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Nonquadratic Controller and Observer Design for Continuous TS Models: A Discrete-Inspired Solution

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Abstract

This work deals with controller and observer design for continuous-time Takagi–Sugeno models based on a new nonquadratic Lyapunov function. The proposed technique intends to attain the level of development reached by its discrete-time counterpart by suitably adapting one of its key ideas: the inclusion of former values of the premise variables. The proposed approach naturally overcome the problem of dealing with the time-derivatives of the membership functions, thus providing a simplified as well as easier alternative to recent results on this matter. Moreover, thanks to the Finsler’s Lemma and a Tustin-like transformation, controller and observer gains are decoupled from the proposed Lyapunov function. The results are expressed as linear matrix inequalities which are efficiently solved by commercially available software implementing convex optimization techniques.

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References

  1. Lu, Y., Arkun, Y.: Quasi-min-max MPC algorithms for LPV systems. Automatica 36(4), 527–540 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Tanaka, K., Wang, H.: Fuzzy Control Systems Design and Analysis. A Linear Matrix Inequality Approach. Wiley, New York (2001)

    Book  Google Scholar 

  3. Lendek, Z., Guerra, T., Babuska, R., De-Schutter, B.: Stability Analysis and Nonlinear Observer Design Using Takagi–Sugeno Fuzzy Models. Springer-Verlag, Amsterdam (2010)

    MATH  Google Scholar 

  4. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybernet. 15(1), 116–132 (1985)

    Article  MATH  Google Scholar 

  5. Tanaka, K., Sugeno, M.: Stability analysis of fuzzy systems using Lyapunov’s direct method. In: Proceedings of the North America Fuzzy Information Processing Society, Toronto, Canada, pp. 133–136 (1990)

  6. Wang, H., Tanaka, K., Griffin, M.: An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4(1), 14–23 (1996)

    Article  Google Scholar 

  7. Tanaka, K., Ikeda, T., Wang, H.: Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. 6(2), 250–264 (1998)

    Article  Google Scholar 

  8. Boyd, S., Ghaoui, L.E., Feron, E., Belakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM: Studies In Applied Mathematics, Philadelphia (1994)

    Book  MATH  Google Scholar 

  9. Taniguchi, T., Tanaka, K., Wang, H.: Model construction, rule reduction and robust compensation for generalized form of Takagi–Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 9(2), 525–537 (2001)

    Article  Google Scholar 

  10. Gao, Q., Zeng, X., Feng, G., Wang, Y., Qiu, J.: T-S-Fuzzy-Model-Based approximation and controller design for general nonlinear systems. IEEE Trans. Syst. Man Cybern. B Cybern. 42(4), 1143–1154 (2012)

    Article  Google Scholar 

  11. Sala, A., Guerra, T., Babuska, R.: Perspectives of fuzzy systems and control. Fuzzy Sets Syst. 156(3), 432–444 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Tuan, H., Apkarian, P., Narikiyo, T., Yamamoto, Y.: Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Trans. Fuzzy Syst. 9(2), 324–332 (2001)

    Article  Google Scholar 

  14. Liu, X., Zhang, Q.: New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica 39(9), 1571–1582 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Sala, A., Arino, C.: Relaxed stability and performance conditions for Takagi–Sugeno fuzzy systems with knowledge on membership function overlap. IEEE Trans. Syst. Man Cybern. B Cybern. 37(3), 727–732 (2007)

    Article  Google Scholar 

  17. Ariño, C., Sala, A.: Extensions to “stability analysis of fuzzy control systems subject to uncertain grades of membership”. IEEE Trans. Syst. Man Cybern. B Cybern. 38(2), 558–563 (2008)

    Article  Google Scholar 

  18. Guelton, K., Bouarar, T., Manamanni, N.: Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi–Sugeno systems: a descriptor redundancy approach. Fuzzy Sets Syst. 160(19), 2796–2811 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.: A sum-of-squares approach to modeling and con- trol of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans. Fuzzy Syst. 7(4), 911–922 (2009)

