Li, D., Xie, Y. (2016). Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications. Iranian Journal of Fuzzy Systems, 13(6), 89-110. doi: 10.22111/ijfs.2016.2823

Dechao Li; Yongjian Xie. "Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications". Iranian Journal of Fuzzy Systems, 13, 6, 2016, 89-110. doi: 10.22111/ijfs.2016.2823

Li, D., Xie, Y. (2016). 'Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications', Iranian Journal of Fuzzy Systems, 13(6), pp. 89-110. doi: 10.22111/ijfs.2016.2823

Li, D., Xie, Y. Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications. Iranian Journal of Fuzzy Systems, 2016; 13(6): 89-110. doi: 10.22111/ijfs.2016.2823

Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications

^{1}School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang, 316022, China and Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhoushan, Zhejiang, 316022, China

^{2}College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China

Abstract

It is firstly proved that the multi-input-single-output (MISO) fuzzy systems based on interval-valued $R$- and $S$-implications can approximate any continuous function defined on a compact set to arbitrary accuracy. A formula to compute the lower upper bounds on the number of interval-valued fuzzy sets needed to achieve a pre-specified approximation accuracy for an arbitrary multivariate continuous function is then presented. In addition, a method to design the interval-valued fuzzy systems based on $R$- and $S$-implications in order to approximate a given continuous function with a required approximation accuracy is represented. Finally, two numerical examples are provided to illustrate the proposed procedure.

[1] O. Castillo and P. Melin, A review on interval type-2 fuzzy logic applications in intelligent control, Information Sciences, 279 (2014), 615–631. [2] C. Cornelis, G. Deschrijver and E. E. Kerre, Implication in intuitionistic and interval-valued fuzzy set theory: construction, classification and application, International Journal of Approximate Reasoning, 35 (2004), 55–95. [3] S. Coupland and R. John, A fast geometric method for defuzzification of type-2 fuzzy sets, IEEE Transaction on Fuzzy Systems, 16(4) (2008), 929–941. [4] T. Dereli, A. Baykasoglu, K. Altun, A. Durmusoglu and I. B. T¨urksen, Industrial applications of type-2 fuzzy sets and systems: a concise review, Computer in Industry, 62 (2011), 125–137. [5] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transaction on Fuzzy Systems, 12(1) (2004), 45–61. [6] G. Deschrijver and E. E. Kerre, Classes of intuitionistic fuzzy t-norms satisfying the residuation principle, International Journal of Uncertainty Fuzziness Knowledge-Based Systems, 11(6) (2003), 691–709. [7] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 133 (2003), 227–235. [8] A. Doostparast, M. H. Fazel Zarandi and H. Zakeri, On type-reduction of type-2 fuzzy sets: A review, Applied Soft Computing, 27 (2015), 614–627. [9] D. Dubois, On ignorance and contradiction considered as truth-values, Logic Journal of the IGPL, 16(2) (2008), 195–216. [10] B. V. Gasse, C. Cornelis, G. Deschrijver and E. E. Kerre, Triangle algebras: A formal logic approach to interval-valued residuated lattices, Fuzzy Sets and Systems, 159 (2008), 1042– 1060. [11] M. B. Gorza lczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21(1) (1987), 1–17. [12] M. B. Gorza lczany, An interval-valued fuzzy inference method-Some basic properties, Fuzzy Sets and Systems, 31(2) (1989), 243–251. [13] S. Greenfield, F. Chiclana, R. I. John and S. Coupland, The sampling method of defuzzification for type-2 fuzzy sets: experimental evaluation, Information Sciences, 189 (2012), 77–92. [14] H. Hagras, A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots, IEEE Transaction on Fuzzy Systems, 12(4) (2004), 524–539

