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An iterative method to design optimal non-fragile \({\varvec{H}}_{\varvec{\infty }}\) observer for Lipschitz nonlinear fractional-order systems

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Abstract

In this paper, the stability of a nonlinear non-fragile \(H_\infty \) fractional-order observer, based on the fractional-order Lyapunov theorem, is investigated in detail. It is the first time to derive the optimal gain of desired observer among a solution set that satisfies the nonlinear robust non-fragile fractional-order observer stability conditions systematically using linear matrix inequality approach. An iterative linear matrix inequality algorithm is introduced while a boundary condition is unknown during the design procedure. Finally, a fractional-order financial system is introduced to show the effectiveness of the proposed method. It has been shown that not only the iterative method is successful to find the proper boundary condition, but also the performance of the proposed observer is satisfying both non-fragility and robustness to external disturbances with an acceptable accuracy.

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References

  1. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, New York (2010)

    Book  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Hilfer, R.: Application of Fractional Calculus in Physics. World Science, Singapore (2000)

    Book  Google Scholar 

  4. Zaslavsky, G.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2006)

    Google Scholar 

  5. Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Science, Singapore (2000)

    Book  Google Scholar 

  6. Dadras, S., Momeni, H.R.: Control of a fractional-order economical system via sliding mode. Phys. A 389, 2434–2442 (2010)

    Article  Google Scholar 

  7. Das, S.: Functional Fractional Calculus for System Identification and Controls, 1st edn. Springer, Germany (2008)

    MATH  Google Scholar 

  8. Zemouche, A., Boutayeb, M.: Nonlinear-observer-based synchronization and unknown input recovery. IEEE T Circuits-I 56, 1720–1731 (2009)

    Article  MathSciNet  Google Scholar 

  9. Reif, K., Sonnemann, F., Unbehauen, R.: Nonlinear state observation using \(H_\infty \)-filtering riccati design. IEEE Trans. Autom. Control 44, 203–208 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mart’ınez, R.M., Mata-Machuca, J.L., Guerra, R.M., Le’on, J.A., Fern’andez-Anaya, G.: A new observer for nonlinear fractional order systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA (2011)

  11. Dadras, S., Momeni, H.R.: Fractional sliding mode observer design for a class of uncertain fractional order nonlinear systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA (2011)

  12. N’Doye, I., Darouach, M., Zasadzinski, M., Radhy, N.E.: Observers design for singular fractional-order systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA (2011)

  13. Jeong, C.S., Yaz, E.E., Bahakeem, A., Yaz, Y.I.: Resilient design of observers with general criteria using LMIs. In: Proceedings of the American Control Conference, Minnesota, USA (2006)

  14. Pourgholi, M., Johari Majd, V.: A novel robust proportional-integral (PI) adaptive observer design for chaos synchronization. Chin. Phys. B 20, 120503 (2011)

    Article  Google Scholar 

  15. Boroujeni, E.A., Momeni, H.R.: Lyapunov based design of a nonlinear resilient fractional order observer. In: Proceedings of the 21st Iranian Conference on Electrical Engineering, Mashhad, Iran (2013)

  16. Boroujeni, E.A., Momeni, H.R.: Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems. Signal Process. 92, 2365–2370 (2012)

  17. Li, C., Wang, J., Lu, J.: Observer-based robust stabilisation of a class of non-linear fractional-order uncertain systems: an linear matrix inequalitie approach. IET Control Theory Appl. 6, 2757–2764 (2012)

    Article  MathSciNet  Google Scholar 

  18. Lan, Y.H., Li, W.J., Zhou, Y., Luo, Y.P.: Non-fragile observer design for fractional-order one-sided Lipschitz nonlinear systems. Int. J Autom. Comput. 10, 296–302 (2013)

    Article  Google Scholar 

  19. N’Doye, I., Voos, H., Darouach, M., Schneider, J.G., Knauf, N.: An unknown input fractional-order observer design for fractional-order glucose-insulin system. In: Proceedings of the IEEE EMBS International Conference on Biomedical Engineering and Sciences, Hong Kong, China (2012)

  20. N’Doye, I., Voos, H., Darouach, M.: Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE J. Emerg. Sel. Top. Circuits Syst. 3, 442–450 (2013)

    Article  Google Scholar 

  21. Senejohnny, D.M., Delvari, H.: Active sliding observer scheme based fractional chaos synchronization. Commun. Nonlinear Sci. Numer. Simul. 17, 4373–4383 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen, WCh.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fract. 36, 1305–1314 (2008)

    Article  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academie Press, New York (1999)

    MATH  Google Scholar 

  24. Efe, M.O.: Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Trans. Syst. Man Cybern. B 38, 1561–1570 (2008)

    Article  Google Scholar 

  25. Efe, M.O., Kasnakoglu, C.: A fractional adaptation law for sliding mode control. Int. J. Adapt. Control 22, 968–986 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, Y., Chen, Y.Q., Podlubny, I.: Mittag–Leffler stability of fractional-order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

  28. Boyd, S., Ghaoui, L., Feron, E.: Linear matrix inequalities in system and control theory. In: Proceedings of the SIAM Studies in Applied Mathematics (1994)

  29. Chen, F., Zhang, W.: LMI criteria for robust chaos synchronization of a class of chaotic systems. Nonlinear Anal. Theory 67, 3384–3393 (2007)

    Article  MATH  Google Scholar 

  30. Cho, Y.M., Rajamani, R.: A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans. Autom. Control 42, 534–537 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Fadiga, L., Farges, Ch., Sabatier, J., Moze, M.: On computation of \(H_\infty \) norm for commensurate fractional order systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA (2011)

  32. Valério, D.: Ninteger v. 2.3, Fractional Control Toolbox for Matlab, Fractional Derivatives and Applications. Universidadetecnica de Lisboainstituto Superior Tecnico (2005)

  33. Lofberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: IEEE International Symposium on Computer Aided Control Systems Design Conference, Taipei, Taiwan (2004)

  34. Gahinet, P., Nemirovski, A., Laub, A., Chilai, M.: LMI Control Toolbox User’s Guide. The Mathworks, Massachusetts (1995)

    Google Scholar 

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Acknowledgments

The authors would like to thank Dr. Mahdi Pourgholi for his suggestions and criticism that helped in improving this paper.

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Authors and Affiliations

  1. Automation and Instruments Laboratory, Electrical Engineering Department, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran

    Elham Amini Boroujeni & Hamid Reza Momeni

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  1. Elham Amini Boroujeni
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  2. Hamid Reza Momeni
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Correspondence to Hamid Reza Momeni.

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Boroujeni, E.A., Momeni, H.R. An iterative method to design optimal non-fragile \({\varvec{H}}_{\varvec{\infty }}\) observer for Lipschitz nonlinear fractional-order systems. Nonlinear Dyn 80, 1801–1810 (2015). https://doi.org/10.1007/s11071-014-1889-9

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  • DOI: https://doi.org/10.1007/s11071-014-1889-9

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