# ARITHMETIC-BASED FUZZY CONTROL

Document Type : Research Paper

Authors

1 Institute of Informatics, University of Szeged, Szeged, Hungary

2 Department of Technical Informatics, University of Szeged, Szeged, Hungary

Abstract

Fuzzy control is one of the most important parts of fuzzy theory for which several approaches exist. Mamdani uses $\alpha$-cuts and builds the union of the membership functions which is called the aggregated consequence function. The resulting function is the starting point of the defuzzification process. In this article, we define a more natural way to calculate the aggregated consequence function via arithmetical operators. Defuzzification is the optimum value of the resultant membership function. The left and right hand sides of the membership function will be handled separately. Here, we present a new ABFC (Arithmetic Based Fuzzy Control) algorithm based on arithmetic operations which use a new defuzzification approach. The solution is much smoother, more accurate, and much faster than the classical Mamdani controller.

Keywords

#### References

[1] S. Assilian, Arti cial intelligence in the control of real dynamical systems, Ph.D. Thesis,
London University, Great Britain, 1974.
[2] J. Dombi, Pliant arithmetics and pliant arithmetic operations, Acta Polytechnica Hungarica,
6(5) (2009), 19{49.
[3] D. Driankov, H. Hellendoorn and M. Reinfrank, An introduction to fuzzy control, Springer
Science & Business Media, Berlin, 2013.
[4] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Systems Science, 9 (1978),
613{626.
[5] D. Dubois and H. Prade, Fuzzy members: An overview, Analysis of Fuzzy Information, Vol.
I., CRC Press, Boca Raton, FL, (1987), 3{39.
[6] D. Dubois and H. Prade, Special issue on fuzzy numbers, Fuzzy Sets and System, 24 (3),
1987.
[7] D. Filev and R.Yager, A generalized defuzzi cation method via BAD distributions, Internat.
J. Intell. Systems, 6 (1991), 689{697.
[8] R. Fuller and R. Mesiar, Special issue on fuzzy arithmetic, Fuzzy Sets and System, 91(2)
(1997).
[9] R. Jain, Tolerance analysis using fuzzy sets, International Journal of Systems Science, 7(12)
(1976), 1393{1401.
[10] T. Jiang and Y. Li, Generalized defuzzi cation strategies and their parameter learning pro-
cedures, IEEE Trans. Fuzzy Systems, 4 (1996), 64{71.
[11] A. Kaufmann and M. M. Gupta, Introduction to fuzzy arithmetic: theory and applications,
Van Nostrand Reinhold, New York, 1985.
[12] A. Kaufmann and M. M. Gupta, Fuzzy mathematical models in engineering and management
science, North-Holland, Amsterdam, 1988.
[13] E. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE,
121 (1974), 1585{1588, .
[14] M. Mares, Computation over fuzzy quantities, CRC Press, Boca Raton, FL, 1994.
[15] M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems
Comput. Controls, 7(5) (1976), 73{81.
[16] M. Mizumoto and K. Tanaka, Algebraic properties of fuzzy numbers, Proc. Int. Conf. On
Cybernetics and Society, Washington, DC, (1976), 559{563.
[17] S. Nahmias, Fuzy variables, Fuzzy Sets and System, 1 (1978), 97{110.
[18] H. T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64
(1978), 369{380.
[19] A. Patel and B. Mohan, Some numerical aspects of center of area defuzzi cation method,
Fuzzy Sets and Systems, 132 (2002), 401{409.
[20] S. Roychowdhury and B.-H.Wang, Cooperative neighbors in defuzzi cation, Fuzzy Sets and
Systems,78 (1996), 37{49.
[21] S. Roychowdhury and W. Pedrycz, A survey of defuzzi cation strategies, Internat. J. Intell.
Systems, 16 (2001), 679{695.
[22] A. Sakly and M. Benrejeb, Activation of trapezoidal fuzzy subsets with di erent inference
methods, International Fuzzy Systems Association World Congress, Springer Berlin Heidel-
berg, (2003), 450{457.
[23] Q. Song and R. Leland, Adaptive learning defuzzi cation techniques and applications, Fuzzy
Sets and Systems, 81 (1996), 321{329.
[24] M. Sugeno, Industrial Applications of Fuzzy Control, Elsevier Science Publishers, New York,
1985.
[25] E. Van Broekhoven and B. De Baets, Fast and accurate centre of gravity defuzzi cation of
fuzzy system outputs de ned on trapezoidal fuzzy partitions, Fuzzy Sets and Systems, 157
(2006), 904{918.
[26] W. Van Leekwijck and E. Kerre, Defuzzi cation: criteria and classi cation, Fuzzy Sets and
Systems, 108 (1999), 159{178.
[27] R. Yager and D. Filev, SLIDE: a simple adaptive defuzzi cation method, IEEE Trans. Fuzzy
Systems, 1 (1993), 69{78.
[28] R. C. Young, The algebra of many-valued quantities, Math. Ann., 104 (1931), 260{290.
[29] L. A. Zadeh, The concept of a linquistic variable and its application to approximate reasoning,
Information Sciences, 1(8) (1975), 199{249.