模糊控制只需要有操作人員的經(jīng)驗(yàn)ppt課件

1、Fuzzy Control SystemsOutline Fuzzy control system Fuzzy Inference Systems Defuzzification method Mamdani fuzzy model Sugeno fuzzy model Tsukamoto fuzzy modelFuzzy Control Systems模糊控制只需求有操作人員的經(jīng)驗(yàn)，就可以設(shè)計(jì)出控制規(guī)則，不需求以數(shù)學(xué)模型來描畫受控系統(tǒng)。模糊控制系統(tǒng)(Fuzzy Control System)的根本架構(gòu)如下圖所示。模糊推論系統(tǒng) 規(guī)則庫(kù)由一些IF. .THEN型式的規(guī)則所組成 資料庫(kù)中定義了控制

2、規(guī)則所運(yùn)用之模糊集合的歸屬函數(shù) 決策邏輯確立推論系統(tǒng)之運(yùn)算元型式 模糊化(Fuzzification)是將輸入數(shù)值轉(zhuǎn)換成所對(duì)應(yīng)語言項(xiàng)之歸屬度(degree) 解模糊化(Defuzzification)將推論結(jié)果轉(zhuǎn)換成輸出數(shù)值 解模糊化的方法將經(jīng)過模糊推論之後產(chǎn)生的結(jié)論，轉(zhuǎn)換為一明確數(shù)值的過程，我們稱之為 解模糊化。由於不同的模糊規(guī)則所採(cǎi)用的後鑑部會(huì)有所不同，因此，經(jīng)過模糊推論後所得到的結(jié)論，有的是以模糊集合來表示(如語意式模糊規(guī)則)，而有的是以明確數(shù)值來表示。一、推論後得到的是模糊集合：令模糊集合 C 為模糊規(guī)則經(jīng)過模糊推論後所得到的結(jié)論，亦即 中的 。1. 重心法 (center of gr

3、avity defuzzifier or center of area defuzzifier) (1) 當(dāng)論域?yàn)檫B續(xù)時(shí)： (2) 當(dāng)論域?yàn)殡x散時(shí)： 解模糊化的方法2. 最大平均法 (mean of maxima defuzzifier) 其中 3. 修正型最大平均法 (modified mean of maxima defuzzifier) 其中4. 中心平均法 (modified center average defuzzifier) 解模糊化的方法5. 修正型重心法 (modified center average defuzzifier) 其中以 j 作為控制歸屬函數(shù)遞減的速率，當(dāng) j

4、 越小，則歸屬函數(shù)遞減的速率越快。 二、推論後得到明確的輸出值：令 j 代表第 j 個(gè)模糊規(guī)則的前鑑部被符合的程度性，亦即“啟動(dòng)強(qiáng)度(firing strength)，yj 為第 j 個(gè)模糊規(guī)則所推論出的結(jié)果，以下的“權(quán)重式平均法(weighted average method)最被廣泛運(yùn)用： 模糊控制範(fàn)例 (1) 模糊規(guī)則一 R1：If x is A1 and y is B1Then z is C1 模糊規(guī)則二 R2：If x is A2 and y is B2Then z is C2 令 x0 與 y0 為感應(yīng)器 x 與 y 之輸入，模糊集合 A1、 A2、 B1 、 B2 、 C1 、以

5、及 C2 運(yùn)用以下之歸屬函數(shù)： 模糊控制範(fàn)例 (2)讀入感應(yīng)器輸入 以及 ，接下來我們將說明如何計(jì)算最後的控制輸出。 首先計(jì)算感應(yīng)器輸入 以及 與兩條模糊規(guī)則的符合程度為： 接下來，兩條模糊規(guī)則的啟動(dòng)強(qiáng)度為： 將 1 對(duì)映至第一條模糊規(guī)則的後件部，可得到如圖8.8中的灰色梯形區(qū)域 ；一樣地，將 2 對(duì)映至第二條模糊規(guī)則的後件部，可得到如圖8.8中的黑色梯形區(qū)域 ；將此兩個(gè)梯形區(qū)域以 “最大運(yùn)算子 (max) 取其最大值，可得最後的歸屬函數(shù)。最後解模糊化可得： 圖8.8：模糊推論過程表示圖。 模糊控制範(fàn)例 (3)以連續(xù)型重心法作為解模糊化機(jī)構(gòu)：首先找出 C 的歸屬函數(shù)為 ： 因此模糊控制範(fàn)例 (

