Fuzzy Parameter and State Estimation

  • S. G. Tzafestas
  • S. Terzakis
  • A. N. Venetsanopoulos
Part of the International Series on Microprocessor-Based and Intelligent Systems Engineering book series (ISCA, volume 11)


The field of parameter estimation goes back to Gauss (1795) who has presented the classical least squares method in his book: “Theoria Motus Coelectium”. Least squares wasthen studied by Legendre (1806) in his book: “Nouvekkes Methodes pour la Determination des Orbits des Cametes”. In Gauss’ own words: “The most probable value of the unknown quantities will be that in which the sum of the squares of the differences between the actually observed and the computed values multiplied by numbers that measure the degree of precusion is a minimum”

Since then, an enormous effort was made by scientists and engineers to develop further the methofd and apply it to sophisticated situations (least squares in functions spaces, least squares filtering, etc.) and applications of industrial and real-life nature. Regarding the fuzzification of the least squares problem, the authors have picked-up the work of the Hungarian mathematician Celmins xc1. He has developed an elegant but quite involved, computatioally, method along with a software package (called COLSACC) that implements it. The authors present an alternative (less general) least-squares method where use is made of the concept and the algebraic properties of the so calles L-R fuzzy numbers xc2. Minimizing their distance with respect to the unknown parameters is a classical minimization problem and has a small computational demad. The chapter continuous by formulating and solving the fuzzy state estimation of a particular discrete time state-space model with fuzzy disturbances and initial conditions xc3–7. Similar problems have been considered by several authors under various mathematical formulations.

For example, it is worthwhile to mention the work of Lee xc8 where a state estimation problem is solved for a class of distributed-parameter systems with uncertain parameters. The parameters are assumed to be arbitrary time functions known to be in a closed and bounded region. The resulting estimation algorithm provides an assured accuracy and a “guaranteed” error estimator which gives an upper bound of the estimation error for any allowed variation of uncertain parameters xc9. Other works on fuzzy estimation include xc10–20.


Membership Function Fuzzy Number Fuzzy Parameter Fuzzy Regression Fuzzy Estimation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Celmins, Least-Squares Model Fitting to Fuzzy Linear Data, Fuzzy Sets and Systems, Vol.22, pp. 245–269, 1987.CrossRefGoogle Scholar
  2. 2.
    D. Dubois and H. Pradé, Operations on Fuzzy Numbers, Int. J. Systems Sci., Vol. 9, No.6, pp. 613–626, 1978.MATHCrossRefGoogle Scholar
  3. 3.
    H. Sira-Ramirez, Fuzzy State Estimation in Linear Dynamic Systems, Proc. IEEE Conf. on Decision and Control, Vol.2, pp. 380–382, 1980.Google Scholar
  4. 4.
    H. Sira-Ramirez, Evolution of Fuzzy Sets in Linear Dynamic Systems, Proc. 1979 Int. Conf. on Cybernetics and Society, Denver, Colorado, Oct. 1979.Google Scholar
  5. 5.
    D.P. Bertsekas, Control of Uncertain Systems with a Set-Membership Description of the Uncertainty, MIT Electr. Syst. Lab. Rept ESL-R-447, Cambridge, June 1971.Google Scholar
  6. 6.
    F.C. Schweppe, Recuirsive State Estimation: Unknown but Bounded Errors and Systems Inputs, IEEE Trans. Auto. Contr. Vol. AC-13, No. 1, pp. 22–28, 1968CrossRefGoogle Scholar
  7. 7.
    H. Witsenhausen, Set of Possible States for Linear Systems Given Perturbed Observations, IEEE Trans. Auto. Contr. Vol. AC-13, No. 5 pp. 556–558, 1968.CrossRefGoogle Scholar
  8. 8.
    K. Lee, Modelling and Estimation of Distributed Systems with Uncertain Parameters, In: Distributed Parameter Systems: Modelling and Identification (A. Ruberti, Ed.), pp. 325–334 1978.Google Scholar
  9. 9.
    S.S.L. Chang, Control and Estimation of Fuzzy Systems, Proc. IEEE Decision and Control Conf., pp. 313–318, 1974.Google Scholar
  10. 10.
    P. Diamond, Higher-Level Fuzzy Numbers Arising from Fuzzy Regression Models, Fuzzy Sets and Systems, Vol. 36, pp. 265–275, 1990.MATHCrossRefGoogle Scholar
  11. 12.
    S. Joszef, On the Effect of Linear Data Transformations in Possibilistic Fuzzy Linear Regression, Fuzzy Sets and Systems, Vol. 45, pp. 185–188, 1992.CrossRefGoogle Scholar
  12. 13.
    H. Tanaka and H. Ishibuchi, Identification of Possibilistic Linear Systems by Quadratic Membership Functions of Fuzzy Parameters, Fuzzy Sets and Systems, Vol.41, pp. 145–160, 1991.MATHCrossRefGoogle Scholar
  13. 14.
    A. Bardossy, Note on Fuzzy Regression, Fuzzy Sets and Systems, Vol. 37, pp. 65–75, 1990.MATHCrossRefGoogle Scholar
  14. 15.
    K. Jajuga, Linear Fuzzy Regression, Fuzzy Sets and Systems, Vol.20, pp. 343–353, 1986.MATHCrossRefGoogle Scholar
  15. 16.
    H. Tanaka and J. Watanabe, Possibilistic Linear Systems and their Applications to the Linear Regression Model, Fuzzy Sets and Systems, Vol.27 pp.275–289, 1988.MATHCrossRefGoogle Scholar
  16. 17.
    H. Tanaka, S. Uejima and K. Asai, Linear Regression Analysis with Fuzzy Model, IEEE Trans. Syst. Man Cybern., Vol.5 SMC-12, No.6, 1982.Google Scholar
  17. 18.
    W. Zhen-Yuan and L. Shou-Mei, Fuzzy Linear Regression; Analysis of Fuzzy Valued Variables, Fuzzy Sets and Systems, Vol.36, pp. 125–136, 1990.CrossRefGoogle Scholar
  18. 19.
    S.-Q. Chen, Analysis of Multiple Fuzzy Regression, Fuzzy Sets and Systems, 1988.Google Scholar
  19. 20.
    D.A. Savic and W. Perdrycz, Evaluation of Fuzzy Linear Regression Models, Fuzzy Sets and Systems, Vol.39, pp. 51–63, 1991.MATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • S. G. Tzafestas
    • 1
  • S. Terzakis
    • 1
  • A. N. Venetsanopoulos
    • 2
  1. 1.Intelligent Robotics and Control Unit Department of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece
  2. 2.Department of Electrical EngineeringUniversity of TorontoTorontoCanada

Personalised recommendations