zbMATH — the first resource for mathematics

Universal approximation theorem for uninorm-based fuzzy systems modeling. (English) Zbl 1040.93043
The fundamental result of this study concerns the universal approximation property of fuzzy systems endowed with arbitrary uninorms. Uninorms are two-argument functions \(U:[0,1]^2\to [0,1]\) satisfying the following properties: (a) identity \(U(g,x)=x,(b)\) commutativity \(U(x,y)=U(y,x)\), (c) associativity \(U(x,U (y,z))= U(U(x, y),z)\), (d) monotonicity \(U(x,y)\leq U(x',y')\) for all \(x\leq x'\) and \(y\leq y'\). For \(g=0\) the uninorm becomes a certain s-norm while for \(g=1\) the definition coincides with the one for any t-norm. Furthermore the authors introduce a relevancy transformation \(h:[0,1]^2\to[0,1]\) such that (a) \(h(1,x)=x\); \(h(0,x)= g\); (b) if \(y<y'\) then \(h(x,y)\leq h(x,y')\), and (c) if \(x<x'\) then \(h(x,y)\leq h(x',y)\) for \(y\leq g\) and \(h(x,y)\geq h(x',y)\) for \(y\geq g\). The property of universal approximation is derived for the Mamdani type of model with the fuzzy relation of the fuzzy system being computed in the following form \[ R(x,y)=U \biggl( h\bigl(A_1(x), B_1(y) \bigr), \dots,h \bigl(A_N(x), B_N(y)\bigr)\biggr). \] Here the model is based on a collection of \(N\)-rules of the form “if \(A_i\) then \(B_i\)” involving fuzzy sets \(A_i\) and \(B_i\) defined in the input and output space, respectively.

93C42 Fuzzy control/observation systems
93A30 Mathematical modelling of systems (MSC2010)
Full Text: DOI
[1] Buchanan, B.G.; Shortliffe, E.H., Rule-based expert systems, (1984), Addison-Wesley Reading, MA, Menlo Park, CA
[2] De Baets, B., Uninorms; the known classes, (), 21-28
[3] De Baets, B.; Fodor, J., Residual operators of uninorms, Soft comput., 3, 89-100, (1999)
[4] D. Dubois, M. Grabisch, H. Prade, Gradual rules and the approximation of functions, Proc. 2nd Internat. Conf. on Fuzzy Logic and Neural Networks, Iizuka, Japan, July 17-22, 1992, pp. 629-632.
[5] D. Dubois, M. Grabisch, H. Prade, Synthesis of real-valued mappings based on gradual rules and interpolative reasoning, Proc. IJCAI’93 Fuzzy Logic in AI Workshop, Chaméry, France, August 28-September 3, 1993, pp. 29-40.
[6] Dubois, D.; Prade, H.; Grabisch, M., Gradual rules and the approximation of control laws, (), 147-181 · Zbl 0897.93037
[7] Fodor, J.; Yager, R.R.; Rybalov, A., Structure of uni-norms, Internat. J. uncertainty, fuzziness, knowledge-based systems, 5, 411-427, (1997) · Zbl 1232.03015
[8] Klir, G.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Upper Saddle River, NJ · Zbl 0915.03001
[9] B. Kosko, Fuzzy systems as universal approximators, Proc. IEEE Internat. Conf. on Fuzzy Systems 1992, San Diego, CA, 1992, pp. 1153-1162.
[10] Kreinovich, V.; Mouzouris, G.C.; Nguyen, H.T., Fuzzy rule based modeling as a universal approximation tool, (), 135-195 · Zbl 0918.93024
[11] H.T. Nguyen, V. Kreinovich, On approximations of controls by fuzzy systems, Fifth International Fuzzy Systems Association World Congress, Seoul, Korea, July 1993, pp. 1414-1417.
[12] Nguyen, H.T.; Kreinovich, V.; Sirisaengtaksin, O., Fuzzy control as a universal control tool, Fuzzy sets and systems, 80, 1, 71-86, (1996) · Zbl 0881.93040
[13] H.T. Nguyen, M. Sugeno (Eds.), Fuzzy Systems: Modeling and Control, Kluwer, Boston, MA, 1998.
[14] Nguyen, H.T.; Walker, E.A., First course in fuzzy logic, (1999), CRC Press Boca Raton, FL
[15] Novák, V.; Perfilieva, I., Evaluating linguistic expressions and functional fuzzy theories in fuzzy logic, (), 383-406 · Zbl 0949.03024
[16] I. Perfilieva, Normal forms for fuzzy logic relations and the best approximation property, Proc. Conf. EUSFLAT’99, Palma de Mallorca, 1999.
[17] Shortliffe, E.H., Computer-based medical consultation: MYCIN, (1976), Elsevier New York
[18] Yager, R.R., Uninorms in fuzzy systems modeling, Fuzzy sets and systems, 122, 1, 167-175, (2001) · Zbl 0978.93007
[19] Yager, R.R.; Rybalov, A., Uninorm aggregation operators, Fuzzy sets and systems, 80, 111-120, (1996) · Zbl 0871.04007
[20] J. Yen, N. Pfluger, Path planning and execution using fuzzy logic, in: AIAA Guidance, Navigation and Control Conference, New Orleans, LA, 1991, vol. 3, pp. 1691-1698.
[21] J. Yen, N. Pfluger, R. Langari, A defuzzification strategy for a fuzzy logic controller employing prohibitive information in command formulation, Proc. IEEE Internat. Conf. on Fuzzy Systems, San Diego, CA, March 1992, pp. 717-723.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.