    Article  Google Scholar 

  20. Tanaka, K., Ohtake, H., Seo, T., Tanaka, M., Wang, H.: Polynomial fuzzy observer designs: a sum-of-squares approach. IEEE Trans. Syst. Man Cybern. B Cybern. 42(5), 1330–1342 (2012)

    Article  Google Scholar 

  21. Johansson, M., Rantzer, A., Arzen, K.: Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Syst. 7(6), 713–722 (1999)

    Article  Google Scholar 

  22. Zhang, H., Feng, G.: Stability analysis and H ∞ controller design of discrete-time fuzzy large-scale systems based on piecewise lyapunov functions. IEEE Trans. Syst. Man Cybern. B Cybern. 38(5), 1390–1401 (2008)

    Article  Google Scholar 

  23. Rhee, B., Won, S.: A new fuzzy Lyapunov function approach for a Takagi–Sugeno fuzzy control system design. Fuzzy Sets Syst. 157(9), 1211–1228 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guerra, T., Vermeiren, L.: LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in Takagi–Sugeno’s form. Automatica 40(5), 823–829 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, L., Su, X., Shi, P., Qiu, J.: A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems. IEEE Trans. Syst. Man Cybern. B Cybern. 41(1), 273–286 (2011)

    Article  Google Scholar 

  26. Guerra, T., Kruszewski, A., Lauber, J.: Discrete Tagaki-Sugeno models for control: where are we? Ann. Rev. Control 33(1), 37–47 (2009)

    Article  Google Scholar 

  27. Delmotte, F., Guerra, T., Kruszewski, A.: Discrete takagi–sugeno’s fuzzy models: reduction of the number of lmi in fuzzy control techniques. IEEE Trans. Syst. Man Cybern. B Cybern. 38(5), 1423–1427 (2008)

    Article  Google Scholar 

  28. Guerra, T., Kruszewski, A., Bernal, M.: Control law proposition for the stabilization of discrete Takagi–Sugeno models. IEEE Trans. Fuzzy Syst. 17(3), 724–731 (2009)

    Article  Google Scholar 

  29. Kerkeni, H., Lauber, J., Guerra, T.: Some results about stabilization of periodic Takagi–Sugeno models. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Jeju, Korea, pp. 814–819 (2009)

  30. Guerra, T., Kerkeni, H., Lauber, J., Vermeiren, L.: An efficient Lyapunov function for discrete T-S models: observer design. IEEE Trans. Fuzzy Syst. 20(1), 187–192 (2012)

    Article  Google Scholar 

  31. Bernal, M., Guerra, T., Jaadari, A.: Non-quadratic stabilization of Takagi–Sugeno models: a local point of view. In: Proceedings of the 2010 IEEE International Conference on Fuzzy Systems, Barcelona, Spain, pp. 2375–2380 (2010)

  32. Blanco, Y., Perruqueti, W., Borne, P.: Stability and stabilization of nonlinear systems and Takagi–Sugeno fuzzy models. Math. Probl. Eng. 7(3), 221–240 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J., Zhou, S., Xu, S.: Fuzzy control system design via fuzzy lyapunov functions. IEEE Trans. Syst. Man Cybern. B Cybern. 38(6), 1657–1661 (2008)

    Article  Google Scholar 

  34. Guelton, K., Guerra, T.M., Bernal, M., Bouarar, T., Manamanni, N.: Comments on fuzzy control systems design via fuzzy lyapunov functions. IEEE Trans. Syst. Man Cybern. B Cybern. 40(3), 970–972 (2010)

    Article  Google Scholar 

  35. Tanaka, K., Hori, T., Wang, H.: A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Trans. Fuzzy Syst. 11(4), 582–589 (2003)

