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Collection Technique .......................................................................... Cahier technique no 191 Fuzzy logic F. Chevrie F. Guély Cahiers Techniques are a collection of documents intended for engineers and technicians, people in the industry who are looking for information in greater depth in order to complement that given in display product catalogues. These Cahiers Techniques go beyond this stage and constitute practical training tools. They contain data allowing to design and implement electrical equipement, industrial electronics and electrical transmission and distribution. Each Cahier Technique provides an in-depth study of a precise subject in the fields of electrical networks, protection devices, monitoring and control and industrial automation systems. The latest publications can be downloaded on Internet from the Schneider server. Code: http://www.schneider-electric.com Section: Mastering electricity Please contact your Schneider representative if you want either a Cahier Technique or the list of available titles. The “Cahiers Techniques” collection is part of the Groupe Schneider’s “Technical series”. Foreword The author disclaims all responsibility further to incorrect use of information or diagrams reproduced in this document, and cannot be held responsible for any errors or oversights, or for the consequences of using information and diagrams contained in this document. Reproduction of all or part of a Cahier Technique is authorised with the prior consent of the Scientific and Technical Division. The statement “Extracted from Schneider Cahier Technique no..... (please specify)” is compulsory. no 191 Fuzzy logic François CHEVRIE After joining Telemecanique in 1987, he joint the Advanced Automation Laboratory of the Research Division in 1993. A CNAM Industrial Automation engineering graduate, his dissertation was based on the integration of fuzzy logic in Schneider programmable controllers. He played an active part in the preparation of the fuzzy logic product offer for the Micro/Premium PC range, and helped implement this technique, particularly in the car and food industries. François GUELY After graduating from the Ecole Centrale de Paris in 1988, he joined Telemecanique in Japan in 1990 and was awarded his PhD in fuzzy logic based automatic control in 1994. He has been in charge of Schneider’s Advanced Automatic Department since 1995 where he has helped prepare the extension to fuzzy logic of the IEC language standard for programmable controllers. ECT 191 first issued, december 1998 Cahier Technique Schneider no 191 / pp.1 Lexicon Activation: the two discrete values 0 (the element does not See degree of truth. belong...) or 1 (...belongs to the set). A fuzzy set Conclusion: is defined by a membership function which can A rule conclusion is a statement combining a take any real values between 0 and 1. linguistic variable and a linguistic term written Inference: after the then of the rule. A conclusion can be Calculation of the degrees of activation of all the made up of a combination of several statements. rules in the base as well as of all the fuzzy sets Condition: of the linguistic variables contained in the See predicate. conclusions of these rules. Data merge: Knowledge base: Data merge consists of extracting, from several Set of membership functions and rules of a fuzzy pieces of data, one or more items of information system containing expertise, knowledge of the which may be different kinds. operator, expert, etc. For example: from variables R, V and B giving Linguistic term: the colour of a biscuit, the cooking state of the Term associated with a membership function biscuit can be deduced. The term “Sensor characterising a linguistic variable. merge” is also used. Linguistic variable: Defuzzification: Numerical variable with a name (pressure, Conversion, after inference, of a fuzzy set of a temperature… to which are associated inguistic linguistic output variable into a numerical value. terms. Degree of activation: Membership function: See degree of truth. Function µA (x) associating to any input value x Degree of membership: its degree of membership to the set A. This An element x belongs to a fuzzy set A with a gradual value belongs to the [0; 1] interval. degree of membership between 0 and 1, given Predicate: by the membership function µ A (x). Also known as premise or condition, a rule Degree of truth: predicate is a statement combining a linguistic The degree of truth, or degree of activation, of a variable and a linguistic term written between rule is a value y between 0 and 1 deduced from the if and the then of the rule. A predicate can the degrees of membership of the rule be made up of a combination of several predicates. It directly affects the value of the statements linked by AND, OR, NOT operators. conclusions of this rule. The rule is also said to Premise: be active at y. See Predicate. Fuzzification: Sensor merge: Conversion of a numerical value into a fuzzy See Data merge. degree of membership by evaluating a Singleton: membership function. Membership function µA (x), equals to zero for all Fuzzy