Universal Triple I Method for Fuzzy Reasoning and Fuzzy Controller

Document Type: Research Paper

Authors

1 Information and Communication Engineering Postdoctoral Research Station, School of Computer and Information, Hefei University of Technology, Hefei 230009, China

2 Institute of Technology and Science, The University of Tokushima, Minami Josanjima, Tokushima, 770-8506, Japan

Abstract

As a generalization of the triple I method, the universal triple I
method is investigated from the viewpoints of both fuzzy reasoning
and fuzzy controller. The universal triple I principle is put
forward, which improves the previous triple I principle. Then,
unified form of universal triple I method is established based on
the (0,1)-implication or R-implication. Moreover, the reversibility
property of universal triple I method is analyzed from expansion,
reduction and other type operators, which demonstrate that its
reversibility property seems fine, especially for the case employing
the (0,1)-implication. Lastly, we analyze the response ability of
fuzzy controllers based on universal triple I method, then the
practicability of triple I method is improved.

Keywords


bibitem{1} S. M. Chen, Y. K. Ko, Y. C. Chang and J. S. Pan, {it Weighted fuzzy
interpolative reasoning based on weighted increment transformation
and weighted ratio transformation techniques},  IEEE Transactions on
Fuzzy Systems, {bf 17}textbf{(6)} (2009), 1412--1427.

bibitem{2} J. C. Fodor, {it On contrapositive symmetry of implications in fuzzy logic}, First European Congress on Fuzzy and Intelligent
Technologies, Aachen, (1993), 1342--1348.

bibitem{3} S. Gottwald, {it A treatise on many-valued logics}, Research Studies Press, Baldock, 2001.

bibitem{4} P. H'ajek, {it  Metamathematics of fuzzy logic}, Kluwer
Academic Publishers, Dordrecht, 1998.

bibitem{5} J. Hou and H. X. Li, {it Reductivity of some fuzzy inference
methods},  Fuzzy Systems and Mathematics, {bf
19}textbf{(4)} (2005), 90--95.

bibitem{6} J. Hou, F. You and H. X. Li, {it Fuzzy systems constructed by
triple I algorithm and their response ability}, Progress in Natural
Science, {bf 15}textbf{(1)} (2005), 29--37.

bibitem{7} S. Kirindis and V. Chatzis, {it A robust fuzzy local information C-means clustering algorithm}, IEEE
Transactions on Image Processing, {bf 19}textbf{(5)}(2010),
1328--1337.

bibitem{8} E. P. Klement, R. Mesiar and E. Pap, {it  Triangular
Norms}, Kluwer Academic Publishers, Dordrecht, 2000.

bibitem{9} H. K. Lam and M. Narimani, {it Quadratic-stability analysis
of fuzzy-model-based control systems using staircase membership
functions}, IEEE Transactions on Fuzzy Systems, {bf
18}textbf{(1)} (2010), 125--137.

bibitem{10} H. X. Li, {it Interpolation mechanism of fuzzy control},
Science in China (Series E), {bf 41}textbf{(3)} (1998), 312--320.

bibitem{11} H. X. Li, {it Probability representations of fuzzy systems},
 Science in China (Series F), {bf 49}textbf{(3)} (2006), 339--363.

 bibitem{12} H. X. Li and E. S. Lee, {it Interpolation representations of
fuzzy logic systems}, Computers and Mathematics with Applications,
{bf 45}textbf{(10)} (2003), 1683--1693.

bibitem{13} D. C. Li, Y. M. Li and Y. J. Xie, {it Robustness of
interval-valued fuzzy inference}, Information Sciences, {bf
181}textbf{(20)} (2011), 4754--4764.

bibitem{14} H. X. Li, F. You and J. Y. Peng, {it Fuzzy controllers based
on some fuzzy implication operators and their response functions},
 Progress in Natural Science, {bf 14}textbf{(1)} (2004), 15--20.

 bibitem{15} H. X. Li, J. Y. Peng, J. Y. Wang, J. Hou and Y. Z. Zhang,
{it Fuzzy systems based on triple I algorithm and their response
ability}, Journal of Systems Science and Complexity, {bf
26}textbf{(5)} (2006), 578--590.

bibitem{16} H. W. Liu and G. J. Wang, {it Continuity of triple I methods
based on several implications}, Computers and Mathematics with
Applications, {bf 56}textbf{(8)} (2008), 2079--2087.

bibitem{17} X. P. Liu, Y. M. Tang, G. T. Shen and X. Chen, {it A formal model
of collaborative discussion for problem-solving}, Chinese Journal
of Electronics, {bf 21}textbf{(3)} (2012), 453--459.

