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Stabilization of Continuous-Time Fractional Positive T–S Fuzzy Systems by Using a Lyapunov Function

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Abstract

The stability and stabilization problems for continuous-time fractional T–S fuzzy systems with the additional condition of nonnegativity of the states are solved. The obtained results are based on a direct Lyapunov function. In particular, the synthesis of state-feedback controllers is obtained by giving conditions in terms of linear programming. An illustrative example is provided to show the usefulness of the results.

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References

  1. M.P. Aghababa, Comments on \(H_\infty \) synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans. 51, 1112 (2012)

    Article  Google Scholar 

  2. M.P. Aghababa, A fractional-order controller for vibration suppression of uncertain structures. ISA Trans. 52, 881887 (2013)

    Article  Google Scholar 

  3. M. Ait Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II 54(2), 151–155 (2007)

    Article  Google Scholar 

  4. M. Araki, B. Kondo, Stability and transient behavior of composite nonlinear systems. IEEE Trans. Automat. Control 17(4), 537–541 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Benzaouia, A. Hmamed, A. El Hajjaji, Stabilization of controlled positive discrete-time T–S fuzzy systems by state feedback control. Int. J. Adapt. Control Signal Process. 24(12), 1091–1106 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Benzaouia, Saturated Switched Systems (Springer, Berlin, 2012)

    Book  MATH  Google Scholar 

  7. A. Benzaouia, A. Hmamed, F. Mesquine, M. Benhayoun, F. Tadeo, Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Trans. Autom. Control 59(8), 2203–2208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Benzaouia, A. El Hajjaji, Delay-dependent stabilization conditions of controlled positive T–S fuzzy systems with time varying. IJICIC 7(4), 1533–1548 (2011)

    Google Scholar 

  9. A. Benzaouia, A. El Hajjaji, Advanced Takagi–Sugeno Systems: Delay and Saturation, Studies in Systems, Decision and Control, vol. 8 (Springer, Berlin, 2014)

    MATH  Google Scholar 

  10. B. Buslowicz, Stability of state space models of linear continuous-time fractional order systems. Acta Mech. Autom. 5(2), 15–22 (2011)

    Google Scholar 

  11. Y.Q. Chen, H.S. Ahn, I. Podlubny, Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Process. 86(10), 2611–2618 (2006)

    Article  MATH  Google Scholar 

  12. M. Efe, Fractional order systems in industrial automation a survey. IEEE Trans. Ind. Inf. 7(4), 582–591 (2011)

    Article  Google Scholar 

  13. A. El Hajjaji, A. Ciocan, D. Hamad, Four wheel steering control by fuzzy approach. J. Intell. Robot. Syst. 41, 141–156 (2005)

    Article  Google Scholar 

  14. G. Feng, Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 12(1), 22–28 (2004)

    Article  Google Scholar 

  15. H. Gassara, A. El Hajjaji, M. Chaabane, Observer-based robust \(H_\infty \) reliable control for uncertain T–S fuzzy systems with time delay. IEEE Trans. Fuzzy Syst. 18(6), 1027–1040 (2010)

    Article  MATH  Google Scholar 

  16. T.M. Guerra, L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form. Automatica 40, 823–829 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Hmamed, A. Benzaouia, F. Tadeo, M. Ait Rami, Positive stabilization of discrete-time systems with unknown delay and bounded controls. in European Control Conference (Kos, Greece, 2–5 July, 2007)

  18. A. Hmamed, A. Benzaouia, F. Tadeo, M.A. Rami, Memoryless control of delayed continuous-time systems within the nonnegative orthant. in IFAC World Congress (Korea, 2008)

  19. A. Hmamed, F. Mesquine, A. Benzaouia, M. Benhayoun, F. Tadeo, Continuous-time fractional bounded positive systems. in 52nd IEEE Conference on Decision and Control (Firenze, Italy, December 10–13, 2013)

  20. A. Hmamed, F. Mesquine, A. Benzaouia, M. Benhayoun, F. Tadeo, Continuous-time fractional positive systems with bounded states. in 52nd IEEE Conference on Decision and Control (Florence, Italy, December 10–13, 2013)

  21. A. Hmamed, M. Ait Rami, A. Benzaouia, F. Tadeo, Stabilization under constrained states and controls of positive systems with time delays. Eur. J. Control 18(2), 182–190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Johansson, A. Rantzer, K. Arzen, Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Syst. 7(6), 713–722 (1999)

    Article  Google Scholar 

  23. T. Kaczorek, Positive 1D and 2D Systems (Springer, Berlin, 2002)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Kaczorek, Selected Problems of Fractional Systems Theory (LNCIS, Springer, Berlin, 2011)

