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Robust H fuzzy control for discrete-time nonlinear systems

Regular Papers Intelligent and Information Systems

Abstract

This paper studies the problem of robust H control for discrete-time nonlinear systems presented as Takagi—Sugeno’s fuzzy models. The generalized non-parallel distributed compensation (non-PDC) law and non-quadratic Lyapunov function is constructed by the proposed homogeneouspolynomially basis-dependent matrix function (HPB-MF for abbreviation). Based on the generalized non-PDC law and non-quadratic Lyapunov function, some linear matrix inequalities (LMIs) are obtained by exploiting the possible combinations of the basis functions. These LMIs ensure the asymptotic stability of the closed-loop system and guarantee a norm bound constraint on disturbance attenuation. In addition, it is shown that the LMIs become less conservative as the degree of HPB-MF increases. The merit of the methods presented in this paper lies in their less conservatism than other methods, as shown by a numerical example borrowed from the literature.

Keywords

Homogeneous polynomially basis-dependent matrix function robust control linear matrix inequality non-quadratic Lyapunov function Takagi—Sugeno’s fuzzy model 

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References

  1. [1]
    T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 15, no. 1, pp. 116–132, 1985.MATHGoogle Scholar
  2. [2]
    M. Sugeno and G. T. Kang, “Structure identification of fuzzy model,” Fuzzy Sets Syst., vol. 28, no. 1, pp. 15–33, 1988.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    T. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stability H control theory, and linear matrix inequalities,” IEEE Trans. Fuzzy Syst., vol. 4, no. 1, pp. 1–13, 1996.CrossRefGoogle Scholar
  4. [4]
    F. H. Hsiao, C. W. Chen, Y. W. Liang, S. D. Xu, and W. L. Chiang, “T-S fuzzy controllers for nonlinear interconnected systems with multiple time delays,” IEEE Trans. Circuits, Systems-I, vol. 52, no. 9, pp. 1883–1893, 2005.CrossRefGoogle Scholar
  5. [5]
    F. H. Hsiao, C. W. Chen, Y. H. Wu, and W. L. Chiang, “Fuzzy controllers for nonlinear interconnected TMD systems with external force,” Journal of the Chinese Institute of Engineers, vol. 28, no. 1, pp. 175–181, 2005.Google Scholar
  6. [6]
    C. W. Chen, “Stability conditions of fuzzy systems and its application to structural and mechanical systems,” Advances in Engineering Software, vol. 37, no. 3, pp. 624–629, 2006.CrossRefGoogle Scholar
  7. [7]
    C. W. Chen, K. Yeh, W. L. Chiang, C. Y. Chen, and D. J. Wu, “Modeling, H control and stability analysis for structural systems using Takagi-Sugeno fuzzy model,” Journal of Vibration and Control, vol. 13, no. 11, pp. 1519–1534, 2007.CrossRefMathSciNetGoogle Scholar
  8. [8]
    E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst, vol. 8, no. 5, pp. 523–534, 2000.CrossRefGoogle Scholar
  9. [9]
    X. D. Liu and Q. L. Zhang, “New approaches to H controller designs based on fuzzy observers for T—S fuzzy systems via LMI,” Automatica, vol. 39, no. 9, pp. 1571–1582, 2003.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    X. D. Liu and Q. L. Zhang, “Approaches to quadratic stability conditions and H control designs for T—S fuzzy systems,” IEEE Trans. Fuzzy Syst, vol. 11, no. 6, pp. 830–839, 2003.CrossRefGoogle Scholar
  11. [11]
    C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong, and C. H. Lee, “A new LMI-Based approach to relaxed quadratic stabilization of T—S fuzzy control systems,” IEEE Trans. Fuzzy Syst, vol. 14, no. 3, pp. 386–397, 2006.CrossRefGoogle Scholar
  12. [12]
    L. Wang and G. Feng, “Piecewise H controller design of discrete time fuzzy systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 682–686, 2004.CrossRefMathSciNetGoogle Scholar
  13. [13]
    W. J. Wang, Y. J. Chen, and C. H. Sun, “Relaxed stabilization criteria for discrete-time T—S fuzzy control systems based on a switching fuzzy model and piecewise Lyapunov function,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 3, pp. 551–559, 2007.CrossRefMathSciNetGoogle Scholar
  14. [14]
    W. J. Wang and C. H. Sun, “A relaxed criterion for T—S fuzzy discrete systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 5, pp. 2155–2158, 2004.CrossRefGoogle Scholar
  15. [15]
