模糊控制只需要有操作人员的经验教学课件

Fuzzy Control SystemsFuzzy Control Systems2OutlineOutline Fuzzy control system Fuzzy Inference Systems Defuzzification method Mamdani fuzzy model Sugeno fuzzy model Tsukamoto fuzzy model3Fuzzy Control Systems模糊控制只需要有操作人員的經驗，就可以設計出控制規則，不需要以數學模型來描述受控系統。模糊控制系統(Fuzzy Control System)的基本架構如下圖所示。4模糊推論系統5p 規則庫由一些IF.THEN型式的規則所組成p 資料庫中定義了控制規則所使用之模糊集合的歸屬函數p 決策邏輯確立推論系統之運算元型式 p 模糊化(Fuzzification)是將輸入數值轉換成所對應語言項之歸屬度(degree)p 解模糊化(Defuzzification)將推論結果轉換成輸出數值 6解模糊化的方法解模糊化的方法將經過模糊推論之後產生的結論，轉換為一明確數值的過程，我們稱之為 解模糊化。由於不同的模糊規則所採用的後鑑部會有所不同，因此，經過模糊推論後所得到的結論，有的是以模模糊糊集集合合來表示(如語意式模糊規則)，而有的是以明明確確數數值值來表示。一一、推推論論後後得得到到的的是是模模糊糊集集合合：令模糊集合 C 為模糊規則經過模糊推論後所得到的結論，亦即 中的 。1.重心法重心法(center of gravity defuzzifier or center of area defuzzifier)(1)當論域為連續時當論域為連續時：(2)當論域為離散時：當論域為離散時：7解模糊化的方法解模糊化的方法2.最大平均法最大平均法(mean of maxima defuzzifier)其中 3.修正型最大平均法修正型最大平均法(modified mean of maxima defuzzifier)其中4.中心平均法中心平均法(modified center average defuzzifier)8解模糊化的方法解模糊化的方法5.修正型重心法修正型重心法(modified center average defuzzifier)其中以 j 作為控制歸屬函數遞減的速率，當 j 越小，則歸屬函數遞減的速率越快。二二、推推論論後後得得到到明明確確的的輸輸出出值值：令 j 代表第 j 個模糊規則的前鑑部被符合的程度性，亦即“啟動強度(firing strength)”，yj 為第 j 個模糊規則所推論出的結果，以下的“權重式平均法權重式平均法(weighted average method)”最被廣泛使用：9模糊控制範例模糊控制範例(1)模糊規則一 R1：If x is A1 and y is B1Then z is C1 模糊規則二 R2：If x is A2 and y is B2Then z is C2 令 x0 與 y0 為感應器 x 與 y 之輸入，模糊集合 A1、A2、B1、B2、C1、以及 C2 使用下列之歸屬函數：10模糊控制範例模糊控制範例(2)讀入感應器輸入 以及 ，接下來我們將說明如何計算最後的控制輸出。首先計算感應器輸入 以及 與兩條模糊規則的符合程度為：接下來，兩條模糊規則的啟動強度為：將 1 對映至第一條模糊規則的後件部，可得到如圖8.8中的灰色梯形區域；相同地，將 2 對映至第二條模糊規則的後件部，可得到如圖8.8中的黑色梯形區域；將此兩個梯形區域以“最大運算子(max)”取其最大值，可得最後的歸屬函數。最後解模糊化可得：11圖8.8：模糊推論過程示意圖。12模糊控制範例模糊控制範例(3)1.以連續型重心法作為解模糊化機構：首先找出以連續型重心法作為解模糊化機構：首先找出 C 的歸屬函數為的歸屬函數為 ：因此13模糊控制範例模糊控制範例(4)(2)以離散型重心法來解模糊化：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 個離散輸出，可得(3)以以“最最大大平平均均法法”作作為為解解模模糊糊化化機機構構：在最後的歸屬函數中，其量化值達到最大歸屬函數值的有 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構：以修正型最大平均法作為解模糊化機構：(5)以中心平均法作為解模糊化機構：以中心平均法作為解模糊化機構：14Fuzzy Inference System(FIS)Fuzzy Inference System(FIS)A computing framework based on the concepts of fuzzy set theory,fuzzy if-then rule,fuzzy reasoning Fuzzy Inference System also called as:Fuzzy Rule_Based System,Fuzzy Expert System,Fuzzy Model,Fuzzy Associative Memory,Fuzzy Logic Controller,Fuzzy System15Three Types of Fuzzy inference system Mamdani fuzzy model Sugeno fuzzy model Tsukamoto fuzzy model16Mamdani Fuzzy ModelsMamdani Fuzzy Models Attempt to control a steam engine 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 boiler a controller of 節流閥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):1-13,1975.17Mamdanis Fuzzy Modelsmax-min T-conorm/normmax-algebraic productT-conorm/norm18Mamdani Fuzzy ModelsMamdani 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/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 dz where Z =z|A(z)=*Smallest of maximum zSOM :the minimum of the maximizing zLargest of maximum zLOM:the maximum of the maximizing z19Mamdani Fuzzy ModelsMamdani 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 Smallest of maximum zSOM Largest of maximum zLOM20Example: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.21Example:Mamdanis Fuzzy Model Two-input single-output Mamdani fuzzy modelIf X is small and Y is small then Z is negative large.If X is small and Y is large then Z is negative smallIf X is large and Y is small then Z is positive small.If X is large and Y is large then Z is positive large.22Variants AND operator(T-norm):for calculating the firing strength of a rule with ANDed antecedents OR 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 operatorfor 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 of the consequent MF.23Theorem: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 MFs area.