set: x, except at a singular point x0. In the classical set theory, the characteristic function defines the set: this function only takes Cahier Technique Schneider n o 191 / pp.2 Fuzzy logic Initially a theory, today fuzzy logic has become an operational technique. Used alongside other advanced control techniques, it is making a discrete but appreciated appearance in industrial control automation systems. Fuzzy logic does not necessarily replace conventional control systems. Rather it completes such systems. Its advantages stem from its ability to: c formalise and simulate the expertise of an operator or designer in process control and tuning, c provide a simple answer for processes which are difficult to model, c continually take into account cases or exceptions of different kinds, and progressively incorporate them into the expertise, c take into account several variables and perform “weighted merging” of influencing into variables. How does this technique contribute to industrial process control? What is the effect on product quality and manufacturing cost? Following a few basic theoretical notions, this Cahier Technique answers the questions asked by automatic control engineers and potential users by means of industrial examples, in terms of implementation and competitive advantages. Contents 1 Introduction 1.1 Fuzzy logic today pp. 4 1.2 The history of fuzzy logic pp. 4 1.3 Value and use of fuzzy logic for control pp. 5 2 Theory of fuzzy sets 2.1 Notion of partial membership pp. 6 2.2 Membership functions pp. 6 2.3 Fuzzy logic operators pp. 8 2.4 Fuzzy rules pp. 9 3 A teaching application example 3.1 Introduction pp. 14 3.2 Presentation of the example pp. 14 3.3 Linguistic variables and terms pp. 15 3.4 Rules and outputs pp. 15 4 Implementation 4.1 when can fuzzy rule bases be used? pp. 16 4.2 Designing an application pp. 16 4.3 Using an application pp. 17 4.4 Choosing the implementation technology pp. 17 4.5 Standards pp. 18 5 Fuzzy application 5.1 Application types pp. 19 5.2 Examples of industrial achievements pp. 20 6 Conclusion pp. 24 Appendix pp. 26 Bibliography pp. 28 Cahier Technique Schneider no 191 / pp.3 1 Introduction 1.1 Fuzzy logic today In the majority of present-day applications, fuzzy In continuous and batch production processes, logic allows many kinds of designer and operator as well as in automation systems (which is the qualitative knowledge in system automation to subject of this Cahier Technique), applications be taken into account. have also increased. Fuzzy logic has developed Fuzzy logic began to interest the media at the in this area as it is an essentially pragmatic, beginning of the nineties. The numerous effective and generic approach. It allows applications in electrical and electronic systematisation of empirical knowledge and household appliances, particularly in Japan, which is thus hard to control. The theory of fuzzy were mainly responsible for such interest. sets offers a suitable method that is easy to Washing machines not requiring adjustment, implement in real time applications, and enables camcorders with Steadyshot (TM) image knowledge of designers and operators to be stabilization and many other innovations brought transcribed into dynamic control systems. the term “fuzzy logic” to the attention of a wide This makes fuzzy logic able to tackle automation public. of procedures such as startup and setting of In the car industry, automatic gear changes, parameters, for which few approaches were injection and anti-rattle controls and air previously available. conditioning can be optimized thanks to fuzzy This Cahier Technique describes fuzzy logic and logic. its application to production processes. 1.2 The history of fuzzy logic Appearance of fuzzy logic Boom The term “fuzzy set” first appeared in 1965 when Fuzzy logic experienced a veritable boom in professor Lotfi A. Zadeh from the university of Japan where research was not only theoretical Berkeley, USA, published a paper entitled but also highly application oriented. At the end of “Fuzzy sets”. Since then he has achieved many the eighties fuzzy logic had taken off in a big major theoretical breakthroughs in this field and way, and consumer products such as washing has been quickly joined by numerous research machines, cameras and camcorders with the workers developing theoretical works. mention “fuzzy logic” were too numerous to be counted. Industrial applications such as Initial applications treatment of water, harbour container cranes, At the same time, some researchers turned their undergrounds and ventilation/air conditioning systems began to use fuzzy logic too. Finally, attention to the resolution by fuzzy logic of applications developed in such other fields such problems considered to be difficult. In 1975 as finance and medical diagnosis. professor Mamdani from London developed a From 1990 onwards, many applications began to strategy for process control and published the emerge in large numbers in Germany, as well encouraging results he had obtained for the as, to a lesser extent, in the USA. control of a steam motor. In 1978 the Danish company, F.L. Smidth, achieved the control of a cement kiln. This was