    Article  Google Scholar 

  36. Mozelli, L., Palhares, R., Avellar, G.: A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems. Inf. Sci. 179(8), 1149–1162 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Guerra, T., Bernal, M.: A way to escape from the quadratic framework. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Jeju, Korea, pp. 784–789 (2009)

  38. Bernal, M., Guerra, T.: Generalized nonquadratic stability of continuous-time Takagi–Sugeno models. IEEE Trans. Fuzzy Syst. 18(4), 815–822 (2010)

    Article  Google Scholar 

  39. Lendek, Z., Guerra, T., Babuska, R.: On non-PDC local observers for TS fuzzy systems. In: Proceedings of the 2010 IEEE International Conference on Fuzzy Systems, Barcelona, Spain, pp. 1–7 (2010)

  40. Pan, J., Guerra, T., Fei, S., Jaadari, A.: Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach. IEEE Trans. Fuzzy Syst. 20(3), 594–602 (2012)

    Article  Google Scholar 

  41. Guerra, T., Bernal, M., Guelton, K., Labiod, S.: Nonquadratic local stabilization for continuous-time Takagi–Sugeno models. Fuzzy Sets Syst. 201, 40–54 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jun-Tao, P., Guerra, T., Shu-min, F., Jaadari, A.: Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach. IEEE Trans. Fuzzy Syst. 20(3), 594–602 (2012)

    Article  Google Scholar 

  43. Jaadari, A., Guerra, T., Sala, A., Bernal, M., Guelton, K.: New controllers and new designs for continuous-time Takagi–Sugeno models. In: Proceedings of the 2012 IEEE International Conference on Fuzzy Systems, Brisbane, Australia, pp. 1–7 (2012)

  44. Marquez, R., Guerra, T., Kruszewski, A., Bernal, M.: Improvements on non-PDC controller design for Takagi–Sugeno models. In: Proceedings of the 2013 IEEE International Conference on Fuzzy Systems, Hyderbad, India, pp. 1–7 (2013)

  45. Oliveira, M., Skelton, R.: Stability tests for constrained linear systems. In Perspectives in Robust Control, ser, pp. 241–257. Lecture Notes in Control and Information Sciencesvol. 268. Springer-Verlag, Berlin (2001)

  46. Yoneyama, J., Nishikawa, M., Katayama, H., Ichikawa, A.: Output stabilization of Takagi–Sugeno fuzzy systems. Fuzzy Sets Syst. 111(2), 253–266 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Guerra, T., Jaadari, A., Pan, J., Sala, A.: Some refinements for non-quadratic stabilization of continuous TS models. In: 2011 IEEE International Conference on Fuzzy Systems (FUZZ), pp. 329–333 (2011)

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Acknowledgments

The authors would like to thank their French sponsors: the Nord-Pas-de-Calais Region, the Ministry of Higher Education and Research, and the National Center for Scientific Research, as well as the Mexican National Council of Science and Technology via the Project Ciencia Básica SEP-CONACYT 168406 and ITSON PROFAPI 00566 and PROFAPI 00468.

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Authors and Affiliations

  1. Department of Electric and Electronics Engineering, Sonora Institute of Technology, Ciudad Obregón, Mexico

    Temoatzin González & Miguel Bernal

  2. LAMIH, Universtiy of Valenciennes and Hainaut-Cambrésis, 59000, Le Mont Houy, France

    Raymundo Márquez & Thierry Marie Guerra

Authors
  1. Temoatzin González
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  2. Raymundo Márquez
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  3. Miguel Bernal
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  4. Thierry Marie Guerra
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Correspondence to Miguel Bernal.

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González, T., Márquez, R., Bernal, M. et al. Nonquadratic Controller and Observer Design for Continuous TS Models: A Discrete-Inspired Solution. Int. J. Fuzzy Syst. 18, 1–14 (2016). https://doi.org/10.1007/s40815-015-0094-4

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  • DOI: https://doi.org/10.1007/s40815-015-0094-4

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