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Stability Analysis of General Polynomial Fuzzy Model-Based Control Systems

  • Hak-Keung LamEmail author
Chapter
  • 593 Downloads
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 64)

Abstract

This chapter investigates the stability of general polynomial fuzzy model-based control systems. In most of the published results, a constraint that the polynomial Lyapunov function matrix is allowed to be dependent on some state variables determined by the structure of the input matrices is required to obtain convex stability conditions. In this chapter, this constraint is removed and a two-step procedure is proposed to search for a feasible solution to the SOS-based stability conditions. Consequently, the stability analysis results can be applied to a wider range of polynomial-fuzzy model-based control systems. Furthermore, three cases of polynomial fuzzy controllers under perfectly, partially and imperfectly matched premises are considered. Their levels of controller complexity, design flexibility and stability analysis results are discussed and compared. Simulations examples are given to compare among the three cases in terms of capability of find feasible solutions and show that the proposed analysis results outperform some published ones.

Keywords

Polynomial Fuzzy Model Stability Analysis Results Premise Membership Functions Lyapunov Stability Theory Greater Design Flexibility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Wang, H.O., Tanaka, K., Griffin, M.F.: An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4(1), 14–23 (1996)CrossRefGoogle Scholar
  2. 2.
    Tanaka, K., Ikeda, T., Wang, H.O.: Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. 6(2), 250–265 (1998)CrossRefGoogle Scholar
  3. 3.
    Kim, E., Lee, H.: New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans. Fuzzy Syst. 8(5), 523–534 (2000)CrossRefGoogle Scholar
  4. 4.
    Teixeira, M.C.M., Assuncão, E., Avellar, R.G.: On relaxed LMI-based designs for fuzzy regulators and fuzzy observers. IEEE Trans. Fuzzy Syst. 11(5), 613–623 (2003)CrossRefGoogle Scholar
  5. 5.
    Liu, X., Zhang, Q.: New approaches to H\(_\infty \) controller designs based on fuzzy observers for Takagi-Sugeno fuzzy systems via LMI. Automatica 39(9), 1571–1582 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu, X., Zhang, Q.: Approaches to quadratic stability conditions and H\(_\infty \) control designs for Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 11(6), 830–839 (2003)CrossRefGoogle Scholar
  7. 7.
    Fang, C.H., Liu, Y.S., Kau, S.W., Hong, L., Lee, C.H.: A new LMI-based approach to relaxed quadratic stabilization of Takagi-Sugeno fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(3), 386–397 (2006)CrossRefGoogle Scholar
  8. 8.
    Sala, A., Ariño, C.: Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya’s theorem. Fuzzy Sets Syst. 158(24), 2671–2686 (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lo, J.C., Wan, J.R.: Studies on linear matrix inequality relaxations for fuzzy control systems via homogeneous polynomials. IET Control Theory Appl. 4(11), 2293–2302 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sala, A., Ariño, C.: Relaxed stability and performance conditions for Takagi-Sugeno fuzzy systems with knowledge on membership function overlap. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 37(3), 727–732 (2007)CrossRefGoogle Scholar
  11. 11.
    Sala, A., Ariño, C.: Relaxed stability and performance LMI conditions for Takagi-Sugeno fuzzy systems with polynomial constraints on membership function shapes. IEEE Trans. Fuzzy Syst. 16(5), 1328–1336 (2008)CrossRefGoogle Scholar
  12. 12.
    Kruszewski, A., Sala, A., Guerra, T., Arino, C.: A triangulation approach to asymptotically exact conditions for fuzzy summations. IEEE Trans. Fuzzy Syst. 17(5), 985–994 (2009)CrossRefGoogle Scholar