the first genuine industrial application of fuzzy logic. Cahier Technique Schneider n o 191 / pp.4 1.3 Value and use of fuzzy logic for control Value Fuzzy rule bases are advantageous in control as Fuzzy logic stems from several observations, they allow: namely: c consideration of existing qualitative expertise, c The knowledge that a human being has of any c consideration of variables the effect of which situation is generally imperfect, would be difficult to model with traditional means, v it can be uncertain (he doubts its validity), but is known in a qualitive way, v or imprecise. c improvement of conventional controller c Human beings often solve complex problems operation by: with approximate data: accuracy of data is often v self-tuning of controller gains off line or on line, useless; for example, in order to choose an v modification of their output (feed forward) apartment he may take into account surface according to events that cannot be taken into area, proximity of shops, distance from the account using a conventional technique. workplace and rent without, however, needing a very precise value for each piece of information. Using knowhow to its best advantage c In industry and technology, operators A vital condition for the use of fuzzy rules is the frequently solve complex problems in a relatively existence of human expertise and knowhow. simple manner without needing to model the Fuzzy rule bases cannot provide a solution when system. Likewise, it is common knowledge that a no-one knows how the system operates or mathematical model is not required to drive a people are unable to manually control it. car, and yet a car is a highly complex system. When such knowhow exists and can be c The more complex a system, the more difficult transcribed in the form of fuzzy rules, fuzzy logic it is to make precise assertions on its behaviour. simplifies its implementation, and operation is then easily understood by the user. The following are naturally deduced from these Fuzzy logic also enables maximum benefit to be observations: derived from practical knowhow, often sought for c rather than modelling the system, it is often in order to prevent loss of knowhow or to share more useful to model the behaviour of a human this knowhow with other people in the company. operator used to control the system; When collecting expertise, unconscious omission c rather than using equations, operation can be of information, the difficulty to explain and the described by qualitatively with an appropriate fear to disclose knowhow are obstacles that are quantitative translation. frequently encountered. This stage must therefore be prepared and conducted with care, Use for control purposes taking into account the human factor. Fuzzy logic is well known by automatic control engineers for its applications in process control If human expertise exists, then fuzzy rules can and monitoring, then commonly referred to as be used, particularly when system knowledge is “fuzzy control”. Just like a conventional tainted by imperfections, when the system is controller, the fuzzy controller is incorporated in complex and hard to model and when the the control loop and computes the control to be method used requires a global view of some of applied to the process according to one or more its aspects. Fuzzy rules do not replace setpoints and one or more measurements taken conventional automatic control methods, rather on the process. they complete these methods. Cahier Technique Schneider no 191 / pp.5 2 Theory of fuzzy sets 2.1 Notion of partial membership In the sets theory, an element either belongs or The notion of a fuzzy set was created in order to does not belong to a set. The notion of a set is take situations of this kind into account. The used in many mathematical theories. This theory of fuzzy sets is based on the notion of essential notion, however, does not take into partial membership: each element belongs account situations which are yet both simple and partially or gradually to the fuzzy sets that have common. Speaking of fruits, it is easy to define been defined. The outlines of each fuzzy set the set of apples. However, it is harder to define (see fig.1 ) are not “crisp”, but “fuzzy” or the set of ripe apples. We understand that an “gradual”. apple ripens progressively... the notion of a ripe apple is thus a gradual one. “Fuzzy” or “gradual” y A B outline t z x “Crisp” outline x belongs neither to A nor B y belongs completely to A z belongs completely to B A: conventional set B: fuzzy set t belongs partially to B Fig. 1 : comparison of a conventional set and a fuzzy set. 2.2 Membership functions A fuzzy set is defined by its “membership range [1.60 m; 1.80 m] and “1” for heights in that function” which corresponds to the notion of a range. The fuzzy set of people of “medium “characteristic function” in classical logic. height” will be defined by a “membership Let us assume that we want to define the set of function” which differs from a characteristic people of “medium height”. In classical logic, we function in that it can assume any value in the would agree for example that people of medium range [0;1]. height are those