bibitem{18} M. Mas, M. Monserrat, J. Torrens and E. Trillas, {it A survey
on fuzzy implication functions}, IEEE Transactions on Fuzzy
Systems, {bf 15}textbf{(1)} (2007), 1107--1121.

bibitem{19} J. M. Mendel, {it Fuzzy logic systems for engineering: a tutorial},
Proceedings of the IEEE, {bf 83}textbf{(3)} (1995), 345--377.

bibitem{20} V. Nov'ak, I. Perfilieva and J. Mov{c}kov{r}, {it
Mathematical principles of fuzzy logic}, Kluwer Academic Publishers,
Boston, 1999.

bibitem{21} D. W. Pei, {it Ideal implications in fuzzy logic and fuzzy control}, Journal of Xi'an Petroleum Institute,
{bf 15}textbf{(6)} (2000), 44--47.

bibitem{22} D. W. Pei, {it  $R_0$ implication: characteristics and
applications}, Fuzzy Sets and Systems, {bf 131}textbf{(3)} (2002),
297--302.

bibitem{23} D. W. Pei, {it Full implication triple I algorithms and their
consistency in fuzzy reasoning}, Journal of Mathematical Research
and Exposition, {bf 24}textbf{(2)} (2004), 359--368.

bibitem{24} D. W. Pei, {it Unified full implication algorithms of fuzzy
reasoning},  Information Sciences, {bf 178}textbf{(2)} (2008),
520--530.

bibitem{25} J. Y. Peng, H. X. Li, J. Hou, F. You and J. Y. Wang,
{it Fuzzy controllers based on pointwise optimization fuzzy
inference and its interpolation mechanism}, Journal of Systems
Science and Complexity, {bf 25}textbf{(3)} (2005), 311--322.

bibitem{26} F. J. Ren, {it Automatic abstracting important sentences },
International Journal of Information Technology and Decision Making,
{bf 4}textbf{(1)} (2005), 141--152.

bibitem{27} S. J. Song, C. B. Feng and E. S. Lee, {it Triple I method of
fuzzy reasoning}, Computers and Mathematics with Applications, {bf
44}textbf{(12)} (2002), 1567--1579.

bibitem{28} Y. M. Tang and X. P. Liu, {it Task partition for function tree
according to innovative functional reasoning}, Twelfth International
Conference on CSCWD, Xian, China, (2008), 189--195.

bibitem{29} Y. M. Tang and X. P. Liu, {it Fuzzy systems constructed by
triple I method or CRI method and their response functions}, Journal
of Hefei University of Technology, {bf 33}textbf{(2)} (2010),
182--187.

bibitem{30} Y. M. Tang and X. P. Liu, {it Differently implicational
universal triple I method of (1, 2, 2) type}, Computers and
Mathematics with Applications, {bf 59}textbf{(6)} (2010),
1965--1984.

bibitem{31} Y. M. Tang, F. J. Ren, X. Sun and Y. X. Chen, {it Reverse
universal triple I method of (1,1,2) type for the Lukasiewicz
implication}, Seventh Conference on NLPKE, Tokushima, Japan, (2011),
23--30.

bibitem{32} I. B. Turksen and Y. Tian, {it Combination of rules or their consequences in fuzzy expert systems}, Fuzzy Sets and Systems,
{bf 58}textbf{(1)} (1993), 3--40.

bibitem{33} G. J. Wang, {it Fully implicational triple I method for fuzzy
reasoning}, Science in China (Series E), {bf 29}textbf{(1)} (1999),
43--53.

bibitem{34} G. J. Wang, {it  Non-classical mathematical logic and
approximate reasoning}, Science in China Press, Beijing, 2000.

bibitem{35} G. J. Wang, {it  Introduction to mathematical logic and
resolution principle}, Science in China Press, Beijing, 2003.

bibitem{36} G. J. Wang and L. Fu, {it Unified forms of triple I method},
 Computers and Mathematics with Applications, {bf 49}textbf{(5)} (2005),
923--932.

bibitem{37} L. A. Zadeh, {it Outline of a new approach to the analysis of
complex systems and decision processes}, IEEE Transactions on
Systems Man and Cybernetics, {bf 3}textbf{(1)} (1973), 28--44.

bibitem{38} J. C. Zhang and X. Y. Yang, {it Some properties of fuzzy
reasoning in propositional fuzzy logic systems},  Information
Sciences, {bf 180}textbf{(23)} (2010), 4661--4671.