摘要
自Zadeh于1965年提出模糊集概念以来，模糊控制技术作为现代工业与新产品开发的高新技术之一，受到国内外普遍重视，而且在应用领域取得了令人瞩目的成功。模糊推理是模糊控制的核心内容之一，而推理又基于逻辑。针对如今在控制领域中广泛应用的CRI等方法的缺陷，王国俊教授提出了一种更为合理、逻辑基础更强的推理算法——三Ⅰ算法。
蕴涵算子的选取与模糊推理的效果密切相关。目前在模糊控制中用得比较多的是R_O和R_L蕴涵算子，它们都是属于特殊的剩余蕴涵算子，具有许多良好的性质。因此研究剩余蕴涵算子对将模糊推理与模糊逻辑和模糊控制相结合具有重要而广泛的意义，进而为实现新型模糊控制器地某些性能指标提供必要的理论依据。本文的目的就是讨论基于剩余蕴涵算子分析讨论模糊推理三Ⅰ算法。
本文首先基于Zadeh提出的CRI等算法中的缺陷与不足，探讨模糊推理三Ⅰ理论，给出基于剩余蕴涵算子的三Ⅰ方法的FMP与FMT算法的计算公式，并考虑算法的还原性：其次，分析了三Ⅰ方法的支持度理论和约束度理论，给出了α—FMP与FMT的计算公式；并进一步讨论了基于剩余蕴涵算子的反向三Ⅰ算法，给出其计算公式；最后，通过在区间值模糊集上引进偏序关系，探讨基于剩余蕴涵算子的区间值模糊推理三Ⅰ算法。
As one of the high and new techniques that develop modern industry and new products, fuzzy control technique has attracted so much attention and obtained the outstanding success in the applied areas, since Zadeh brought up the concept of fuzzy sets in 1965. Professor Wang Guojun set forth a new triple-I algorithm for the defect of CRI method applied extensively in control areas. This new method is more reasonable and has better logic foundation.
The selection of implication has closely related with the result of fuzzy reasoning. Now R0 and Rl operators, which belong to special residual implication operator and have many fine properties, are used mostly in fuzzy control. So this research has important and extensive significance to combine fuzzy reasoning with fuzzy logic and fuzzy control. The textual purpose is to discuss the triple-I algorithm of fuzzy reasoning based on residual implication operator.
Firstly, this text further develops the triple-I theory of fuzzy reasoning based on the defect of CRI algorithm, gives the formula of triple-I FMP and FMT based on residual implication operator and discusses their reducible properties. Secondly, the sustaining and restraining theories of triple-I method are discussed and the formula of a-FMP and FMT are provided. Then the reverse triple-I algorithm are discussed based on residual implication operator. Finally, because common fuzzy reasoning is special interval fuzzy reasoning, we set up the interval fuzzy reasoning based on residual implication operator by introducing the relation of partial order.
引文
[1] Dubois D, Prade H, Lang J. Fuzzy sets in approximately reasoning[J]. Fuzzy Sets and Systems, 1991,40:143～244.
[2] Dubois D, Prade H. Fuzzy sets in approximate reasoning. Part Ⅰ: Inference with possibility distributions. Fuzzy Sets and Systems, 1991, 41 (1): 153～203.
[3] János C Fodor. Contra-positive symmetry of fuzzy implications [J], Fuzzy sets and systems, 1995, 69:141～156.
[4] L.T.Kóczy, etc, Approximately reasoning by linear rule interpolation and general approximation, International Journal of Approximate Reasoning, 1993,9.
[5] Mizumoto M, et al. Comparison of fuzzy reasoning methods. Fuzzy Sets and Systems, 1982, 8:253.
[6] M. Mizumoto, etc, Some Considerations on K6czy Linear Interpolative Reasoning Method, Journal. of Japan Society for Fuzzy Theory and Systems, 1996, 8(1).
[7] Pavelka J. On fuzzy logic(Ⅱ)[J]. Z.für Mathematic Logic u Grundlagen d Mathematic, 1979, 25:119～134.
[8] Q. Chen, etc, On Fuzzy Reasoning by Kóczy's Linear Rule Interpolation, The Journal of Fuzzy Mathematics, 1998,6(2).
[9] Wang G. J. On the logic foundations of FMP and FMT. Int J Fuzzy Mathematics, 1997, 5(1): 229～250.
[10] Wang G. J. On the logic foundation of fuzzy reasoning, Lecture notes in fuzzy mathematics and computer science. Omaha: Creighton University, 1997, 4:1～48.
[11] Wu W M. Fuzzy reasoning and fuzzy relational equation. Fuzzy sets and systems, 1986, 20:67.
[12] Zadeh L.A. Outline of a new approach to the analysis of complex systems and decision process. IEEE Trans Systems Man Cybernetics, 1973, 3:28～44.
