Linearization of Fractional Nonlinear Systems by State-Feedbacks

  • Tadeusz KaczorekEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1140)


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.


Fractional Nonlinear System Lie derivative Linearization State-feedback 



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


  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)zbMATHGoogle Scholar
  2. 2.
    Ostalczyk, P.: Epitome of the Fractional Calculus: Theory and Its Applications in Automatics. Wydawnictwo Politechniki Łódzkiej, Łódź (2008). (in Polish)Google Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  4. 4.
    Kaczorek, T.: Fractional positive continuous-time systems and their reachability. Int. J. Appl. Comput. Sci. 18(2), 223–228 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. 58(6), 1203–1210 (2011)MathSciNetCrossRefGoogle Scholar
  6. 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. 7.
    Dzieliński, A., Sierociuk, D.: Stability of discrete fractional order state-space systems. J. Vib. Control 14(9/10), 1543–1556 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2012)zbMATHGoogle Scholar
  9. 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. 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. 11.
    Jumerie, G.: The Leibniz rule for fractional derivatives holds with non-differentiable functions. Math. Stat. 1(2), 50–52 (2013)Google Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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. 14.
    Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Press, Dordrecht (1991)zbMATHGoogle Scholar
  15. 15.
    Klamka, J.: Minimum energy control of 2D systems in Hilbert spaces. Syst. Sci. 9(1–2), 33–42 (1983)MathSciNetzbMATHGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kaczorek, T., Klamka, J.: Minimum energy control of 2D linear systems with variable coefficients. Int. J. Control 44(3), 645–650 (1986)CrossRefGoogle Scholar
  19. 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. 20.
    Brockett, R.W.: Nonlinear systems and differential geometry. Proc. IEEE 64(1), 61–71 (1976)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Isidori, A.: Nonlinear Control Systems. Springer, Berlin (1989)CrossRefGoogle Scholar
  22. 22.
    Marino, R., Tomei, P.: Nonlinear Control Design – Geometric, Adaptive, and Robust. Prentice Hall, London (1995)zbMATHGoogle Scholar
  23. 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. 24.
    Charlet, B., Levine, J., Marino, R.: Sufficient conditions for dynamic state feedback linearization. SIAM J. Contr. Optim. 29(1), 38–57 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Daizhan, C., Tzyh-Jong, T., Isidori, A.: Global external linearization of nonlinear systems via feedback. IEEE Trans. Autom. Control 30, 808–811 (1985)MathSciNetCrossRefGoogle Scholar
  26. 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. 27.
    Jakubczyk, B.: Introduction to Geometric Nonlinear Control; Controllability and Lie Bracket. Summer Schools on Mathematical Control Theory, Trieste (2001)zbMATHGoogle Scholar
  28. 28.
    Jakubczyk, B., Respondek, W.: On linearization of control systems. Bull. Pol. Acad. Sci. Tech. 28, 517–521 (1980)MathSciNetzbMATHGoogle Scholar
  29. 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. 30.
    Taylor, J.H., Antoniotti, A.J.: Linearization algorithms for computer-aided control engineering. Control Syst. Mag. 13, 58–64 (1993)Google Scholar
  31. 31.
    Khalil, R., Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Bialystok University of TechnologyBialystokPoland

Personalised recommendations