Skip to main content
SpringerLink
Log in

Linearization of Fractional Nonlinear Systems by State-Feedbacks

  • Conference paper
  • First Online:
Automation 2020: Towards Industry of the Future (AUTOMATION 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1140))

Included in the following conference series:

Abstract

Using the conformable fractional derivative the notions of Lie derivative and full relative degree are extended to the fractional nonlinear systems. The canonical form of the fractional nonlinear systems is introduced and sufficient conditions for the existence of the canonical form for the systems are established. A method for finding nonlinear state-feedbacks linearizing the fractional nonlinear system is proposed.

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 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
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. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Ostalczyk, P.: Epitome of the Fractional Calculus: Theory and Its Applications in Automatics. Wydawnictwo Politechniki Łódzkiej, Łódź (2008). (in Polish)

    Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  4. Kaczorek, T.: Fractional positive continuous-time systems and their reachability. Int. J. Appl. Comput. Sci. 18(2), 223–228 (2008)

    Article  MathSciNet  Google Scholar 

  5. Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. 58(6), 1203–1210 (2011)

    Article  MathSciNet  Google Scholar 

  6. Busłowicz, M.: Stability of linear continuous time fractional order systems with delays of the retarded type. Bull. Pol. Acad. Sci. Tech. 56(4), 319–324 (2008)

    Google Scholar 

  7. Dzieliński, A., Sierociuk, D.: Stability of discrete fractional order state-space systems. J. Vib. Control 14(9/10), 1543–1556 (2008)

    Article  MathSciNet  Google Scholar 

  8. Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2012)

    MATH  Google Scholar 

  9. Kaczorek, T.: Practical stability of positive fractional discrete-time linear systems. Bull. Pol. Acad. Sci. Tech. 56(4), 313–317 (2008)

    Google Scholar 

  10. Dzieliński, A., Sierociuk, D., Sarwas, G.: Ultracapacitor parameters identification based on fractional order model. In: Proceedings of ECC 2009, Budapest (2009)

    Google Scholar 

  11. Jumerie, G.: The Leibniz rule for fractional derivatives holds with non-differentiable functions. Math. Stat. 1(2), 50–52 (2013)

    Google Scholar 

  12. Radwan, A.G., Soliman, A.M., Elwakil, A.S., Sedeek, A.: On the stability of linear systems with fractional-order elements. Chaos Solitones Fractals 40(5), 2317–2328 (2009)

    Article  Google Scholar 

  13. Kaczorek, T.: An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs. Bull. Pol. Acad. Tech. 62(2), 227–231 (2014)

    Google Scholar 

  14. Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Press, Dordrecht (1991)

    MATH  Google Scholar 

  15. Klamka, J.: Minimum energy control of 2D systems in Hilbert spaces. Syst. Sci. 9(1–2), 33–42 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Klamka, J.: Controllability and minimum energy control problem of fractional discrete-time systems. In: Baleanu, D., Guvenc, Z.B., Tenreiro Machado, J.A. (eds.) New Trends in Nanotechnology and Fractional Calculus Applications, pp. 503–509. Springer, Dordrecht (2010)

    Chapter  Google Scholar 

  17. Klamka, J.: Relative controllability and minimum energy control of linear systems with distributed delays in control. IEEE Trans. Autom. Contr. 21(4), 594–595 (1976)

    Article  MathSciNet  Google Scholar 

  18. Kaczorek, T., Klamka, J.: Minimum energy control of 2D linear systems with variable coefficients. Int. J. Control 44(3), 645–650 (1986)

    Article  Google Scholar 

  19. Solteiro Pires, E.J., Tenreiro Machado, J.A., Moura Oliveira, P.B.: Functional dynamics in genetic algorithms. In: Workshop on Fractional Differentiation and Its Application, vol. 2, pp. 414–419 (2006)

    Google Scholar 

  20. Brockett, R.W.: Nonlinear systems and differential geometry. Proc. IEEE 64(1), 61–71 (1976)

    Article  MathSciNet  Google Scholar 

  21. Isidori, A.: Nonlinear Control Systems. Springer, Berlin (1989)

    Book  Google Scholar 

  22. Marino, R., Tomei, P.: Nonlinear Control Design – Geometric, Adaptive, and Robust. Prentice Hall, London (1995)

    MATH  Google Scholar 

  23. Aguiller, J.L.M., Garcia, R.A., D’Attellis, C.E.: Exact linearization of nonlinear systems: trajectory tracking with bounded control and state constrains. In: Proceedings of the 38th Midwest Symposium on Circuits and Systems, Rio de Janeiro, pp. 620–622 (1995)

    Google Scholar 

  24. Charlet, B., Levine, J., Marino, R.: Sufficient conditions for dynamic state feedback linearization. SIAM J. Contr. Optim. 29(1), 38–57 (1991)

    Article  MathSciNet  Google Scholar 

  25. Daizhan, C., Tzyh-Jong, T., Isidori, A.: Global external linearization of nonlinear systems via feedback. IEEE Trans. Autom. Control 30, 808–811 (1985)

    Article  MathSciNet  Google Scholar 

  26. Fang, B., Kelkar, A.G.: Exact linearization of nonlinear systems by time scale transformation. In: Proceedings of the American Control Conference, Denver-Colorado, pp. 3555–3560 (2003)

    Google Scholar 

  27. Jakubczyk, B.: Introduction to Geometric Nonlinear Control; Controllability and Lie Bracket. Summer Schools on Mathematical Control Theory, Trieste (2001)

    MATH  Google Scholar 

  28. Jakubczyk, B., Respondek, W.: On linearization of control systems. Bull. Pol. Acad. Sci. Tech. 28, 517–521 (1980)

    MathSciNet  MATH  Google Scholar 

  29. Melham, K., Saad, M., Abou, S.C.: Linearization by redundancy and stabilization of nonlinear dynamical systems: a state transformation approach. In: IEEE International Symposium on Industrial Electronics, pp. 61–68 (2009)

    Google Scholar 

  30. Taylor, J.H., Antoniotti, A.J.: Linearization algorithms for computer-aided control engineering. Control Syst. Mag. 13, 58–64 (1993)

    Google Scholar 

  31. Khalil, R., Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

This work was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.

Author information

Authors and Affiliations

  1. Bialystok University of Technology, Bialystok, Poland

    Tadeusz Kaczorek

Authors
  1. Tadeusz Kaczorek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tadeusz Kaczorek .

Editor information

Editors and Affiliations

  1. ŁUKASIEWICZ Research Network, Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Roman Szewczyk

  2. ŁUKASIEWICZ Research Network, Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Cezary Zieliński

  3. ŁUKASIEWICZ Research Network, Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Małgorzata Kaliczyńska

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kaczorek, T. (2020). Linearization of Fractional Nonlinear Systems by State-Feedbacks. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2020: Towards Industry of the Future. AUTOMATION 2020. Advances in Intelligent Systems and Computing, vol 1140. Springer, Cham. https://doi.org/10.1007/978-3-030-40971-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation