Fuzzy Logic Basic Concepts and Applications1 1 Introduction 1.1 Why fuzzy control systems ? Human experience often cannot be described in terms of numeric values. This makes it difficuul to introduce such experience into computer based control systems as these rely on mathematical algorithms and logic relations using "sharp" numeric or boolean values. Converrsel there is an urgent need to incorporate human decision making competence into automaati process control systems. Many real world processes are modeled only poorly by mathematical methods but may very well be controlled by human "common sense". Fuzzy control uses this intuitive view on real world phenomena and aims at designing automatic decision making machines by imitating "intuitive" human methods. Fuzzy control should be used when more formal methods either are not available or are too complicated for practical applications. Some well known applications: Camcorders with sharpness correction Elevator control with optimized timing Heating control considering number of persons in a room Intelligent traffic control systems Money investment decision making systems Pattern and speech recognition systems Fuzzy control systems may be designed using special hardware devices or computers and software. In both cases the technical requirements are pretty high and large scale applications have been realized only since some years. Today special fuzzy processors are available in quantity from stock and new applications are announced on short terms. Many fuzzy systems use inexpensive personal computers as their hardware platforms. 1.2 History First paper: Lofti A. Zadeh: Fuzzy Sets. Information and Control, 8(1965) University of Berkeley, Cal. An outlook: Ebrahim Mamdami and Seto Assilian, Queen Mary College, London, 1975: Control of steam engines In this early phase practical applications did hardly seem possible A technical breakthrough: Automatic control of cement production furnace by fuzzy methods. Despite many efforts process control by conventional methods before had turned out to be impossible. No adequate process model could be designed. Copenhagen (Denmark), F. L. Smidth company.2 Another spectacular application: Metropolitan subway control in Sendai, Japan An important German investigation: Reusch: Potential der Fuzzy-Technologie in Nordrhein-Westfalen. (Potential of fuzzy technology in the country of Nordrhein-Westfalen.) Studie der Fuzzy-Initiative NRW, Ministerium für Wirtschaft, Mittelstand und Technologie, 4( 1993) (Study of the Fuzzy Initiative of NRW, Ministry of Economics and Technology, 4, 1993)3 2 Entities, Values, Member Function 2.1 Abstract entities, base values, fuzzy values Abstract entity: A process variable which is significant in the context of a fuzzy control system is called an "abstract entity". Examples: amount_of_money significant in financial decision processes temperature, air_pressure significant in technical processes time_of_day, date significant in real time processes average_of_marks significant in education processes number_of_population significant in social processes Abstract entities become concrete by assigning them values in the widest sense Examples: The amount_of_money is ... big small 1000 dollars growing not sufficient The date is ... today Wednesday morning beginning of next week January 3rd The air_pressure increases remains as it was before is extremely low is 1023 hPa Base values To each abstract entity in a fuzzy control system a basic set of measuring values is assigned. Which set of base values is chosen in a special application cannot be decided upon in general but depends on usefulness. Following conditions should be met: Uniqueness In any possible situation an abstract entity accepts one and only one of its base values Measurability In any possible situation one can determine by measurement the actual momentary base value of an abstract entity.4 Examples: Abstract entity: date Possible set of base values: B1={ Mon, Tue, Wed, Thur, Fri, Sat, Sun } Another set of base values: B2={1.1.96, ... ,31.12.1999 } Yet another set: B3={Jan, Feb, ... ,Dec } All three sets are numerable Abstract entity: temperature Possible set of base values: B1={ t °K; 0.0 ≤ t } Also possible: B2={ t °C; -273.16 ≤ t } Both sets are continuous Abstract entity: amount_of_money Possible set of base values: B1={ x$; x=positive integer number } Also possible: B2={ x.yy$; x=positive integer number, y=00...99 } Both sets are numerable Abstract entity: vector Possible set of base values: B1={ (x,y,z); x,y,z are real numbers } Also possible: B2={r ,cos(α),cos(β); r=positive real, 0≤ α, β <2π } Both sets are continuous triples of numbers Fuzzy values In colloquial speech abstract entities are often quantified in a different, less formal way. Insttea of using "sharp" terms from an agreed set of base values "linguistic" terms are emplooyed In contrast to its base values an abstract entity may have several linguistic values simultanneousl and it is not always certain which of its possible linguistic values it possesses at the moment. Linguistic values are "fuzzy" in some sense. Similar to the base values of an abstrrac entity also its agreed fuzzy values form a set. Examples: Abstract entity: date Some linguistic values: beginning_of_week, later, earlier, end_of_this_year, at_night Abstract entity: temperature Some linguistic values: hot, warm, cold, much_to_cold, rather_hot Abstract entity: amount_of_money Some linguistic values: sufficient, small, very_big, poor, insufficient Abstract entity: vector Some linguistic values: small_by_amount, perpendicular, big_z_component, long Fuzzy systems rely on the description of relations between abstract entities by means of linguistic "fuzzy" values. In contrast, conventional process control systems use sets of base values ("sharp" valuees to describe processes and their time behavior.5 Just as the base values of an abstract entity also its fuzzy values have to be agreed upon, depenndin on the considered application and on usefulness in a certain environment. In contrast to base values fuzzy values must not necessarily be mutually exclusive. Most important: Fuzzy values should be words of everyday language or at least be closely related to such words. So one can describe human experience and human knowledge about a process using fuzzy values and fuzzy relations between the abstract entities which are characteristic for the process under consideration. Examples: Abstract entity: date Possible set of fuzzy values: F1 = { beginning_of_week, mid_of_week, end_of_week } Another set of fuzzy values: F2 = {spring, summer, autumn, winter } Yet another set: F3 = {at_easter, at_whitsun, at_christmas } Abstract entity: temperature Possible set of fuzzy values: F1 = { cold, lukewarm, warm, hot, boiling } Also possible: F2 = { low, high } Also possible: F3 = { below_zero, about_zero, above_zero } Abstract entity: amount_of_money Possible set of fuzzy values: F1 = { small, medium, sufficient, big } Also possible: F2 = { nearly nothing, about_100, about_200, more_than_300 } Abstract entity: vector Possible set of fuzzy values: F1 = { nearly_in_x_direction, nearly_in_y_direction, nearly_in_z_direction, nearly_perpendicular, rather_long, rather_short } 2.2 Member function 2.2.1 Definition and meaning In any possible situation an abstract entity has one and only one sharp value from its agreed set of base values. In that same situation it has one ore more fuzzy values from its agreed set of fuzzy values. Example: Abstract entity: date Set of base values: B = { Mon, Tue, Wed, Thur, Fri, Sat, Sun } Set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week } If entity "date" possesses the sharp value "Wed" (respective Wednesday) this certainly correspoond to the fuzzy value mid_of_week but hardly to beginning_of_week or to end_of_week.6 One can construct a lookup table: base value does not mean does perhaps mean does probably mean does certainly mean Mon mid_of_week, end_of_week beginning_of_week Tue end_of_week mid_of_week beginning_of_week Wed beginning_of_week, end_of_week mid_of_week Thur beginning_of_week, end_of_week mid_of_week Fri beginning_of_week, mid_of_week end_of_week end_of_week Sat beginning_of_week, mid_of_week end_of_week end_of_week Sun beginning_of_week, mid_of_week end_of_week Table 1 Above table describes the degree of membership of a sharp base value to one or more of the fuzzy values of the abstract entity "date". The base value "Thur" does "not" belong to the fuzzy value "beginning_of_week" , does "perhaps" and also does "probably" belong to "end_of_week" and does not "certainly" belong to any fuzzy value at all. This intuitive relation between base and fuzzy values of an abstract entity is expressed more accurate and quantitatively by the member function, mostly marked with Greek letter µ. µ(b,f) is a real value function of the two variables b of type "base value" and f of type "fuzzy value". Per definition: 0 ≤ µ(b,f) ≤ 1 for any possible choice of b and f If we have µ(b0,f0)=0 for a certain pair of values (b0,f0) this means that the sharp value b0 does not at all belong to the fuzzy value f0, it is not a "member" of f0. If, conversely, we have µ(b0,f0)=1 the sharp value b0 belongs completely to the fuzzy value f0. Partial or uncertain memberships are expressed by function values between 0 and 1. Example: Lookup table of µ(b,f) for the above (qualitative) dependency fuzzy value base value beginning_of_week mid_of_week end_of_week sum Mon 1.0 0.0 0.0 1.0 Tue 0.6 0.2 0.0 0.8 Wed 0.0 1.0 0.0 1.0 Thur 0.0 0.6 0.3 0.9 Fri 0.0 0.1 0.7 0.8 Sat 0.0 0.0 0.9 0.9 Sun 0.0 0.0 1.0 1.0 Table 27 The sum of the numbers in the horizontal lines is often but not necessary close to 1,0. More formally: ∑ µ(b,f) ≈ 1 for any fuzzy value f A sum of exactly 1 really means that the fuzzy values uniformly cover the whole range of the base values. In practice the member function is often constructed such that this condition is met. For a fixed fuzzy value f=f0 the member function µ(b,f0) is a real valued function of the singgl variable b alone, called partial member function. 2.2.2 Common representations 2.2.2.1 Lookup tables This kind of representation is used mainly if the sets of base values and fuzzy values both contain only some few discrete elements. Example: Base values: B = {1, 2, 3, 4, 5, 6 } Fuzzy values: F = {unacceptable, acceptable, good, best } fuzzy value base value unacceptable acceptable good best 1 0.0 0.0 0.3 1.0 2 0.0 0.1 0.5 0.2 3 0.1 0.4 0.2 0.0 4 0.5 0.3 0.0 0.0 5 0.8 0.2 0.0 0.0 6 1.0 0.0 0.0 0.0 Table 3 We read from the table: µ(3, acceptable) = 0,4 µ(2, optimum) = 0,2 2.2.2.2 Mathematical formula representation This kind of representation is used mainly if the set of base values is continuous. Example: Base values: B = { x; x=real number, -20 ≤ x ≤+35 } Fuzzy values: F = { spring, summer, autumn, winter } µ(x, spring) = 0,0 if x<0 = 0,05*x if 0≤x≤20 = 1-0,1*(x-20) if 20≤x≤30 = 0,0 else µ(x, summer) = 0,0 if x<10 = 0,067*(x-10) if 10≤x≤25 = 1,0 if 25≤x Similarly for µ(x, autumn) and µ(x, winter)8 2.2.2.3 Computer program representation General purpose representation for many different applications. Example: Base values: B = { 1, 2, 3, 4, 5, 6 } Fuzzy values: F = { unacceptable, acceptable, good, optimum } //List of fuzzy values enum fuzzy{ unacceptable, acceptable, good, optimum }; //definition of member function //b: base value; f: fuzzy value double mue(short b, fuzzy f) { switch(b) { case 1: switch(f) { case unacceptable: return 0.0; case acceptable: return 0.0; case good: return 0.3; case optimum: return 1.0; } case 2: switch(f) { case unacceptable: return 0.0; case acceptable: return 0.1; case good: return 0.5; case optimum: return 0.3; } case 3: switch(f) { case unacceptable: return 0.1; case acceptable: return 0.4; case good: return 0.2; case optimum: return 0.0; } case 4: switch(f) { case unacceptable: return 0.5; case acceptable: return 0.3; case good: return 0.0; case optimum: return 0.0; } //continued on next page9 //continuation case 5: switch(f) { case unacceptable: return 0.8; case acceptable: return 0.2; case good: return 0.0; case optimum: return 0.0; } case 6: switch(f) { case unacceptable: return 1.0; case acceptable: return 0.0; case good: return 0.0; case optimum: return 0.0; } default: return 0.0; } } //Calling program void main(void) { short base_value = 3; fuzzy fuzzy_value = good; double z; z=mue(base_value, fuzzy_value); }10 2.2.2.4 Graphic representation This kind of representation is mostly used if the set of base values is continuous. Advantage: highly descriptive, easy to understand. Disadvantage: hardly usable in automatic processing applications Example: B = { v; 0 ≤ v ≤ 200 km/h} F = { stopping, slow, normal, fast, racing } 0 40 80 120 160 200 0,0 0,2 0,4 0,6 0,8 1,0 m(b,f) Speed in km/h Member function m(b,f) stopping slow normal fast racing Picture 111 2.3 Fuzzy numbers Often abstract entities are on one hand described by sharp numbers as their base values, on the other hand by fuzzy values like "about_1", "about_2.5", "roughly_6.0". The latter terms are called "fuzzy" because they are in some way "unsharp". The corresponding member functions are most often chosen as Gaussian distributions (especially where measuring problems are encountered) or as triangular functions. Example: Abstract entity: temperature/°K Set of base values : B = { t; t=positive real number } Set of fuzzy values: F = { about_T; T=positive real number } Both sets are innumerable, set B is also continuous, for set F continuity is not defined. Member function (Gauss): µ(t, about_T) = exp[ -a*(t-T)² ] a > 0, constant Member function (Triangle): µ(t, about_T) = 0 if t ≤ T-a a > 0, constant = (t-T+a)/a if T-a ≤ t ≤ T = (t-T+b)/a if T ≤ t ≤ T+b b > 0, constant = 0 if t ≥ T+b Here you see how both member functions look like for a=b=1 Member function m(b,about_0) 0 0,2 0,4 0,6 0,81 -1,8 -1,4 -1 -0,6 -0,2 0 0,6 1 1,4 1,8 Base values b m(b,f) Gauss triangle Picture 212 2.4 Fuzzy intervals /Fuzzy flat numbers Consider an abstract entity with base values which are ordinary real numbers and fuzzy values similar to the following: roughly_between_1_and_5 approximately_in_the_range_from_100_to_120 probably_between_ 5_and_6 Fuzzy values of that kind are called "fuzzy flat numbers" or "fuzzy intervals". The member function is usually chosen to be of trapezoidal shape. Example: Abstract entity: temperature Set of base values: B = { t; t=positive real number } Set of fuzzy values: F = { seemingly_below_(-10), approximately_in_(-10 to 0), approximately_in_(0 to +10), approximately_in_(+10 to +20), seemingly_above_(+20) } Trapezoidal member function: µ(t, seemingly_below_(-10) ) = 1 if t ≤ -15 = 1-(t+15)/10 if -15 ≤ t≤ -5 = 0 if -5 ≤ t ( trapezoid, left side open ) µ(t, approximately_in(-10 to 0) ) = 1 if t ≤ -12 = (t+12)/4 if -12 ≤ t ≤ -8 = 1 if -8 ≤ t ≤ -2 = 1-(t+2)/5 if -2 ≤ t ≤ +3 = 0 if +3 ≤ t ( trapezoid with different inclinations on both sides ) Similar: µ(t, approximately_in(0 to +10) ) = .... µ(t, approximately_in(+10 to +20) ) = .... µ(t, approximately_in(0 to +10) ) = .... µ(t, seemingly_above_(+20) ) = .... 2.5 Fuzzy sets Consider an abstract entity with agreed sets B={b} of base values and F={f} of fuzzy values. To each fuzzy value f from F a new set M(f) may be attached which is defined as follows: M(f) = { [b, µ(b,f)]; b ∈ B }13 Colloquially: M(f) consists of all base values b ∈ B of the abstract entity, each value b being "paired" with its degree of membership to the fuzzy value f under consideration. M(f) is called "fuzzy set of fuzzy value f over base value set B " or simply "fuzzy set of f over B" or even more simple "fuzzy set of f". Example: Lookup table of µ(b,f) fuzzy value base value beginning_of_week mid_of_week end_of_week sum Mon 1.0 0.0 0.0 1.0 Tue 0.6 0.2 0.0 0.8 Wed 0.0 1.0 0.0 1.0 Thur 0.0 0.6 0.3 0.9 Fri 0.0 0.1 0.7 0.8 Sat 0.0 0.0 0.9 0.9 Sun 0.0 0.0 1.0 1.0 Table 4 From that the following three fuzzy sets are derived: M(beginning_of_week) = {(Mon,1.0), (Tue, 0.6), (Wed,0.0), (Thur,0.0), (Fri, 0.0), (Sat,0.0), (Sun,0.0) } M(mid_of_week) = {(Mon,0.0), (Tue, 0.2), (Wed,1.0), (Thur,0.6), (Fri, 0.1), (Sat,0.0), (Sun,0.0) } M(end_of_week) = {(Mon,0.0), (Tue, 0.0), (Wed,0.0), (Thur,0.3), (Fri, 0.7), (Sat,0.9), (Sun,1.0) } In principle each fuzzy set contains all agreed base values b with their respective degrees of membership to the fuzzy value f0 under consideration. But rather often base values with µ(b,f0)=0 are omitted. Usual reasoning: "they do not belong to the considered fuzzy value" or, alternatively, "they are not contained in the fuzzy set". 2.6 Generalized fuzzy sets All fuzzy sets are of the general form M = { (b, m); b in some set of base values, 0.0≤m≤1.0 } A certain pair (b, m) is contained only once in set M, that means the elements of M are differeen by pairs. Besides no two pairs (b1,m1) and (b2,m2) do exist with b1=b2 but m1≠m2. Each element b therefor has a unique attached membership m. The following definitions are useful: Unity set: E = { (b,1); b in set of base values } Zero set: Z = { (b,0); b in set of base values }14 Subset: Set M1 = { (b,m1); b in set of base values, 0≤m1≤1 } is a subset of set M2 = { (b,m2); b in set of base values, 0≤m2≤1 } if for each value of b holds m1≤ m2 True subset of a set: Set M1 = { (b,m1); b in set of base values, 0≤m1≤1 } is a true subset of set M2 = { (b,m2); b in set of base values, 0≤m2≤1 } if for each value of b holds m1 ≤ m2 and for at least one value of b really m1 < m2 Equality of sets: Sets M1 = { (b,m1); b in set of base values, 0≤m1≤1 } and M2 = { (b,m2); b in set of base values, 0≤m2≤1 } are equal if for each value of b holds m1 = m2 Each fuzzy set over base value set B is contained in the unity set over B, that means is a subsse of the unity set over B. Each fuzzy set over base value set B contains the zero set over B, that means the zero set is a subset B.15 3 Operations with fuzzy sets 3.1 Complement set /NOT Conventional set theory definition Given a set M of elements from a base value set B the complementary set ⌐M is defined to be the set of those elements from B which do not belong to set M: ⌐M={ b; b∈B, b∉M } Fuzzy set theory definition Given a fuzzy set M = { [b,m]; b ∈ B, 0 ≤ m ≤ 1 } over a base value set B the complementary fuzzy set ⌐M is defined as ⌐M = { [b,(1-m)] } Example: M = { (Mon, 0.2), (Tue, 0.5), (Wed, 0.4), (Thur, 0.0), (Fri, 1.0), (Sat, 0.3), (Sun, 1.0) } ⌐M = { (Mon, 0.8), (Tue, 0.5), (Wed, 0.6), (Thur, 1.0), (Fri, 0.0), (Sat, 0.7), (Sun, 0.0) } A certain value b0 which is completely contained in M is not at all contained in ⌐M m(b0) = 1 → 1-m(b0) = 0 A certain value b0 which is partially contained in M is also partially contained in ⌐M 0 < m(b0) < 1 → 0 < 1-m(b0) < 1 A certain value b0 which is not at all contained in M is completely contained in ⌐M m(b0) = 0 → 1-m(b0) = 1 What is that good for ? The prefix NOT Starting from a given set F = { f } of fuzzy values of an abstract entity A it often proves useffu to add new colloquial terms to F, that is to add new fuzzy values. We begin with negation. Each fuzzy value f from F is prefixed by the term NOT_ so that we get the new terms NOT_f. These are considered to be new additional fuzzy values of abstract entity A and are adjoined to the original set F. Accordingly the membership function of abstract entity A is redefined: µ(b,NOT_f) = 1-µ(b,f) for all new fuzzy values NOT_f of A The respective fuzzy sets are M(NOT_f) = { [b,µ(b,NOT_f)] } = { [ b, ( 1-µ(b,f) ) ] } = ⌐M(f) In words: The fuzzy set belonging to the fuzzy value NOT_f is the complement of the fuzzy set belonngin to f. Example: Abstract entity: date Set of base values: B = { Mon, Tue, Wed, Thur, Fri, Sat, Sun }16 Original set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week } Enlarged set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week, NOT_beginning_of_week, NOT_mid_of_week, NOT_end_of_week } Original fuzzy sets: M(beginning_of_week) = {(Mon, 1.0), (Tue, 0.6), (Wed, 0.0), (Thur, 0.0), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } M(mid_of_week) = {(Mon, 0.0), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M(end_of_week) = {(Mon, 0.0), (Tue, 0.0), (Wed, 0.0), (Thur, 0.3), (Fri, 0.7), (Sat, 0.9), (Sun, 1.0) } Additional fuzzy sets: M(NOT_beginning_of_week) = { (Mon, 0.0), (Tue, 0.4), (Wed, 1.0), (Thur, 1.0), (Fri, 1.0), (Sat, 1.0), (Sun, 1.0) } M(NOT_mid_of_week) = { (Mon, 1.0), (Tue, 0.8), (Wed, 0.0), (Thur, 0.4), (Fri, 0.9), (Sat, 1.0), (Sun, 1.0) } M(NOT_end_of_week) = { (Mon, 1.0), (Tue, 1.0), (Wed, 1.0), (Thur, 0.7), (Fri, 0.3), (Sat,0.1), (Sun,0.0) } Instead of prefix NOT_ other prefixes with similar meanings may be used if required. 3.2 Intersection /AND Conventional set theory definition Given two sets M1 and M2 of elements from a base set B the intersection set M= M1∩ M2 is defined to be the set of exactly those elements from B which belong to both sets M1 and M2 simultaneously: M = M1∩ M2 = { b; b both in M1 and M2} Fuzzy set theory definition Given two fuzzy sets M1 = { [b,m1]; b ∈ B, 0 ≤ m1 ≤ 1 } and M2 = { [b,m2]; b ∈ B , 0 ≤m2 ≤1 } over a base value set B the intersection set is defined as M = M1∩ M2 = { [b,min(m1,m2)] } Example: M1 = { (Mon,1.0), (Tue, 0.6), (Wed, 0.4), (Thur, 0.2), (Fri, 0.0), (Sat, 0.8), (Sun, 0.0) } M2 = { (Mon, 0.7), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M1∩ M2 = { (Mon, 0.7), (Tue, 0.2), (Wed, 0.4), (Thur, 0.2), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } A certain value b0 which is completely contained in both M1 and M2 is also completely contaiine in M1∩ M2: m1(b0) =1 and m2(b0)=1 → min(m1,m2) = 117 A certain value b0 which is only partially contained in M1 or in M2 is also only partially contaiine in M1∩ M2: 0 < m1(b0) < 1 or 0 < m2(b0) < 1 → 0 < min(m1,m2) < 1 A certain value b0 which is not at all contained in M1 or in M2 is also not at all contained in M1∩ M2: m1(b0)=0 or m2(b0)=0 → min(m1,m2) = 0 What is that good for ? The infix AND As before we start with a given set F = { f } of fuzzy values of an abstract entity A and add new colloquial terms to it, this time using the connecting term "and". To do so any two fuzzy values f1 and f2 from F are connected by the infix _AND_ . By that we get the new terms f1_AND_f2 which are considered to be new additional fuzzy values of abstract entity A. They are adjoined to the original set F of fuzzy values. The membership function of abstract entity A is redefined: µ(b,f1_AND_f2) = min[µ(b,f1), µ(b,f2) ] for all new fuzzy values f1_AND_f2 of A Respective fuzzy sets are M(f1_AND_f2) = { [b, µ(b,f1_AND_f2)] } = { ( b,min[µ(b,f1), µ(b,f2) ]) } = M(f1) ∩ M(f1) In words: The fuzzy set belonging to the fuzzy value f1_AND_f2 is the fuzzy intersection set of the fuzzy sets belonging to f1 and to f2. Example: Abstract entity: date Set of base values: B = { Mon, Tue, Wed, Thur, Fri, Sat, Sun } Original set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week } Enlarged set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week, beginning_of_week_AND_mid_of_week, beginning_of_week_AND_end_of_week, mid_of_week_AND_end_of_week } As infix _AND_ is symmetric, that means as µ(b,f1_AND_f2)=µ(b,f2_AND_f1), other possible fuzzy values like "mid_of_week_AND_beginning_of_week" need not be considered. "mid_of_week_AND_beginning_of_week" is similar to "begining_of_week_AND_mid_of_week".18 Original fuzzy sets: M(beginning_of_week) = {(Mon,1.0), (Tue, 0.6), (Wed, 0.0), (Thur, 0.0), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } M(mid_of_week) = {(Mon, 0.0), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M(end_of_week) = {(Mon, 0.0), (Tue, 0.0), (Wed, 0.0), (Thur, 0.3), (Fri, 0.7), (Sat, 0.9), (Sun,1.0) } Additional fuzzy sets: M(beginning_of_week_AND_mid_of_week) = M(beginning_of_week) ∩ M(mid_of_week) = { (Mon, 0.0), (Tue, 0.2), (Wed, 0.0), (Thur, 0.0), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } M(beginning_of_week_AND_end_of_week) = M(beginning_of_week) ∩ M(end_of_week) = { (Mon, 0.0), (Tue, 0.0), (Wed, 0.0), (Thur, 0.0), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } M(mid_of_week_AND_end_of_week) = M(mid_of_week) ∩ M(end_of_week) = { (Mon, 0.0), (Tue, 0.0), (Wed, 0.0), (Thur, 0.3), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } Instead of infix _AND_ other infixes with similar meaning may be used if required. 3.3 Junction /OR Conventional set theory definition Given two sets M1 and M2 of elements from a base set B the junction set M= M1∪ M2 is defiine to be the set of exactly those elements from B which either belong to set M1 or to set M2 or to both sets simultaneously: M = M1∪ M2 = { b; b ∈ M1 or b ∈ M2 } Fuzzy set theory definition Given two fuzzy sets M1={ [b,m1]; b in B, 0 ≤ m1 ≤ 1 } and M2 = { [b,m2]; b in B, 0 ≤ m2 ≤1 } over a base value set B the junction set M= M1∪ M2 is defined as M = { [ b , max(m1,m2) ] } Example: M1 = { (Mon, 1.0), (Tue, 0.6), (Wed, 0.4), (Thur, 0.2), (Fri, 0.0), (Sat, 0.8), (Sun, 0.0) } M2 = { (Mon, 0.7), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M1∪ M2 = { (Mon, 1.0), (Tue, 0.6), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.8), (Sun, 0.0) } A certain value b0 which is completely contained in at least one of the sets M1 or M2 is also completely contained in M1∪ M2: m1(b0) = 1 or m2(b0) = 1 → max(m1,m2) = 119 A certain value b0 which is at least partially contained in M1 or in M2 is always at least partiaall contained in M1∪ M2: 0 < m1(b0) or 0 < m2(b0) → 0 < max(m1,m2) A certain value b0 which is neither contained in M1 nor in M2 also not at all is contained in M1 ∪ M2: m1(b0) = 0 and m2(b0) = 0 → max(m1,m2) = 0 What is that good for ? The infix OR This time we start with a given set F = { f } of fuzzy values of an abstract entity A and add new colloquial terms to it using the connecting term "or". Any two fuzzy values f1 and f2 from F are connected by the infix _OR_ which results in the new terms f1_OR_f2 . They are adjoined as new fuzzy values to the original set F. The membership function of abstract entity A is redefined: µ(b,f1_OR_f2) = max [ µ(b,f1), µ(b,f2) ] for all new fuzzy values f1_OR_f2 of A The respective fuzzy sets are M(f1_OR_f2) = { [b, µ(b,f1_OR_f2)] } = { ( b,max[µ(b,f1), µ(b,f2) ]) } = M(f1) ∪ M(f1) In words: The fuzzy set belonging to the fuzzy value f1_OR_f2 is the fuzzy junction set of the fuzzy sets belonging to f1 and to f2. Example: Abstract entity: date Set of base values: B = { Mon, Tue, Wed, Thur, Fri, Sat, Sun } Original set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week } Enlarged set of fuzzy values: F = { beginning_of_week, mid_of_week, end_of_week, beginning_of_week_OR_mid_of_week, beginning_of_week_OR_end_of_week, mid_of_week_OR_end_of_week } As infix _OR_ is symmetric, that means as µ(b,f1_OR_f2)=µ(b,f2_OR_f1), other possible fuzzy values like "mid_of_week_OR_beginning_of_week" need not be considered. "mid_of_week_OR_beginning_of_week" is similar to "beginning_of_week_OR_mid_of_week".20 Original fuzzy sets: M(beginning_of_week) = {(Mon, 1.0), (Tue, 0.6), (Wed, 0.0), (Thur, 0.0), (Fri, 0.0), (Sat, 0.0), (Sun, 0.0) } M(mid_of_week) = {(Mon, 0.0), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M(end_of_week) = {(Mon, 0.0), (Tue, 0.0), (Wed, 0.0), (Thur, 0.3), (Fri, 0.7), (Sat,0.9), (Sun,1.0) } Additional fuzzy sets: M(beginning_of_week_OR_mid_of_week) = M(beginning_of_week) ∪ M(mid_of_week) = { (Mon,1.0), (Tue, 0.6), (Wed, 1.0), (Thur, 0.6), (Fri, 0.1), (Sat, 0.0), (Sun, 0.0) } M(beginning_of_week_OR_end_of_week) = M(beginning_of_week) ∪ M(end_of_week) = { (Mon, 1.0), (Tue, 0.6), (Wed, 0.0), (Thur, 0.3), (Fri, 0.7), (Sat, 0.9), (Sun, 1.0) } M(mid_of_week_OR_end_of_week) = M(mid_of_week) ∪ M(end_of_week) = { (Mon, 0.0), (Tue, 0.2), (Wed, 1.0), (Thur, 0.6), (Fri, 0.7), (Sat, 0.9), (Sun,1.0) } Instead of infix _OR_ other infixes with similar meaning may be used if required. 3.4 More general definitions Above we defined the operations of intersection and junction of two fuzzy sets as follows: Intersection: M1∩ M2 = { [b,min(m1,m2)] } Junction: M1∪ M2 = { [b,max(m1,m2)] } Similarly we defined "derived" fuzzy values from original ones by the infixes _AND_: µ(b,f1_AND_f2) = min[µ(b,f1), µ(b,f2) ] _OR_: µ(b,f1_OR_f2) = max[µ(b,f1), µ(b,f2) ] In fact, these definitions are only special cases of more general ones using so-called "T-Norm functions" and "S-Norm functions". These expressions mean real-valued functions with speciia properties which will be discussed later in detail. Here are the two most important properties of T-Norm functions T(x,y) and S-Norm functions S(x,y): T(x,y) and S(x,y) are functions of two real variables x and y defined in the argument ranges 0≤x≤1 and 0≤y≤1, respectively21 The function values of T(x,y) and S(x,y) also are in the ranges 0≤T(x,y)≤1 and 0≤S(x,y)≤1, respectively Using any T-Norm T(x,y) and S-Norm S(x,y) we can now generalize definitions of fuzzy set operations: Intersection: M1∩ M2 = { [b, T(m1,m2)] } Junction: M1∪ M2 = { [b, S(m1,m2)] } _AND_: µ(b,f1_AND_f2) = T( µ(b,f1), µ(b,f2) ) _OR_: µ(b,f1_OR_f2) = S( µ(b,f1), µ(b,f2) ) As will be seen later for 0≤x≤1 and 0≤y≤1 min(x,y) is a T-Norm function and max(x,y) an SNoor function Following two tables list some well-known T-and S-Norms. Function definition: T(x,y) = Function name 0 if max(x,y) < 1 min(x,y) if max(x,y) = 1 Drastic product max(0,x+y-1) Limited difference x*y/[1+(1-x)*(1-y)] Einstein product x*y Algebraic product x*y/[1-(1-x)*1-y)] Hamacher product min(x,y) Minimum Table 5 Some T-Norm functions Function definition: S(x,y) = Function name max(x,y) Maximum 1-(1-x)*(1-y)/(1-x*y) Hamacher sum 1-(1-x)*(1-y) Algebraic sum 1-(1-x)*(1-y)/(1+x*y) Einstein sum min(1,x+y) Limited sum 1 if min(x,y) > 0 max(x,y) if min(x,y) = 0 Drastic sum Table 6 Some S-Norm functions 3.5 Common properties of T-Norm functions and S-Norm functions The above T-Norm and S-Norm functions have common properties which are listed here for reference. Proofs have been omitted but are given in the appendix. Range of definition: 0 ≤ x ≤ 1 0 ≤ y ≤ 1 Range of values: 0 ≤ T(x,y) ≤ 1 0 ≤ S(x,y) ≤ 1 Commutivity T(x,y) = T(y,x) S(x,y) = S(x,y)22 Associativity T[ T(x,y),z ] = T[ x,T(y,z) ] S[ S(x,y),z ] = S[ x,S(y,z) ] Monotony From x ≤ u and y ≤ v always follows T(x,y) ≤ T(u,v) S(x,y) ≤ S(u,v) Special values T(0,y) = 0 S(0,y) = y T(1,y) = y S(1,y) = 1 The above equations serve as definitions: Each function with properties in the left column is called a T-Norm function or Triangular Norm function, each function with properties in the right column an S-Norm function or Trianggula Conorm function. Generalized De-Morgan relations Consider the following six pairs of T-Norms and S-Norms: T-Norm: T(x,y) S-Norm: S(x,y) drastic product(x,y) drastic sum(x,y) limited difference(x,y) limited sum(x,y) Einstein product(x,y) Einstein sum(x,y) algebraic product(x,y) algebraic sum(x,y) Hamacher product(x,y) Hamacher sum(x,y) minimum(x.y maximum(x,y) Table 7 Pairs of Norm and Conorm functions For each of these pairs the following equations are valid: T(x,y) = 1 -S(1-x, 1-y) S(x,y) = 1 -T(1-x, 1-y) Each pair of Triangular Norm and Conorm functions which obey the generalized De-Morgan relations is called a pair of "related" Triangular Norm functions. All membership functions both for AND -and OR -connections of fuzzy values treated so far are pairs of related Trianggula Norms and Conorms. The T-Norm functions hold for the AND-connection, the S-Norm functions for the OR-connection. Why do we call these functions "triangular" ? Because of T(x,y) = T(y,x) and S(x,y) = S(y,x) both functions have do be defined only in the triangular area 0 ≤ x ≤ 1, 0 ≤ y ≤ x23 Ascending sequence of function values For any given values of x and y the following ascending sequence of function values is valid: 0 ≤ drastic product(x,y) ≤ limited difference(x,y) ≤ Einstein product(x,y) ≤ algebraic product(x,y) ≤ Hamacher product(x,y) ≤ minimum(x,y) ≤ maximum(x,y) ≤ Hamacher sum ≤ algebraic sum ≤ Einstein sum ≤ limited sum ≤ drastic sum ≤ 1 For any two T-Norm and S-Norm functions (not only for related ones) the following inequallitie are valid: 0 ≤ T(x,y) ≤ T(1,y) ≤ y ≤ S(0,y) ≤ S(x,y) ≤ 1 An important additional inequality For any pair x,y between 0 and 1, inclusively, there is 1 -x -y + x*y = 1 -x*(1-y) -y ≥ 1 -1* (1-y) -y = 1 -(1-y) -y = 0 or x + y -x*y ≤ 1 More exactly: If max(x,y) = 1 then x + y -x*y = 1 If max(x,y) < 1 then x + y -x*y < 1 3.6 Additional formula As the associativity rule is valid for all Norm and Conorm functions the following generalizze definitions make sense T(x,y,z) = T(T(x,y),z) = T(x,T(y,z)) S(x,y,z) = S(S(x,y),z) = S(x,S(y,z)) or for N parameters T(x1,x2,...,xN) = T(T (x1,x2,...,xN-1),xN) = T(x1,T(x2,...,xN)) S(x1,x2,...,xN) = S(S (x1,x2,...,xN-1),xN) = S(x1,S(x2,...,xN)) Limited difference /Limited sum T(x1,x2,...,xN) = max [0, ∑xn -(N-1) ] S(x1,x2,...,xN) = min [1, ∑xn ] Algebraic product /Algebraic sum T(x1,x2,...,xN) = ∏xn S(x1,x2,...,xN) = 1 -∏(1-xn) Minimum /Maximum T(x1,x2,...,xN) = min(x1,x2,...,xN) S(x1,x2,...,xN) = max(x1,x2,...,xN) For other triangular functions no simple generalization formula do exist24 3.7 Numeric example Suppose x=0.6 and y=0.2. Drastic product T(0.6, 0.2) = 0 as max (x,y) = max (0.6,0.2) = 0.6 < 1 Limited difference T(0.6, 0.2) = max (0,x+y-1) = max (0, -0.2) = 0 Einstein product T(0.6, 0.2) = x*y /[1+(1-x)*(1-y)] = 0.6*0.2 /(1+0.4*0.8) = 0.12 /1.32 = 0.091 Algebraic product T(0.6, 0.2) = x*y = 0.6*0.2 = 0.12 Hamacher product T(0.6, 0.2) = x*y /[1-(1-x)*1-y)] = 0.6*0.2 /(1-0.4*0.8) = 0.12 /0.68 = 0.176 Minimum T(0.6, 0.2) = min (x,y) = min (0.6, 0.2) = 0.2 Maximum S(0.6, 0.2) = max (x,y) = max (0.6, 0.2) = 0.6 Hamacher sum (0.6, 0.2) = 1-(1-x)*(1-y) /(1-x*y) = 1-0.4*0.8 /(1-0.6*0.2) = 1 -0.32 /0.88 = 0.636 Algebraic sum S(0.6, 0.2) = 1-(1-0.6)*(1-0.2) = 1-0.4*0.8 = 1 -0.32 = 0.68 Einstein sum S(0.6, 0.2) = 1-(1-x)*(1-y) /(1+x*y) = 1-0.4*0.8 /(1+ 0.6*0.2) = 1 -0.32 /1.12 = 0.714 Limited sum S(0.6, 0.2) = min (1,x+y) = min (1, 0.6+0.2) = min (1,0.8) = 0.8 Drastic sum S(0.6, 0.2) = 1 as min (x,y) = min (0.6,0.2) = 0.2 > 0 The sequence of ascending numeric values is clearly seen.25 3.8 Connecting a fuzzy set to itself 3.8.1 Intersection In contrast to traditional set theory the intersection of fuzzy set M = { (b,m) } with itself is not necessary equal to M. M∩ M = { [b, T(m,m)] } ≠ M if T(m,m) ≠ m (which is normally the case). Depending on the choice of T(x,y) the following table gives respective memberships T(m,m) of base values b in fuzzy set M∩ M. Function name Function definition: T(x,y) = T(m,m) = Drastic product 0 if max(x,y) < 1 min (x,y) if max(x,y) = 1 0 if m < 1 1 if m = 1 Limited differennc max (0,x+y-1) 0 if m ≤ 0.5 2*m -1 if m > 0.5 Einstein product x*y /[1+(1-x)*(1-y)] m² /(2-2*m+ m²) Algebraic product x*y m² Hamacher produuc x*y /[1-(1-x)*1-y)] m /(2-m) Minimum min (x,y) m Table 8 Membership functions of fuzzy set intersection with itself For all T-Norms in the table there is T(m,m) ≤ min(m,m) = m and therefore M ∩ M = { [b, T(m,m)] } is a subset of M. With exception of T(x,y)=min(x,y) M∩ M even is a true subset of M as one easily can prove that at least one value of m exists for which is definitely T(m,m) < m. If T(x,y)=min(x,y) then T(m,m) = m so that M and M∩ M are identical.26 3.8.2 Junction Also the junction of fuzzy set M ={ (b,m) } with itself is not necessary equal to M. M ∪ M = { [b, S(m,m)] } ≠ M if S(m,m) ≠ m (which is normally the case). Depending upon the choice of S(x,y) the following table gives respective memberships S(m,m) of base values b in fuzzy set M∪M. Function name Function definition: S(x,y) = S(m,m) = Maximum max(x,y) m Hamacher sum 1-(1-x)*(1-y) /(1-x*y) 2*m /(1+m) Algebraic sum 1-(1-x)*(1-y) m*(2-m) Einstein sum 1-(1-x)*(1-y) /(1+x*y) 2*m /(1+m²) Limited sum min (1,x+y) 2*m if m ≤ 0.5 1 if m > 0.5 Drastic sum 1 if min (x,y) > 0 max (x,y) if min (x,y) = 0 1 if m > 0 0 if m = 0 Table 9 Membership functions of fuzzy set junction with itself For all S-Norms in the table there is S(m,m) ≥ max(m,m) = m and therefore M ∪ M = { [b, S(m,m)] } is a superset of M. With exception of S(x,y)=max(x,y) M∪M even is a true superset of M as one easily can prove that at least one value m exists for which definitely is S(m,m) > m. If S(x,y)=max(x,y) then S(m,m) = m which means that M and M∪M are identiccal 3.9 Connecting a fuzzy set to its complement set 3.9.1 Intersection Different to traditional set theory the intersection of fuzzy set M = { (b,m) } with its complemeen set ⌐M = { (b,1-m) } is not necessary an empty set M∩ (⌐M) = { [b, T(m,1-m)] } ≠ ∅ if T(m,1-m) ≠ 0 The following table gives respective memberships of base values b in fuzzy set M ∩ (⌐M) . Only the "drastic product" and the "limited difference" functions result in M∩ (⌐M) = ∅ as traditionally expected, the other T-Norm functions do not27 Function name Function definition: T(x,y) = T(m,m) = Drastic product 0 if max(x,y) < 1 min (x,y) if max(x,y) = 1 0 Limited difference max (0,x+y-1) 0 Einstein product x*y /[1+(1-x)*(1-y)] m*(1-m) /[1+(1-m)*m] Algebraic product x*y m*(1-m) Hamacher product x*y /[1-(1-x)*1-y)] m *(1-m) /[1-(1-m)*m] Minimum min (x,y) min (m,1-m) Table 10 Membership functions of fuzzy set intersection with complement set 3.9.2 Junction The junction of fuzzy set M = { (b,m) } with its complement set ⌐M = { (b,1-m) } is not necesssar the unity set M∪ (⌐M) = { [b, S(m,1-m)] } ≠ 1 if S(m,1-m) ≠ 1. The following table gives respective memberships of base values b in fuzzy set M∪(⌐M). Only the "limited sum" and the "drastic sum" functions result in M∪(⌐M) = 1 as traditionally expected, the other S-Norm functions do not Function name Function definition: S(x,y) = S(m,m) = Maximum max(x,y) max(m, 1-m) Hamacher sum 1-(1-x)*(1-y) /(1-x*y) [1-2*m*(1-m)] /[ (1-m*(1-m)] Algebraic sum 1-(1-x)*(1-y) 1 -m*(1-m) Einstein sum 1-(1-x)*(1-y) /(1+x*y) 1 /[1+m*(1-m)] Limited sum min(1,x+y) 1 Drastic sum 1 if min(x,y) > 0 max(x,y) if min(x,y) = 0 1 Table 11 Membership functions of fuzzy set junction complement set 3.10 Scalable triangular functions With any pair of related Norm and Conorm functions a "weighted average" may be formed: R(x,y) = (1-α)*T(x,y) + α*S(x,y) arithmetic average, 0 ≤ α ≤ 1 R(x,y) = T(x,y) (1-α) + S(x,y) α geometric average, 0 ≤ α ≤ 1 Many scalable functions are discussed in literature, some of which are given here: Hamacher intersection function P(x,y) = x*y /[α -(α -1)*(x+y-x*y)] 0 ≤ α ≤ ∞28 For special selections of parameter α this goes over into the following T-Norm functions: α = 0: Hamacher product α = 1: Algebraic product α = 2: Einstein product α = ∞: Drastic product Hamacher junction function Q(x,y) = [ (α -1)*x*y + x + y ] /(1 + x*y) -1 ≤ α ≤ ∞ For special selections of parameter α this goes over into the following S-Norm functions: α = -1: Hamacher sum α = 0: Algebraic sum α = 1: Einstein sum α = ∞: Drastic sum Yager intersection function ] ) 1 ( ) 1 ( , 1 min[ 1 y) P(x, α α α y x − + − − = For special selections of parameter α this goes over into the following T-Norm functions: α = 1: Limited difference α = ∞: Minimum Yager junction function ] , 1 min[ y) Q(x, α α α y x + = For special selections of parameter α this goes over into the following S-Norm functions: α = 1: Limited sum α = ∞: Maximum Werner functions P(x,y) = α*min(x,y) + (1-α )*(x+y)/2 AND-connecting function Q(x,y) = α*max(x,y) + (1-α )*(x+y)/2 OR-connecting function 3.11 Set product /two-dimensional member functions 3.11.1 General definitions In practical applications it is often useful to join two standard fuzzy sets into one combined fuzzy set called the "set product" or "product set" of both. A similar concept is also widely used in traditional set theory. Suppose we have two abstract entities A and B with two sets of base values {a} and { b}, two sets of fuzzy values {f } and {g} and two member functions p(a,f) and q(b,g), respectively. Entities A and B may be truly different but also may in fact be identical.29 Now look at two fuzzy sets P(f) = { [a,p(a,f)] } of abstract entity A and Q(g) = { [b,q(b,g)] } of abstract entity B. n(x,y) be an arbitrary Norm or Conorm function. We then define the set product of fuzzy sets P(f) and Q(g) as follows: Z(f,g) = P(f)*Q(g) = { ( a, b, n[ p,(a,f),q(b,g) ] ); [a,p(a,f)]∈P(f); [b,q(b,g)])∈Q(g) } In other words: Z is the set of all triples a,b,n where a and b are base values of A and B, f and g are fuzzy values of A and B and n[ p,(a,f),q(b,g) ]=µ (a,b,f,g) denotes the combined memberrshi of the pair (a,b) of base values to the pair (f,g) of fuzzy values under consideration. If the pair (f,g) of fuzzy values is considered to be _AND_ connected, that is (f,g) = f_AND_g, one normally chooses the Norm function n(x,y) to be a T-Norm function. If, conversely, (f,g) = f_OR_g then the connecting function n(x,y) should be an S-Norm function. In general the selection of n(x,y) depends upon usefulness. 3.11.2 Projections of two-dimensional member functions If in a two-dimensional member function µ(a,b,f,g) = n[ p(a,f),q(b,g) ] variables b and g are held to fixed values b=b0 and g=g0 of abstract entity B then the function ν(a,f) = µ(a,b0,f,g0) is called "the projection of membership function µ(a,b,f,g) on the value pair (b0,g0) of abstract entity B". ν(a,f) depends only upon the two variables a and f and represents the membership of base values a to fuzzy values f of abstract entity A when only those cases are considered where abstract entity B has its sharp value b0 and its fuzzy value g0 . There are as many projecttion ν(a,f) of two-dimensional membership function µ(a,b,f,g) as there are possible combinaation (b0,g0). Similarly projections ν(b,g) = µ(a0,b,f0,g) on the value pairs (a0,,f0) of abstract entity A are defined. 3.11.3 Two-dimensional fuzzy sets The triples (a,b,m) with a and b from the sets of base values of abstract entities A and B, respecttively and 0 ≤ m ≤ 1 are the elements of the "two-dimensional fuzzy set" of abstract entittie A and B: M = { (a,b,m); a,b base values of A respective B; 0 ≤ m ≤1 } In this set every pair (a,b) of sharp values is contained with its membership m to both abstrrac entities A and B simultaneously -obviously a generalization of the one-dimensional fuzzy sets discussed earlier . Consequently we can also define set operations for two-dimensional fuzzy sets. Complement set ⌐M = { (a,b,1-m) Intersection M1∩ M2 = { [a,b, T(m1,m2)]; } with a suitable T-Norm function30 Junction M1∪M2 = { [a,b, S(m1,m2)] } with a suitable S-Norm function 3.12 Multi dimensional member functions 3.12.1 Definition The above reasoning may be easily generalized for an arbitrary number of abstract entities The n-tupel of base values a1,...,aN of abstract entities A1,...AN simultaneously belong to the n-tuels of fuzzy values f1,...,fN of the same abstract entities A1,...AN with membership µ(a1,...,aN,f1,...fN) = n[p1(a1,f1),...,pN(aN,fN) ] where n(x1,...,xN) is any agreed T-Norm or S-Norm function. If the multiple (f1,...fN) of fuzzy values is AND-connected one normally chooses n(x1,...,xN) to be a T-Norm function. If, conversely, (f1,...fN) is OR -connected the connecting function n(x1,...,xN) should be an S-Norm function. 3.12.2 Projections Projections of multi-dimensional membership functions are defined similarly as for twodimennsiona membership functions. One or more of the sharp and related fuzzy variables are held constant, the others may vary freely. Projected functions give the memberships of base values to fuzzy values of certain abstract entities when only those cases are considered where the other abstract entities have definite fixed sharp and fuzzy values . 3.12.3 Fuzzy sets Also this is a straightforward generalization of the two-dimensional case. The combinations (a1,...,aN,m) with a1,...aN from the sets of base values of the abstract entities A1,...AN and m between 0 and 1 are the elements of an "N-dimensional fuzzy set": M = { (a1,...,aN,m); a1,...aN are base values of A1,...AN; 0 ≤ m ≤1 } M may be a product set of several one-dimensional fuzzy sets: M1(f1) = { (a1,m1); a1 ∈ set of base values of abstract entity A1; m1 membership of a1 to f1} M2 (f2)= { (a2,m2); a2 ∈ set of base values of abstract entity A2; m2 membership of a2 to f2} .... MN (fN)= { (aN,mN); aN∈set of base values of abstract entity AN;mN membership of aN to fN} M(f1, f2,...,fN) = { (a1,...,aN,T(m1,...,mN) } = M1(f1)*...*MN(fN) respectively M(f1, f2,...,fN) = { (a1,...,aN,S(m1,...,mN) } = M1(f1)*...*MN(fN)31 3.13 Example The following example shows how two-dimensional member functions are constructed and used. The notations are similar to those of the preceding chapter. Abstract entities A: temperature B: pressure Physical units Kelvin (°K) 105 Pascal (105Pa) Sets of base values {a} = { a; 0 ≤ a ≤ 1000 } {b} = { b; 1 ≤ b ≤ 10 } Sets of fuzzy values {f} = { cold, warm, hot } {g} = { low, middle, high } Membership functions p(a,f) q(b,g) p(a, cold) = 1 if 0 ≤ a ≤ 300 1 -(a-300)/400 if 300 ≤ a ≤ 700 0 if 700 ≤ a ≤ 1000 p(a, warm) = a/300 if 0 ≤ a ≤ 300 1 if 300 ≤ a ≤ 700 1 -(a-700)/300 if 700 ≤ a ≤ 1000 p(a, hot) = 0 if 0 ≤ a ≤ 300 (a-300)/400 if 300 ≤ a ≤ 700 1 if 700 ≤ a ≤ 1000 q(b, low) = 1 if 1 ≤ b ≤ 3 1 -(b-3)/4 if 3 ≤ b ≤ 7 0 if 7 ≤ b ≤ 10 q(b, middle) = b/3 if 1 ≤ b ≤ 3 1 if 3 ≤ b ≤ 7 0 -(b-7)/3 if 7 ≤ b ≤ 10 q(b, high) = 0 if 1 ≤ b ≤ 3 (b-3)/4 if 3 ≤ b ≤ 7 1 if 7 ≤ b ≤ 10 Fuzzy sets P(f) = { (a, p(a,f) } Q(g) = { (b, q(b,g) } P(cold) = { [ a, p(a,cold) ]; 0 ≤ a ≤ 1000 } P(warm) = { [ a, p(a,warm) ]; 0 ≤ a ≤ 1000 } P(hot) = { [ a, p(a,hot) ]; 0 ≤ a ≤ 1000 } Q(low) = { [ b, q(b,low) ]; 0 ≤ b ≤ 10 } Q(middle) = { [ b, q(b,middle) ]; 0 ≤ b ≤ 10 } Q(high) = { [ b, q(b,high) ]; 0 ≤ b ≤ 10 } Table 12 Membership functions p(a,f) of temperature and q(b,g) of pressure; Fuzzy sets P(f) of temperature and Q(g) of pressure32 Infix _AND_ _OR_ Two-dimensional fuzzy values cold_AND_low; cold_AND_middle; cold_AND_high; warm_AND_low; warm_AND_middle; warm_AND_high; hot_AND_low; hot_AND_middle; hot_AND_high; cold_OR_low; cold_OR_middle; cold_OR_high; warm_OR_low; warm_OR_middle; warm_OR_high; hot_OR_low; hot_OR_middle; hot_OR_high; Norm function min(x,y) max(x,y) Table 13 Two-dimensional fuzzy values with _AND_ respective _OR_ connection a 0 100 200 300 400 500 600 700 800 900 1000 p(a,warm) 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 b q(b,high) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0.25 0 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0 5 0.50 0 0.33 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.33 0 6 0.75 0 0.33 0.67 0.75 0.75 0.75 0.75 0.75 0.67 0.33 0 7 1 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 8 1 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 9 1 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 10 1 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 Table 14 Partial two-dimensional membership function µ(a,b,warm_AND_high) = min[ p(a,warm), q(b,high) ]33 a 0 100 200 300 400 500 600 700 800 900 1000 p(a,hot) 0 0 0 0 0.25 0.50 0.75 1 1 1 1 b q(b,low) 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 4 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 1 1 1 1 5 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.75 1 1 1 1 6 0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.75 1 1 1 1 7 0 0 0 0 0 0.25 0.50 0.75 1 1 1 1 8 0 0 0 0 0 0.25 0.50 0.75 1 1 1 1 9 0 0 0 0 0 0.25 0.50 0.75 1 1 1 1 10 0 0 0 0 0 0.25 0.50 0.75 1 1 1 1 Table 15 Partial two-dimensional membership function µ(a,b,hot_OR_low) = max [ p(a,hot),q(b,low) ] The other two-dimensional partial membership functions have similar value tables a 0 100 200 300 400 500 600 700 800 900 1000 p(a,cold) 1 1 1 1 0.75 0.50 0.25 0 0 0 0 p(a,warm) 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 p(a,hot) 0 0 0 0 0.25 0.50 0.75 1 1 1 1 ν(a,cold) 0.75 0.75 0.75 0.75 0.75 0.50 0.25 0 0 0 0 ν(a,warm) 0 0.33 0.67 0.75 0.75 0.75 0.75 0.75 0.67 0.33 0 ν(a,hot) 0 0 0 0 0.25 0.50 0.75 0.75 0.75 0.75 0.75 Table 16 Projection ν(a,f) of two-dimensional membership function µ(a,b,f_AND_g) on value pair b = b0 = 6*105Pa (sharp) and g = g0 = "high" (fuzzy) of abstract entity pressure ν(a,f) = µ( a, 6, f_AND_high ) = min[ p(a,f), q(6,high) ] = min [ p(a,f), 0.75 ]34 a 0 100 200 300 400 500 600 700 800 900 1000 p(a,cold) 1 1 1 1 0.75 0.50 0.25 0 0 0 0 p(a,warm) 0 0.33 0.67 1 1 1 1 1 0.67 0.33 0 p(a,hot) 0 0 0 0 0.25 0.50 0.75 1 1 1 1 ν(a,cold) 1 1 1 1 0.75 0.75 0.75 0.75 0.75 0.75 0.75 ν(a,warm) 0.75 0.75 0.75 1 1 1 1 1 0.75 0.75 0.75 ν(a,hot) 0.75 0.75 0.75 0.75 0.75 0.75 0.75 1 1 1 1 Table 17 Projection ν(a,f) of two-dimensional membership function µ(a,b,f_OR_g) on value pair b = b0 = 6*105Pa (sharp) and g = g0 = "high" (fuzzy) of abstract entity pressure. ν(a,f) = µ( a, 6, f_OR_high ) = max[ p(a,f), q(6,high) ] = max [ p(a,f), 0.75 ] Fuzzy sets: M(f_AND_g) = { (a, b, m); a,b base values of temperature respectively. pressure; 0 ≤ m ≤1 } = { (a, b, µ(a,b,f_AND_g) } = { (a, b, min[ p(a,f), q(b,g) ] } M(f_OR_g) = { (a, b, m); a,b base values of temperature respectively. pressure; 0 ≤ m ≤1 } = { (a, b, µ(a,b,f_OR_g) } = { (a, b, max[ p(a,f), q(b,g) ] }35 4 Fuzzy Controllers 4.1 Introduction As described earlier processes in real world can be characterized by time-dependant values of certain abstract entities. Those entities may be classified as "input variables" and "output variables" to the process. Generally spoken input variables determine the time dependant procees flow whereas output variables describe the momentary process states. Some of the input variables can be influenced more or less accurately by operating staff persons or by computerrs These are the "control variables" of the process. Other input variables either cannot easily be controlled at all (that is their values are accidental) or are not considered being important in the context. Process control in its widest sense means operating a process in a "closed loop" with the aim of producing best results in some sense, that is of producing optimum values of some or all output variables. The basic principle is pretty simple: • determine the present values of all relevant output variables • compare these present output values to given target values • adjust values of suitable control variables such as to approach the target values better than before • repeat above steps forever (or until target values have been met sufficiently exact) In the example below the process has three input variables among which there are two control variables (input 2 and input 3) and one accidental variable (input 1). The process also has three output variables which are all used for control. The controller (might be a fuzzy controlller uses the three output variables, transforms them according to some algorithm into two control variables which are then "fed back" as control inputs to the process. Comparison of output values to target values is done within the controller itself.36 A fuzzy controller performs exactly as described. Its only peculiarity is its method of transforrmin output values into control values. Most standard controllers do that by pretty compliccate mathematical computations. A fuzzy controller uses "rules" similar to those used by human operators. Examples IF pressure is low THEN open valve considerably A chemical production process IF interests are high THEN reduce obligations greatly A financial planning process IF energy consumption is low THEN reduce generator input power to near standby A power plant process Process Output 1 Output 3 Input 1 Input 2 Input 3 Output 2 Controller37 Without loss of clearness these phrases may be reduced to IF pressure.low THEN valve.open_considerably IF interests.high THEN obligations.reduce_greatly IF energy consumption.low THEN generator.reduce_ input_ power_ to_ near_ standby The underlined terms are names of abstract entities, the italicized terms are fuzzy values of respective abstract entities, the cancelled terms are redundant and may be omitted. Rules like these are suitable for human process control. An experienced operator knows what terms like "pressure is low" or "open valve considerably" really do mean. But automatic contrro devices deal with sharp numbers and not with fuzzy values: the output variables of many technical process are treated by analog-to-digital converters, the control variables by digitalttoanalog converters, both kinds of devices are restricted to numeric value processing. So to set up a fuzzy controller basically we have to do three things: • transform sharp values of process output variables into fuzzy values • use rules to transform fuzzy output variable values into fuzzy values of the control variabble • transform fuzzy values of the control variables into sharp values The first task, transformation of sharp values to fuzzy values, was treated in Chapter 2. For each abstract entity A of the process we define a member function µ(a,f) which attaches any sharp value a of A to each of the agreed fuzzy values f of A. The second and third tasks will be discussed now 4.2 Rules, implications, conclusions 4.2.1 Rules A rule in a fuzzy control system has the general form IF f0 THEN g0 where f0 and g0 are any two fixed fuzzy values of some output variable A and some control variable B of the process under consideration. Example Consider a process with • 3 output variables: pressure with fuzzy values { pressure.low, pressure.middle, pressure.high } temperature with fuzzy values { temp.cold, temp.lukewarm, temp.warm, temp.hot } moisture with fuzzy values { moisture.dry, moisture.moist, moisture.wet }38 • 2 control variables: valve with fuzzy values { valve.closed, valve.half_open, valve.open } heater with fuzzy values { heater.off, heater.low, heater.middle, heater.max } There might be two rules: • IF pressure.low THEN valve.half_open f0 = pressure.low fuzzy value of output variable "pressure" g0 = valve.half_open fuzzy value of control variable "valve" • IF temp.cold_AND_moisture.wet THEN heater.max f0 = temp.cold_AND_moisture.wet combined fuzzy value of output variables "temperature" and "moisture" g0 = heater.max fuzzy value of control variable "heater" The value f0 in the IF-part of a rule (the precondition) may either be a simple fuzzy value of one output variable or a combination of fuzzy values of several output variables. The fuzzy value g0 in the THEN-part of a rule (the conclusion part) is nearly always a simple fuzzy value of only one control variable.39 4.2.2 Implications Terms like "IF f0 THEN g0" also occur in boolean logic and there are called implications. f 0 and g0 are considered being boolean entities with attached truth values 0 or 1. Formally an implication is written f0→g0 and for each pair of truth values of f 0 and g0 returns a result according to following table: f0 g0 f0 → g0 (⌐f0) ⋁g0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 Table 18 Truth table of boolean functions f0 → g0 and (⌐f0) ⋁g0 Clearly f0 →g0 is equivalent to (⌐f0)⋁g0 In fuzzy logic the implication f0 →g0 may be defined for fuzzy values f0 and g0 as a new combined fuzzy value Definition of fuzzy implication: (NOT_f0)_OR_g0 Membership function: µ(a,b, f0 →g0) = µ [ a,b, (NOT_f0)_OR_g0 ] = S[ 1-m(a, f0), n(b,g0 ) ] Here a and b are any two base values of the respective abstract entities A and B to which f0 and g0 belong. m(a, f0) and n(b,g0 ) are respective partial member functions of A and B. 4.2.3 Combination of process states and rules As an introduction consider a process with only one output variable A, one control variable B and one rule connecting two selected fuzzy values f 0 of A and g0 of B. Output variable Control variable Name: A B Base values: {a} {b} Fuzzy values: {f} {g} Member functions: m(a,f) n(b,g) Selected fuzzy values: f0 ∈ {f} b0 ∈ {g} Present sharp values: α(t) ∈{a } β ∈{b } Rule: IF f0 THEN b040 The situation may by described as follows. • At any time t sharp value α(t) of output variable A belongs to fuzzy value f0 with memberrshi m(α,f0) AND At any time t and for any sharp value b of control variable B the pair (α , b) belongs to the fuzzy value f0 →g0 = (NOT_f0)_OR_g0 with membership µ(α , b, f0 →g0) = S[ 1-m(α, f0), n(b,g0 ) ] Regarding the situation as a whole we should look at the AND_combined fuzzy values from both phrases, that is at fuzzy value • f0_AND_[f0 →g0] = f0 _AND_[(NOT_f0)_OR_g0 ] At any time t and for any sharp value b of B the pair (α , b) belongs to this new fuzzy value with membership • r(α, b, f0 _AND_ [(NOT_f0)_OR_g0 ] = T{ m(α, f0), S[ 1-m(α, f0), n(b,g0 ) ] } For fixed time t and therefore fixed value α(t) this function depends solely on b. 4.2.4 How to select a sharp value of control variable B After the foregoing considerations the answer is obvious: For any fixed time t=t0 the sharp value b=β(t0) of control variable B should be chosen such as to maximize the membership of the pair [α(t0),b] to fuzzy value f0 AND_ [(NOT_f0)_OR_g0 ]: T{ m(α, f0), S[ 1-m(α, f0), n(b,g0 ) ] } = max. for b=β(t)=bmax This formalism contains the basic concept of fuzzy control. It permits the determination of control variable values β(t) for a process from the output values α(t) by means of colloquial rules. 4.2.5 Details Let us discuss the behavior of the function ρ(b) = T{ m(α(t0), f0 ) , S[1-m(α(t0) ), f0 ), n(b,g0 )] } = T{ m0 , S[1-m0, n(b,g0 )] } for a fixed time t=t0 , that is for a fixed value m0 of m. ρ(b) gives the membership of the pair of sharp values [α(t0), b] of output variable A at time t=t0 and control variable B to fuzzy value f0 _AND_[(NOT_f0)_OR_g0]. As said before should be chosen such that ρ(b) is maximmu for b=β(t0)=bmax.41 4.2.5.1 Precondition of rule highly fulfilled (m≈1) This means m≈1 and therefore ρ(b) ≈ T{1, S[0, n(b,g0 )] } = T{1, n(b,g0 )] } = n(b,g0 ). Accorddingl control value b should be chosen such as to let membership n(b,g0 ) become a maximum. This seems reasonable: if the preconditions of the rule are nearly fully met then the rule should be executed as effectively as possible. 4.2.5.2 Precondition of rule partly fulfilled (m≤0.5) Now we have S[1-m, n(b,g0 )] ≥ max [1-m, n(b,g0 )] ≥ 0.5 ≥ m and therefore ρ(b) = T{m, S[1-m, n(b,g0 )] } ≤ min {m, S[1-m, n(b,g0 )]} ≤ m. The lower the value of m, the lower membership ρ(b) for any choice of b so that the actual selection of b becomes less significant. 4.2.5.3 Precondition of rule hardly or not fulfilled (m ≈ 0) Now ρ(b) ≈ T{0, S[1, n(b,g0 )] } = T{0, 1} = 0 independent of b and the choice of b is irreleevan as any value will suffice. This corresponds to everyday experience that rules whose preconditions are not met need not be fulfilled. 4.2.6 Two examples 4.2.6.1 T(x,y) = drastic product(x,y) /S(x,y) = drastic sum(x,y) This gives ρ(b) = 0 if m<1 ρ(b) = n(b,g0) if m=1 Only if the process output value α(t) fully belongs to the precondition f0 of the rule the calculattio of an optimum value β=bmax is meaningful, otherwise choice of b is irrelevant.42 4.2.6.2 T(x,y) = min(x,y) /S(x,y) = max(x,y) This results in ρ(b) = min{m, max[1-m, n(b,g0 )] } the values of which are given in following table. m n(b,g0 ) ρ(b) = 0 ≤ m ≤ 0.5 irrelevant m 0.5 ≤ m ≤ 1.0 0 ≤ n(b,g0 ) ≤ 1-m 1-m 1-m ≤ n(b,g0 ) ≤ m n(b,g0 ) m ≤ n(b,g0 ) ≤ 1 m Table 19 The table shows: • If preconditions of the rule are poorly satisfied ( m ≤ 0.5) function ρ(b) does not at all depeen on variable b and therefore the choice of b is irrelevant ... as common sense predicts • If preconditions of the rule are better satisfied then we have to distinguish: for values of b for which n(b,g0 ) is small (0 ≤ n(b,g0 ) ≤ 1-m) function ρ(b) does not depend on b but has the small constant value ρ(b) = 1-m for values of b for which n(b,g0 ) is medium (1-m ≤ n(b,g0 ) ≤ m) function ρ(b) = n(b,g0 ) can be used to determine β=bmax for values of b for which n(b,g0 ) is big (m ≤ n(b,g0 ) ≤ 1) function ρ(b) does not depend on b but has the big constant value ρ(b) = m Summary: • If n(b,g0 ) is too small ( 0 ≤ n(b,g0 ) ≤ 1-m for all values of b) all values of b equally conform to the rule and to the momentary process state. There is no significant choice b=β • Also if n(b,g0 ) is too big (m ≤ n(b,g0 ) ≤ for all values of b ) all values of b equally confoor to the rule and to the momentary process state and again there is no significant choice β=bmax . • Only if n(b,g0 ) is neither too small nor too big (1-m ≤ n(b,g0 ) ≤ m for all values of b ) then ρ(b)=n(b,g0) depends sufficiently on b to permit the calculation of an optimum value b=β. 4.2.7 Final modification The above examples cannot easily be generalized to other Triangular Norm and Conorm functions because the mathematical representations of ρ(b) = T{m, S[1-m, n(b,g0 )] } tend to become rather complex. So a simplification seems necessary. The most interesting cases are those where memberships m(α(t),f0 ) and n(b,g0 ) are both close to 1, resulting in S[1-m, n(b,g0) ] ≈ S[0, n(b,g0 )] = n(b,g0 ). In these cases we may approximately set ρ(b) ≈ T{m, n(b,g0 ) } We will now treat ρ ( b ) = T { m( α(t), f0 ) , n( b, g0 ) }43 not only as an approximation but as a new simplified definition for ρ(b) valid for all values of m(α(t),f0 ) and n(b,g0) regardless of their magnitudes. The new definition in fact gives the membership of base value pair [α(t), b] to fuzzy value f0_AND_g0 instead to original fuzzy value f0_AND_[(NOT_f0)_OR_g0]. In boolean logic this would not make a difference since we have f0 ⋀ (f0 →g0) = f0 ⋀( (¬f0) ⋁ g0) = (f0 ⋀ (¬f0)) ⋁ (f0 ⋀ g0)) = f0 ⋀ g0 but for fuzzy values f0 and g0 similar relations do not apply. From now on we will use only the simplified definition ρ( b ) = T { m( α, f0 ) , n( b, g0 ) } if not otherwise said 4.2.8 Combined preconditions Rules like IF pressure.low_OR_pressure.medium THEN valve.open or IF temp.cold_AND_moisture.wet THEN heater.full with combined preconditions are treated similarly as before. It makes no difference whether the combined preconditions refer to only one or to more than one output variables. In both cases from the sharp output value(s) α1(t), ... , αN(t) at time t the degree of membership m( α1(t), ... , αN(t), f0 ) to the combined fuzzy value f0 is determined. After that we may proceee as before, that is we consider function ρ ( b ) = T { m(α1(t), ... , αN(t), f0 ) , n( b, g0 ) } and try to find an optimum value for b. 4.3 Accumulation and De-Fuzzification 4.3.1 Several simultaneous rules Consider a set of rules like the following for car driving IF distance.near THEN speed.slow IF time.short THEN speed.fast IF date.weekend THEN speed. fast IF fuel.empty THEN speed.stop IF money.short THEN speed.economy All rules refer to the same output variable "speed" with its fuzzy values "stop", "slow", "economy" and "fast" (among others). The rules may lead to coincident or to contradictory conclusions depending on how the preconditions actually are fulfilled in a certain driving situation. If, for example, time and money happen to be "short" simultaneously, should speed then be "fast" or "economy" ?44 If there are several rules simultaneously we have to decide which of them to fulfill: rule1 OR rule2 OR rule3 OR rule4 OR rule5 OR in this context means "inclusively-OR" as several rules may ( and if possible should ) be fulfilled simultaneously. A rule with precondition f0 and conclusion g0 is equivalent to fuzzy value f0_AND_g0 . At any time t the sharp process output value(s) together with value b of the control variable to which the rule refers should result in optimum membership ρ0(b) = T{ m(α1(t), ... , αN(t), f0 ), n( b, g0 ) } to f0_AND_g0. Several rules together one or more of which should be fulfilled simultaneously are equivalent to the combined fuzzy value (f1_AND_g1)_OR_(f2_AND_g2)_OR_....._OR_(f3_AND_g3) with membership function ρ( b ) = S { ρ1( b ), ρ2( b ),....., ρN( b ) } where ρ1 ( b ) = T { m(α1(t), ... , αN(t), f0 ) , n( b, g1 ) } rule #1 ρ2 ( b ) = T { m(α1(t), ... , αN(t), f0 ) , n( b, g2 ) } rule #2 ρ3 ( b ) = T { m(α1(t), ... , αN(t), f0 ) , n( b, g3 ) } rule #3 .......................................................................... ........ ρN ( b ) = T { m(α1(t), ... , αN(t), f0 ) , n( b, gN) } rule #N are respective membership functions from each of the rules. As before ρ( b ) = S { ρ1( b ), ρ2( b ),....., ρN( b ) } at any fixed time t is a function of b alone from which the optimum value of b=β may be determined. 4.3.2 De-Fuzzification The foregoing chapters discussed how to derive from the control rules and the momentary process state an "optimum" value β of control variable b. This procedure called defuzzifficatio is the last step of each control cycle in a fuzzy system. Generally said b=β should make partial membership function ρ(b) = S { ρ1(b), ρ2(b),.., ρN(b) } = S {T[m1,n( b, g1 )], T[m2,n( b, g2 )],..,T[mN,n( b, gN )] } "optimal" or "maximal" or anything similar. But what does that really mean ? In fact three different methods are used for de-fuzzyfication which will be described now45 4.3.2.1 Method of maximum height This method is always applicable if function ρ( b ) has a definite absolute maximum at some value b=bmax that is if ρ( bmax ) ≥ ρ( b ) for all values of b. Value b = bmax may then be used. The method has the advantage of being very simple and straightforward, a disadvantage lies in the fact that function ρ( b ) is not considered in its whole area of definition but only at its maximum position. This might lead to inappropriate results. 4.3.2.2 Method of mean of maximum If function ρ( b ) has several relative maxima the (weighted) average of these can be taken. Thereby the general form of ρ( b ) is better taken into account than before. The method is also applicable if ρ( b ) is constant over certain intervals of b which happens rather often in practiice In this case one takes the centers of each such interval as "relative local maximum" and these values are then averaged. 4.3.2.3 Method of center of gravity A graphical representation of function ρ( b ) in a Cartesian coordinate system is a curve which together with the abscissa (the b-axis) and the upper and lower bounds for b borders a flat area of the coordinate plain. Then the abscissa β of its center of gravity is taken as the desired "optimum" F M b db b F db b b Mopt dc dc /* ) ( * ) ( * = == ρ ρ If ρ( b ) is defined only for discrete values of b (so called singletons ) computation or the center of gravity is modified somewhat: F M b n b F b n b Mopt n n n/) ( ) ( * = == ρ ρ46 5 Fuzzy Numbers and Fuzzy Intervalls 5.1 Problem When evaluating scientific experiments often the following situation arises: Suppose the outcome of an experiment is a set of real numbers. Because of inadequate experimmenta design, measuring inaccuracies or other disturbing effects the result values are not fully accurate but always are more or less faulty. Now consequences from the results shall be derived. If the outcome values were accurate the same would hold for the consequences: they were definitely certain. But as in fact the results are inherently faulty also the consequences will be to some degree uncertain. So the question of reliability comes up: If the inaccuracies of the result values are known, what degree of uncertainty is to be expected for the consequences ? How can one calculate the degree of uncertainty ? What are the limits of reliability ? All these questions and more may be treated in depth by means of probability theory, general statistics, sampling theory and related techniques. The Gaussian method of least squares togetthe with error propagation law is a good example. Below we give another approach using fuzzy methods. By doing so the aim is not to replace the well-proven traditional methods but to offer them a supplement. 5.2 Relations 5.2.1 One independent variable As shown in chapter 2 numbers can be given either as sharp numbers x from some set of base values {x} or as fuzzy numbers f from some set {f}of fuzzy values. Both representations are connected to each other by the membership function m(x,f) respectively by the fuzzy sets X(f) = { (x, m[x,f]) }. A function w=fun(x) is an algorithm by which to each sharp number x from {x} exactly one sharp number w from a certain set {w} of real numbers is connected. w = fun(x) is the "pictuur " of x under function fun(x): x: →w. Can we give similar definition for functions of fuzzy numbers ? Let f0 be some fuzzy number (e.g. f0 =about_3.4) and X(f0) = {(x,m(x,f0) } the related fuzzy set. We now construct another fuzzy set W as follows: • Let (x0,m0) be any element of fuzzy set X(f0). Then from x0 a sharp value w0 is calculaate by w0 = fun(x0) • A membership r(w0) is calculated as follows: 1.) All elements (x,m)∈X(f0) with fun(x)=w0 are collected (besides x0 there might be other numbers x with fun(x)=w0 ) Let these elements be (x0,m0); (x1,m1); ...; (xN,mN) 2.) r0 is calculated as r0= S(m0, m1,...,mN) with some formerly agreed Conorm S.47 • Let W = {(w,r ) } be the set of all pairs (w,r ) = (w0,r0 ) calculated in that way for all possiibl elements (x0,m0)∈ X(f0) We may consider W as being the fuzzy set of a certain (yet unnamed) fuzzy number p0 defiine on the set {w} of base values w. This fuzzy number is regarded as the "picture" p0 = fun(f0) of f0 under function fun(f) just as w0=f(x0) is the picture of x0 under function fun(x). So W becomes W(fun(f0)). Generally for any fuzzy number from {f}: x: → w=fun(x), f: → p=fun(f), X(f): → W(p) Definition If fuzzy set W is constructed from fuzzy set X as described above we say there is a relation between X and W or symbolically X: →W But there is a difficulty: If X contains only a finite number of elements (x,m) then we have also only a finite number of values m0, m1,...,mN and expression S(m0, m1,...,mN) is well defined. But if X contains an infinite number of elements then also m1,...,mN might be an infinite sequence for which till now S is not defined. So at first the definition of S should be extended to an infinite number of arguments. Unfortunately this cannot be done for all Conorm functions in general but only for some special ones. One of these is S(x0,x1,...) = max(x0,x1,...) and so for an infinite set X (practically the most important case) we restrict ourselves to that definition of S. For finite fuzzy sets X any definition shall be allowed. Example } w x/5 -3/x ; 2 ) 4 . 3 ( * 20 { max ) r(w : Membership x/5 -3/x fun(x) w(x) : Function } about_3.4) m(x, (x, { 4) X(about_3. : set Fuzzy 2 ) 4 . 3 ( * 20 about_3.4) m(x, : Membership about_3.40 : f number Fuzzy number} real any x {x; : values base of Set 0 0 0 = − − = = = = − − = x e x e Which values of x should be considered in determining r(w0) ? ) 15 4 /* 25 ( 2 /* 5 0 15 * * 5 x 0 x * w -/5 x -3 w x/5 -3/x 2 0 0 2 /1 0 2 0 2 0 + ± − = = − + = = w w x x w With these values x1 and x2 we finally get } 2 ) 4 . 3 ( * 20 , 2 ) 4 . 3 ( * 20 { max ) r(w : Membership 2 1 0 x e x e − − − − =48 The following tables give respective values which are visualized in the diagram. The values x0 and x1 respectively m(x0,f0) and m(x1,f0) should be identical. The small occasional differences are due to rounding inaccuracies. x0 2.50 2.80 2.90 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 m(x0,f0) 0.00 0.00 0.01 0.04 0.09 0.17 0.29 0.45 0.64 0.82 0.95 1.00 w0 0.70 0.51 0.45 0.40 0.37 0.34 0.32 0.30 0.27 0.25 0.23 0.20 x1 2.50 2.80 2.91 3.00 3.06 3.12 3.15 3.19 3.26 3.30 3.34 3.41 x2 -6.00 -5.35 -5.16 -5.00 -4.91 -4.82 -4.75 -4.49 -4.61 -4.55 -4.49 -4.41 m(x1,f0) 0.00 0.00 0.01 0.04 0.10 0.21 0.29 0.41 0.68 0.82 0.93 1.00 m(x2,f0) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 r(w0) 0.00 0.00 0.01 0.04 0.10 0.21 0.29 0.41 0.68 0.82 0.93 1.00 Table 20 Example of a relation x0 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.9 4.0 4.5 m(x0,f0) 1.00 0.95 0.82 0.64 0.45 0.29 0.17 0.09 0.04 0.01 0.00 0.00 w0 0.20 0.18 0.16 0.14 0.11 0.09 0.07 0.05 0.03 -0.01 -0.05 -0.23 x1 3.41 3.45 3.49 3.54 3.61 3.65 3.70 3.75 3.80 3.90 4.00 4.49 x2 -4.41 -4.35 -4.29 -4.24 -4.16 -4.10 -4.05 -4.00 -3.95 -3.85 -3.75 -3.34 m(x1,f0) 1.00 0.95 0.85 0.68 0.41 0.29 0.17 0.09 0.04 0.01 0.00 0.00 m(x2,f0) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 r(w0) 1.00 0.95 0.85 0.68 0.41 0.29 0.17 0.09 0.04 0.01 0.00 0.00 Table 21 Example of a relation (continued) -0.1 +0.35 .... +2.75 +3.2 +3.65 0 0,2 0,4 0,6 0,81 m(x), r(w) w resp. x Relation X(about_3.4):-->W r(w) m(x) Picture 349 5.2.2 Several independant variables Relations may be generalized to more than one independent variable. Consider x and y being sharp numbers from two sets of base values {x} and {y } and f and g being fuzzy number from two sets of fuzzy values {f } and {g }. The related memebership functions are m(x,f) resp. n(y,g), the fuzzy sets are X(f) = {(x,m(x,f) } and Y(g) = { (y,n(y,g) }. w=fun(x,y) be a unique function which for any (allowed) pair (x,y) gives w as the "picture" of pair (x,.y): (x,y)→w Let now f0 and g0 be two fuzzy numbers (e.g. f0 = about_3.4 and g0 = about_5.9 ). X(f0) = {(x,m(x,f0) } and Y(g0) = {(y,n(y,g0) } are the related fuzzy sets. We construct fuzzy set W as follows: • Let (x0,m0) and (y0,n0) be any elements of fuzzy sets X(f0) and Y(g0) . Then from x0 and y0 a sharp value w0 is calculated by w0 = fun(x0,y0) • A membership r(w0) is calculated as follows: 1.) All elements (x,m)∈X(f0) and (y,n)∈Y(g0) with fun(x,y)=w0 are collected (besides (x0,y0) there might be other pairs (x,y) with fun(x,y)=w0 ) Let these elements be (x0,m0),(y0,n0); (x1,m1),(y1,n1); ... ;(xN,mN), (yN,nN) 2.) r0 is calculated as r0= S(T(m0,n0), T(m1,n1),...,T(mN,nN) ) with some formerly agreed Norm function T and Conorm function S. • Let W = {(w,r ) } be the set of all pairs (w,r ) = (w0,r0 ) calculated in that way The construction of W can be described equivalently using the concept of fuzzy set product. • Construct the set product Z(f0,g0) = X(f0) * Y(g0) with a suitably chosen T-Norm T(x,y): Z(f0,g0) = { (x,y, µ (x,y) } = Z{ (x,y,T[m(x,f0), n(y,g0)] ) } • To any element ( x0,y0, µ [x0,y0] ) of Z(f0,g0) calculate w0 = fun(x0,y0) • Calculate membership r(w0) as follows 1. ) Collect all elements (x,y, µ [x,y] ) of Z(f0,g0) with fun(x,y)=w0 Let these elements be (x0,y0,µ0); (x1,y1,µ1); ... ;(xN,yN,µN) 2.) r0 is calculated as r0= S(µ0, µ1,...,µN) with an agreed Conorm function S. • Let W = {(w,r ) } be the set of all pairs (w,r ) = (w0,r0 ) calculated in that way We may consider W as being the fuzzy set of a certain (yet unnamed) fuzzy number p0 defiine on the set {w} of base values w. If necessary we give that fuzzy set p0 a suitable name and regard it as the "picture" p0=fun(f0, g0) of pair (f0,g0) under function fun(x,y) similarly as sharp number w=fun(x,y) is the "picture" of sharp number pair (x,y) under the same function: (f0,g0): → p0 . Generally for any fuzzy number pair (f,g) : (f,g): → p or p=fun(f,g). Again the same restrictions as before apply if the number of elements of Z is infinite. In that case we delimit ourselves to the special cases T(x,y) = min(x,y) and S(x,y) = max(x,y). Generalization to an arbitrary number of variables is obvious and does not need discussion.50 Example y * x y) fun(x, y) w(x, : Function } about_5.9) n(y, (y, { 9) Y(about_5. } about_3.4) m(x, (x, { 4) X(about_3. : sets Fuzzy 2 ) 9 . 5 ( * 10 about_5.9) n(y, 2 ) 4 . 3 ( * 20 about_3.4) m(x, : s Membership about_5.9 g about_3.40 f : numbers Fuzzy number} real any of pair y) (x, y); {(x, : values base of Set 00 = = == − − = − − = == y e x e Which values of x should be considered in determining r(w0) ? /x w y w y * x 0 0 = = → The following table shows for combinations (x0,y0) in the ranges 3.0≤x0≤3.8 and 5.4≤y0≤6.4 the resulting values w0 = x0*y0 together with memberships µ(w0) = min [m(x0,f0), n(y0,g0) ]. For product values w0 ≈16 to 24 in steps of ≈0.5 the positions with maximum membership µ(w0) are underlined.51 A B C D E F G H I x0 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 m(x0,f0) 0.04 0.17 0.45 0.82 1.00 0.82 0.45 0.17 0.04 y0 n(y0,g0) 5.4 16.2 16.7 17.3 17.8 18.4 18.9 19.4 20.0 20.5 a 0.08 0.04 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.04 b 5.5 16.5 17.1 17.6 18.2 18.7 19.3 19.8 20.4 20.9 c 0.20 0.04 0.17 0.20 0.20 0.20 0.20 0.20 0.17 0.04 d 5.6 16.8 17.4 17.9 18.5 19.0 19.6 20.2 20.7 21.3 e 0.41 0.04 0.17 0.41 0.41 0.41 0.41 0.41 0.17 0.04 f 5.7 17.1 17.7 18.2 18.8 19.4 20.0 20.5 21.1 21.7 g 0.67 0.04 0.17 0.45 0.67 0.67 0.67 0.45 0.17 0.04 h 5.8 17.4 18.0 18.6 19.0 19.7 20.3 20.9 21.5 22.0 i 0.90 0.04 0.17 0.45 0.82 0.90 0.82 0.45 0.17 0.04 j 5.9 17.7 18.3 18.9 19.5 20.1 20.7 21.2 21.9 22.4 k 1.00 0.04 0.17 0.45 0.82 1.00 0.82 0.45 0.17 0.04 l 6.00 18.0 18.6 19.2 19.8 20.4 21.0 21.6 22.2 22.8 m 0.90 0.04 0.17 0.45 0.82 0.90 0.82 0.45 0.17 0.04 n 6.1 18.3 18.9 19.5 20.2 20.7 21.4 22.0 22.6 23.2 o 0.67 0.04 0.17 0.45 0.67 0.67 0.67 0.45 0.17 0.04 p 6.2 18.6 19.2 19.8 20.5 21.1 21.7 22.3 22.9 23.6 q 0.41 0.04 0.17 0.41 0.41 0.41 0.41 0.41 0.17 0.04 r 6.3 18.9 19.5 20.2 20.8 21.4 22.1 22.7 23.3 23.9 s 0.20 0.04 0.17 0.20 0.20 0.20 0.20 0.20 0.17 0.04 t 6.4 19.2 19.8 20.5 21.1 21.8 22.4 23.0 23.7 24.3 u 0.08 0.04 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.04 v Table 22 Products w0 = x0*y0 and memberships r(w0) Example for using the table: w0 ≈ 19 is obtained by x0=3.5 y0=5.4 m(x0,f0) = 0.82 n(y0,g0) = 0.08 µ(w0) = min [0.82, 0.08) = 0.08 Col. F Lines a/b x0=3.5 y0=5..5 m(x0,f0) = 0.82 n(y0,g0) = 0.20 µ(w0) = min [0.82, 0.20) = 0.20 Col. F Lines c/d x0=3.4 y0=5.6 m(x0,f0) = 1.00 n(y0,g0) = 0.41 µ(w0) = min [1.00, 0.41) = 0.41 Col. E Lines e/f x0=3.3 y0=5.7 m(x0,f0) = 0.82 n(y0,g0) = 0.67 µ(w0) = min [0.82, 0.67) = 0.67 Col. D Lines g/h x0=3.3 y0=5.8 m(x0,f0) = 0.82 n(y0,g0) = 0.90 µ(w0) = min [0.82, 0.90) = 0.82 Col. D Lines i/j x0=3.2 y0=5.9 m(x0,f0) = 0.45 n(y0,g0) = 1.00 µ(w0) = min [0.45, 1.00) = 0.45 Col. C Lines k/l x0=3.2 y0=6.0 m(x0,f0) = 0.45 n(y0,g0) = 0.90 µ(w0) = min [0.45, 0.90) = 0.45 Col. C Lines m/n52 x0=3.1 y0=6.1 m(x0,f0) = 0.17 n(y0,g0) = 0.67 µ(w0) = min [0.17, 0.67) = 0.17 Col. B Lines o/p x0=3.1 y0=6.2 m(x0,f0) = 0.17 n(y0,g0) = 0.41 µ(w0) = min [0.17, 0.41) = 0.17 Col. B Lines q/r x0=3.0 y0=6.3 m(x0,f0) = 0.04 n(y0,g0) = 0.20 µ(w0) = min [0.04, 0.20) = 0.04 Col. A Lines s/t x0=3.0 y0=6.4 m(x0,f0) = 0.04 n(y0,g0) = 0.08 µ(w0) = min [0.04, 0.08) = 0.04 Col. A Lines u/v So we finally get r0(19) = S(µ0, µ1,...,µN) = max (0.08, 0.20, 0.41, 0.67, 0.82, 0.45, 0.45, 0.17, 0.17, 0.04, 0.04 ) = 0.82. as approximate membership of w0 = 19 in fuzzy set fun(f0, g0 ) = f0 * g0 = about_3.4 * about_5.9 Next two tables give memberships r(w0) for product values w0 ≈16 to 24 in steps of ≈0.5 as read from above table. w0 16.2 16.5 17.1 17.6 17.9 18.5 19.0 19.5 20.1 r(w0) 0.04 0.04 0.17 0.20 0.41 0.41 0.82 0.82 1.00 Table 23 Membership p(w0, about_3.4 * about_5.9 ) w0 20.4 21.0 21.4 22.0 22.3 22.9 23.3 23.9 24.3 r(w0) 0.90 0.82 0.67 0.45 0.41 0.17 0.17 0.04 0.04 Table 24 Membership p(w0, about_3.4 * about_5.9 ), continued53 2,53,5 4,5 5,5 6,5 .... 16 21 26 m(x) r(w) 0 0,2 0,4 0,6 0,81 m(x), n(y), r(w) x, y, wRelation w=x*y m(x) n(y) r(w) Picture 454 6 Appendix 6.1 Proofs of common properties 6.1.1 Sequence of magnitude We have to show 0 ≤ 0 if max(x,y) < 1 Drastic product (1) 0 ≤ min(x,y) if max(x,y) = 1 (2) ≤ max(0,x+y-1) Limited difference (3) ≤ x*y /{ 1+(1-x)*(1-y) } Einstein product (4) ≤ x*y Algebraic product (5) ≤ x*y /{1-(1-x)*(1-y) } Hamacher product (6) ≤ min(x,y) Minimum (7) ≤ max(x,y) Maximum (8) ≤ 1 -(1-x)*(1-y) /(1-x*y) Hamacher sum ( 9) ≤ 1 -(1-x)*(1-y) Algebraic sum (10) ≤ 1 -(1-x)*(1-y) /(1+x*y) Einstein sum (11) ≤ min(1,x+y) Limited sum (12) min(1,x+y) ≤ max(x,y) if min(x,y) = 0 (13) 1 ≤ 1 if min(x,y) > 0 Drastic sum (14) Here are the proves 1) 0 ≤ 0 is always true 2) 0 ≤ min[x,y] is always true 3) if max(x,y) < 1 then drastic product(x,y) = 0 ≤ max[0,x+y-1] = limited difference (x,y) if max(x,y) = 1 then x = 1 or y=1 if x =1 then x+y-1 = y; hence drastic product (x,y) = y = max(0,y) = max(0,x+y-1); if y=1 then similar; 4) if x+y≤1 then max[0,x+y-1] = 0 ≤ xy/{1+[1-x][1-y]} if x+y>1 then 2-x-y ≥ 0 2-x-y+x*y ≥ x*y {1+[1-x]*[1-y]} ≥ x*y 1 ≥ x*y /{1+(1-x)*(1-y) } (1-x)*(1-y) ≥ x*y*(1-x)*(1-y) /{1+(1-x)*(1-y) } -(1-x)*(1-y) ≤ -x*y*(1-x)*(1-y) /{1+(1-x)*(1-y) } -(1-x)*(1-y) ≤ { -x*y-xy*(1-x)*(1-y)+xy } /{1+(1-x)*(1-y) }55 -xy+x+y-1 ≤ x*y /{1+(1-x)*(1-y) } -x*y x+y-1 ≤ x*y /{1+(1-x)*(1-y) } max[0,x+y-1] ≤ x*y /{1+(1-x)*(1-y) } 5) 1 ≤ 1+(1-x)*(1-y) 1 /{ 1+(1-x)*(1-y) } ≤ 1 x*y /{ 1+(1-x)*(1-y) } ≤ x*y 6) 1-(1-x)*(1-y) ≤ 1 1 ≤ 1 /{1-(1-x)*(1-y) } x*y ≤ x*y /{ 1-(1-x)*(1-y) } 7) if x ≤ ythen 0 ≤ x+(1-y) 0 ≤ x-x*y y ≤ x+y-x*y y ≤ 1-(1-x)*(1-y) y /{ 1-(1-x)*(1-y)} ≤ 1 x*y /{ 1-(1-x)*(1-y) } ≤ x x*y /{ 1-(1-x)*(1-y) } ≤ min[x,y] if y ≤ xthen interchange of names x ↔ y in above inequalities leads to the same final result as the last inequality is invariant against such interchange 8) min[x,y] ≤ max[x,y] 9) if x ≤ ythen 0 ≤ x*(1-y)2 0 ≤ x*(1-2*y+y2) 0 ≤ x-2*x*y+x*y2 y-x*y*y ≤ x+y-2x*y y*(1-x*y) ≤ x+y-2x*y y ≤ { x+y-2x*y }/(1-x*y) max[x,y] ≤ { x+y-2x*y }/(1-x*y) max[x,y] ≤ { (1-x*y)-(1-x)*(1-y) }/(1-x*y) max[x,y] ≤ 1 -(1-x)*(1-y) /(1-x*y) if y ≤ xthen interchange of names x ↔ y in above inequalities leads to the same final result as the last inequality is invariant against such interchange 10) 1 ≥ (1-x*y) 1 /(1-x*y) ≥ 1 (1-x)*(1-y) /(1-x*y) ≥ (1-x)*(1-y) -(1-x)*(1-y) /(1-x*y) ≤ -(1-x)*(1-y) 1 -(1-x)*(1-y) /(1-x*y) ≤ 1 -(1-x)*(1-y)56 11) 1+x*y ≥ 1 1 ≥ 1 /(1+x*y) -1 ≤ -1 /(1+x*y) 1 -(1-x)*(1-y) ≤ 1 -(1-x)*(1-y) /(1+x*y) 12) if x+y ≤ 1 then { x+y } /(1+x*y) ≤ x+y {1+x*y-(1-x)*(1-y) } /(1+x*y) ≤ x+y 1 -(1-x)*(1-y) /(1+x*y) ≤ x+y 1 -(1-x)*(1-y) /(1+x*y) ≤ min(1,x+y) if x+y>1then 1-y+y ≤ 1 x*(1-y)+y ≤ 1 x-x*y+y ≤ 1 x+y ≤ (1+xy) { 1+x*y-(1-x)*(1-y) } /(1+x*y) ≤ 1 1 -(1-x)*(1-y) /(1+x*y) ≤ 1 1 -(1-x)*(1-y) /(1+x*y) ≤ min(1,x+y) 13) if x=0 then min(1,x+y) = y = max(x,y) if y=0 then similar; 14) 1 ≤ 1 is always true57 6.1.2 Commutivity Drastic product If max(x,y) < 1 then T(x,y) = T(y,x) = 0 If max(x,y) = 1 then if x =1 then T(x,y) = min(x,y) = min(1,y) = y = min(y,1) = min(y,x) = T(y,x) if y = 1 similar Limited difference T(x,y) = max(0,x+y-1) = max(0,y+x-1) = T(y,x) Einstein product T(x,y) = x*y /[1+(1-x)*(1-y) ] = y*x /[1+(1-y)*(1-x) ] = T(y,x) Algebraic product T(x,y) = x*y = y*x = T(y,x) Hamacher product T(x,y) = x*y /[ 1-(1-x)*(1-y) ] = y*x /[ 1-(1-y)*(1-x) ] = T(y,x) Minimum T(x,y) = min(x,y) = min(y,x) = T(y,x) Maximum S(x,y) = max(x,y) = max(y,x) = S(y,x) Hamacher sum S(x,y) = 1-[1-x]*[1-y] /[1-x*y] = 1-[1-y]*[1-x] /[1-y*x] = S(y,x) Algebraic sum S(x,y) = 1 -[1-x]*[1-y] = 1 -[1-y]*[1-x] = S(y,x) Einstein sum S(x,y) = 1-[1-x]*[1-y]/[1+x*y] = 1-[1-y]*[1-x]/[1+y*x] = S(y,x) Limited sum S(x,y) = min(1,x+y) = min(1,y+x) = S(y,x)58 Drastic sum: if min(x,y) > 0 then S(x,y) = S(y,x) = 1 if min(x,y) = 0 then if x = 0 then S(x,y) = max(x,y) = max(0,y) = y = max(y,0) = max(y,x) = S(y,x) if y = 0 similar 6.1.3 Associativity We have to show T[T(x,y),z] = T[x,T(y,z)] Drastic product x y z T(x,y) T(y,z) T[T(x,y),z] T[x,T(y,z)] --------------------------------------------------------------------------------------------0 0 0 0 0 0 0 0 0 r 0 0 0 0 0 0 1 0 0 0 0 0 r 0 0 0 0 0 0 r r 0 0 0 0 0 r 1 0 r 0 0 0 1 0 0 0 0 0 0 1 r 0 r 0 0 0 1 1 0 1 0 0 r 0 0 0 0 0 0 r 0 r 0 0 0 0 r 0 1 0 0 0 0 r r 0 0 0 0 0 r r r 0 0 0 0 r r 1 0 r 0 0 r 1 0 r 0 0 0 r 1 r r r 0 0 r 1 1 