6、4)(2) 以離散型重心法來解模糊化：我們將輸出量化成 1,2,.,9 等 9 個(gè)離散輸出，可得 (3) 以 “最大平均法 作為解模糊化機(jī)構(gòu)：在最後的歸屬函數(shù)中，其量化值達(dá)到最大歸屬函數(shù)值的有 3、4、以及 5，因此我們可以得到： (4) 以修正型最大平均法作為解模糊化機(jī)構(gòu)： (5) 以中心平均法作為解模糊化機(jī)構(gòu)： Fuzzy Inference System (FIS) A computing framework based on the concepts of fuzzy set theory, fuzzy if-then rule, fuzzy reasoning Fuzzy Infer

7、ence System also called as:Fuzzy Rule_Based System, Fuzzy Expert System, Fuzzy Model, Fuzzy Associative Memory, Fuzzy Logic Controller, Fuzzy SystemThree Types of Fuzzy inference system Mamdani fuzzy model Sugeno fuzzy model Tsukamoto fuzzy modelMamdani Fuzzy Models Attempt to control a steam engine

8、 and boiler combination by a set of linguistic control rules obtained from human operators. Usemax-algebraic product for T-conorm/T-norm andmax-product composition The overall output: defuzzification 2 FISs: a controller to generate the heat input to the boiler to regulate the steam pressure in the

9、boiler a controller of 節(jié)流閥opening of the engine cylinder to 控制 the speed of the engine E.H. Mamdani and S. Assilian. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 7(1):1-13, 1975.Mamdanis Fuzzy Modelsmax-min T-conorm/normmax-algebr

10、aic productT-conorm/normMamdani Fuzzy ModelsDefuzzification: a method to extract a representative crisp value from a fuzzy set. defuzzification of a fuzzy set A of a universe of disourse Z: A(z): the aggregated output MF.Centroid of area zCOA : expected values of prob. Distribution.zCOA =Z A(z) z dz

11、 / Z A(z) dz Bisector of area zBOA : the vertical line z=zBOA partitions the region b/t z=, z=, y=0, y=A(z) into 2 regions with the same area.zBOA A(z) dz = zBOA A(z) dz where = min z| z Z, = max z| z ZMean of maximum zMOM : average of the maximizing z at which MF reach a maximum *zMOM =Z z dz / Z d

12、z where Z = z| A(z)=* Smallest of maximum zSOM : the minimum of the maximizing zLargest of maximum zLOM: the maximum of the maximizing zMamdani Fuzzy ModelsCentroid of area zCOA zCOA = Z A(z) z dz / Z A(z) dz Bisector of area zBOA zBOA A(z) dz = zBOA A(z) dzMean of maximum zMOM zMOM =Z z dz / Z dz S

13、mallest of maximum zSOM Largest of maximum zLOMExample: Mamdanis Fuzzy Model Single-input single-output Mamdani fuzzy modelIf X is smallthen Y is small.If X is mediumthen Y is medium.If X is largethen Y is large.Example: Mamdanis Fuzzy Model Two-input single-output Mamdani fuzzy modelIf X is small a

14、nd Y is smallthen Z is negative large.If X is small and Y is largethen Z is negative smallIf X is large and Y is smallthen Z is positive small.If X is large and Y is largethen Z is positive large.Variants AND operator (T-norm): for calculating the firing strength of a rule with ANDed antecedents OR

15、operator (T-conorm):the calculating the firing strength of a rule with ORed antecedents Implication operatorfor calculating qualified consequent MFs based on given firing strength Aggregation operatorfor aggregating qualified consequent MFs to generate an overall output MF. Defuzzification operatorf

16、or transforming an output MF to a crisp single output value. Sum-product composition (aggregation implication operator) The final crisp output via centroid defuzzification = the weighted average of the centroids of consequent MFS, wherethe weighting factor for each rule =its firing strength the area

17、 of the consequent MF.Theorem: Computation Shortcut for Mamdani Fuzzy Inference SystemsUnder sum-product composition,the output of a Mamdani FIS with centroid defuzzification =the weighted average of the centroids of consequent MFS, where each of the weighting factors =firing strength the consequent

18、 MFs area. (wi ai)Pf) Use product for implication, and sum for aggregation operator. Then, C (z) = w1C1 (z) + w2C2 (z) The crisp output under centroid defuzzification isSugeno Fuzzy Models (TSK model) Takagi, Sugeno and Kang a systematic approach to generate fuzzy rules from a given input/output dat

