Advertisement

Stabilization of T–S Fuzzy Systems with Constrained Controls

  • Abdellah BenzaouiaEmail author
  • Ahmed El Hajjaji
Chapter
  • 1k Downloads
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 8)

Abstract

This chapter deals with the extension of the positive invariance approach to nonlinear systems modeled by Takagi–Sugeno fuzzy systems. The saturations on the control signal are taken into account during the design phase. Sufficient conditions of asymptotic stability are given ensuring at the same time that the control is always admissible inside the corresponding polyhedral set. Both a common Lyapunov function and piecewise Lyapunov function are used.

Keywords

Fuzzy Systems Common Lyapunov Function Positive Innovation Positive Invariance Approach Asymptotic Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man Cybernetics 15:116–132CrossRefzbMATHGoogle Scholar
  2. 2.
    Wang HO, Tanaka K, Griffin MF (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans Fuzzy Syst 4:14–23CrossRefGoogle Scholar
  3. 3.
    Chang WJ, Huang WH, Ku CC (2011) Robust fuzzy control for discrete perturbed time-delay affine Takagi-Sugeno fuzzy models. Int J Cont Aut Syst 9:86–97Google Scholar
  4. 4.
    EL Hajjaji A, Benzaouia A, Naib M (2006) Stabilization of fuzzy systems with constrained controls by using positively invariant sets. Math Prob Eng 2006:1–17 (Article ID. 013832)Google Scholar
  5. 5.
    Lee DH, Park JB, Joo YH, Lin KC, Ham CH (2010) Robust \(H_\infty \) control for uncertain nonlinear active magnetic bearing systems via Takagi-Sugeno fuzzy models. Int J Control Autom Syst 8:636–646CrossRefGoogle Scholar
  6. 6.
    Nachidi M, Tadeo F, Hmamed A, Benzaouia A (2007) Static output-feedback stabilization for time-delay Takagi-Sugeno fuzzy systems. In: 46th conference decision control, New Orleans-LA, USA, 12–14 Dec 2007, pp 1634–1639Google Scholar
  7. 7.
    Nachidi M, Benzaouia A, Tadeo F, Ait Rami M (2008) LMI-based approach for output feedback stabilization for discrete-time Takagi-Sugeno systems. IEEE Trans Fuzzy Syst 16:1188–1196Google Scholar
  8. 8.
    Wang LK, Liu XD (2010) Robust \(H_\infty \) fuzzy control for discrete-time nonlinear systems. Int J Cont Auto Syst 8:118–126Google Scholar
  9. 9.
    Benzaouia A, Burgat C (1988) Regulator problem for linear discrete-time systems with non symmetrical constrained control. Int J Control 48:2441–2451CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Benzaouia A, Hmamed A (1993) Regulator problem for linear continuous-time systems with nonsymmetrical constrained control. IEEE Trans Aut Control 38:1556–1560CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Benzaouia A (2012) Saturated switching systems. Springer, SLNC, New YorkCrossRefGoogle Scholar
  12. 12.
    Blanchini F (1999) Set invariance in control. Automatica 35:1747–1767CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Mesquine F, Tadeo F, Benzaouia A (2004) Regulator problem for linear systems with constraints on the control and its increments or rate. Automatica 40:1378–1395CrossRefMathSciNetGoogle Scholar
  14. 14.
    Benzaouia A, El Hajjaji A, Naib M (2006) Stabilization of a class of constrained non linear systems by fuzzy control. IJICIC 2:49–760Google Scholar
  15. 15.
    Benzaouia A, Tadeo F, Mesquine F (2006) The regulator problem for linear systems with saturations on the control and its increments or rate: an LMI approach. IEEE Trans Circuits Syst I fundam Theory Appl 53:2681–2691Google Scholar
  16. 16.
    Hu T, Lin Z, Chen BM (2002) An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38:351–359CrossRefzbMATHGoogle Scholar
  17. 17.
    Cao YY, Lin Z (2003) Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation. IEEE Trans Fuzzy Syst 11:57–67Google Scholar
  18. 18.
    Benzaouia A, Baddou A (1999) Piecewise linear constrained control for continuous-time systems. IEEE Trans Aut Control 44:1477–1481CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Cao SG, Rees NW, Feng G (1996a) Stability analysis and design for a class of continuous-time fuzzy control systems. Int J Control 64:1069–1088CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Johansson M, Rantzer A, Arzen KE (1999) Piecewise quadratic stability of fuzzy systems. IEEE Trans Fuzzy Syst 7:713–722CrossRefGoogle Scholar
  21. 21.
    Benzaouia A, Gounane S, Tadeo F, EL Hajjaji A (2011) Stabilization of saturated discrete-time fuzzy systems. Int J Control Autom Syst 9:581–587Google Scholar
  22. 22.
    Naib M (2006) Commande Contrainte des Systèmes Dynamique par la Logique Floue et la Norme \(l_1\). PhD Thesis of University Cadi Ayyad, Marrakesh, MoroccoGoogle Scholar
  23. 23.
    Benzaouia A (2002) Further results on the saturated controller design for linear continuous-time systems. In: Mediterranean Conference on Control and Automation, Lisbon, Portugal, 9–13 July 2002Google Scholar
  24. 24.
    Benzaouia A (1994) The resolution of equation \(XA + XBX = HX\) and the pole assignment problem. IEEE Trans Aut Control 39:2091–2095CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Porter B (1977) Eigenvalue assignment in linear multivariable systems by output feedback. Int J Control 25:483–490CrossRefzbMATHGoogle Scholar
  26. 26.
    Gutman PO, Hagander P (1985) A new design of constrained controllers for linear systems. IEEE Trans Aut Control 30:22–33CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hindi H, Boyd S (1998) Analysis of linear systems with saturating using convex optimization. In: Proceeding of \(37^{th}\) IEEE conference decision control, Florida-Tampa, USA, 16–18 Dec 1998, pp 903–908Google Scholar
  28. 28.
    Wang Y, Li S, Cao Y, Sun Y, Shou T (2006) Invariant approximations and disturbance attenuation for constrained linear discrete-time systems. Int J Inf Technol 12:88–96Google Scholar
  29. 29.
    De Olivera MC, Skelton RE (2001) Stability tests for constrained linear systems. In: Perspective in robust control. Springer, LNCIS, Berlin, pp 241–257Google Scholar
  30. 30.
    Boyd SP, EL Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaGoogle Scholar
  31. 31.
    Tanaka K, Sano M (1994) A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer. IEEE Trans Fuzzy Syst 2:119–134CrossRefGoogle Scholar
  32. 32.
    Bentalba S, El Hajjaji A (1999) Parking controller design for vehicle dynamics using fuzzy logic. J Aut Control A 40:26–29Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of Cadi AyyadMarrakechMorocco
  2. 2.Laboratoire de Modélisation, Information et SystèmesUniversité de Picardie Jules VerneAmiensFrance

Personalised recommendations