    K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control system,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582–589, 2003.CrossRefGoogle Scholar
  16. [16]
    S. S. Zhou, J. Lam, and W. X. Zheng, “Control design for fuzzy systems based on relaxed nonquadratic stability and H performance conditions,” IEEE Trans. Fuzzy Syst, vol. 15, no. 2, pp. 188–199, 2007.CrossRefGoogle Scholar
  17. [17]
    G. Feng, C.-L. Chen, D. Sun, and Y. Zhu, “H controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities,” IEEE Trans. Fuzzy Syst., vol. 13, no. 1, pp. 94–103, 2005.CrossRefGoogle Scholar
  18. [18]
    S. S. Zhou, G. Feng, J. Lam, and S. Y. Xu, “Robust H control for discrete fuzzy systems via basisdependent Lyapunov functions,” Inf. Sci, vol. 174, no. 3–4, pp. 197–217, 2005.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    S. Y. Xu and J. Lam, “Robust H control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,” IEEE Trans. Fuzzy Syst., vol. 13, no. 1, pp. 82–93, 2005.CrossRefGoogle Scholar
  20. [20]
    H.-N. Wu, “Reliable LQ fuzzy control for continuous-time nonlinear systems with actuator faults,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 4, pp. 1743–1752, 2004.CrossRefGoogle Scholar
  21. [21]
    H.-N. Wu and H.-Y. Zhang, “Reliable mixed L2 /H fuzzy static output feedback control for nonlinear systems with sensor faults,” Automatica, vol. 41, no. 11, pp. 1925–1932, 2005.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    B. Chen, X. P. Liu, S. C. Tong, and C. Lin, “Guaranteed cost control of T—S fuzzy systems with state and input delays,” Fuzzy Sets Syst., vol. 158, no. 20, pp. 2251–2267, 2007.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Y. Wang, Z. Q. Sun, and F. C. Sun, “Robust fuzzy control of a class of nonlinear descriptor systems with time-varying delay,” International Journal of Control, Automation, and Systems, vol. 2, no. 1, pp. 76–82, 2004.Google Scholar
  24. [24]
    Y. W. Cho, K. S. Seo, and H. J. Lee, “A direct adaptive fuzzy control of nonlinear systems with application to robot manipulator tracking control,” International Journal of Control, Automation, and Systems, vol. 5, no. 6, pp. 630–642, 2007.Google Scholar
  25. [25]
    N. Essounbouli and A. Hamzaoui, “Direct and indirect robust adaptive fuzzy controllers for a class of nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 4, no. 2, pp. 146–154, 2006.Google Scholar
  26. [26]
    T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi—Sugeno’s form,” Automatica, vol. 40, no. 5, pp. 823–829, 2004.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    B. C. Ding, H. X. Sun, and P. Yang, “Further studies on LMI-Based relaxed stabilization conditions for nonlinear systems in Takagi—Sugeno’s form,” Automatica, vol. 42, no. 3, pp. 503–508, 2006.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    B. C. Ding and B. Huang, “Reformulation of LMIbased stabilization conditions for non-linear systems in Takagi-Sugeno’s form,” International Journal of Systems Science, vol. 39, no. 5, pp. 487–496, 2008.MATHMathSciNetGoogle Scholar
  29. [29]
    L. K. Wang and X. D. Liu, “Comments on Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno’s form,” Automatica, vol. 44, no. 11, pp. 2292–2293, 2008.CrossRefGoogle Scholar
  30. [30]
    L. K. Wang and X. D. Liu, “New relaxed stabilization conditions for fuzzy control systems,” International Journal of Innovative Computing, Information and Control, vol. 5, no. 5, pp. 1451–1460, 2009.Google Scholar
  31. [31]
    C. E. De Souza and X. Li, “Delay-dependent robust H control of uncertain linear state-delayed systems,” Automatica, vol. 35, no. 7, pp. 1313–1321, 1999.MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach,” IEEE Trans. Automat. Control, vol. 50, no. 3, pp. 365–370, 2005.CrossRefMathSciNetGoogle Scholar
  33. [33]
    A. Sala and C. Ariño, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya’s theorem,” Fuzzy Sets Syst, vol. 158, pp. 2671–2686, 2007.MATHCrossRefGoogle Scholar
  34. [34]
    R. C. L. F. Oliveira and P. L. D. Peres, “LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions,” Systems Control Lett., vol. 55, no. 1, pp. 52–61, 2006.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceNanchang Hangkong UniversityNanchangChina
  2. 2.Research Center of Information and ControlDalian University of TechnologyDalianChina

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