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International Journal of Fuzzy Systems

, Volume 20, Issue 3, pp 741–749 | Cite as

Relaxed Stability Conditions for Discrete-Time T–S Fuzzy Systems via Double Homogeneous Polynomial Approach

  • Jun ChenEmail author
  • Shengyuan Xu
  • Qian Ma
  • Guangming Zhuang
Article

Abstract

This paper deals with the problem of stability analysis for discrete-time Takagi–Sugeno (T–S) fuzzy systems. The double homogeneous polynomially parameter-dependent (DHPPD) Lyapunov function is proposed, which is expressed in the double homogeneous polynomial form not only of the membership functions but also of the state variables. Based on the DHPPD Lyapunov function, a relaxed stability condition is derived. Additionally, the complete square matricial representation of homogeneous polynomials is considered to further reduce the conservatism. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Keywords

Homogeneous polynomials Lyapunov functions Stability Takagi–Sugeno (T–S) fuzzy systems 

Notes

Acknowledgements

This work was supported in part by the National Nature Science Foundation under Grants 61673215, 61403199, 61403178, the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025.

References

  1. 1.
    Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. SMC–15(1), 116–132 (1985)CrossRefzbMATHGoogle Scholar
  2. 2.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2001)CrossRefGoogle Scholar
  3. 3.
    Tanaka, K., Hori, T., Wang, H.O.: A multiple Lyapunov function approach to stability of fuzzy control systems. IEEE Trans. Fuzzy Syst. 11(4), 582–589 (2003)CrossRefGoogle Scholar
  4. 4.
    Lee, D.H., Park, J.B., Joo, Y.H.: A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems. Inf. Sci. 185(1), 230–248 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sala, A., Arino, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Peng, C., Yue, D., Fei, M.R.: Relaxed stability and stabilization conditions of networked fuzzy control systems subject to asynchronous grades of membership. IEEE Trans. Fuzzy Syst. 22(5), 1101–1112 (2014)CrossRefGoogle Scholar
  7. 7.
    Chen, J., Xu, S., Li, Y., Qi, Z., Chu, Y.: Improvement on stability conditions for continuous-time T–S fuzzy systems. J. Frankl. Inst. 353(10), 2218–2236 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, J., Xu, S., Li, Y., Chu, Y., Zou, Y.: Further studies on stability and stabilization conditions for discrete-time T–S systems with the order relation information of membership functions. J. Frankl. Inst. 352(12), 5796–5809 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, J., Xu, S., Jia, X., Zhang, B.: Novel summation inequalities and their applications to stability analysis for systems with time-varying delay. IEEE Trans. Autom. Control. 62(5), 2470–2475 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Su, X., Wu, L., Shi, P., Song, Y.: A novel approach to output feedback control of fuzzy stochastic systems. Automatica 50(12), 3268–3275 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Su, X., Shi, P., Wu, L., Song, Y.: A novel control design on discrete-time Takagi–Sugeno fuzzy systems with time-varying delays. IEEE Trans. Fuzzy Syst. 21(4), 655–671 (2013)CrossRefGoogle Scholar
  12. 12.
    Wu, L., Su, X., Shi, P., Qiu, J.: A new approach to stability analysis and stabilization of discrete-time TS fuzzy time-varying delay systems. IEEE Trans. Syst. Man Cybern. B Cybern. 41(1), 273–286 (2011)CrossRefGoogle Scholar
  13. 13.
    Guerra, T.M., Miguel, M.: Strategies to exploit non-quadratic local stability analysis. Int. J. Fuzzy Syst. 14(3), 372–379 (2012)MathSciNetGoogle Scholar
  14. 14.