r 1 r r 1 0 0 0 0 0 0 1 0 r 0 0 0 0 1 0 1 0 0 0 0 1 r 0 r 0 0 0 1 r r r 0 0 0 1 r 1 r r r r 1 1 0 1 0 0 0 1 1 r 1 r r r 1 1 1 1 1 1 159 Limited difference T[T(x,y),z] = max [ 0,T(x,y)+z-1 ] = max [ 0,max(0,x+y-1)+z-1 ] = max[0,x+y+z-2] T[x,T(y,z)] = max [ 0,x+T(y,z)-1 ] = max [ 0,x+max(0,y+z-1)-1 ] = max[0,x+y+z-2] Einstein product T(x,y) * z T[T(x,y),z] = --------------------------1 + {1-T(x,y) }*[1-z] x*y /[1+(1-x)*(1-y) ] * z = -------------------------------------------(A) 1 + {1-xy /[ 1+(1-x)*(1-y) ] }*[1-z] x*y*z = -----------------------------------------------------------[ 1+(1-x)*(1-y) ] + { 1+(1-x)*(1-y)-x*y } * [1-z] x*y*z = ------------------------------------1+(1-x)*(1-y) + {2-x-y}*[1-z] x*y*z = ----------------------------------------------1+(1-x)*(1-y) + { (1-x)+(1-y) }*[1-z] x*y*z = ---------------------------------------------------(B) 1 + (1-x)*(1-y) + (1-y)*(1-z) + (1-z)*(1-x) x * T(y,z) T[x,T(y,z)] = -------------------------1 + [1-x]*{ 1-T(y,z) } x * y*z /[ 1+(1-y)*(1-z) ] = ---------------------------------------------1 + [1-x]*{ 1-y*z /[ 1+(1-y)*(1-z) ] } y*z /[ 1+(1-y)*(1-z) ] * x = ---------------------------------------------(C) 1 + { 1-y*z /[ 1+(1-y)*(1-z) ] }*[1-x] (C) results from (A) by cyclic interchanging x→y→z→x. Therefore continuation of the calculaatio leads to a result similar to (B) also with cyclic interchange of variables. But as (B) is invariant against such an interchange the prove is complete.60 Algebraic product T[T(x,y),z] = T(x,y)*z = x*y*z = x*T(y,z) = T[x,T(y,z)] Hamacher product T(x,y) * z T[T(x,y),z] = --------------------------1 -{ 1-T(x,y) }*(1-z) x*y /[ 1-(1-x)*(1-y) ] * z = -------------------------------------------(A) 1 -{ 1-xy /[ 1-(1-x)*(1-y) ] }*(1-z) x*y*z = --------------------------------------------------------[ 1-(1-x)*(1-y) ] -{ 1-(1-x)*(1-y)-x*y }*(1-z) x*y*z = -------------------------------------x+y-x*y -{ x+y-2x*y }*(1-z) x*y*z = -----------------------------(B) x*y+y*z+z*x-2*x*y*z x*y*z = ----------------------------------------------------------------------1-(1-x)*(1-y)-(1-y)*(1-z)-(1-z)*(1-x)+2*(1-x)*(1-y)*(1-z) x * T(y,z) T[x,T(y,z)] = --------------------------1 -(1-x)*{ 1-T(y,z) } x * y*z /[ 1-(1-y)*(1-z) ] = -------------------------------------------1 -(1-x)*{1-y*z /[1-(1-y)*(1-z) ] } y*z /[ 1-(1-y)*(1-z) ] * x = ---------------------------------------------(C) 1 -{ 1-y*z /[ 1-(1-y)*(1-z) ] }*(1-x) (C) results from (A) by cyclic interchanging x→y→z→x. Therefore continuation of the calculaatio leads to a result similar to (B) also with cyclic interchange of variables. But as (B) is invariant against such an interchange the prove is complete.61 Minimum T[T(x,y),z] = min[T(x,y),z] = min[min(x,y),z] = min[x,y,z] = min[x,min(y,z)] = min[x,T(y,z)] = T[x,T(y,z)] Maximum S[S(x,y),z] = max[S(x,y),z] = max[max(x,y),z] = max[x,y,z] = max[x,max(y,z)] = max[x,S(y,z)] = S[x,S(y,z)] Hamacher sum [1-S(x,y)] * (1-z) S[S(x,y),z] = 1 -----------------------1 -S(x,y) * z { 1-[ 1-(1-x)*(1-y) /(1-xy) ] } * (1-z) = 1 ----------------------------------------------(A) 1 -{ 1-(1-x)*(1-y) /(1-x*y) } * z (1-x)*(1-y) /(1-x*y) * (1-z) = 1 ---------------------------------------------1 -[ 1-x*y-(1-x)*(1-y) ] /(1-x*y) * z (1-x)*(1-y)*(1-z) = 1 ---------------------------------------1-x*y -[ 1-x*y-(1-x)*(1-y) ] * z (1-x)*(1-y)*(1-z) = 1---------------------------------------1-x*y -[ 1-x*y-1+x+y-x*y ] * z (1-x)*(1-y)*(1-z) = 1 ------------------------------(B) 1-x*y-y*z-z*x+2*x*y*z (1-x) * [1-S(y,z)] S[x,S(y,z)] = 1 ----------------------1 -x * S(y,z) (1-x) * { 1-[ 1-(1-y)*(1-z) /(1-y*z) ] } = 1 -----------------------------------------------1 -x * { 1-(1-y)*(1-z) /(1-y*z) } { 1-[ 1-(1-y)*(1-z) /(1-y*z) ] } * (1-x) = 1 -----------------------------------------------(C) 1 -{ 1-(1-y)*(1-z) /(1-y*z) } * x62 (C) results from (A) by cyclic interchanging x→y→z→x. Therefore continuation of the calculaatio leads to a result similar to (B) also with cyclic interchange of variables. But as (B) is invariant against such an interchange the prove is complete. Algebraic sum S[S(x,y),z] = 1-{1-S(x,y)}*(1-z) = 1-{1-[ 1-(1-x)*(1-y) ] }*(1-z) = 1-(1-x)*(1-y)*(1-z) S[x,S(y,z)] = 1-(1-x)*{1-S(y,z) } = 1-(1-x)*{ 1-[ 1-(1-y)*(1-z) ] }= 1-(1-x)*(1-y)*(1-z) Einstein sun S[S(x,y),z] = 1 -{ 1-S(x,y) }*(1-z) /{ 1+S(x,y)*z } { 1 -[ 1-(1-x)*(1-y) /(1+x*y) ] } * (1-z) = 1 ---------------------------------------------------1 + [ 1-(1-x)*( 1-y) /(1+x*y) ] * z (1-x)*( 1-y)/(1+x*y) * (1-z) = 1 ------------------------------------------------(A) 1 + [ 1+x*y-(1-x)*( 1-y) ] /(1+x*y) * z (1-x)*( 1-y)*( 1-z) = 1 --------------------------------------------1+x*y + [ 1+x*y-(1-x)*( 1-y) ] * z (1-x)*( 1-y)*( 1-z) = 1 ------------------------------------------1+x*y + [ 1+x*y-1+x+y-x*y ] * z (1-x)*( 1-y)*( 1-z) = 1 -------------------------1+x*y + [ x+y ] * z (1-x)*( 1-y)*( 1-z) = 1 -----------------------(B) 1+x*y+yz+zx S[ x,S(y,z) ] = 1 -(1-x)*{ 1-S(y,z) } /{ 1+x*S(y,z) } (1-x) * { 1 -[ 1-(1-y)*( 1-z) /(1+yz) ] } = 1 -------------------------------------------------1 + x * [ 1-(1-y)*( 1-z) /(1+yz) ] (1-y)*( 1-z) /(1+yz) * (1-x) = 1 ---------------------------------------------(C) 1 + [ 1+yz-(1-y)*( 1-z) ] /(1+yz) * x63 (C) results from (A) by cyclic interchanging x→y→z→x. Therefore continuation of the calculaatio leads to a result similar to (B) also with cyclic interchange of variables. But as (B) is invariant against such an interchange the prove is complete. Limited sum S[S(x,y),z] = min[ 1,S(x,y)+z ] = min[ 1,min(1,x+y)+z ] = min[ 1,x+y+z ] S[x,S(y,z)] = min[ 1,x+S(y,z) ] = min[ 1,x+min(1,y+z) ] = min[ 1,x+y+z ] Drastic sum x y z S(x,y) S(y,z) S[S(x,y),z] S[x,S(y,z)] --------------------------------------------------------------------------------------------0 0 0 0 0 0 0 0 0 r 0 r r r 0 0 1 0 1 1 1 0 r 0 r r r r 0 r r r 1 1 1 0 r 1 r 1 1 1 0 1 0 1 1 1 1 0 1 r 1 1 1 1 0 1 1 1 1 1 1 r 0 0 r 0 r r r 0 r r r 1 1 r 0 1 r 1 1 1 r r 0 1 r 1 1 r r r 1 1 1 1 r r 1 1 1 1 1 r 1 0 1 1 1 1 r 1 r 1 1 1 1 r 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 r 1 r 1 1 1 0 1 1 1 1 1 1 r 0 1 r 1 1 1 r r 1 1 1 1 1 r 1 1 1 1 1 1 1 0 1 1 1 1 1 1 r 1 1 1 1 1 1 1 1 1 1 1 64 6.1.4 Monotony Assume 0≤x≤u≤1, 0≤y≤v≤1 Drastic product T(x,y) = 0 if max(x,y)<1 = min(x,y) if max(x,y)=1 If max(u,v) <1 then max(x,y) < 1 hence T(x,y) = 0 and T(u,v) = 0 If max(u,v) = 1 and max(x,y) < 1 then T(x,y) = 0 ≤ min(u,v) = T(u,v) If max(u,v) = 1 and max(x,y) = 1 then T(x,y) = min(x,y) ≤ min(u,v) = T(u,v) Limited difference T(x,y) = max(0,x+y-1) ≤ max(0,u+v-1) = T(u,v) Algebraic product T(x,y) = x*y ≤ u*v = T(u,v) Hamacher product T(x,y) = x*y /[1-(1-x)*(1-y) ] dT(x,y) /dx = { y*[ 1-(1-x)*( 1-y) ] -(1-y)*x*y } /N2 = { y*(x+y-x*y) -x*y + x*y2 } /N2 = { x*y+y2-x*y2-x*y + x*y2 } /N2 = y2 /N2 ≥ 0 dT(x,y)/dy = { x*[ 1-(1-x)*( 1-y) ] -(1-x)*x*y } /N2 = { x*(x+y-x*y)-x*y + x2y } /N2 = { x2+x*y-x2y-x*y + x2y } /N2 = x2 /N2 ≥ 0 Minimum T(x,y)=min(x,y) ≤ min(u,v)=T(u,v) Maximum S(x,y)=max(x,y) ≤ max(u,v)=S(u,v) Hamacher sum S(x,y) = 1 -(1-x)*( 1-y) /(1-x*y)65 dS(x,y)/dx = -[ -(1-y)*( 1-x*y) + y(1-x)*( 1-y) ] /N2 = -[ -1+y+x*y-x*y2 + y-x*y-y2+x*y2 ] /N2 = -[ -1+2*y-y2 ] /N2 = (1-2*y+y2) /N2 = (1-y)2 /N2 ≥ 0 dS(x,y)/dy = -[ -(1-x)*( 1-x*y)+x(1-x)*( 1-y) ] /N2 = -[ -1+x+x*y-x2y +x-x2-x*y+x2y ] /N2 = -[ -1+2*x-x2 ] /N2 = (1-2*x+x2) /N2 = (1-x)2 /N2 ≥ 0 Algebraic sum S(x,y) = 1-(1-x)*(1-y) ≤ 1-(1-u)*(1-v) = S(u,v) Limited sum S(x,y) = min(1,x+y) ≤ min(1,u+v) = S(u,v) Drastic sum S(x,y) = 1 if min(x,y) > 0 = max(x,y) if min(x,y) = 0 If 0 0 = max[x,y] falls min[x,y] == 0 If max(x,y) < 1 then min(1-x,1-y) > 0 hence T(x,y) = 0 and S(1-x,1-y) = 1 hence T(x,y) = 0 = 1 -1 = 1 -S(1-x,1-y)66 If max(x,y) = 1 then min(1-x,1-y) = 0 if x=1 then 1-x=0 hence T(x,y) = y and S(1-x,1-y) = 1-y hence T(x,y) = y = 1-(1-y) = 1 -S(1-x,1-y) if y=1 similar If min(x,y) > 0 then max(1-x,1-y) < 1 hence S(x,y) = 1 and T(1-x,1-y) = 0 hence S(x,y) = 1 = 1 -0 = 1 -T(1-x,1-y) If min(x,y) = 0 then max(1-x,1-y) = 1 if x=0 then 1-x=1 hence S(x,y) = y and T(1-x,1-y) = 1-y hence S(x,y) = y = 1-(1-y) = 1 -T(1-x,1-y) if y=1 similar Limited sum /Limited difference T(x,y) = S(x,y) = min (1,x+y ) If x+y≤1then 2 -(x+y) ≥1 hence T(x,y) = max (0,x+y-1 ) = 0 S(1-x,1-y) = min[1,(1-x)+(1-y)] = min[1, 2-(x+y)] = 1 hence T(x,y) = 0 = 1-1 = 1-S(1-x,1-y) If x+y>1then 2 -(x+y) < 1 hence T(x,y) = max (0,x+y-1 ) = x+y-1 S(1-x,1-y) = min[1,(1-x)+(1-y)] = min[1, 2-(x+y)] = 2-(x+y) hence T(x,y) = x+y-1 = 1-[ 2-(x+y) ] = 1-S(1-x,1-y) If x+y≤1then 1 -(x+y) ≥0 hence S(x,y) = min (1,x+y ) = x+y T(1-x,1-y) = max[ 0, (1-x)+(1-y)-1 ] = max[ 0, 1-(x+y) ] = 1-(x+y) hence S(x,y) = x+y = 1-[ 1-(x+y) ] = 1-T(1-x,1-y) If x+y>1then 1 -(x+y) < 0 hence S(x,y) = min (1,x+y ) = 1 T(1-x,1-y) = max [0,(1-x)+(1-y)-1] = max[0, 1-(x+y)] = 0 hence S(x,y) = 1 = 1-0 = 1 = 1-T(1-x,1-y)67 Einstein product /Einstein sum x*y /[ 1+(1-x)*(1-y) ] = 1 -[ 1-x]*[1-y] /[1+x*y] x*y /[ 1+(1-x)*( 1-y) ] = x*y /[ 1+(1-x)*( 1-y) ] x*y /[ 1+(1-x)*( 1-y) ] = 1 -{ 1-x*y /[ 1+(1-x)*( 1-y) ] } x*y /[ 1+(1-x)*( 1-y) ] = 1 -{ 1-[1-(1-x)]*[1-(1-y)] /[ 1+(1-x)*( 1-y) ] } T(x,y) = 1 -S(1-x,1-y) 1 -(1-x)*(1-y) /( 1+x*y ) = 1-(1-x)*(1-y) /( 1+x*y ) 1 -(1-x)*(1-y) /( 1+x*y ) = 1-(1-x)*(1-y) /{1+[ 1-(1-x) ]*[ 1-(1-y) ] } Algebraic product /Algebraic sum T(x,y) = x*y S(x,y) = 1 -[1-x]*[1-y] x*y = 1 -(1 -x*y) x*y = 1 -{ 1 -[ 1-(1-x) ]*[ 1-(1-y) ] } T(x,y) = 1 -S(1-x,1-y) 1-(1-x)*(1-y) = 1 -(1-x)*(1-y) S(x,y) = = 1 -T(1-x,1-y) Hamacher product /Hamacher sum T(x,y) = x*y /[ 1-(1-x)*(1-y) ] S(x,y) = 1 -(1-x)*[ (1-y] /[1-x*y ] xy /[ 1-(1-x)(1-y) ] = xy /[ 1-(1-x)*(1-y) ] xy /[ 1-(1-x)(1-y) ] = 1 -{ 1-[ 1-(1-x) ]*[ 1-(1-y) ] /[ 1-(1-x)*(1-y) ] } T(x,y) = 1 -S(1-x,1-y) (1-x)*(1-y) /(1-x*y) = (1-x)*(1-y) /(1-x*y) 1 -(1-x)*(1-y) /(1-x*y) = 1 -(1-x)*(1-y) /{ 1-[1 -(1-x) ]*[ 1-(1-y) ] } S(x,y) = 1 -T(1-x,1-y) Minimum /Maximum T(x,y) = min(x,y) S(x,y) = max(x,y) If x≤y then 1-x ≥ 1-y hence T(x,y) = min(x,y) = x S(1-x,1-y) = max(1-x,1-y) = 1-x hence T(x,y) = x = 1 -(1-x) = 1 -S(1-x,1-y) If x>y then similar68 6.1.6 Special values of scalable functions 6.1.6.1 Hamacher intersection P(x,y) = x*y /[ α -(α -1)*(x+y-x*y) ] α=0 P(x,y) = x*y /(x+y-x*y) = x*y /[ 1-(1-x)*(1-y) ] Hamacher product α=1 P(x,y) = x*y Algebraic product α=2 P(x,y) = x*y /[ 2-(x+y-x*y) ] = x*y /[ 1+(1-x)*(1-y) ] Einstein product α=∞ If max(x,y)<1 then x+y-x*y<1 hence x+y-x*y = 1-z with z>0 P(x,y) = x*y /[ α -(α -1)*(1-z) ] = x*y /( α -α +1+ α *z-z) = x*y /(1+ α* z-z) P(x,y)→0 for α→∞ because z>0 If max(x,y)=1 then if x=1 then P(x,y) = y /[ α -( α -1) ] = y if y=1 then P(x,y) = x /[ α -( α -1) ] = x Summary: lim P(x,y) = 0 for α→∞ if max(x,y)<1 lim P(x,y) = min(x,y) for α→∞ if max(x,y)=1 6.1.6.2 Hamacher junction Q(x,y) = [ (α -1)*x*y+x+y ] /(1+ α*x*y) α =-1 Q(x,y) = (-2*x*y+x+y) /(1-x*y) = [ 1-x*y-(1-x)*(1-y) ] /(1-x*y) = 1-[ (1-x)*(1-y) ] /(1-x*y) Hamacher sum α =0 Q(x,y) = -x*y+x+y = 1-(1-x)*(1-y) Algebraic sum α =1 Q(x,y) = (x+y) /(1+x*y) = [ 1+x*y-(1-x)*(1-y) ] /(1-x*y) = 1-[ (1-x)*(1-y) ] /(1+x*y) Einstein sum69 α →∞ If min(x,y)>0 then xy>0, Q(x,y) → (α*x*y) /(α*x*y) = 1 If min(x,y)=0 then xy=0, Q(x,y) = x+y = max(x,y) Summary: lim Q(x,y) = 1 for α→∞ if min(x,y)>0 lim Q(x,y) = max(x,y) for α→∞ if min(x,y)=0 6.1.6.3 Yager intersection P(x,y) = 1 -min{1, α √[(1-x)α + (1-y)α ]} α =1 P(x,y) = 1 -min {1,(1-x)+(1-y) } = max(0,x+y-1) Limited difference α→∞ If x>y then (1-x)<(1-y) hence P(x,y) → 1-min{1, α √ (1-y)α } = 1-min(1,1-y) = y = min(x,y) If x(1-y) hence P(x,y) → 1-min{1, α√(1-x)α ] } = 1-min(1,1-x) = x = min(x,y) If x=y then P(x,y) = 1 -min{1, α√2*(1-x)α] } = 1 -min{1, α√2*(1-x) } hence P(x,y) → 1 -min{ 1,1*(1-x) } = 1-(1-x) = x = min(x,y) Summary: lim P(x,y) = min(x,y) for α→∞ Minimum 6.1.6.4 Yager junction Q(x,y) = min{1, α√[x α + y α ]} α =1 Q(x,y) = min(1,x+y) Limited sum α→∞ If x>y then Q(x,y) → min{1, α√[x α ]} = min(1,x) = x = max(x,y) If x

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