    Book  MATH  Google Scholar 

  26. T. Kaczorek, Necessary and sufficient stability conditions of fractional positive continuous-time linear systems. Acta Mech. Autom. 5(2), 52–54 (2011)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Li, K. Su, J. Zhang, D. Wei, Robust control for fractional-order four-wing hyperchaotic system using LMI. Optik 124, 5807–5810 (2013)

    Article  Google Scholar 

  29. T.-C. Lin, C.-H. Kuob, \(H_\infty \) synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans. 50, 548–556 (2011)

    Article  Google Scholar 

  30. C.P. Li, F.R. Zhang, A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193, 27–47 (2011)

    Article  Google Scholar 

  31. J. Lu, G. Chen, Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans. Autom. Control 54(6), 1294–1299 (2009)

    Article  MathSciNet  Google Scholar 

  32. D. Matignon, Stability results on fractional differential equations with application to control processing. in Computational Engineering in Systems Applications, vol 2 (Lille, France, 1996)

  33. F. Mesquine, A. Hmamed, M. Benhayoun, A. Benzaouia, F. Tadeo, Robust stabilization of constrained uncertain continuous-time fractional positive systems. J. Franklin Inst. 352(1), 259–270 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Willey, New York, 1993)

    MATH  Google Scholar 

  35. M. Nachidi, A. Benzaouia, F. Tadeo, Based approach for output-feedback stabilization for discrete time Takagi-Sugeno systems. IEEE Trans. Fuzzy Syst. 16(5), 1188–1196 (2008)

    Article  Google Scholar 

  36. M. Nachidi, F. Tadeo, A. Benzaouia, M. Ait Rami, Static output-feedback for Takagi-Sugeno systems with delays. Int. J. Adapt. Control Signal Process. 25(4), 295–312 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. I. N’Doye, M. Zasadzinski, M. Darouach, N. E. Radhy, Robust stabilization of linear and nonlinear fractional-order systems with nonlinear uncertain parameters. in 49th IEEE Conference on Decision and Control Atlanta (USA, 2010)

  38. K. Nishimoto, Fractional Calculus (Descartes Press, Koriyama, 1984)

    MATH  Google Scholar 

  39. K.B. Oldham, J. Spanier, Fractional Calculus (Academic Press, Cambridge, 1974)

    MATH  Google Scholar 

  40. A. Oustaloup, Commande CRONE (Hermes, Paris, 1983)

    MATH  Google Scholar 

  41. M.A. Rami, F. Tadeo, Linear programming approach to impose positiveness in closed-loop and estimated states. in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (Kyoto, Japan, 2006)

  42. J. Sabatier, O.P. Agrawal, J.A.T. Machado (eds.), Advances in Fractional Calculus, Theoretical Development and Applications in Physics and Engineering (Springer, London, 2007)

  43. T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 15(9), 116–132 (1985)

    Article  MATH  Google Scholar 

  44. Y. Tang, X. Zhang, D. Zhang, G. Zhao, X. Guan, Fractional order sliding mode controller design for antilock braking systems. Neurocomputing 111, 122–130 (2013)

    Article  Google Scholar 

  45. J.C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91, 437–445 (2001)

    Article  MATH  Google Scholar 

  46. B.M. Vinagre, C.A. Monje, A.J. Calderon, Fractional order systems and fractional order control actions. in Proceedings of the IEEE CDC Conference Tutorial Workshop: Fractional Calculus Applications in Automatic Control and Robotics (Las Vegas Nevada, pp. 15 – 38, 2002)

  47. H.O. Wang, K. Tanaka, M.F. Griffin, An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4(1), 14–23 (1996)

    Article  Google Scholar 

  48. Y. Zheng, Y. Nian, D. Wanga, Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375, 125129 (2010)

    Article  Google Scholar 

  49. H.W. Zhou, C.P. Wang, B.B. Han, Z.Q. Duan, A creep constitutive model for salt rock based on fractional derivatives. Int. J. Rock Mech. Min. Sci. 48, 116–121 (2011)

    Article  Google Scholar 

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Authors and Affiliations

  1. LAEPT, Faculty of Science Semlalia, University Cadi Ayyad, BP 2390, Marrakech, Morocco

    Abdellah Benzaouia

  2. MIS, University of Picardie Jules-Verne, 33 rue Saint Leu, 80039, Amiens Cedex 1 Amiens, France

    Ahmed El Hajjaji

Authors
  1. Abdellah Benzaouia
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  2. Ahmed El Hajjaji
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Correspondence to Abdellah Benzaouia.

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Benzaouia, A., El Hajjaji, A. Stabilization of Continuous-Time Fractional Positive T–S Fuzzy Systems by Using a Lyapunov Function. Circuits Syst Signal Process 36, 3944–3957 (2017). https://doi.org/10.1007/s00034-017-0507-8

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  • DOI: https://doi.org/10.1007/s00034-017-0507-8

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