  13. 13.
    Narimani, M., Lam, H.K.: Relaxed LMI-based stability conditions for Takagi-Sugeno fuzzy control systems using regional-membership-function-shape-dependent analysis approach. IEEE Trans. Fuzzy Syst. 17(5), 1221–1228 (2009)CrossRefGoogle Scholar
  14. 14.
    Lam, H.K., Leung, F.H.F.: Stability analysis of fuzzy control systems subject to uncertain grades of membership. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 35(6), 1322–1325 (2005)CrossRefGoogle Scholar
  15. 15.
    Lam, H.K., Leung, F.H.F.: LMI-based stability and performance design of fuzzy control systems: fuzzy models and controllers with different premises. In: Proceedings of the International Conference on Fuzzy Systems 2006 (FUZZ-IEEE 2006), pp. 9499–9506. Vancouver, BC, Canada (2006)Google Scholar
  16. 16.
    Ariño, C., Sala, A.: Extensions to “stability analysis of fuzzy control systems subject to uncertain grades of membership”. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38(2), 558–563 (2008)CrossRefGoogle Scholar
  17. 17.
    Lam, H.K., Narimani, M.: Stability analysis and performance design for fuzzy-model-based control system under imperfect premise matching. IEEE Trans. Fuzzy Syst. 17(4), 949–961 (2009)CrossRefGoogle Scholar
  18. 18.
    Lam, H.K., Narimani, M.: Quadratic stability analysis of fuzzy-model-based control systems using staircase membership functions. IEEE Trans. Fuzzy Syst. 18(1), 125–137 (2010)CrossRefGoogle Scholar
  19. 19.
    Lam, H.K.: LMI-based stability analysis for fuzzy-model-based control systems using artificial T-S fuzzy model. IEEE Trans. Fuzzy Syst. 19(3), 505–513 (2011)CrossRefGoogle Scholar
  20. 20.
    Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.O.: A sum of squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans. Fuzzy Syst. 17(4), 911–922 (2009)CrossRefGoogle Scholar
  21. 21.
    Tanaka, K., Ohtake, H., Wang, H.O.: Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 39(2), 561–567 (2009)CrossRefGoogle Scholar
  22. 22.
    Sala, A., Ariño, C.: Polynomial fuzzy models for nonlinear control: a Taylor-series approach. IEEE Trans. Fuzzy Syst. 17(6), 284–295 (2009)CrossRefGoogle Scholar
  23. 23.
    Narimani, M., Lam, H.K.: SOS-based stability analysis of polynomial fuzzy-model-based control systems via polynomial membership functions. IEEE Trans. Fuzzy Syst. 18(5), 862–871 (2010)CrossRefGoogle Scholar
  24. 24.
    Lam, H.K.: Polynomial fuzzy-model-based control systems: stability analysis via piecewise-linear membership functions. IEEE Trans. Fuzzy Syst. 19(3), 588–593 (2011)CrossRefGoogle Scholar
  25. 25.
    Lam, H.K., Seneviratne, L.D.: Stability analysis of polynomial fuzzy-model-based control systems under perfect/imperfect premise matching. IET Control Theory Appl. 5(15), 1689–1697 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lam, H.K., Narimani, M., Li, H., Liu, H.: Stability analysis of polynomial-fuzzy-model-based control systems using switching polynomial Lyapunov function. IEEE Trans. Fuzzy Syst. 21(5), 800–813 (2013)CrossRefGoogle Scholar
  27. 27.
    Ebenbauer, C., Renz, J., Allgower, F.: Polynomial feedback and observer design using nonquadratic Lyapunov functions. In: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05), Seville, Spain, pp. 7587–7592. IEEE (2005)Google Scholar
  28. 28.
    Lo, J.C., Lin, Y.T., Chang, W.S., Lin, F.Y.: SOS-based fuzzy stability analysis via homogeneous Lyapunov functions. In: Proceedings of the 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2014), pp. 2300–2305. IEEE (2014)Google Scholar
  29. 29.
    Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control, vol. 1, pp. 741–746. Las Vegas, Nevada, USA (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of InformaticsKing’s College LondonLondonUK

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