Stabilization of Two-Dimensional T–S Fuzzy Systems

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


This chapter deals with sufficient conditions of asymptotic stability for nonlinear discrete-time 2D systems represented by a Takagi-Sugeno fuzzy model of Roesser type with state feedback control. This work is based on common and multiple Lyapunov functions. The results are presented in the form of LMIs.


Fuzzy Systems Multiple Lyapunov Functions (MLF) Roesser Type State Feedback Controller 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.


  1. 1.
    Fornasini E, Marchesini G (1976) State-space realization theory of two-dimensional filters. IEEE Trans Aut Cont 21:484–492CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Givone DD, Roesser RP (1972) Multidimensional linear iterative circuits-general properties. IEEE Trans Comp 21:1067–1073CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Roesser R (1975) A discrete state-space model for linear image processing. IEEE Trans Aut Cont 20:1–10CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Marszalek W (1984) Two dimensional state-space discrete models for hyperbolic partial differential equations. Appl Math Models 8:11–14CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fornasini E, Marchesini G (1978) Doubly-indexed dynamical systems: state-space models and structural properties. Math Syst Theor 12:59–72CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kaczorek T (1997) Realization problem, reachability and minimum energy control of positive 2D Roesser model. In: Proceeding 6th annual international conference on advances in communication and control, Corfu, 23–27 June, pp 765–776 (1997)Google Scholar
  7. 7.
    Brian D, Anderson O, Agathoklis P, Jury EI, Mansour M (1986) Stability and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans Circuits Syst 33:261–266CrossRefzbMATHGoogle Scholar
  8. 8.
    Wu-Sheng L, Lee EB (1985) Stability analysis for two-dimensional systems via a Lyapunov approach. IEEE Trans Circuits Syst 32:61–68CrossRefGoogle Scholar
  9. 9.
    Galkowski K, Rogers E, Xu S, Lam J, Owens DH (2002) LMIs-a fundamental tool in analysis and controller design for discrete linear repetitive process. IEEE Trans Circuits Syst 49: 768–778Google Scholar
  10. 10.
    Hmamed A, Alfidi M, Benzaouia A, Tadeo F (2008) LMI conditions for robust stability of 2D linear discrete-time systems, mathematical problems in engineering. 2008 (Article ID 356124)Google Scholar
  11. 11.
    Lee EB, Lu WS (1985) Stabilization of two-dimensional systems. IEEE Trans Aut Cont 30:409–411CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lu WS (1994) Some new results on stability robustness of two-dimensional discrete systems. Multidimension Syst Signal Proc 5:345–361CrossRefzbMATHGoogle Scholar
  13. 13.
    Yaz E (1985) On state-feedback stabilization of two-dimensional digital systems. IEEE Trans Circuits Syst 32:1069–1070CrossRefzbMATHGoogle Scholar
  14. 14.
    Hmamed A, El Hajjaji A, Benzaouia A (2009) Stabilization of discrete-time 2D T-S fuzzy systems by state feedback control. In: 17th IEEE Mediterranean conference on control automation, Thessaloniki, 24–26 June 2009Google Scholar
  15. 15.
    Benhayoun M, Benzaouia A, Mesquine F, Tadeo F (2011) Stabilization of 2D continuous Takagi-Sugeno systems with non-PDC state feedback control. In: 12th International conference on science and technology aut. control and computer engineering. Sousse, 18–20 Dec 2011Google Scholar
  16. 16.
    Benhayoun M (2010) Contribution à la Commande des Systèmes 2D Retardés avec Contraintes sur la Commande, Ph.D. thesis, University Cadi Ayyad, MarrakeshGoogle Scholar
  17. 17.
    Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man Cybern 15:116–132CrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Hmamed A, Mesquine F, Tadeo F, Benhayoun M, Benzaouia A (2010) Stabilization of 2D saturated systems by state feedback control. Multidimen Syst Signal Proc 21:277–292CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nachidi M (2009) Stabilization of Takagi–Sugeno fuzzy systems with application on a greenhouse. Ph.D. thesis of University of ValladolidGoogle Scholar
  21. 21.
    Guerra TM, Vermeiren L (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form. Automatica 40:823–829CrossRefzbMATHMathSciNetGoogle 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