[13] 袁和军．模糊推理的形式化理论．陕西师范大学．硕士学位论文．2002．4．
[14] 陈永义．模糊控制技术及应用实例．北京：北京师范大学出版社，1993．
[15] 胡淑礼．模糊数学及其应用[M]．成都：四川大学出版社，1994．
[16] 李洪兴．模糊控制的插值机理[J]．中国科学，E辑，1998，28(3)：259～267．
[17] 李洪兴．从模糊控制的数学本质看模糊逻辑的成功[J]．模糊系统与数学，1995，4：1～4．
[18] 孙长银等．模糊推理三Ⅰ算法的几个结果．湖北三峡学院学报．2000，22(5)：17～20．
[19] 宋士吉．冯纯伯．吴从圻．关于模糊推理的全蕴涵三Ⅰ算法的约束度理论．自然科学进展．2000，10(10)：884～889．
[20] 王国俊．模糊推理的一个新方法．模糊系统与数学．1999，13(3)：1～10．
[21] 王国俊．模糊推理的全蕴涵三Ⅰ算法[J]．中国科学，E辑，1999，29(1)：43～53．
[22] 王国俊．三Ⅰ方法与区间值模糊推理．中国科学E辑．2000．30(4)：331～340．
[23] 王国俊．非经典数理逻辑与近似推理[M]．北京：科学出版社，2000
[24] 王国俊．模糊推理与模糊逻辑．系统工程学报．1998，13(2)：1～16．
[25] 张乃尧．典型模糊控制器的结构分析[J]．模糊系统与数学．1997，2：10～21．
[26] 张文修，梁怡．不确定性推理原理．西安：西安交通大学出版社，1996：57～76．
[27] 张文修，T模运算下F关系方程最大解，科学通报，1982，3．
[28] 张尧庭，张能胜，杜劲松．专家系统中不确定性方法学．广西师范大学出版社，1994．
[29] 周保魁．王国俊．不同蕴涵算子下的三Ⅰ算法．陕西师范大学学报(自然科学版)．1998，26(4)：1～5．
[30] 袁和军，李骏．推广形式下的三Ⅰ算法．陕西师范大学学报(自然科学版)．2002，30(1)：22～26．
[31] 陈启浩．模糊集及其在模糊推理中的应用．北京：北京师范大学出版社，2000：31～56．
[32] 邹开其、徐扬编著．模糊系统与专家系统．成都：西南交通大学出版社．1991：266～339．
[33] 王向云．区间值模糊推理的三Ⅰ算法．陕西师范大学．硕士学位论文．
2001．4．
[34] 孙长银，宋士吉等．模糊推理的三Ⅰ逢法的RL型支持度分析．应用数学．2001，14(1)：126～130．
[35] 宋士吉、吴澄．模糊推理的反向三Ⅰ约束算法．自然科学进展．2002，12(1)：95～100．
[36] Mamdami E H. Application of fuzzy logic to approximate reasoning using linguistic systems. IEEE Trans Compute, 1997,26:1182.
[37] 宋士吉，吴澄，模糊推理的反向三Ⅰ算法，中国科学E辑，2002，32(2)：230～246．
[38] 乔全喜．关于模糊推理相容性的研究．西南交通大学．硕士学位论文．1994：7～27．
[39] 热合木江、古丽·吐尔逊．基于不同代数算子的模糊推理方法的比较．新疆石油学院学报．2000，12(3)：64～66．
[40] 汪培庄．模糊集理论及其应用．上海：上海科技出版社．1983．
[41] 陈永义．三角模的生成与粘结．模糊推理的原理与方法论文集．1989年度工作报告：35～40．
[42] 胡长流、宋振明．格论基础．河南大学出版社．1990．
[43] [日]水本雅晴．模糊数学及其应用．科学出版社．1988．
[44] D.S.Yeung, E.C.C.Tsang. A comparative study on similarity-based fuzzy reasoning methods. IEEE Trans Systems Man Cybernetics. 1997,27(2):216～227.
[45] Bo Yuan, Yin Pan. Generalized fuzzy method-of-cases. International Journal of Approximate Reasoning. 1993(9): 129～138.
[46] 张师超、严小卫等．不确定性推理技术．桂林：广西师范大学出版社．1996：26～58．
[47] 裴道武．FMT问题的两种三Ⅰ算法及其还原性．模糊系统与数学．2001，15(4)：2～6．
[48] Ying M S. Reasoning of the compositional rule of fuzzy inference. Fuzzy sets and systems, 1990, 36:305.
[49] 张文修、王国俊等著．模糊数学引论．西安：西安交通大学出版社．1991：22～27．
[50] 孙长银．基于蕴涵算子R＜，L＞的模糊推理三Ⅰ方法．东南大学．硕士学位论文．2001．4．