20.模糊数学与模糊逻辑基础

1 模糊控制模糊控制 第一讲：模糊数学与模糊逻辑基础第一讲：模糊数学与模糊逻辑基础 第二讲：模糊系统的构造及万能逼近特性第二讲：模糊系统的构造及万能逼近特性 第三讲：基于数据的模糊系统设计及在金融建模中的应用第三讲：基于数据的模糊系统设计及在金融建模中的应用 第四讲：模糊自适应控制第四讲：模糊自适应控制 模糊控制第一讲：模糊数学与模糊逻辑基础模糊控制第一讲：模糊数学与模糊逻辑基础 一、一、 模糊控制简介模糊控制简介 二、二、 模糊集合及其基本运算模糊集合及其基本运算 三、三、 模糊集合的广义运算模糊集合的广义运算 四、四、 模糊关系及推广原理模糊关系及推广原理 五、五、 语言变量及模糊规则语言变量及模糊规则 2 Linguistic Information What is Information? Numerical Information Human world Physical world Fuzzy systems (brain software) Neural networks (brain hardware) Intelligent Systems 3 一、一、模糊控制简介模糊控制简介 Why Fuzzy Systems? Reason 1: The principle of Incompatibility (Zadeh,1972) As the complexity of a system increases, our ability to make precise and yet significant statement about its behaviour diminishes until a threshold is reached beyond which precision and significance became almost mutually exclusive characteristics. Reason 2: As we move into the information area, human knowledge becomes increasingly important. We need a theory to formulate human knowledge in a systematic manner and put it into engineering systems, together with other information like mathematical model and sensory measurements. 4 What are Fuzzy Systems? Fuzzy systems are knowledge-based or rule-based systems. Example 1: A fuzzy system used to control a car may contain the following rules: IF the speed is high, THEN apply less force to the accelerator. IF the speed is low, THEN more Three types of Fuzzy Systems - Pure fuzzy systems Fuzzy Rule Base Fuzzy Inference Engine fuzzy sets fuzzy sets 5 - Takagi-Sugeno-Kang (TSK) fuzzy system Fuzzy Rule Base Weighted Average real number real number - Standard fuzzy systems Fuzzy Rule Base Fuzzy Inference Engine fuzzy sets fuzzy sets Fuzzifier Defuzzifier real number real number 6 What Are the Major Applications of Fuzzy Systems? Fuzzy systems are most suitable for problems where: - no precise mathematical model, and - there are human experts who can provide important information about the system. Major application fields: - Industrial process control, including cement kiln, steel plant, waste water treatment process, etc. - Complex mechanical systems, including unmanned helicopter, subway train, etc. - Home electronics, including camcorder, washing machine, air conditioner, etc. - Financial and social systems 7 Example 2: Digital Image Stabilizer The hand is not shaking. The hand is shaking. Rule 1: IF all points in the picture are moving in the same direction, THEN the hand is shaking; Rule 2: IF only some points in the picture are moving, THEN the hand is not shaking 8 Example 3: Fuzzy Control of A Subway Train For safety: IF the speed is approaching the limit, THEN select the maximum brake notch. For riding comfort: IF the train will stop in the allowed zone, THEN do not change the control notch. For riding comfort and safety: IF the train is in the allowed zone, THEN gradually change the control notch from acceleration to braking. 9 Example 4: 股票价格股票价格动态模型动态模型 Heuristic 1: A buy (sell) signal is generated if a shorter moving average of the price is crossing a longer moving trend average of the price from below (above). Usually, the following larger the difference between the two moving averages, the stronger the buy (sell) signal. But, if the difference between the two moving averages is too large, the stock may be over-bought (over-sold), contrarian so a small sell (buy) order should be placed to safeguard the investment. What are the price dynamics when Heuristic 1 is used by technical traders? 10 Major Research Fields in Fuzzy Theory Fuzzy Theory Fuzzy Mathematics Fuzzy Systems Fuzzy Logic , 1 )( Axif Axif x A Example: 1 10% 80% students in a class 10% Top Students Average Students Poor Students 14 Fuzzy Set: A fuzzy set in a universe of discourse U is characterized by a membership function )(x A that take values in the interval 0,1. Example: 2 A =“top students” )(x A 1 Top 10% 20% x Example: 3 A = “numbers close to zero” 2 )( x A ex )(x A )(x A 1 1 or 0 x -1 0 1 x 15 2.2 Basic Concepts Associated with Fuzzy Set Support of a fuzzy set: 0)(|)(xUxASupp A Fuzzy singletion: A fuzzy set whose support is a single point in U. Center of a fuzzy set: The mean of all points in U at which )(x A achieves its maximum values; if the mean equals infinite, center is defined as follows: )(x A )(x A )(x A 1 0.8 0.5 center center center Height of a fuzzy set: The largest membership value attained by any point. Normal fuzzy set: The height equals one. cut : )(| xUxA A , 10. Convex fuzzy set: A is convex iff its cut A is a convex set for any 1 , 0(. 16 LEMMA 2.1 A is a fuzzy set. A is convex )(),(min()1 ( 2121 xxxx AAA , 1 , 0, 21 n Rxx. 2.3 Operations on Fuzzy Sets Equality ： BA iff )()(xx BA , for all Ux. Containment: BA iff )()(xx BA , for all Ux. Complement: )(1)(xx A A . Union: )(),(max)(xxx BABA . Intersection: )(),(min)(xxx BABA . LEMMA 2.2: The De Morgans Laws are true for fuzzy sets: BABA BABA 17 Projection of a fuzzy set Let A be a fuzzy set in 2 RU , the projection of A on x1 is a fuzzy set 1 A with ),(max)( 211 2 1 xxx A Rx A . Similarly, ),(max)( 212 1 2 xxx A Rx A . x2 A1E A2 A1 x1 A 18 三、模糊集合的广义运算模糊集合的广义运算 Basic Operations: )(1)(xx A A . )(),(max)(xxx BABA . )(),(min)(xxx BABA . 3.1 Fuzzy Complement Let )()(xxc A A , c is a fuzzy complement if: Axiom c1. 10 c and 01 c (boundary condition) Axiom c2. For all 1 , 0(, ba, ba implies )()(bcac. (nonincreasing condition) Example 1：Sugeno class of fuzzy complements )( C 1 1 , , 1. Example 2：Yager class of fuzzy complements )( w C= w w /1 1, , 0w 19 3.2 Fuzzy unionThe S-Norms Let )()(),(xxxS BABA . S is a fuzzy union or S-norm if: Axiom s1. 11 , 1s , aasas0 , 0. (boundary condition) Axiom s2. absbas,. (commutative condition) Axiom s3. If ,bbaathen basbas,. (nonincreasing condition) Axiom s4. cbsascbass,. (associative condition) Examples: Dombi class /1 1 1 1 1 1 1 ),( ba bas, , 0. Dubios-Prade class ,1 ,1max 1 ,min ),( ba baabba bas , 1 , 0. 20 Yager class w ww w babas /1 , 1min),(, , 0w. Drastic Sum . , 1 ; 0 , ; 0 , ),( otherwise aifb bifa basds Algebraic Sum abbabasas),(. Maximum ),max(),( max babas. Theorem 3.1 ),(),(),max(basbasba ds . 21 3.3 Fuzzy Intersection The T-Norms Let )()(),(xxxt BABA , t is a fuzzy intersection or t-norm if: Axiom s1. 00 , 0t, aatat1 , 1. (boundary condition) Axiom s2. abtbat,. (commutative condition) Axiom s3. If ,bbaathen batbat,. (nonincreasing condition) Axiom s4. cbtatcbatt,. (associative condition) Examples: Dombi class /1 1 1 1 1 1 1 ),( ba bat, , 0. Dubios-Prade class ,max ),( ba ab bat, 1 , 0. 22 Yager class w ww w babat /1 11, 1min1),(, , 0w. Drastic product . , 0 , 1 , , 1 , ),( otherwise aifb bifa batdp Algebraic product abbatap),(. Minimum ),min(),( min babat. Theorem 3.2 ),min(),(),(babatbatdp. 23 3.4 Averaging Operators a,b 1 , 0 0 a 2 ba b 1 t min(a,b) max(a,b) sds(a,b) t-norms averaging s-norms operators Max-min averages: ),max()1 (),min(),(bababaV , 1 , 0. Generalized averages: /1 2 ),( ba baV, )0( ,R. “Fuzzy and: 2 )1 (),min(),( ba pbapbaVp , 1 , 0p. “Fuzzy or: 2 )1 (),max(),( ba babaV , 1 , 0. 24 四、模糊关系及推广原理模糊关系及推广原理 4.1 From Classical Relations to Fuzzy Relations Classical Relation: Let 1 U, 2 U, , n U be universe of discourse and n UUU 21 be the cartesian product, i.e., nnnn UuUuuuUUU,.|,. 11121 then a relation among 1 U, 2 U, , n U is a subset of the cartesian product n UUU 21 , i.e., nn UUUUUUQ 2121 ),(. Example: 3 , 2 , 1U, 4 , 3 , 2V. )3 , 3(),2 , 3(),2 , 2(),(VUQ In matrix form: V U 0113 0012 0001 432 ),2 , 2(),4 , 1 (),3 , 1 (),2 , 1(VU. 25 Fuzzy Relation: A fuzzy relation is a fuzzy set defined in n UUU 21 . Example: U=Hong Kong, Tokyo, V=New York, Hong Kong, Q=“very far”. V U 2 . 