Fuzzy Stability Analysis of Fuzzy Systems: A Lyapunov Approach

  • J. P. Marin
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 16)


The modelling phase in the analysis of the dynamic behavior of complex systems is crucial. The formalism of the model must satisfy a number of requirements. First, it must be flexible enough to capture all system properties, e.g., nonlinearities and uncertainties). Then it must allow the use of all available sources of information about the system. Finally, it must allow the analysis of the system behavior in a convenient, systematic way.


Fuzzy System Fuzzy Rule Fuzzy Model Fuzzy Relation Quadratic Constraint 
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.


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  1. [Zadeh, 73]
    L. A. Zadeh: “Outline of a new approach to the analysis of complex systems and decision processes”, IEEE Trans. Syst. Man. Cybern., vol 3, pp 28–44, 1973.MathSciNetMATHCrossRefGoogle Scholar
  2. [Zadeh, 75]
    L. A. Zadeh: “The concept of linguistic variable and its application to approximate reasoning”, Information Science, vol 8, pp 199–250, pp 301–358, vol 9, pp 43–80, 1975.CrossRefGoogle Scholar
  3. [Mamdani, 75]
    E. H. Mamdani and S. Assilian: “Application of fuzzy algorithms for control of simple dynamic plant”, Proc IEE, vol 121, pp 1585–1588, 1974.Google Scholar
  4. [Takagi, 85]
    T. Takagi and M. Sugeno: “Fuzzy identification of systems and its application to modelling and control”, IEEE Trans. Syst. Man. Cybern., vol 15, pp 116–132, 1985.MATHCrossRefGoogle Scholar
  5. [Aracil, 91]
    J. Aracil & al: “Fuzzy Control of Dynamical Systems. Stability Analysis based on Conicity Criterion”, Proc IFSA’91, Vol Engineering, Brussel, pp 5–8, 1991.Google Scholar
  6. [Opitz, 93]
    H.P. Opitz: “ Fuzzy Control and Stability criteria”, Proc EUFIT’93, Aachen, pp 130–136,1993.Google Scholar
  7. [Wang, 94]
    L. Wang and R. Langari: “Fuzzy controller design via hypersta-bility theory”, Proc FUZZ-IEEE’94, pp 178–182, Orlando, 1994.Google Scholar
  8. Marin, 96a]
    [ J.P Marin, A. Titli: “Robust performances of closed-loop fuzzy systems: A global Lyapunov approach”, Proc FUZZ-IEEE’96, pp 732–737, New-Orleans, 1996.Google Scholar
  9. [Boyd, 94]
    S. Boyd, L. El Gahoui, E. Feron, V. Balakrishnan: “Linear Matrix Inequalities in Systems and Control Theory”, Volume 15 of SIAM Studies in Applied Mathematics. SIAM, 1994.CrossRefGoogle Scholar
  10. [Tanaka, 96]
    K. Tanaka, K. Ikeda and H.O. Wang: “Robust Stabilization of a class of Uncertain Nonlinear systems via fuzzy control: Quadratic Sta-bilizability, H Control Theory and Linear Matrix Inequalities”, IEEE Trans on Fuzzy Systems 4 (1), pp 1–13, 1996.CrossRefGoogle Scholar
  11. [Zhao, 96]
    J. Zhao, V. Wertz and R. Gorez: “Fuzzy gain scheduling based on fuzzy models”, Proc FUZZ-IEEE’96, pp 1670–1676, New-Orleans, 1996.Google Scholar
  12. [Tanaka, 94]
    K. Tanaka, N. Sano: “A robust stabilization problem of fuzzy control systems and its application to backing up truck trailer”, IEEE Trans on Fuzzy Systems, pp 29–34, 1994.Google Scholar
  13. [Tanaka, 92]
    K. Tanaka, M. Sugeno: “Stability analysis and design of fuzzy control system”, Fuzzy sets and systems 45, pp 135–156, 1992.MathSciNetMATHCrossRefGoogle Scholar
  14. [Marin, 95a]
    J.P. Marin, A. Titli: “Necessary and Sufficient conditions for Quadratic Stability of a class of Tagaki-Sugeno Fuzzy Systems”, Proc EUFIT’95, Aachen, 1995.Google Scholar