[15] H. R. Hassanzadeh, M. T. A. Akbarzadeh and A. Rezaei, An interval-valued fuzzy controller for complex dynamical systems with application to a 3-PSP parallel robot, Fuzzy Sets and Systems, 235(16) (2014), 83–100. [16] M. Y. Hsiao, T. S. Li, J. Z. Lee, C. H. Chao and S. H. Tsai, Design of interval type-2 fuzzy sliding-mode controller, Information Sciences, 178(6) (2008), 1696–1716. [17] C. F. Juang and Y. W. Tsao, A type-2 self-organizing neural fuzzy system and its FPGA implementation, IEEE Transaction on System Man Cybernet. Part B: Cybernet, 38(6) (2008), 1537–1548. [18] H. K. Lam, H. Li, C. Deters, E. L. Secco, H. A. Wurdemann and K. Althoefer, Control design for interval type-2 fuzzy systems under imperfect premise matching, IEEE Transactions on Industrial Electronics, 61(2) (2014), 956–968, art. no. 6480840. [19] D. C. Li, Y. M. Li and Y. J. Xie, Robustness of interval-valued fuzzy inference, Information Science, 181 (2011), 4754–4764. [20] Y. M. Li and Y. J. Du, Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model, Applied Mathematical Modelling, 36(4) (2012), 1558–1569. [21] Y. M. Li, Z. K. Shi and Z. H. Li, Approximation theory of fuzzy systems based upon genuine many-valued implications: SISO cases, Fuzzy Sets and Systems, 130 (2002), 147–157. [22] Y. M. Li, Z. K. Shi and Z. H. Li, Approximation theory of fuzzy systems based upon genuine many-valued implications: MIMO cases, Fuzzy Sets and Systems, 130 (2002), 159–174. [23] Q. Liang and J. M. Mendel, Interval type-2 fuzzy logic systems: theory and design, IEEE Transaction on Fuzzy Systems, 8 (2000), 535–550. [24] O. Linda and M. Manic, Uncertainty-robust design of interval type-2 fuzzy logic controller for delta parallel robot, IEEE Trans. Ind. Inf. 7(4) (2011), 661–670. [25] X. Liu and J. Mendel, Connect Karnik-Mendel algorithms to root-finding for computing the centroid of an interval type-2 fuzzy set, IEEE Transaction on Fuzzy Systems, 19(4) (2011), 652–665. [26] S. Mandal and B. Jayaram, SISO fuzzy relational inference systems based on fuzzy implications are universal approximators, Fuzzy Sets and Systems, 277 (2015), 1–21. [27] V. Nov´ak and S. Lehmke, Logical structure of fuzzy IF-THEN rules, Fuzzy Sets and Systems, 157(15) (2006), 2003–2029. [28] V. Nov´ak, I. Perfilieva and J. Mˇckˇcr, Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Boston, 1999. [29] I. Perfilieva, Normal forms in BL-algebra off unctions and their contribution to universal approximation, Fuzzy Sets and Systems, 143(1) (2004), 111–127. [30] I. Perfilieva and V. Kreinovich, A new universal approximation result for fuzzy systems, which reflects CNF-DNF duality, Int. J. Intell. Syst. 17(12) (2002), 1121–1130. [31] Y. M. Tang and X. P. Liu, Differently implicational universal triple I method of (1, 2, 2) type, Computers and Mathematics with Applications, 59(6) (2010), 1965–1984. [32] I. B. T¨urksen, Type 2 representation and reasoning for CWW, Fuzzy Sets and Systems, 127 (2002), 17–36. [33] I. B. T¨urksen and Y. Tian, Interval-valued fuzzy sets representation on multiple antecedent fuzzy S-implications and reasoning, Fuzzy Sets and Systems, 52(2) (1992), 143–167. [34] G. Wang and X. Li, Correlation and information energy of interval-valued fuzzy numbers, Fuzzy Sets and Systtem, 103(1) (1999), 169–175. [35] D. Wu, On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers, IEEE Transactions on Fuzzy Systems, art. no. 6145645, 20(5) (2012), 832–848. [36] D. Wu and W. W. Tan, A type-2 fuzzy logic controller for the liquid-level process, in: 2004 IEEE International Conference on Fuzzy Systems, (2004), Proceedings. 2 (2004), 953–958. [37] H. Ying, Sufficient conditions on general fuzzy systems as function approximators, Automatic, 30(3) (1994), 521–525. [38] H. Ying, General interval type-2 Mamdani fuzzy systems are universal approximators, Proceedings of North American Fuzzy Information Processing Society Conference, New York, NY, May 19–22, 2008.

[39] H. Ying, Interval type-2 Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators, The 28th North American Fuzzy Information Processing Society Annual Conference, Cincinnati, Ohio, June 14–17, 2009. [40] L. A. Zadeh, The concepts of a linguistic variable and its application to approximate reasoning (I), (II), Information Science, 8 (1975), 199–249; 301–357. [41] L. A. Zadeh, The concepts of a linguistic variable and its application to approximate reasoning (III), Information Science, 9 (1975), 43–80. [42] W. Y. Zeng and S. Feng, Approximate reasoning algorithm of interval-valued fuzzy sets based on least square method, Information Sciences, 272 (2014), 73–83. [43] H. Zhou and H. Ying, A method for deriving the analytical structure of a broad class of typical interval type-2 mamdani fuzzy controllers, in: IEEE Transactions on Fuzzy Systems, art. no. 6341818, 21(3) (2013), 447–458.