    Guelton, K., Manamanni, N., Duong, C.C., Koumba-Emianiwe, D.L.: Sum-of-squares stability analysis of Takagi–Sugeno systems based on multiple polynomial Lyapunov functions. Int. J. Fuzzy Syst. 15(1), 1–8 (2013)MathSciNetGoogle Scholar
  15. 15.
    Li, H., Sun, X., Wu, L., Lam, H.: State and output feedback control of a class of fuzzy systems with mismatched membership functions. IEEE Trans. Syst. 23(6), 1943–1957 (2015)Google Scholar
  16. 16.
    Li, H., Sun, X., Shi, P., Lam, H.: Control design of interval type-2 fuzzy systems with actuator fault: sampled-data control approach. Inf. Sci. 302, 1–13 (2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lendek, Z., Guerra, T.M., Lauber, J.: Controller design for TS models using delayed nonquadratic Lyapunov functions. IEEE Trans. Cybern. 45(3), 453–464 (2015)CrossRefGoogle Scholar
  18. 18.
    Guerra, T.M., Vermeiren, L.: LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form. Automatica 40(5), 823–829 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guerra, T.M., Bernal, M., Guelton, K., Labiod, S.: Non-quadratic local stabilization for continuous-time Takagi–Sugeno models. Fuzzy Sets Syst. 201, 40–54 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Feng, G.: Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 12(1), 22–28 (2004)CrossRefGoogle Scholar
  21. 21.
    Ding, B.: Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi–Sugeno systems via nonparallel distributed compensation law. IEEE Trans. Fuzzy Syst. 18(5), 994–1000 (2010)CrossRefGoogle Scholar
  22. 22.
    Lee, D.H., Park, J.B., Joo, Y.H.: Improvement on nonquadratic stabilization of discrete-time Takagi–Sugeno fuzzy systems: multiple-parameterization approach. IEEE Trans. Fuzzy Syst. 18(2), 425–429 (2010)Google Scholar
  23. 23.
    Xie, X., Ma, H., Zhao, Y., Ding, D.W., Wang, Y.: Control synthesis of discrete-time T–S fuzzy systems based on a novel non-PDC control scheme. IEEE Trans. Fuzzy Syst. 21(1), 147–157 (2013)CrossRefGoogle Scholar
  24. 24.
    Xie, X., Yue, D., Zhu, X.: Further studies on control synthesis of discrete-time T–S fuzzy systems via useful matrix equalities. IEEE Trans. Fuzzy Syst. 22(4), 1026–1031 (2014)CrossRefGoogle Scholar
  25. 25.
    Shen, H., Su, L., Park, J.H.: Reliable mixed \(H_{\infty }\)/passive control for T–S fuzzy delayed systems based on a semi-Markov jump model approach. Fuzzy Sets Syst. 314, 79–98 (2017)CrossRefzbMATHGoogle Scholar
  26. 26.
    Shen, H., Park, J.H., Wu, Z.G.: Finite-time reliable \(\cal{L}_2-\cal{L}_{\infty }/\cal{H}_{\infty }\) control for Takagi–Sugeno fuzzy systems with actuator faults. IET Control Theory Appl. 8(9), 688–696 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Chen, J., Xu, S., Zhang, B., Qi, Z., Li, Z.: Novel stability conditions for discrete-time T–S fuzzy systems: a Kronecker-product approach. Inf. Sci. 337–338, 72–81 (2016)CrossRefGoogle Scholar
  28. 28.
    Chen, J., Xu, S., Zhang, B., Chu, Y., Zou, Y.: New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models. Inf. Sci. 329, 447–460 (2016)CrossRefGoogle Scholar
  29. 29.
    Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica 39(6), 1027–1035 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Chesi, G.: Establishing robust stability of discrete-time systems with time-varying uncertainty: the Gram-SOS approach. Automatica 50(11), 2813–2821 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Oliveira, R.C.L.F., Peres, P.L.D.: Parameter-dependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations. IEEE Trans. Autom. Control. 52(7), 1334–1340 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of AutomationNanjing University of Science and TechnologyNanjingChina
  2. 2.School of Mathematical SciencesLiaocheng UniversityLiaochengChina

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