between 1.60 m and 1.80 m Each possible height will be assigned a “degree tall. The characteristic function of the set of membership” to the fuzzy set of “medium (see fig. 2 ) gives “0” for heights outside the heights” (see fig. 3 ) between 0 and 1. Degree of membership µ Degree of membership µ 1 1 Characteristic function Characteristic function “medium height” “medium height” 0 0 1m60 1m80 Variable: height 1m72 Variable: height Fig. 2 : characteristic function. Fig. 3 : membership function. Cahier Technique Schneider n o 191 / pp.6 A number of fuzzy sets can be defined on the c they are simple, same variable, for example the sets “small c they contain points allowing definition of areas height”, “medium height” and “tall height”, each where the notion is true and areas where it is notion being explained by a membership function false, thereby simplifying the gathering of (see fig. 4 ). expertise. We have chosen to use membership functions of this kind in the rest of this document. µ In some cases, membership functions may be Small Medium Tall equal to 1 for a single value of the variable, and 1 equal to 0 elsewhere. They are then known as 0.7 “singleton membership functions”. A fuzzy singleton (see fig. 6 ) defined on a real variable (height) is the expression in the fuzzy field of a 0.3 specific value (Paul’s height) of this variable 0 (see appendix). 1.60 1.80 2 Height (m) Fig. 4 : membership function, variable and linguistic term. µ This example shows the graduality that enables 1 fuzzy logic to be introduced. A 1.80 m tall person belongs to the “tall” set with a degree of 0.3, and to the set “medium height” with a degree of 0.7. In classical logic, the change from average to tall 0 would be sudden. A 1.80 m person would then 1.78 m Paul's height be of medium height, whereas a 1.81 m person would be tall, an assertion which shocks Fig. 6 : singleton membership function. intuition. The variable (for example: height) as well as the terms (for example: medium, tall) defined by the membership functions, are known Fuzzification - Degree of membership as linguistic variable and linguistic term Fuzzification enables a real value to be respectively. converted into a fuzzy one. As we shall see further on, both linguistic It consists of determining the degree of variables and terms can be used directly in rules. membership of a value (measured by example) to a fuzzy set. For example (see fig. 7 ), if the Membership functions can assume any shape. current value of the “input” variable is 2, the However they are often defined by straight degree of membership to the “low input” segments and said to be “piece-wise linear” membership function is equal to 0.4 which is the (see fig. 5 ). result of the fuzzification. “Piece-wise linear” membership functions are We can also say that the “low input” proposal is frequently used as: true at 0.4. We then talk of degree of truth of the proposal. Degree of membership and degree of truth are therefore similar notions. µ “Totally” medium height µ Small Medium Tall Low 1 Height 0.4 Small “not at all” Tall “not at all” 0 medium medium 2 Input Fig. 5 : piece-wise linear membership functions. Fig. 7 : fuzzification. Cahier Technique Schneider no 191 / pp.7 2.3. Fuzzy logic operators These operators are used to write logic NB: this fuzzy OR is compatible with classical combinations between fuzzy notions, i.e. to logic: 0 OR 1 yields 1. perform computations on degrees of truth. Just as for classical logic, AND, OR and NOT Complement operators can be defined. The logic operator corresponding to the For example: Interesting Apartment = complement of a set is the negation. Reasonable Rent AND Sufficient Surface Area. µ(NOT A) = 1 - µ(A) Choice of operators For example: These operators have many variants (see “Low Temperature” is true at 0.7 appendix). However the most common are the “NOT Low Temperature” that we will normally “Zadeh” operators described below. write as “Temperature NOT Low” is therefore true at 0.3. The degree of truth of a proposal A will be noted µ(A). NB: the negation operator is compatible with classical logic: NOT(0) yields 1 and NOT(1) Intersection yields 0. The logic operator corresponding to the Fuzzy ladder intersection of sets is AND. The degree of truth of the proposal “A AND B” is the minimum value Ladder language or contact language is of the degrees of truth of A and B: commonly used by automatic control engineers to write logic combinations, as it enables their µ(A AND B) = MIN(µ(A),µ(B)) graphic representation. Schneider has For example: introduced the use of ladder representation to “Low Temperature” is true at 0.7 describe fuzzy logic combinations. “Low Pressure” is true at 0.5 Below is an example dealing with the comfort of “Low Temperature AND Low Pressure” is ambient air: therefore true at 0.5 = MIN(0.7; 0.5). hot, damp air is uncomfortable (excessive NB: this fuzzy AND is compatible with classical perspiration); likewise breathing is difficult in air logic 0 and 1, yelds 0. that is cold and too dry. The most comfortable thermal situations are those in which air is hot Union and dry, or cold and damp. This can be The logic operator corresponding to the union