Skip to main content
Log in

Stabilization and Robust Stabilization of Polynomial Fuzzy Systems: A Piecewise Polynomial Lyapunov Function Approach

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

This paper presents a piecewise polynomial Lyapunov function (PPLF) approach to stabilization and robust stabilization of polynomial fuzzy systems that are a generalized form of the well-known Takagi–Sugeno fuzzy systems. Both stabilization and robust stabilization conditions are formulated as sum of squares (SOS) optimization problems which can be solved by an SOS solver. A switching polynomial fuzzy controller based on switching information on the PPLF is designed to stabilize both polynomial fuzzy systems and polynomial fuzzy systems with uncertainties. In particular, by fully considering the properties of the piecewise and polynomial Lyapunov function, some relaxation are carried out in the derivation of SOS robust stabilization conditions for polynomial fuzzy systems with uncertainties. Four design examples are demonstrated to show the effectiveness of the proposed design approach in comparison with the existing approaches, i.e., linear matrix inequalities approaches and SOS stabilization and robust stabilization approaches.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  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)

    Article  MATH  Google Scholar 

  2. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2002)

    Google Scholar 

  3. Wang, H.O., Tanaka, K., Griffin, M.F.: Parallel distributed compensation of nonlinear system by Takagi–Sugeno fuzzy model. In: Proceedings of the FUZZ-IEEE/IFES’95, pp. 531–538. Yokohama, Japan (1995)

  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), 14–22 (1996)

    Article  Google Scholar 

  5. Tanaka, K., Ikeda, T., Wang, H.O.: Robust stabilization of a class of uncertain nonlinear system via fuzzy control: quadratic stabilizability, \(H^{\infty }\) control theory, and linear matrix inequality. IEEE Trans. Fuzzy Syst. 4(1), 1–12 (1996)

    Article  Google Scholar 

  6. Tanaka, K., Kosaki, T.: Design of stable fuzzy controller for an articulated vehicle. IEEE Trans. Syst. Man Cybern. B Cybern. 27(3), 552–558 (1997)

    Article  Google Scholar 

  7. Cao, S.G., Rees, N.W., Feng, G.: Analysis and design for a class of complex control systems part II: fuzzy controller design. Automatica 33(6), 1029–1039 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ying, H.: Design of a general class of Takagi–Sugeno fuzzy control systems. In: Proceedings of the American Control Conference, pp. 3746–3750. Albuquerque, New Mexico (1997)

  9. Xiaodong, L., Qingling, Z.: New approaches to \(H_{\infty }\) controller designs based on fuzzy observer for T–S fuzzy system via LMI. Automatica 39, 1571–1582 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, W.-J., Sun, C.-H.: A relaxed stability criterion for T–S fuzzy discrete systems. IEEE Trans. Syst. Man Cybern. B Cybern. 34(5), 2155–2158 (2004)

    Article  Google Scholar 

  11. 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 

  12. Gao, Z., Shi, X., Ding, S.X.: Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation. IEEE Trans. Syst. Man Cybern. B Cybern. 38(3), 875–880 (2008)

    Article  Google Scholar 

  13. Zhao, X., Yin, Y., Niu, B., Zheng, X.: Stabilization for a class of switched nonlinear systems with novel average dwell time switching by T–S fuzzy modeling. IEEE Trans. Cybern. 46(8), 1952–1957 (2016)

    Article  Google Scholar 

  14. Wang, L.K., Zhang, H.G., Liu, X.D.: \(H_{\infty }\) observer design for continuous-time Takagi–Sugeno fuzzy model with unknown premise variables via nonquadratic Lyapunov function. IEEE Trans. Cybern. 46(9), 1986–1996 (2016)

    Article  Google Scholar 

  15. Nguyen, A.-T., Márquez, R., Guerra, T.M.: Improved LMI conditions for local qudratic stabilization of constrained Takagi–Sugeno fuzzy systems. Int. J. Fuzzy Syst. 19(1), 225–237 (2017)

    Article  MathSciNet  Google Scholar 

  16. Yang, J., Tong, S.: An observer-based robust fuzzy stabilization control design for switched nonlinear systems with immeasurable premise. Int. J. Fuzzy Syst. 18(6), 1019–1030 (2016)

    Article  MathSciNet  Google Scholar 

  17. Sun, C.-H.: Relaxed stabilization conditions for the T–S fuzzy system with input constraint. Int. J. Fuzzy Syst. 18(2), 168–176 (2016)

    Article  MathSciNet  Google Scholar 

  18. 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 

  19. Papachristodoulou, A., Anderson, J., Valmorbida, G., Prajna, S., Seiler, P., Parrilo, P.A.: SOSTOOLS: sum of squares optimization toolbox for MATLAB, Version 3.00 (2013)

  20. 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. B Cybern. 39(2), 561–567 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  22. Cao, K., Gao, X.Z., Vasilakos, T., Pedrycz, W.: Analysis of stability and robust stability of polynomial fuzzy model-based control system using a sum-of-squares approach. J. Soft Comput. 18(3), 433–442 (2013)

    Article  MATH  Google Scholar 

  23. 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 

  24. Chae, S., Nguang, S.K.: SOS based robust \({\cal{H}}_{\infty }\) fuzzy dynamic output feedback control of nonlinear networked control systems. IEEE Trans. Cybern. 44(7), 1204–1213 (2014)

    Article  Google Scholar 

  25. Seiler, P.: SOSOPT: a toolbox for polynomial optimization, Version 2.00 (2016)

  26. Chen, Y.-J., Ohtake, H., Tanaka, K., Wang, W.-J., Wang, H.O.: Relaxed stabilization criterion for T–S fuzzy systems by minimum-type piecewise-Lyapunov-function-based switching Fuzzy controller. IEEE Trans. Fuzzy Syst. 20(6), 1166–1173 (2012)

    Article  Google Scholar 

  27. Chen, S.-H., Juang, J.-C.: A switching controller design via sum-of-squares approach for a class of polynomial T–S fuzzy model. Int. J. Innov. Comput. Inf. Control 7(7b), 4363–4376 (2011)

    Google Scholar 

  28. Lam, H.K., Narimani, M., Li, H., Liu, H.: Stability analysis of polynomial-fuzzy-model-based control systems using switching polynomial Lyapunov function. IEEE Trans. Fuzzy Syst. 21(5), 800–814 (2013)

    Article  Google Scholar 

  29. Chen, Y.-J., Tanaka, M., Tanaka, K., Wang, H.O.: Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: polynomial fuzzy model approach. IEEE Trans. Fuzzy Syst. 23(4), 1314–1322 (2015)

    Article  Google Scholar 

  30. Furqon, R., Chen, Y.-J., Tanaka, M., Tanaka, K., Wang, H.O.: An SOS-based control Lyapunov function design for polynomial fuzzy system control of nonlinear systems. IEEE Trans. Fuzzy Syst. early access article in IEEE Xplore

  31. Campos, V.C.S., Souza, F.O., Torres, L.A.B., Palhares, R.M.: New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations. IEEE Trans. Fuzzy Syst. 21(4), 748–760 (2013)

    Article  Google Scholar 

  32. González, T., Bernal, M.: Progessively better estimates of the domain of the attraction for nonlinear systems via piecewise Takagi–Sugeno models: stability and stabilization issues. Fuzzy Sets Syst. 297, 73–95 (2016)

    Article  MATH  Google Scholar 

  33. González, T., Sala, A., Bernal, M., Robles, R.: Piecewise–Takagi–Sugeno asymptotically exact estimation of the domain of attraction of nonlinear systems. J. Frankl. Inst. 353(3), 1514–1541 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lam, H.K.: Stability analysis of polynomial fuzzy model-based control systems using switching polynomial Lyapunov function. In: Polynomial fuzzy model-based control systems: stability analysis and control synthesis using membership function dependent techniques, pp. 223–258. Springer International Publisher (2016)

  35. Parrilo, P.A.: Structure semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. Dissertation, Calif. Tech, Pasadena, CA (2000)

  36. Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P. A.: SOSTOOLS: sum of squares optimization toolbox for MATLAB, Version 2.00 (2004)

  37. Boyd, S.: Lecture 15 linear matrix inequality and the \({\cal{S}}\)-procedure. Stanford Univ., California, CA, USA, Sept. 2008. [Online]. Available: http://stanford.edu/class/ee363/lectures/lmi-s-proc.pdf

  38. Topcu, U., Packard, A.: Linearized analysis versus optimization based nonlinear analysis for nonlinear systems. In: American Control Conference, pp. 790–795. St. Louis, MO, USA (2009)

  39. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  40. Tanaka, K., Tanaka, M., Chen, Y.-J., Wang, H.O.: A new sum-of-squares design framework for robust control of polynomial fuzzy systems with uncertainties. IEEE Trans. Fuzzy Syst. 24(1), 94–110 (2016)

    Article  Google Scholar 

  41. 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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tan, W., Packard, A.: Searching for control Lyapunov function using sums of squares programming. In: 42nd Annu. Allerton Conf. Commun. Control Comput., pp. 210–219. Monticello, Illinois, USA (2004)

  43. Hu, T.: Nonlinear control design for linear differential inclusions via convex hull of quadratics. Automatica 43(4), 685–692 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tan, W., Packard, A.: Stability region analysis using sum of squares Programming. In: American Control Conference, pp. 2297–2302. Minneapolis, Minnesota, USA (2006)

  45. Chen, Y.-J., Wang, W.-J., Tanaka, K.: Sum-of-squares based copositive relaxation for the double fuzzy summation problem of fuzzy-model-based control systems. In: 2015 Int. Conf. Fuzzy Theory Its Appl., pp. 94–98. Yilan, Taiwan (2015)

  46. Sala, A., Arino, 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 

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

    Article  Google Scholar 

  48. Teixeira, M.C.M., Assuncao, E., Avellar, R.G.: On relaxed LMI-based design for fuzzy regulators and fuzzy observers. IEEE Trans. Fuzzy Syst. 11(5), 613–623 (2003)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Mr. Daisuke Ogura for his contribution to this research. The first author gratefully acknowledges a scholarship for her Ph. D. study from Indonesia Endowment Fund for Education (LPDP).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alissa Ully Ashar.

Appendix A

Appendix A

Table 4 shows the list of the shortened notations used in this paper.

Table 4 List of shortened notations used in this paper

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ashar, A.U., Tanaka, M. & Tanaka, K. Stabilization and Robust Stabilization of Polynomial Fuzzy Systems: A Piecewise Polynomial Lyapunov Function Approach. Int. J. Fuzzy Syst. 20, 1423–1438 (2018). https://doi.org/10.1007/s40815-017-0435-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-017-0435-6

Keywords

Navigation