) Nonlinear control systems, an introduction. Springer-Verlag, Berlin

    Book  Google Scholar 

  3. Utkin V (1977) Variable structure systems with sliding modes. IEEE Trans Autom Control 22(2):212–222

    Article  MathSciNet  Google Scholar 

  4. Freeman RA, Kokotovic PV (1992) Backstepping design of robust controllers for a class of nonlinear systems. IFAC nonlinear control series design, IFAC symposium, Bordeaux, France, pp 431–436

  5. Franco ALD, Bourlès H, De Pieri ER, Guillard H (2006) Robust nonlinear control associating robust feedback linearization and H∞ control. IEEE Trans Autom Control 51(7):1200–1207

    Article  MathSciNet  Google Scholar 

  6. Sastry SS, Isidori A (1989) Adaptive control of linearizable systems. IEEE Trans Autom Control 34(11):1123–1131

    Article  MathSciNet  Google Scholar 

  7. Slotine J-JE, Hedrick JK (1993) Robust input-output feedback linearization. Int J Control 57(5):1133–1139

    Article  MathSciNet  Google Scholar 

  8. Karimi HR, Motlagh MRJ (2006) Robust feedback linearization control for a non-linearizable MIMO nonlinear system in the presence of model uncertainties. In: Proceedings of IEEE International Conferenece on SOLI, Shangai, China, pp 965–970

  9. Chen JS, Chen YH (1991) Robust control of nonlinear uncertain systems: a feedback linearization approach. In: Proceedings of 30th conference on decision and control, Brighton, England, pp 2515–2520

  10. Calvet JP, Y. Arkun (1989) Robust control design for uncertain nonlinear systems under feedback linearization. In: Proceedings of 28th conference on decision and control, Tampa, FL, pp 102–106

  11. Kanellakopoulos I, Kokotovict PV, Marino R (1991) An extended direct scheme for robust adaptive nonlinear control. Automatica 27(2):247–255

    Article  MathSciNet  Google Scholar 

  12. Taylor DG, Kokotovic PV, Marino R, Kanellakopoulos I (1988) Adaptive regulation of nonlinear systems with unmodeled dynamics. In: Proceedings of ACC, Atlanta, GA, pp 360–365

  13. Spong MW, Sira-Ramirez H (1986) Robust control design techniques for a class of nonlinear systems. In: Proceedings of ACC, Seattle, WA, pp 1515–1522

  14. Pfeiffer F, Edgar TF, Fernandez B (1999) Robust feedback linearization and fuzzy control. In: Proceedings of ACC, San Diego, CA, pp 1508–1514

  15. Kaheni1 M, Zarif MH, Kalat AA, Chisci L (2018) Robust feedback linearization for input-constrained nonlinear systems with matched uncertainties. In: Proceedings of European control conference (ECC), Limassol, Cyprus

  16. Zaidabadi nejad M, Ansarifar GR (2020) Observer based adaptive robust feedback-linearization control for VVER-1000 nuclear reactors with bounded axial power distribution based on the validated multipoint kinetics reactor model. Ann Nucl Energy 142:107380

  17. Oliveira L, Bento A, Leite VJS, Gomide F (2020) Comparison of robust methods on feedback linearization through experimental tests. In: Proceedings of 21st IFAC World Congress, Berlin, Germany

  18. Oliveira L, Bento A, Leite VJS, Gomide F (2020) Evolving granular feedback linearization: design, analysis, and applications. Appl Soft Comput 86:105927

    Article  Google Scholar 

  19. Accetta A, Alonge F, Maurizio M, D’Ippolito F, Pucci M, Rabbeni R, Sferlazza A (2019) Robust control for high performance induction motor drives based on partial state-feedback linearization. IEEE Trans Ind Appl 55(1):490–503

    Article  Google Scholar 

  20. Oh S-Y, Choi H-L (2021) On robust approximate feedback linearization: control with two gain-scaling factors. Int J Control Autom Syst 19(2):1151–1157

    Article  Google Scholar 

  21. Yang P, Yang Y, Liu Z, Yang Y, Qin Y (2023) Robust control strategy of VSC-HVDC systems based on feedback linearization and disturbance compensation method. Energy Rep 9(7):637–645