09 . 0 01 Tokyo HK HKNY Projections: Let Q be a fuzzy relation in n UUU 21 and , 1k ii be a subsequence of , 2 , 1n. Then the projection of Q on ikk UU 1 is a fuzzy relation P Q in ikk UU 1 with nQ UuUu iiQ uuuu knj kn jjj kp ,.,max,., 1 ,., )( )(11 1 where kkn iijj uuuu,.,., 1)(1 = n uu ,., 1 . 26 Cylindric extension: Let P Q be a fuzzy relation in ikk UU 1 and , 1k ii is a subsequence of , 2 , 1n. Then the cylindric extension of P Q to n UUU 21 is a fuzzy relation PE Q in n UUU 21 with kppE iiQnQ uuuu,.,., 1 1 . Cartesian Product of Fuzzy Sets Let 1 A, 2 A , n A be fuzzy sets in 1 U, 2 U, , n U, respectively. The cartesian product of 1 A, 2 A, , n A is a fuzzy relation in n UUU 21 with 1 A1 A1 AA 11n1 ,.,uuuu n where denotes any t-norm. 27 4.2 Composition of Fuzzy Relations P Q U V W For nonfuzzy P and Q, the composition QP is a relation in WU such that QPzx),( iff Vy s.t. Pyx),(, Qzy),(. Lemma 4.2 QP is a composition of ),(VUP and ),(WVQ iff ),(),(sup),(zyyxtzx QPVyQP (4.28) for any WUzx),(. Let ),(VUP and ),(WVQ be fuzzy relations. Their composition QP is a fuzzy relation in WU with ),(),(max),(zyyxtzx QPVyQP . 28 max-min composition: ),(),(minmax),(zyyxzx QPVyQP max-product composition: ),(),(max),(zyyxzx QPVyQP Example: Let V W P: U 2 . 09 . 0 01 9 . 03 . 0 3 2 1 21 u u u vv Q: V 9 . 01 . 0 1 . 08 . 0 2 1 21 v v ww Compute QP using max-min and max-product compositions. 29 4.3 The Extension Principle f U V Given VUf:and fuzzy set A in U, how to determine B=f(A)? If f is one-to-one, then )()( 1 yfy AB , Vy. If f is not one-to-one, then )(max)( )( 1 xy A yfx B , Vy where )( 1 yf is the set of all points x in U with yxf)(. This is the extension principle. B A 30 五、语言变量五、语言变量及模糊规则及模糊规则 5.1 From Numerical Variables to Linguistic Variables numerical variable linguistic variable U U If a variable can take words as its values, it is a linguistic variable, where the words are represented by fuzzy sets. Example: x = speed of a car， xslow, medium, fast, x0,Vmax 5.2 Linguistic Hedges Hedges: “very”, “more or less”, 2)()(xx AveryA 2/1 A lessor more )()(xx A 31 5.3 Fuzzy IF-THEN Rules Fuzzy Propositions Atomic fuzzy proposition: x is A )(x A ，where x: linguistic variable, A: fuzzy set Compound fuzzy proposition: Combination of atomatic fuzzy propositions with “and”,”or”,”not”. x is S and x is not M or x is F x is F and y is L. “and” = fuzzy intersection x is A and y is B )(),(),(yxtyx BABA A t B “or” = fuzzy union x is A or y is B )(),(),(yxsyx BABA “not” = fuzzy complement x is not A )()(xcx A A 32 Fuzzy IF-THEN Rules IF , THEN In classical logic p q qp q p pqp T T T F F T F F T F T T T F T T T F T T qp qpqp Consider IF , THEN = Q where FP1 is a fuzzy proposition (relation) in n UUU 1and FP2 is a fuzzy proposition in n VVV 1 , so Q is a fuzzy relation in VU . 33 Dienes-Rescher Implication )(),(1max),( 21 yxyx FPFPQD Lukasiewicz Implication )()(1min),( 21 yxyx FPFPQL Zadeh Implication )(1),(),(maxmin(),( 121 xyxyx FPFPFPQz Lemma 5.1: For all VUyx),(, it is true that ),(),(),(yxyxyx LD QQQz 34 IF , THEN IF , THEN ELSE qp qp Mamdani Implication )(),(min),( 21 yxyx FPFPQM or )()(),( 21 yxyx FPFPQM 35 模糊模糊控制控制任课教师任课教师简介简介：王立新，分别于 1984 和 1987 年毕业于西北工业大学电子工程系和计算机 系，获学士和硕士学位；1987 年 4 月至 1989 年 8 月西北工业大学计算机系助教；1989 年 9 月至 1992 年 5 月美国南加州大学（University of Southern California）电机工程系博士研究生，1992 年 8 月获博士学位（Ph.D.）；1992 年 6 月至 1993 年 10 月美国加州大学（University of California） Berkeley分校电机工程与计算机科学系博士后；1993 年 11 月至 2007 年 8 月任教于香港科技大学电子 与计算机工程系；2007 年 9 月至 2013 年 7 月为独立研究者与投资人；2013 年 8 月至 2016 年 12 月任 西安交通大学自动化科学与技术系教授； 2017 年 1 月起任教于中国科学院大学资源与环境学院。 主要研究方向：资产价格动态模型，市场微观结构，交易策略，模糊系统，舆情网络。在以下四 个方面为模糊系统及模糊控制领域作出开拓性贡献： 1. 提出第一个由数据产生语言规则的方法:Wang-Mendel(WM)方法，将语言信息和数据信息统一在同 一个数学框架之下，从而促成并引领模糊神经网络模糊神经网络领域的形成与发展。作为模糊神经网络领域的 经典算法，WM 方法是后续算法性能比较的标杆，在广泛领域取得成功应用。WM 方法的原始论文 （L.X. Wang and J.M. Men