of transcribed by the fuzzy ladder in figure 8 sets is OR. The degree of truth of the proposal corresponding to the following combination: “A OR B” is the maximum value of the degrees Good comfort = (Low Temperature AND High of truth of A and B: Humidity) OR (High Temperature AND Low µ(A OR B) = MAX(µ(A),µ(B)) Humidity). For example: It represents a possible definition of the “Low Temperature” is true at 0.7 sensation of comfort felt by a person in a thermal “Low Pressure” is true at 0.5 environment in which air does not move. “Low Temperature OR Low Pressure” is therefore true at 0.7. µ µ Low High Low High Low High Good temperature humidity comfort 10 20 30 °C 50 100 % High Low Temperature Humidity temperature humidity Fig. 8 : fuzzy ladder. Cahier Technique Schneider n o 191 / pp.8 Fuzzy classification belongs to a varying degree to the class of “fresh Classification normally consists of two steps: lettuces”. c preparation: determining the classes to be Classification methods, whether they produce a considered, gradual, boolean or probabilistic result, can be c on line: assigning the elements to classes. developed from: The notions of class and set are identical c an experiment (case of “fuzzy ladder” theoretically. mentioned above), There are three types of assignment methods c examples used for learning purposes (e.g. for according to the result produced: neuron network classifiers), c boolean: the elements either belong or do not c mathematical or physical knowledge of a belong to the classes, problem (for example, the comfort of a thermal situation can be evaluated from thermal balance c probabilistic: the elements have a probability of equations). belonging to boolean classes, such as for example the probability that a patient has measles given Gradual (or fuzzy) classification methods can be the symptoms that he shows (diagnosis), used in control loops. This is the case of the c gradual: the elements have a degree of industrial cooking example for biscuits described membership to the sets; for example a lettuce later on. 2.4. Fuzzy rules Fuzzy logic and artificial intelligence The purpose of fuzzy rule bases is to formalise Inputs Outputs and implement a human being’s method of reasoning. As such it can be classed in the field of artifical intelligence. The tool most commonly used in fuzzy logic Fuz- Defuz- Inferences applications is the fuzzy rule base. A fuzzy rule zification zification base is made of rules which are normally used in parallel but which can also be concatenated in some applications. A rule is of the type: Numerical Fuzzy Numerical values area values IF “predicate” THEN “conclusion”. For example: IF “high temperature and high Fig. 10 : fuzzy processing. pressure” THEN “strong ventilation and wide open valve”. Predicate Fuzzy rule bases, just like conventional expert A predicate (also known as a premise or systems, rely on a knowledge base derived from condition) is a combination of proposals by AND, human expertise. Nevertheless, there are major OR, NOT operators. differences in the characteristics and processing of this knowledge (see fig. 9 ). The “high temperature” and “high pressure” proposals in the previous example are combined A fuzzy rule comprises three unctional parts by the AND operator to form the predicate of the summarised in figure 10 . rule. Fuzzy rule base Conventional rule base (expert system) Few rules Many rules Gradual processing Boolean processing Concatenation possible but scarcely used Concatenated rules A OR B ⇒ C, C ⇒ D, D AND A ⇒ E Rules processed in parallel Rules used one by one, sequentially Interpolation between rules that No interpolation, no contradiction may contradict one another Fig. 9 : fuzzy rule base and conventional rule base. Cahier Technique Schneider no 191 / pp.9 Inference conclusions are uncertain. The theory of The most commonly used inference mechanism possibilities, invented by Lotfi Zadeh, offers an is the “Mamdani” one. It represents a appropriate methodology in such cases. simplification of the more general mechanism Likewise, negation is not used in conclusions for based on “fuzzy implication” and the Mamdani rules. This is because if a rule were to “generalised modus ponens”. These concepts have the conclusion “Then ventilation not are explained in the appendix. Only the average”, it would be impossible to say whether “Mamdani” rule bases are used below. this means “weak ventilation” or “strong ventilation”. This would be yet another case of Conclusion uncertainty. The conclusion of a fuzzy rule is a combination of proposals linked by AND operators. In the Mamdani inference mechanism previous example, “strong ventilation” and “wide c Principle open valve” are the conclusion of the rule. A Mamdani fuzzy rule base therefore contains “OR” clauses are not used in conclusions as they linguistic rules using membership functions to would introduce an uncertainty into the describe the concepts used (see fig. 11 ). knowledge (the expertise would not make it The inference mechanism is made up of the possible to determine which decision should be following steps: made). This uncertainty is not taken into account by the Mamdani inference