Advertisement

Global Stability Analysis of Second-Order Fuzzy Systems

  • J. Aracil
  • F. Gordillo
  • T. Álamo
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 16)

Abstract

The design of fuzzy controllers (FCs) is an open problem in which the use of heuristics is one of the main design methodologies. In this way, available design methods do not give fully satisfactory results because they lack an analytical background which would allow a thorough analysis. Only analytical study will assure a good performance in all situations (or at least, in a large class of them). To introduce such analytical results, a concrete class of fuzzy control systems will be studied in this article. This class of systems has been chosen such that is at the same time simple in structure and general in the richness of the results obtained. This class is the one shown in Fig. 1 where there is a linear plant, represented by its transfer function G(s), and a FC with the associated nonlinearity u = Φ(e, ẏ) = Φ(-y, ẏ). The main objective of the analysis developed is to study an elementary system with a well known mathematical structure and of which all the behaviours it can exhibit can be fully analyzed. To that end a linear model for the plant has been adopted. All the nonlinearities in the system stand in the fuzzy controller. As will be seen, the qualitative analysis of this system allows to display a full variety of behaviours. For the sake of simplicity, in this article it will be assumed that (1) G(s) is second-order; (2) Φ(-y, -ẏ) = -Φ( y, ẏ); and (3) out of the region of normal operation, the controller saturates. This unavoidable saturation will have important consequences in the following.

Keywords

Equilibrium Point Hopf Bifurcation Operating Point Fuzzy Controller Global 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abdelnour, G., Cheung, J.Y., Chang, C.-H. and Tinetti, G. (1993), “Steady-state analysis of a three-term fuzzy controller”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 23, No 2, 607–610.MATHCrossRefGoogle Scholar
  2. [2]
    Abdelnour, G., Cheung, J.Y., Chang, C.-H. and Tinetti, G. (1993), “Application of describing funtions in the transient response analysis of a three-term fuzzy controller”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 23, No 2, 603–606.MATHCrossRefGoogle Scholar
  3. [3]
    Abed, E.H., Wang, H.O. and Tesi, A. (1996), “Control of bifurcations and chaos”, in W.S. Levine (Ed.) The Control Handbook, IEEE Press.Google Scholar
  4. [4]
    Aracil, J., Ollero A. and García-Cerezo, A. (1989), “Stability indices for the global analysis of expert control systems”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 19, 988–1007.CrossRefGoogle Scholar
  5. [5]
    Aracil, J., Ponce, E. and Álamo, T. A Frequency Domain Approach to Bifurcations in Control Systems with Saturation, Internal Report GAR 1997–01.Google Scholar
  6. [6]
    Cook, P.A. (1994), Nonlinear Dynamical Systems, 2nd. Ed., Prentice Hall.Google Scholar
  7. [7]
    Driankov, D., Hellendoorm, H. and Reinfrank, M. (1993), An Introduction to Fuzzy Control, Springer-Verlag.MATHCrossRefGoogle Scholar
  8. [8]
    Genesio, R. and Tesi, A. (1992), “Harmonic Balance Methods for the Analysis of Chaotic Dynamics in Nonlinear Systems”, Automatica, Vol. 28, No. 3, 531–548.MATHCrossRefGoogle Scholar
  9. [9]
    Glover, J.N. (1989), “Hopf bifurcations at infinity”, Nonlinear Analysis, Vol. 13, 1393–1398.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer Verlag, New York.MATHGoogle Scholar
  11. [11]
    Hale, J.K. and Koçak, H. (1991), Dynamics and Bifurcations, Springer-Verlag.MATHCrossRefGoogle Scholar
  12. [12]
    Khalil, H.K. (1996), Nonlinear Systems, Macmillan.Google Scholar
  13. [13]
    Kickert, W.J.M. and Mandani, E.H. (1978), “Analysis of a fuzzy logic controller”, Fuzzy Sets and Systems, 29–44Google Scholar
  14. [14]
    Kuznetsov, Y.A. (1995), Elements of Applied Bifurcation Theory, Springer-Verlag.MATHCrossRefGoogle Scholar
  15. [15]
    Llibre, J. and Ponce, E. (1996), “Global first harmonic bifurcation diagram for odd piecewise linear control systems”, Dynamics and Stability of Systems, Vol. 11, No. 1, 49–88.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Llibre, J. and Sotomayor, J. (1996), “Phase portraits of planar control systems”, Nonlinear Analysis, Theory, Methods and Applications., Vol. 27, No. 10, 1177–1197.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    MacFarlane, A.G.J. (Ed.) (1979), Frequency-Response Methods in Control Systems, IEEE Press.Google Scholar
  18. [18]
    Mees, A.I. (1981), Dynamics of Feedback Systems, Wiley.MATHGoogle Scholar
  19. [19]
    Nayfeh, A.H. and Balachandran, B. (1995), Applied Nonlinear Dynamics, Weley.MATHCrossRefGoogle Scholar
  20. [20]
    Ponce, E., Álamo, T. and Aracil, J. (1996), “Robustness and bifurcations for a class of piecewise linear control systems”, 1996 IFAC World Congress, paper 2b-02 5, Vol. E, 61–66.Google Scholar
  21. [21]
    Strogatz, S.H. (1995), Nonlinear Dynamics and Chaos, Addison-Wesley.Google Scholar
  22. [22]
    Yager, R.R., and Filev, D.P. (1994), Essentials of Fuzzy Modeling and Control, Wiley.Google Scholar
  23. [23]
    Wang, L-X (1997), A Course in Fuzzy Systems and Control, Prentice-Hall.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • J. Aracil
    • 1
  • F. Gordillo
    • 1
  • T. Álamo
    • 1
  1. 1.Escuela Superior de IngenierosUniversidad de SevillaSevillaSpain

Personalised recommendations