模糊数学课件

1、FUZZY MATHEMATICS,Copyright 2010 By Hu Bao Qing Wuhan University All rights reserved,Teaching Material,模糊理论基础 胡宝清 编著 武汉大学出版社 2004年10月(第二次印刷) 2010年06月(第二版),I. Science Research Issue,Certainty Classical Mathematics Uncertainty Stochastic Probability 1974年,L.A.Zadeh发表模糊推理的研究报告,从此,模糊理论成了一个热门的课题. 1974年,英

2、国的E.H.Mamdani首次用模糊逻辑和模糊推理实现了世界上第一个实验性的蒸汽机控制,并取得了比传统的直接数字控制算法更好的效果,从而宣告模糊控制的诞生. 1980年丹麦的L.P.Holmblad和Ostergard在水泥窑炉采用模糊控制并取得了成功,这是第一个商业化的有实际意义的模糊控制器.,IV. Fuzzy Maths in China,Threshold the second half of 1970s Prosperity 1980s World leading in Fuzzy topology and algebra Fuzzy Math (模糊数学, 原华中理工大学) has

3、 international influence,Downfall 1990s Cause: 1. Theory Application 2. It is not regarded by mathematician 3. Journal was shifted from Wuhan to Hunan and its name was changed as Fuzzy System and Mathematics (模糊系统与数学) Prosperity again 2000s,V. Applications of Fuzzy Maths,The word “fuzzy” is perhaps

4、no longer fuzzy to many engineers today. Control Theory System Theory Decision Analysis Information Searches Medicine Biology .,e.g. Household Electrical Wares,Fuzzy Water Heater Fuzzy Electric Cooker Fuzzy Air-conditioner Fuzzy Washer Fuzzy Cleaner Fuzzy Refrigerator Fuzzy Microwave Oven Fuzzy Colo

5、ur TV Set Fuzzy Electric Fan ,e.g. Fuzzy Washer,These,e.g. Fuzzy Washer,These,e.g. Fuzzy Washer,These,e.g. Fuzzy Electric Fan,看一则广告,minutes.,These,电风扇系列 落地扇 原产地：中国 公司： 广东祥立实业有限公司（国际销售部） 五星级供应商会员 流畅时尚外观 模糊逻辑控制技术. 详细信息,高管为什么要他们学模糊数学,e.g. 2008年8月应邀到深圳为富士康(Foxconn)高层人员讲模糊理论36学时,minutes.,These,VI. Benefi

6、ts of Fuzzy Mathematics,The ability to model highly complex business problems Stephen Haking（斯蒂芬.霍金） 2000年指出 我认为，下个世纪将是复杂性的世纪(I think the next century will be the century of complexity),Improved cognitive modeling of expert system,What makes an expert an expert? With their ability to directly model

7、such imprecise information, fuzzy systems reduce overall cognitive dissonance (不一致) in the modeling process.,The ability to model systems involving multiple experts Fuzzy systems are well suited to representing multiple cooperating, collaborating, and even conflicting experts.,Reduced model complexi

8、ty Fuzzy models, as we will see, require fewer rules than conventional system, and these rules are closer to the way knowledge is expressed in natural language.,Improved handling of uncertainty and possibilities The representation of uncertainty in expert and decision support systems is an area of c

9、ontinual debate. In this aspect, fuzzy logic provides a better, more consistent, and more mathematically sound method of handling uncertainties. .,VII. Nave Thoughts on Fuzzy,Fuzzy phenomena and fuzzy concepts e.g. “rain” is a nature phenomenon. mizzle downpour,Fuzzy partition: light rain，moderate r

10、ain，heavy rain 比如，解释“大雨”，专业的天气预报书籍可能解释为“二十四小时内雨量累计(在雨量收集器内)达40至79.9毫米（mm）雨”，这是十分精确的。 Fuzzy judgment: What kind of rain is it now？ Fuzzy inference: predicting a good, fair, or poor harvest this year based on the results of fuzzy judging.,e.g.1 电脑不如人脑 两岁的小孩 图像识别能力 语音识别能力,IQ,EQ,Science：猴子可通过思维代替“手”操作电脑

11、,e.g. 2 “请你十点左右在行政大楼附近等我一下” Where When,听不懂?,Examples on fuzzy concepts,Consider the following statements,The engine temperature is hot.,Current inflation rates are rising rapidly.,CPI消费者价格指数 PPI生产者价格指数,Examples on fuzzy concepts,Tom is rather tall, but Judy is short. If you are tall, than you are qu

12、ite likely heavy.,Examples on fuzzy concepts,The description of a human characteristic such as healthy. The classification of patients as depressed. The classification of certain objects as large. The classification of people by age such as old. A rule for driving such as “if an obstacle is close, t

13、hen brake immediately”.,Examples on fuzzy concepts,Examples on fuzzy concepts,Yes or No,10 years and 6 months of imprisonment,Death penalty,如果有一天我们能建立法律公式,Concept and set,intension (内涵)：attributes of the object concept extension (外延)：all of the objects defined by the concept（set） G. Cantor (1887):,I

14、f a fuzzy concept can be rigidly described by Cantors notion of sets or the bivalent (true/false or two-valued) logic？,模糊概念能否用Cantor集合来刻画？,秃头悖论 一位已经谢顶的老教授与他的学生争论他是否为秃头问题。 教授：我是秃头吗？ 学生：您的头顶上已经没有多少头发，确实应该说是。 教授：你秀发稠密，绝对不算秃头，问你，如果你头上脱落了一根头发之后，能说变成了秃头了吗？ 学生：我减少一根头发之后，当然不会变成秃头。 教授：好了，总结我们的讨论，得出下面的命题：如果一个人不是秃头，那么他减少一根头发仍不是秃头，你说对吗？ 学生：对！ 教授：我年轻时代也和你一样一头秀发，当时没有人说我秃头，后来随着年龄的增高，头发一根根减少到今天的样子。但每掉一根头发，根据我们刚才的命题，我都不应该称为秃头，这样经有限次头发的减少，用这一命题有限次，结论是：我今天仍不是秃头。,Postulate: If a man with n (a nature number) hairs is baldheaded, then so is a man with n+1 hairs. Baldh