  15. [Kim, 95]
    W.C Kim, C.S. Ahn, W.H. Kwon: “Stability Analysis and stabilization of fuzzy state space models”, Fuzzy sets and systems, 71, pp 131–142, 1995.MathSciNetMATHCrossRefGoogle Scholar
  16. [Marin, 95b]
    J.P Marin: “Conditions nécessaires et suffisantes de stabilité quadratique d’une classe de systèmes flous”, Proc LFA’95, pp 240–247, Paris, 1995.Google Scholar
  17. [Marin, 96b]
    J.P Marin: “H performance analysis of fuzzy system using quadratic storage function”, Proc AADECA’96, pp 7–11, Buenos-Aires, 1996.Google Scholar
  18. [Kiska, 85]
    J.B. Kiska, M.M. Gupta and P.N. Nikiforuk “Energistic Stability of fuzzy dynamic systems”, IEEE Trans. Syst. Man. Cybern., 5 (15), pp 783–792, 1985.CrossRefGoogle Scholar
  19. [Czogala, 82]
    E. Czogala, W. Pedrycz: “Control problems in fuzzy systems”, Fuzzy sets and systems 7, pp 257–273, 1982.MathSciNetMATHCrossRefGoogle Scholar
  20. [Jianquin, 93]
    C. Jianqin, C. Laijiu: “Study on stability of fuzzy closed-loop systems”, Fuzzy sets and systems 57, pp 159–168, 1993.MathSciNetMATHCrossRefGoogle Scholar
  21. [Deglas, 82]
    M. De Glas: “A mathematical theory of fuzzy systems”, Fuzzy Information and Decision Process and E. Sanchez (eds.),@North-Holland Publishing Company, 1982.Google Scholar
  22. [Deglas, 84]
    M. De Glas: “Invariance and Stability of Fuzzy Systems”, Journal of Mathematical Analysis and Application, vol 99, pp 299–319, 1984.MATHCrossRefGoogle Scholar
  23. [Safonov, 94]
    M. G. Safonov, K.J. Goh, J.H. Ly: “Control Systems synthesis via Bilinear Matrix Inequalities”, Proc ACC, pp 45–49, Baltimore, Maryland, June 1994.Google Scholar
  24. [Rotea, 93]
    M.A. Rotea, M. Corless, D. Da and I.R. Petersen: “System with structured uncertainty: Relation between Quadratic and Robust Stability”, IEEE. Trans. Aut. Cont., vol 38, n 5, pp 799–803, 1993.MathSciNetMATHCrossRefGoogle Scholar
  25. [Vidyasagar, 92]
    M. Vidyasagar: “Nonlinear Systems Analysis”, second edition, Prentice Hall, Englewood Cliffs, New Jersey 07632, 1992.MATHGoogle Scholar
  26. [Van der Shaft, 96]
    A. Van der Schaft: “L 2-gain and Passivity Techniques in Nonlinear Control”, Lecture Notes in Control and information Sciences, Vol 218, Springer, 1996.MATHCrossRefGoogle Scholar
  27. [Lu, 95]
    W.M. Lu, J.C. Doyle: “H Control of Nonlinear Systems: A Convex Characterization”, IEEE. Trans. Aut. Cont., vol 40, n 9, pp 1668–1675, 1995.MathSciNetMATHCrossRefGoogle Scholar
  28. [Cao, 96]
    S.G. Cao, N.W. Rees and G. Feng: “Analysis and Design of Uncertain Fuzzy Control Systems, Part II: Fuzzy Controller Design”, Proc FUZZ-IEEE’96, pp 647–653, New-Orleans, 1996.Google Scholar
  29. [Marin, 97a]
    J.P. Marin, A. Titli: “Robust Quadratic Stabilizability of Non-Homogeneous Sugeno Systems Ensuring Completeness of the Closed-loop systems”, in Proc FUZZ-IEEE’97, Barcelona, pp 185–192, 1997.Google Scholar
  30. [Marin, 97b]
    J.P. Marin, A. Titli: “ Robust Quadratic Stabilizability of fuzzy systems using fuzzy dynamic output feedback: a Matrix Inequality approach”, in Proc IFSA’97, Prague, vol 4, pp 451–456, 1997.Google Scholar
  31. [S.P. Wu, 96]
    S.P. Wu, L. Vandenberghe ans S. Boyd: “Software for Determinant Maximization Problems, User’s Guide”, Version alpha, Stand-ford University, May 1996Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • J. P. Marin
    • 1
  1. 1.LAAS-CNRSToulouse CedexFrance

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