(wi ai)Pf)Use product for implication,and sum for aggregation operator.Then,C (z)=w1 C1(z)+w2 C2(z)The crisp output under centroid defuzzification is24Sugeno Fuzzy Models(TSK model)Takagi,Sugeno and Kang a systematic approach to generate fuzzy rules from a given input/output data 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 rules 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 constraints(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 1.25Sugeno Fuzzy Models(TSK model)dont strictly follow Compositional Rule of Inference,but still employ the matching of fuzzy 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 modeling and control.IEEE Transactions on Systems,Man,and Cybernetics,15:116-132,1985.26Sugeno Fuzzy Models(TSK model)27Example: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 vs.Input-output curve28Example: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 surface29Tsukamoto 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 output is always crisp.30Example: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 C3.31Other ConsiderationCommon Issues concerning 3 FISs:How to partition an input space?How to construct a FIS for a particular application?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 partitionScatter partition32Input 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 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 desired i-o data pairs,thus orthogonality doesnt hold in X,Y,or X Y.33Fuzzy 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 input/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 target system b.m.o.linguistic terms.2.identification of deep structure:-Determine the MFs of each linguistic term.34Fuzzy ModelingA process for constructing a FIS:1.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.1.Select relevant input-output variables.2.Choose a specific type of FIS.3.Determine the number of linguistic terms associated with each input-output variables(and the order of consequent equation for Sugenos model).4.Design a collection of fuzzy if-then rules.35Fuzzy ModelingA process for constructing a FIS:2.identification of deep structure:-Meaning of the linguistic terms are determined by the MFs of each linguistic term(and the coefficients of each rules output polynomial in Sugenos model).1.Choose an appropriate family of parameterized MFs.2.Interview human experts familiar with the target systems to determine the parameters of the MFs used in the RB.3.Refine the parameters of the MFs using regression and optimization techniques.-the desired input-output data set is assumed.36倒單擺的模糊控制37倒單擺的模糊控制語言變數：夾角angle 角速度 施力y五個模糊集合:(1)positive medium(PM);(2)positive small(PS);(3)zero(ZE);(4)negative small(NS);(5)negative medium(NM)38模糊控制規則39模糊控制40模糊控制41每一語言變數的模糊集合N3:Large negative.N2:Medium negative.N1:Small negative.Z:Zero.P1:Small positive.P2:Medium positive.P3:Large positive.42應用模糊規則rule 2:IF temperature IS cool AND pressure IS low,THEN throttle is P2.43應用模糊規則rule 3:IF temperature IS cool AND pressure IS ok,THEN throttle is Z.44模糊推論谢谢