Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Polynomial Fuzzy Model-Based Control Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 64))

  • 869 Accesses

Abstract

This chapter investigates the stability of polynomial fuzzy model-based control systems by treating the membership functions and system states as symbolic variables. The information of membership functions is considered in the stability analysis and brought to the SOS-based stability conditions. Techniques are proposed to introduce slack matrix variables carrying the information of membership functions, including the property of membership functions, boundary information of membership functions and boundary information of premise variables, to the SOS-based stability conditions without increasing much the computational demand. Details of mathematical derivation are shown to help readers follow easily the analysis. A simulation example is given to show how to apply the obtained stability conditions during the control design and demonstrate the merits of the proposed stability analysis results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Feng, G.: A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006)

    Article  Google Scholar 

  2. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modelling and control. IEEE Trans. Syst. Man Cybern. smc–15(1), 116–132 (1985)

    Article  MATH  Google Scholar 

  3. Sugeno, M., Kang, G.T.: Structure identification of fuzzy model. Fuzzy Sets Syst. 28(1), 15–33 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Chen, C.L., Chen, P.C., Chen, C.K.: Analysis and design of fuzzy control system. Fuzzy Sets Syst. 57(2), 125–140 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kim, E., Lee, H.: New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans. Fuzzy Syst. 8(5), 523–534 (2000)

    Article  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. 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)

    Article  MATH  Google Scholar 

  13. Montagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D.: Convergent LMI relaxations for quadratic stabilizability and control of Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 17(4), 863–873 (2009)

    Article  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  Google Scholar 

  20. Lam, H.K.: Design of stable fuzzy controller for non-linear systems subject to imperfect premise matching based on grid-point approach. IET Control Theory Appl. 4(12), 2770–2780 (2010)

    Article  MathSciNet  Google Scholar 

  21. 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)

    Article  Google Scholar 

  22. 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)

    Article  Google Scholar 

  23. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  25. 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)

    Article  Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. 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)

    Article  Google Scholar 

  28. Sala, A., Ariño, C.: Polynomial fuzzy models for nonlinear control: a Taylor-series approach. IEEE Trans. Fuzzy Syst. 17(6), 284–295 (2009)

    Article  Google Scholar 

  29. Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Nonlinear control synthesis by sum-of-squares optimization: a Lyapunov-based approach. In: Proceedings of the Asian Control Conference (ASCC), vol. 1, pp. 157–165. Melbourne, Australia (2004)

    Google Scholar 

  30. 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 

  31. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley-Interscience, Hoboken (2001)

    Book  Google Scholar 

  32. Lam, H.K., Tsai, S.H.: Stability analysis of polynomial-fuzzy-model-based control systems with mismatched premise membership functions. IEEE Trans. Fuzzy Syst. 22(1), 223–229 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, King’s College London, London, UK

    Hak-Keung Lam

Authors
  1. Hak-Keung Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hak-Keung Lam .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Lam, HK. (2016). Stability Analysis of Polynomial Fuzzy Model-Based Control Systems with Mismatched Premise Membership Functions Through Symbolic Variables. In: Polynomial Fuzzy Model-Based Control Systems. Studies in Systems, Decision and Control, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-319-34094-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-34094-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-34092-0

  • Online ISBN: 978-3-319-34094-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation