International Journal of Fuzzy Systems

, Volume 19, Issue 5, pp 1392–1405 | Cite as

LMI Relaxations for Quadratic Stabilization of Guaranteed Cost Control of T–S Fuzzy Systems

  • Bo Pang
  • Yunze Cai
  • Weidong Zhang


Less conservative condition is provided in this paper for quadratic stabilization of guaranteed cost control (GCC) of Takagi–Sugeno fuzzy systems with parallel distributed compensation (PDC). To derive the condition, firstly a parameter-dependent linear matrix inequality (PD-LMI) is established to find quadratically stable PDC controller gains of GCC. Secondly, by applying Pólya’s theorem, evaluation of the PD-LMI is transformed into an equivalent problem of evaluation of a sequence of LMI relaxations. Different from other existing conditions, the LMI relaxations are sufficient and asymptotically reach necessity for evaluating the PD-LMI as a related scalar parameter, d, increases. The resulting guaranteed costs of PDC controllers are non-increasing with respect to the increase in the parameter d and converge to the global optimal value under quadratic stability at the limiting case. In addition, for input-affine nonlinear systems, the proposed condition is extended with the consideration of modeling errors, which helps to reduce the computational complexity of the LMI relaxations. Finally, simulations of two examples demonstrate the efficiency and feasibility of the proposed condition.


Takagi–Sugeno (T–S) fuzzy systems Guaranteed cost control (GCC) Linear matrix inequalities (LMIs) Quadratic stabilization Parallel distributed compensation (PDC) 



This paper is partly supported by the National Science Foundation of China (61473183, 61521063, U1509211), Program of Shanghai Subject Chief Scientist (14XD1402400).


  1. 1.
    Babuka, R.: Fuzzy Modeling for Control. Springer Science & Business Media, Berlin (2012)Google Scholar
  2. 2.
    Boyd, S.P., El, G.L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, Philadelphia (1994)CrossRefMATHGoogle Scholar
  3. 3.
    Chen, B., Liu, X.: Fuzzy guaranteed cost control for nonlinear systems with time-varying delay. IEEE Trans. Fuzzy Syst. 13(2), 238–249 (2005)CrossRefGoogle Scholar
  4. 4.
    Chen, B., Liu, X., Tong, S., et al.: Guaranteed cost control of T–S fuzzy systems with state and input delays. Fuzzy Set Syst. 158(20), 2251–2267 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fang, C.H., Liu, Y.S., Kau, S.W., et al.: A new LMI-based approach to relaxed quadratic stabilization of TS fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(3), 386–397 (2006)CrossRefGoogle Scholar
  6. 6.
    Feng, G.: A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006)CrossRefGoogle Scholar
  7. 7.
    Guan, X.P., Chen, C.L.: Delay-dependent guaranteed cost control for TS fuzzy systems with time delays. IEEE Trans. Fuzzy Syst. 12(2), 236–249 (2004)CrossRefMATHGoogle Scholar
  8. 8.
    Han, D., Shi, L.: Guaranteed cost control of input-affine nonlinear systems via partition of unity method. Automatica 49(2), 660–666 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jadbabaie, A., Abdallah, C.T., Jamshidi, M., et al.: Guaranteed-cost control of the nonlinear benchmark problem using model-based fuzzy systems. IEEE Intl. Conf. Contr. 2, 792–796 (1998)Google Scholar
  10. 10.
    Khalil, H.K., Grizzle, J.W.: Nonlinear Syst. Prentice-Hall, New Jersey (1996)Google Scholar
  11. 11.
    Kim, E., Lee, H.: New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans. Fuzzy Syst. 8(5), 523–534 (2000)CrossRefGoogle Scholar
  12. 12.
    Kruszewski, A., Sala, A., Guerra, T.M., et al.: A triangulation approach to asymptotically exact conditions for fuzzy summations. IEEE Trans. Fuzzy Syst. 17(5), 985–994 (2009)CrossRefGoogle Scholar
  13. 13.
    Lee, D.H., Park, J.B., Joo, Y.H.: A new fuzzy Lyapunov function for relaxed stability condition of continuous-time TakagiSugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 19(4), 785–791 (2011)CrossRefGoogle Scholar
  14. 14.
    Li, Y.X., Yang, G.H.: Fuzzy adaptive output feedback fault-tolerant tracking control of a class of uncertain nonlinear systems with nonaffine nonlinear faults. IEEE Trans. Fuzzy Syst. 24(1), 223–234 (2016)CrossRefGoogle Scholar
  15. 15.
    Li, Y.X., Yang, G.H.: Adaptive fuzzy decentralized control for a class of large-scale nonlinear systems with actuator faults and unknown dead zones. IEEE Trans. Syst. Man Cybern. Syst. PP(99), 1–12 (2016)Google Scholar
  16. 16.
    Liu, X.D., Zhang, Q.L.: New approaches to \(H_{\infty }\) controller designs based on fuzzy observers for TS fuzzy systems via LMI. Automatica 39(9), 1571–1582 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    de Loera, J.A., Santos, F.: An effective version of Pólya’s theorem on positive definite forms. J. Pure Appl. Algebra 108(3), 231–240 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: 2004 IEEE International Symposium on Computer Aided Control Systems Design, IEEE, pp: 284–289 (2004)Google Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Mosek APS, The MOSEK optimization software,, 54 (2010)
  21. 21.
    Mrquez, R., Guerra, T.M., Bernal, M. et al.: Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameter-dependent controller design. Fuzzy Set Syst. 306, 46–62 (2015)Google Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    de Oliveira, M.C., Skelton, R.E.: Stability Tests for Constrained Linear Systems. Perspectives in Robust Control. Springer, London (2001)Google Scholar
  24. 24.
    Powers, V., Reznick, B.: A new bound for Pólya’s theorem with applications to polynomials positive on polyhedra. J. Pure Appl. Algebra 164(1), 221–229 (2001)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sala, A.: On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems. Annu. Rev. Control 33(1), 48–58 (2009)CrossRefGoogle Scholar
  26. 26.
    Sala, A., Ariño, C.: Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya’s theorem. Fuzzy Set Syst. 158(24), 2671–2686 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 1, 116–132 (1985)CrossRefMATHGoogle Scholar
  28. 28.
    Tanaka, K., Ikeda, T., Wang, H.: Design of fuzzy control systems based on relaxed LMI stability conditions. IEEE Decis. Contr. Proc. 1, 598–603 (1996)CrossRefGoogle Scholar
  29. 29.
    Tanaka, K., Taniguchi, T., Wang, H.O.: Fuzzy control based on quadratic performance function-a linear matrix inequality approach. IEEE Decis. Contr. Proc. 3, 2914–2919 (1998)Google Scholar
  30. 30.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2004)Google Scholar
  31. 31.
    Teixeira, M., Assunção, E., Avellar, R.G.: On relaxed LMI-based designs for fuzzy regulators and fuzzy observers. IEEE Trans. Fuzzy Syst. 11(5), 613–623 (2003)CrossRefGoogle Scholar
  32. 32.
    Tuan, H.D., Apkarian, P., Narikiyo, T., et al.: Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Trans. Fuzzy Syst. 9(2), 324–332 (2001)CrossRefGoogle Scholar
  33. 33.
    Wang, Z.P., Wu, H.N.: Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems. Inf. Sci. 327, 21–39 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wang, Z.P. Wu, H.N.: Fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost. Fuzzy Set Syst. 302, 143–162 (2015)Google Scholar
  35. 35.
    Wu, H.N., Zhu, H.Y., Wang, J.W.: Fuzzy control for a class of nonlinear coupled ODE-PDE systems with input constraint. IEEE Trans. Fuzzy Syst. 23(3), 593–604 (2015)CrossRefGoogle Scholar
  36. 36.
    Xie, X., Yue, D., Ma, T., et al.: Further studies on control synthesis of discrete-time TS fuzzy systems via augmented multi-indexed matrix approach. IEEE Trans. Cybern. 44(12), 2784–2791 (2014)CrossRefGoogle Scholar
  37. 37.
    Xie, X., Yue, D., Zhang, H., et al.: Control synthesis of discrete-time TS fuzzy systems via a multi-instant homogenous polynomial approach. IEEE Trans. Cybern. 46(3), 630–640 (2016)CrossRefGoogle Scholar
  38. 38.
    Yin, P., Yu, L., Zheng, K.: TS model-based non-fragile guaranteed cost fuzzy control for nonlinear time-delay systems. Control Theory Appl. 1, 016 (2008)CrossRefGoogle Scholar
  39. 39.
    Ying, H.: General SISO Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators. IEEE Trans. Fuzzy Syst. 6(4), 582–587 (1998)CrossRefGoogle Scholar
  40. 40.
    Zhang, H., Xie, X.: Relaxed stability conditions for continuous-time T-S fuzzy-control systems via augmented multi-indexed matrix approach. IEEE Trans. Fuzzy Syst. 19(3), 478–492 (2011)CrossRefGoogle Scholar
  41. 41.
    Zhao, Y., Zhang, C., Gao, H.: A new approach to guaranteed cost control of TS fuzzy dynamic systems with interval parameter uncertainties. IEEE Trans. Syst. Man Cybern B 39(6), 1516–1527 (2009)CrossRefGoogle Scholar
  42. 42.
    Zhao, Q., Hautamaki, V., Frnti, P.: Knee point detection in BIC for detecting the number of clusters. In: International Conference on Advanced Concepts for Intelligent Vision Systems. Springer Berlin Heidelberg, pp. 664–673 (2008)Google Scholar
  43. 43.
    Zou, T., Yu, H.: Asymptotically necessary and sufficient stability conditions for discrete-time Takagi-Sugeno model: extended applications of Polya’s theorem and homogeneous polynomials. J. Frankl. Inst. 351(2), 922–940 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of AutomationShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Key Laboratory of System Control and Information ProcessingMinistry of EducationShanghaiPeople’s Republic of China

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