19、a set. if x is A and y is B then z = f(x,y)z=f(x,y): a crisp function in the consequent.f(x,y): a polynomial fn; but it can be any fn.1st-order Sugeno fuzzy model: f(x, y) is a 1st order polynomial.Zero-order Sugeno fuzzy model: f(x,y) is a constantA special case of Mamdani model, in which each rule

20、s consequent is specified by a fuzzy single (or a pre-defuzzified consequent)A special case of Tsukamoto fuzzy model, in which each rules consequent is specified by an MF of a step function center at the constant.Functionally, equivalent to a Radial Basis Function network under certain minor constra

21、ints (Chap.12) The overall output:weighted average z = (w1z1+w2z2) / (w1+w2) - no defuzzification.Orweighted sum z = w1z1+w2z2 - the loss of MF linguistic meanings unless I wi 1.Sugeno Fuzzy Models (TSK model) dont strictly follow Compositional Rule of Inference, but still employ the matching of fuz

22、zy sets in the antecedent part. Most popular candidate for sample-data-based fuzzy modeling, w/o defuzzification. M. Sugeno and G.T. Kang. Structure identification of fuzzy model. Fuzzy Sets and Systems, 28:15-33, 1988 T. Takagi and M. Sugeno. Fuzzy identification of systems and its applications to

23、modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15:116-132, 1985.Sugeno Fuzzy Models (TSK model)Example: Sugeno Fuzzy Models Comparison of Fuzzy and Nonfuzzy Rules SetIf X is smallthen Y = 0.1X + 6.4If X is mediumthen Y = - 0.5X + 4If X is largethen Y = X 2.Antecedent MFs v

24、s. Input-output curveExample: Sugeno Fuzzy Model Two-input single-output Sugeno fuzzy modelIf X is small and Y is smallthen z=-x+y+1.If X is small and Y is largethen z=-y+3.If X is large and Y is smallthen z=-x+3.If X is large and Y is largethen z=x+y+2.Antecedent/consequent MFOverall input-output s

25、urfaceTsukamoto Fuzzy Model the consequent of each fuzzy if-then rule: a fuzzy set with a monotonical MF. Overall output: the weighted average of each rules output. No defuzzification. Not as transparent as mamdanis or Sugenos fuzzy model. Not follow strictly the compositional rule of inference: the

26、 output is always crisp.Example: Tsukamoto Fuzzy Model Single-input Tsukamoto fuzzy modelIf X is smallthen Y is C1 .If X is mediumthen Y is C2 .If X is largethen Y is C3 .Other ConsiderationCommon Issues concerning 3 FISs:How to partition an input space?How to construct a FIS for a particular applic

27、ation?In 3 FISs, the same Antecedent in 3 FISs - defines a local fuzzy region vs. different Consequent (MF, a constant, a polynomial) describes the behavior within the regionMethods of partitioning input spaces: to form the antecedents - applicable to all 3 types of FISsGrid partitionTree partitionS

28、catter partitionInput Space Partitioning Grid partition:Often chosen in a fuzzy controller which involves only several state variables as the inputsNeeds only a small # of MFSs for each input.For large # of inputs? - exponential # of rules.- curse of dimensionality. Tree partitionEach region can be

29、uniquely specified along a corresponding decision tree. relieves an exponential increase in # of rules. Scatter partitionLimit the # of rules to a reasonable amount by covering a subset of the whole input space which characterizes a region of possible occurrence of the input vectors.Dictated by desi

30、red i-o data pairs, thus orthogonality doesnt hold in X,Y, or X Y.Fuzzy Modeling A process for constructing a FIS Features: The rule structure of FIS makes it easy to incorporate human expertise a/t the target system directly into the modeling process take advantage of domain knowledge When the inpu

31、t/output data of a target system is available, conventional system identification techniques can be used. the important role of the use of numerical data in fuzzy modeling. A process for constructing a FIS1. identification of the surface structure:Obtain rule base which describe the behavior of the

32、target system b.m.o. linguistic terms.2. identification of deep structure:Determine the MFs of each linguistic term.Fuzzy ModelingA process for constructing a FIS:identification of the surface structure:Obtain rule base which describe the behavior of the target system b.m.o. linguistic terms.Rely on the knowledge of the target system whose information provided by human experts or trial & error.Select relevant input-output variables.Choose a specific type of FIS.Determine the number of linguistic terms associated with each input-output variables (an