Stability Analysis of Polynomial Fuzzy Model-Based Control Systems with Mismatched Premise Membership Functions Through Taylor Series Membership Functions

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


This chapter investigates the stability of polynomial fuzzy model-based control systems by bringing the approximated membership functions into the SOS-based stability conditions. Various approximation methods of membership functions are reviewed and their characteristics are discussed. Using the Taylor series expansion, the original membership functions are represented by approximated membership functions which are a weighted sum of local polynomials in a favorable form for stability analysis. SOS-based stability conditions are obtained which guarantee the system stability if the fuzzy model-based control system is stable at all chosen Taylor series expansion points. A simulation example is presented to illustrate the influence of the density of expansion points to the capability of stability conditions finding a feasible solution and demonstrate the effectiveness of the proposed stability conditions over some published results.


  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.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley-Interscience, New York (2001)Google Scholar
  4. 4.
    Feng, G.: A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Tanaka, K., Ohtake, H., Wang, H.O.: Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach. IEEE Trans. Syst. Man and Cybern.-Part B: Cybern. 39(2), 561–567 (2009)Google Scholar
  7. 7.
    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
  8. 8.
    Narimani, M., Lam, H.K., Dilmaghani, R., Wolfe, C.: LMI-based stability analysis of fuzzy-model-based control systems using approximated polynomial membership functions. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 41(3), 713–724 (2011)Google Scholar
  9. 9.
    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
  10. 10.
    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
  11. 11.
    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
  12. 12.
    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
  13. 13.
    Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists. Elsevier, New York (2005)Google Scholar
  14. 14.
    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 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of InformaticsKing’s College LondonLondonUK

Personalised recommendations