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Polynomial Fuzzy Model-Based Control Systems pp 39–58Cite as

Preliminaries

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

Abstract

This chapter provides the technical and mathematical background for the fuzzy model-based control which offers the equations of the fuzzy model and closed-loop systems, definition of variables, published stability conditions in terms of linear matrix inequalities (LMIs) and sum of squares (SOS). Numerical examples are given to demonstrate the motivation using polynomial fuzzy model over T-S fuzzy model. State-feedback fuzzy controller and polynomial fuzzy controller are introduced to close the feedback loop. Three main types of control design including perfectly, partially and imperfectly matched premises are discussed and compared. LMI/SOS-based stability conditions in the literature are reviewed, which will be used in other chapters for comparison purposes.

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References

  1. Tanaka, K., Iwasaki, M., Wang, H.O.: Switching control of an R/C hovercraft: stabilization and smooth switching. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 31(6), 853–863 (2001)

    Google Scholar 

  2. 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 

  3. 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)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. 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 

  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 Theor. Appl. 4(11), 2293–2302 (2010)

    Article  MathSciNet  Google Scholar 

  15. 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 

  16. 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)

    Google Scholar 

  17. 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 

  18. Lam, H.K., Narimani, M.: Sum-of-squares-based stability analysis of polynomial fuzzy-model-based control systems. In: Proceedings of the 2009 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2009), pp. 234–239. IEEE, ICC Jeju, Jeju Island, Korea (2009)

    Google Scholar 

  19. 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 

  20. 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 

  21. Lam, H.K.: Stabilization of nonlinear systems using sampled-data output-feedback fuzzy controller based on polynomial-fuzzy-model-based control approach. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 41(1), 258–267 (2012)

    Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. 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)

    Google Scholar 

  26. 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 

  27. Lam, H.K., Seneviratne, L.D.: Stability analysis of polynomial fuzzy-model-based control systems under perfect/imperfect premise matching. IET Control Theor. Appl. 5(15), 1689–1697 (2011)

    Article  MathSciNet  Google Scholar 

  28. 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 

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

    Article  MathSciNet  Google Scholar 

  30. 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)

    Article  Google Scholar 

  31. 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 

  32. 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 

  33. 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 

  34. 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 

  35. Papachristodoulou, A., Prajna, S.: A tutorial on sum of squares techniques for system analysis. In: Proceedings of the American Control Conference (ASCC), pp. 2686–2700. Portland, OR, USA (2005)

    Google Scholar 

  36. Prajna, S., Papachristodoulou, A., Parrilo, P.A.: SOSTOOLS - sum of squares optimization toolbox, user’s guide (2002)

    Google Scholar 

  37. 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 

  38. Kim, E., Park, M., Ji, S., Park, M.: A new approach to fuzzy modeling. IEEE Trans. Fuzzy Syst. 5(3), 328–337 (1997)

    Article  Google Scholar 

  39. Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Set. Syst. 45(2), 135–156 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Khalil, H.K., Grizzle, J.W.: Nonlinear Systems. Prentice hall, Englewood Cliffs (1996)

    Google Scholar 

  41. Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: The LMI control toolbox. In: Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 3, pp. 2038–2041. IEEE, Lake Buena Vista, FL, USA (1995)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Hak-Keung Lam

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Lam, HK. (2016). Preliminaries. 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_2

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  • DOI: https://doi.org/10.1007/978-3-319-34094-4_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-34092-0

  • Online ISBN: 978-3-319-34094-4

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