    Article  Google Scholar 

  22. Guillard H, Bourlès H (2000) Robust feedback linearization. In: Proceedings of 14th international sympoisum on mathematical theory of networks and systems, Perpignan, France

  23. Pop CI, Dulf EH (2011) Robust feedback linearization control for reference tracking and disturbance rejection, in nonlinear systems. Recent advances in robust control—novel approaches and design methods, Dr. Andreas Mueller (Ed), InTech, ISBN 978-953-307-339-2

  24. Westenbroek T, Fridovich-Keil D, Mazumdar E, Arora S, Prabhu V, Sastry SS, Tomlin CJ (2020) Feedback linearization for uncertain systems via reinforcement learning. In: Proceedings of IEEE international conference on robotics and automation (ICRA), Paris, France

  25. Mahmoodabadi MJ, Soleymani T (2020) Optimum fuzzy combination of robust decoupled sliding mode and adaptive feedback linearization controllers for uncertain underactuated nonlinear systems. Chin J Phys 64:241–250

    Article  Google Scholar 

  26. Kim WH, Groves FR Jr (1993) Robustness analysis of feedback linearization for parametric and structural uncertainties. Chem Eng Commun 119:71–94

    Article  Google Scholar 

  27. Slotine J-JE, Li W (1991) Applied nonlinear control, Prentice Hall, Englewood Cliffs, pp 149–151

  28. Khalil HK (1996) Nonlinear systems, 2nd Ed, Prentice Hall, Upper Saddle River, pp 63–64

  29. Cunha S (2019) Moderate gain Luenberger-like observer for Lipschitz nonlinear dynamics. J Appl Nonl Dyn 9(4):551–566

    Article  MathSciNet  Google Scholar 

  30. Man Y-K (2017) On computing the Vandermonde matrix inverse. In: Proceedings of World congress on engineering, London

  31. Hosseini MS, Chen A, Plataniotis KN (2019) On the closed form expression of elementary symmetric polynomials and the inverse of Vandermonde matrix. arXiv preprint arXiv:1909.08155

  32. Lyapunov AM (1992) The general problem of the stability of motion. Int J Control 55(3):531–773

    Article  MathSciNet  Google Scholar 

  33. Apostol TM (1965) Mathematical analysis. Addison-Wesley, Reading

  34. Lang S (1969) Analysis I. Addison-Wesley, Reading

  35. Krasovskii NN (1959) Problems of the theory of stability of motion, Mir. English translation by Stanford University Press, Moskow, p 1963

    Google Scholar 

  36. Lohmiller W, Slotine J-JE (1998) On contraction analysis for non-linear systems. Automatica 34(6):683–696

    Article  MathSciNet  Google Scholar 

  37. Slotine J-JE (2003) Modular stability tools for distributed computation and control. Int J Adapt Control 17:397–416

    Article  Google Scholar 

  38. Spong MW (1989) Adaptive control of flexible joint manipulators. Syst Control Lett 15–21

  39. Ghorbel F, Hung JY, Spong MW (1989) Adaptive control of flexible-joint manipulators. IEEE Control Syst Mag

  40. Raghavan S, Hedrick JK (1994) Observer design for a class of nonlinear systems. Int J Control 59:515–528

    Article  MathSciNet  Google Scholar 

Download references

Funding

The author received no financial support for the research, authorship and/or publication of this article.

Author information

Authors and Affiliations

  1. Rio de Janeiro State University, R. São Francisco Xavier, 524-Maracanã, Rio de Janeiro, RJ, 20550-013, Brazil

    Sérgio Barros da Cunha

Authors
  1. Sérgio Barros da Cunha
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The author is the intellectual responsible for all developments presented in the manuscript, which presents original material.

Corresponding author

Correspondence to Sérgio Barros da Cunha.

Ethics declarations

Competing interests

The author declares no potential competing interests with respect to the research, authorship and/or publication of this article.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

da Cunha, S.B. On the robustness of feedback linearization. Int. J. Dynam. Control 12, 3318–3331 (2024). https://doi.org/10.1007/s40435-023-01375-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-023-01375-3

Keywords

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy