Takagi-Sugeno Fuzzy Modeling For Process Control
Takagi-Sugeno Fuzzy Modeling for Process ControlKamyar MehranIndustrial Automation, Robotics and Artificial Intelligence (EEE8005)School of Electrical, Electronic and Computer EngineeringNewcastle University· 2008 ·
Contents1 Introduction12 Takagi-Sugeno fuzzy modeling2.1 Construction of Fuzzy Models . . . . . . . . . . . . . .2.1.1 Sector Nonlinearity . . . . . . . . . . . . . . . .2.2 Basic Fuzzy Mathematics for Modeling . . . . . . . . .2.2.1 Local Approximation in Fuzzy Partition Spaces.2. 3. 4. 5. 103 Appendix17References18ii
Chapter 1IntroductionAlmost all of the physical dynamical systems in real life cannot be represented by linear differential equations and have a nonlinear nature. At the same time, linear controlmethods rely on the key assumption of small range of operation for the linear model,acquired from linearizing the nonlinear system, to be valid. When the required operationrange is large, a linear controller is prone to be unstable, because the nonlinearities inthe plant cannot be properly dealt with. Another assumption of linear control is that thesystem model is indeed linearizable and the linear model is accurate enough for buildingup the controller. However, the highly nonlinear and discontinuous nature of many, forinstance, mechanical and electrical systems does not allow linear approximation. It is alsonecessary, in the design process of controllers, that the system model is well achievablethrough a mathematical model and the parameters of the system model are reasonablywell-known. Nevertheless, for many nonlinear plans i.e. chemical processes, building amathematical model is very difficult and only the input-output data yielded from runningthe process is accessible for an estimation. Many control problems involve uncertaintiesin the model parameters. A controller based on inaccurate or obsolete values of the modelparameters may show significant performance degradation or even instability. There aresome complicated approaches like auto-regressive model based on the input-output data tocompensate model uncertainties, which usually use to design a process control. Howeverdue to the high nonlinearity of the process, the order of the model often becomes veryhigh so that past effects are taken into account, even if that is physically unrealistic.One way to cope with such difficulty is to develop a nonlinear model composing of anumber of sub-models which are simple, understandable, and responsible for respectivesub-domains. The idea of multi-model approach [1] is not new, but the idea of fuzzymodeling [2] using the concept of the fuzzy sets theory [3] offers a new technique to buildmulti-models of the process based on the input-output data or the original mathematicalmodel of the system. Facing complex and nonlinear systems, we have to recognize thatmodeling is an art and it is important to realize system modeling is generally an act tounderstand things directly rather than by computer. At most a linear combination like afuzzy model is clearly understandable.The fuzzy model proposed by Takagi and Sugeno [2] is described by fuzzy IF-THENrules which represents local input-output relations of a nonlinear system. The main featureof a Takagi-Sugeno fuzzy model is to express the local dynamics of each fuzzy implication(rule) by a linear system model. The overall fuzzy model of the system is achieved byfuzzy ”blending” of the linear system models. In this tutorial, the reader will find, bysome examples, that almost all nonlinear dynamical systems can be represented by TakagiSugeno fuzzy models to high degree of precision. In fact, it is proved that Takagi-Sugenofuzzy models are universal approximators of any smooth nonlinear system [4, 5].1
Chapter 2Takagi-Sugeno fuzzy modelingA fuzzy controller or model uses fuzzy rules, which are linguistic if-then statements involving fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representingexpert control/modeling knowledge and experience and in linking the input variables offuzzy controllers/models to output variable (or variables). Two major types of fuzzy rulesexist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy rules.Lets first start with the familiar Mamdani fuzzy systems. A simple but representativeMamdani fuzzy rule describing the movement of a car is:IF Speed is High AND Acceleration is Small THEN Braking is (should be) Modest,where Speed and Acceleration are input variables and Braking is an output variable.”High,” ”Small,” and ”Modest” are fuzzy sets, and the first two are called input fuzzysets while the last one is named the output fuzzy set.The variables as well as linguistic terms, such as ”High”, can be represented by mathematical symbols. Thus, a Mamdani fuzzy rule for a fuzzy controller involving three inputvariables and two output variables can be described as follows:IF x1 is M1 AND x2 is M2 AND x3 is M3 THEN u1 is M4 , u2 is M5 ,(2.1)where x1 , x2 , and x3 are input variables (e.g., error, its first derivative and its secondderivative), and u1 and u2 are output variables (e.g., valve openness). In theory, thesevariables can be either continuous or discrete; practically speaking, however, they shouldbe discrete because virtually all fuzzy controllers and models are implemented using digitalcomputers. M1 , M2 , M3 , M4 , and M5 are fuzzy sets, and AND are fuzzy logic ANDoperators. ”IF x1 is M1 AND x2 is M2 AND x3 is M3 ” is called the rule antecedent,whereas the remaining part is named the rule consequent.The structure of Mamdani fuzzy rules for fuzzy modeling is the same. The variablesinvolved, however, are different. An example of a Mamdani fuzzy rule for fuzzy modelingisIF y(n) is M1 AND y(n 1) is M2 AND y(n 2) is M3 AND u(n) is M4 AND u(n 1) is M5THEN y(n 1) is M6(2.2)where M1 , M2 , M3 , M4 , M5 , and M6 are fuzzy sets, y(n), y(n 1), and y(n 2) are theoutput of the system to be modeled at sampling time n, n 1 and n 2, respectively.And, u(n) and u(n 1) are system input at time n and n 1, respectively; y(n 1) issystem output at the next sampling time, n 1.Now, let us look at the so-called TS fuzzy rules. Unlike Mamdani fuzzy rules, TSrules use functions of input variables as the rule consequent. For fuzzy control, a TS rule2
corresponding to the Mamdani rule (2.1) isIF x1 is M1 AND x2 is M2 AND x3 is M3 THEN u1 f (x1 , x2 , x3 ), u2 g(x1 , x2 , x3 ),where f () and g() are two real functions of any type. Similarly, for fuzzy modeling, a TSrule analogous to the Mamdani rule (2.2) is in the following form:IF y(n) is M1 AND y(n 1) is M2 AND y(n 2) is M3 AND u(n) is M4 AND u(n 1) is M5THEN y(n 1) F (y(n), y(n 1), y(n 2), u(n), u(n 1)),where F () is an arbitrary function.Fuzzy inference systems also known as fuzzy rule-based systems or fuzzy models areschematically shown in Figure 2.1. They are composed of 5 conventional block: a rulebase containing a number of fuzzy if-then rules, a database which defines the membershipfunctions of the fuzzy sets used in the fuzzy rules, a decision-making unit which performs the inference operations on the rules, a fuzzification interface which transform thecrisp inputs into degrees of match with linguistic values, a defuzzification interface whichtransform the fuzzy results of the inference into a crisp output.Figure 2.1: Fuzzy inference system.Figure 2.2 utilizes a two-rule two-input fuzzy inference system to show different typesof fuzzy system mentioned above. Type 2 is the widely-used Mamdani type fuzzy systemwhich the output function is determined based on overall fuzzy output; some of them arecentroid of area, min of maxima, maximum of maxima, etc. Type 3 is the Takagi-Sugenotype fuzzy system.In this tutorial, we focus only on fuzzy models that use the T-S rule consequent.2.1Construction of Fuzzy ModelsFigure 2.3 illustrates the model-based fuzzy control design approach. To design a T-S fuzzycontroller, we need a T-S fuzzy model for a nonlinear system. Therefore the constructionof a fuzzy model represent an important and basic procedure in this approach. In generalthere are two approaches for constructing fuzzy models:1. Identification (fuzzy modeling) using input-output data and2. Derivation from given nonlinear system equations.There has been an extensive literature of fuzzy modeling using input-output data followingTakagi’s, Sugeno’s and Kang’s excellent work [6, 7]. The procedure mainly consist of twoparts: structure identification and parameter identification. The identification approachto fuzzy modeling is suitable for plants that are unable or too difficult to be represented3
Consequent partPremise partType 1M 12M 11Type 2M4Mwultiplication(o wrmin)M4x1x2M 22u1ux2x1M5u u2 ax1 bx2 rw2u2uWeightedaveragex2uM52x1u1 ax1 bx2 cw11M 21Type 3w1 u1 w2 u 2w1 w2uWeightedaveragemaxu w1 u1 w2 u 2w1 w2u ( centroid of area )Figure 2.2: Commonly used fuzzy if-then rules and fuzzy mechanism.analytical and/or physical models. On the other hand, nonlinear dynamical models formechanical systems can be readily obtained by, for example, the Lagrange method andthe Newton-Euler method. In such cases, the second approach, which derives a fuzzymodel from given nonlinear dynamical models, is more appropriate. This tutorial focuses on second approach. This approach utilizes the idea of ”sector nonlinearity”, ”localapproximation,” or a combination of them to construct fuzzy models.2.1.1Sector NonlinearityThe idea of using sector nonlinearity in fuzzy model construction first appeared in [8].Sector nonlinearity is based on the following idea. Consider a simple nonlinear system ẋ f (x(t)), where f (0) 0. The aim is to find the global sector such thatẋ f (x) [a1 a2 ]x(t). Figure 2.4a illustrates the sector nonlinearity approach. Thisapproach guarantees an exact fuzzy model construction. However, it is sometimes difficultto find global sector for general nonlinear systems. In this case, we consider local sector nonlinearity. This is reasonable as variables of physical systems are always bounded.Figure 2.4b shows the local sector nonlinearity, where two lines become the local sectors under d x(t) d. The fuzzy model exactly represents the local region, that is, d x(t) d. The following two examples illustrate the concrete steps to constructfuzzy models.4
co Pmarepn alesa ldtio isn rtbi(P uD etCd)Figure 2.3: Model-based fuzzy control design.a1 x(t )f ( x(t ))f ( x(t ))a2 x(t )x(t ) ddx(t )(a)(b)Figure 2.4: (a) Global sector nonlinearity. (b) Local sector nonlinearity.2.2Basic Fuzzy Mathematics for ModelingLets consider the nonlinear system below:½ẋ1 x2ẋ2 x21 x22 u(2.3)The goal is to derive a T-S fuzzy model from the above given nonlinear system equations bythe sector nonlinearity approach as if the response of the T-S fuzzy model in the specifieddomain exactly match with the response of the original system with the same input u.The following steps should be taken to derive the T-S fuzzy model of (2.3). Forsimplicity, we assume that x1 [0.5, 3.5] and x2 [ 1, 4]. Here x1 and x2 are nonlinearterms in the equations in the last equations so we make them as our fuzzy variables.Generally they are denoted as z1 , z2 and are known as premise variables that may befunctions of state variables, input variables, external disturbances and/or time. Therefore5
z1 x1 and z2 x2 . Equation (2.3) can be written as· 01ẋ(t) x(t),x1 x2where x(t) [x1 (t) x2 (t)]T . The first step for any kind of fuzzy modeling is to determinethe fuzzy variables and fuzzy sets or so-called membership functions. Although there isno general procedure for this step and it can be done by various methods predominantlytrial and error, in exact fuzzy modeling using sector nonlinearity, it is quite routine. It isassumed in this tutorial that the premise variables are just functions of the state variablesfor the sake of simplicity. This assumption is needed to avoid a complicated defuzzificationprocess of the fuzzy controllers [9].To acquire membership functions, we should calculate the minimum and maximumvalues of z1 (t) and z2 (t) which under x1 [0.5, 3.5] and x2 [ 1, 4], they are obviouslyobtained as follows:max z1 (t) 3.5,min z1 (t) 0.5,max z2 (t) 4,min z2 (t) 1.Therefore x1 and x2 can be represented by for membership functions M1 , M2 , N1 and N2as follows:z1 (t) x1 (t) M1 (z1 (t)) · 3.5 M2 (z1 (t)) · 0.5,z2 (t) x2 (t) N1 (z2 (t)) · 4 N2 (z2 (t)) · ( 1),and because M1 , M2 , N1 and N2 are actually fuzzy sets according to fuzzy mathematicsM1 (z1 (t)) M2 (z1 (t)) 1,N1 (z2 (t)) N2 (z2 (t)) 1.We name the membership functions ”Positive”, ”Negative,” ”Big,” and ”Small,” respectively. Figure 2.5 shows these membership functions. Here, we can generalize that the ith1M 2 ( z1 (t ))Negative00.5M 1 ( z1 (t ))1N 2 ( z2 (t ))N1 ( z2 (t ))SmallPositivez1 (t ) 03.5 1Bigz2 (t )2.54Figure 2.5: Membership functions M1 (z1 (t)), M2 (z2 (t)), N1 (z2 (t)) and N2 (z2 (t)).rule of the continuous T-S fuzzy models are of the following forms:Model Rule i:IF z1 (t) is Mi1 and . . . and zp (t) is Mip ,½ẋ Ai x(t) Bi u(t),THENy(t) Ci x(t),i 1,2,. . . ,r ;i 1,2,. . . ,r.(2.4)Here, Mij is the fuzzy set and r is the number of model rules; x(t) is the state vector, u(t)is the input vector, y(t) is the output vector, Ai is the square matrix with real elements andz1 (t), . . . , zp (t) are known premise variables as mentioned before. Each linear consequentequation represented by Ai x(t) Bi u(t) is called a subsystem.Therefore, the nonlinear system (2.3) is modeled by the following fuzzy rules (we don’t6
consider input u(t) in this stage):Model Rule 1: IF z1 (t) is ”Positive” and z2 (t) is ”Big,” THEN ẋ(t) A1 x(t).Model Rule 2: IF z1 (t) is ”Positive” and z2 (t) is ”Small,” THEN ẋ(t) A2 x(t).Model Rule 3: IF z1 (t) is ”Negative” and z2 (t) is ”Big,” THEN ẋ(t) A3 x(t).Model Rule 4: IF z1 (t) is ”Negative” and z2 (t) is ”Small,” THEN ẋ(t) A4 x(t).where the subsystems are determined as:"#01A1 max z1 max z2 ,z1 Positive"A3 0maxA2 z2 Bigz10maxz1 Positive#z1 Negativewhich is""1max z2 ,A4 z2 Bigz10maxz1 Negative· 0 1,3.5 4· 0 1A3 ,0.5 4·A1 A2 ·A4 03.500.5z1#1max z2 ,z2 Small#1max z2 ,z2 Small 1, 1 1, 1Now, ẋ can be derived out of defuzzification process as:ẋ(t) h1 (z(t))A1 x(t) h2 (z(t))A2 x(t) h3 (z(t))A3 x(t) h4 (z(t))A4 x(t)whereh1 (z(t)) M1 (z1 (t)) N1 (z2 (t)),h2 (z(t)) M1 (z1 (t)) N2 (z2 (t)),h3 (z(t)) M2 (z1 (t)) N1 (z2 (t)),h4 (z(t)) M2 (z1 (t)) N2 (z2 (t)).This T-S fuzzy model can exactly represents the nonlinear system in the region [0.5, 3.5] [ 1, 4] on the x1 x2 space. To have a clear picture of the fuzzy modeling procedureabove, we calculate the final output of ẋ for the specific given values of x1 and x2 .According to the model rules above½ẋ1 x2Model Rule 1: IF z1 (t) is ”Positive” and z2 (t) is ”Big,” THEN.ẋ2 3.5x1 4x2½ẋ1 x2Model Rule 2: IF z1 (t) is ”Positive” and z2 (t) is ”Small,” THEN.ẋ2 3.5x1 x2½ẋ1 x2Model Rule 3: IF z1 (t) is ”Negative” and z2 (t) is ”Big,” THEN.ẋ2 0.5x1 4x2½ẋ1 x2Model Rule 4: IF z1 (t) is ”Negative” and z2 (t) is ”Small,” THEN.ẋ2 0.5x1 x2Therefore, if z1 x1 2.75 and z2 x2 0.25, according to (2.6), the T-S fuzzymodeling implication can be derived as:7
M 2 ( z1 (t ))M 1 ( z1 (t ))N 2 ( z2 (t ))0.78120.75PositiveNegativeSmallN1 ( z2 (t ))Big0.252.218z1 (t )z1 2.75 3.50.5 1z2 0.25z2 (t )4Figure 2.6: Given the value of z1 2.75 and z2 0.25 to the membership functionsImplicationPremiseRule 1M1 (z1 ) 0.7812, N1 (z2 ) 0.25Rule 2M1 (z1 ) 0.7812, N2 (z2 ) 0.75Rule 3M2 (z1 ) 0.218, N1 (z2 ) 0.25Rule 4M2 (z1 ) 0.218, N2 (z2 ) 0.75Consequenceẋ1 0.25½ẋ2 3.5 2.75 4 0.25ẋ1 0.25½ ẋ2 3.5 2.75 0.25ẋ1 0.25½ẋ2 0.5 2.75 4 0.25ẋ1 0.25ẋ2 0.5 2.75 0.25Truth value½0.7812 0.25 0.250.7812 0.75 0.750.218 0.25 0.2180.218 0.75 0.218Now, the final values for ẋ1 and ẋ2 , in T-S fuzzy defuzzification process, can be calculatedas: 0.25 0.25 0.25 0.75 0.25 0.218 0.25 0.218 0.25 ẋ1 0.25 0.75 0.218 0.218 ẋ 10.625 0.25 9.375 0.75 2.375 0.218 1.125 0.218 7.277520.25 0.75 0.218 0.218Comparing, the values of ẋ1 0.25 and ẋ2 7.6225, we can see that the T-S fuzzyapproximation does the good job and small value difference of ẋ2 , actually comes fromrounding error of the premise fuzzy variables.Generally, given a pair of (x(t), u(t)) for the model rule (3.6), the final outputs ofthe fuzzy model for the Continuous Fuzzy System (for the general rule of Discrete FuzzySystems see the appendix) are inferred as follows:Prwi (z(t)){Ai x(t) Bi u(t)}Prẋ i 1i 1 wi (z(t)) rXhi (z(t)){Ai x(t) Bi u(t)},i 1y(t) (2.5)Prw (z(t))Ci x(t)i 1Pr ii 1 wi (z(t))rXhi (z(t))Ci x(t).i 1wherez(t) [z1 (t)z2 (t) . . . zp (t)],wi (z(t)) pYj 18Mij (zj (t)),(2.6)
and weighting functions wi should be normalized aswi (z(t))hi (z(t)) Pri 1 wi (z(t))(2.7)for all t. The term Mij (zj (t)) is the grade of membership of zj (t) in Mij . Since½ Pri 1 wi (z(t)) 0,wi (z(t)) 0,i 1,2,. . . ,r,we have(2.8)½ Pri 1 hi (z(t)) 1,hi (z(t)) 0,i 1,2,. . . ,r,(2.9)for all t.Example 1 Consider the following nonlinear system:µ¶¶ µẋ1 (t) x1 (t) x1 (t)x32 (t) ẋ2 (t) x2 (t) (3 x2 (t))x31 (t)(2.10)For simplicity, we assume that x1 [ 1, 1] and x2 [ 1, 1]. Of course, we can assumeany range for x1 (t) and x2 (t) to construct a fuzzy model. Equation (2.10) can be writtenas · 1x1 (t)x22 (t)x(t),ẋ(t) 1(3 x2 (t))x21 (t)where x(t) [x1 (t) x2 (t)]T and x1 (t)x22 (t) and (3 x2 (t))x21 (t) are nonlinear terms. Forthe nonlinear terms, define z1 (t) x1 (t)x22 (t) and z2 (t) (3 x2 (t))x21 (t). Then, we have· 1 z1 (t)ẋ(t) x(t).z2 (t) 1Next, we should calculate the minimum and maximum values of z1 (t) and z2 (t) underx1 (t) [ 1, 1] and x2 (t) [ 1, 1]. They are obtained as follows:maxx1 (t),x2 (t)z1 (t) 1,maxx1 (t),x2 (t)minx1 (t),x2 (t)z2 (t) 4,minz1 (t) 1,x1 (t),x2 (t)z2 (t) 0.From the maximum and minimum values, z1 (t) and z2 (t) can be represented byz1 (t) x1 (t)x22 (t) M1 (z1 (t)) · 1 M2 (z1 (t)) · ( 1),z2 (t) (3 x2 (t))x21 (t) N1 (z2 (t)) · 4 N2 (z2 (t)) · 0,whereM1 (z1 (t)) M2 (z1 (t)) 1,N1 (z2 (t)) N2 (z2 (t)) 1.Therefore the membership functions can be calculated asM1 (z1 (t)) z1 (t) 1,2M2 (z1 (t)) 1 z1 (1),2z2 (t)4 z2 (t),N2 (z2 (t)) ,44We name the membership functions ”Positive,” ”Negative,” ”Big,” and ”Small,” respectively. Then, the nonlinear system (2.10) is represented by the following fuzzy model.N1 (z2 (t)) 9
Model Rule 1: IF z1 (t) is ”Positive” and z2 (t) is ”Big,” THEN ẋ(t) A1 x(t).Model Rule 2: IF z1 (t) is ”Positive” and z2 (t) is ”Small,” THEN ẋ(t) A2 x(t).Model Rule 3: IF z1 (t) is ”Negative” and z2 (t) is ”Big,” THEN ẋ(t) A3 x(t).Model Rule 4: IF z1 (t) is ”Negative” and z2 (t) is ”Small,” THEN ẋ(t) A4 x(t).Here,· 14· 1A3 4A1 1, 1 1, 1·A2 ·A4 10 10 1, 1 1, 1Figures 2.7a and 2.7b illustrates the above membership functions. The defuzzification is1M 2 ( z1 (t ))M 1 ( z1 (t ))PositiveNegative0N1 ( z2 (t ))Smallz1 (t )0 1N 2 ( z2 (t ))11Bigz2 (t )0(a)024(b)Figure 2.7: Membership functions M1 (z1 (t)), M2 (z1 (t)), N1 (z2 (t)) and N2 (z2 (t)).carried out asẋ(t) 4Xhi (z(t))Ai x(t),i 1whereh1 (z(t)) M1 (z1 (t)) N1 (z2 (t)),h2 (z(t)) M1 (z1 (t)) N2 (z2 (t)),h3 (z(t)) M2 (z1 (t)) N1 (z2 (t)),h4 (z(t)) M2 (z1 (t)) N2 (z2 (t)).This fuzzy model exactly represents the nonlinear system in the region [ 1, 1] [ 1, 1]on the x1 x2 space.Figure 2.8 shows the implementation of the above fuzzy model in Matlab/Simulink. Asit is evident in Figure 2.9a, the time responses of the fuzzy model can exactly follow theresponses of the original differential eq
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
dynamics of each fuzzy inference (rule) by a linear system model is the main characteristics of the T-S fuzzy model. Particularly, description of the Takagi-Sugeno fuzzy systems is done by fuzzy IF-THEN rules, which linear input-output relations of a system is locally represented by. The fuzzy system is of the following form [29, 30]: Rule i IF q 1
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
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