mechanism which c Fuzzification only manages imprecisions. Therefore the Fuzzification consists of evaluating the “Mamdani” fuzzy rules are not in theory suitable membership functions used in rule predicates, as for a diagnosis of the “medical” kind for which is illustrated in figure 12 : IF “high pressure” AND “high temp.” THEN “valve wide open” µ µ µ High High Wide Pressure Temperature Valve opening IF “average pressure” AND “high temp.” THEN “average valve opening” µ µ µ Average High Average Pressure Temperature Valve opening Fig. 11 : implication. Cahier Technique Schneider n o 191 / pp.10 IF “high pressure” AND “high temp.” THEN “valve wide open” µ µ µ High High Wide 0.5 0.3 2.5 bar 17°C Pressure Temperature Valve opening Fig. 12 : fuzzification. c Degree of activation The degree of activation of a rule is the (see section 2.3.), as shown in figure 13 . The evaluation of the predicate of each rule by logic “AND” is performed by realising the minimum combination of the predicate proposals between the degrees of truth of the proposals. IF “high pressure” AND “high temp.” THEN “valve wide open” µ µ µ Min High 0.5 0.3 } = 0.3 2.5 bar 17°C Pressure Temperature Valve opening Fig. 13 : activation. c Implication The degree of activation of the rule is used to The conclusion fuzzy set is built by realising the determine the conclusion of the rule: this minimum between the degree of activation and operation is called the implication. There are the membership function, a sort of “clipping” several implication operators (see appendix), but of the conclusion membership function the most common is the “minimum” operator. (see fig. 14 ). IF “high pressure” AND “high temp.” THEN “valve wide open” µ µ µ Wide Min 0.5 0.3 } = 0.3 2.5 bar 17°C Pressure Temperature Valve opening Fig. 14 : implication. Cahier Technique Schneider no 191 / pp.11 c Aggregation The output global fuzzy set is built by output. The rules are considered to be linked by aggregation of the fuzzy sets obtained by each a logic “OR”, and we therefore calculate the rule concerning this output. The example below maximum value between the resulting shows the case when two rules act on an membership functions for each rule (see fig. 15 ). IF “high pressure” AND “high temp.” THEN “valve wide open” µ µ µ High High Wide 0.3 2.5 bar 17°C Pressure Temperature Valve opening IF “average pressure” AND “high temp.” THEN “valve wide open” µ µ µ Average High Average 2.5 bar 17°C Pressure Temperature Valve opening µ Aggregation: MAXIMUM Valve opening Fig. 15 : aggregation of rules. Defuzzification “Free” and “able” rules At the end of inference, the output fuzzy set is Fuzzy rule bases, in their general case, use determined, but cannot be directly used to membership functions on system variables, and provide the operator with precise information or rules that can be written textually. Each rule uses control an actuator. We need to move from the its own inputs and outputs, as shown by the “fuzzy world” to the “real world”: this is known as example below: defuzzification. R1: IF “high temperature” A number of methods can be used, the most THEN “high output” common of which is calculation of the “centre of gravity” of the fuzzy set (see fig. 16 ). R2: IF “average temperature” AND “low pressure” THEN “average output” µ ∫xµ(x)dx R3: IF “average temperature” AND “high pressure” ∫ µ(x)dx THEN “low output” R4: IF “low temperature” AND “high pressure” 35.6° Valve opening THEN “very low output” Fig. 16 : defuzzification by centre of gravity. Cahier Technique Schneider n o 191 / pp.12 In diagram form, the “areas of action” of the rules it does not interest us. It is best to verify it as this and their overlapping can be represented in the may be an omission; table in figure 17 . c the first rule only takes temperature into account: this situation is normal in that it reflects the existing expertise. Pressure However, many applications define rule “tables”. In this context, the space is “gridded” and each “box” in the grid is assigned a rule. This approach has the advantage of being systematic, but: Very c it does not always allow simple expression (in Low High low output a minimum number of rules) of the existing output expertise, c it can be applied only for two or three inputs, High output whereas ”free” rule bases can be built with a large number of variables. Average Remarks Low output c The behaviour of a fuzzy rule base is static and non-linear with respect to its inputs. c Fuzzy rule bases are not themselves dynamic, Low Average High Temp. although they often use as inputs variables Fig. 17 : implication represented in a table. expressing system dynamics (derivatives, integrals, etc. ) or time. c The main advantage of the “fuzzy PID” We can make the following observations: controller, often presented as a teaching c not all the space is necessarily covered: the example to give an idea of fuzzy logic, is to make combination “low temperature and low pressure” a non-linear PID, which rarely justifies its use in is not taken into account in this case. The the place of a conventional PID. Moreover it explanation is for example that this combination would be hard to incorporate an existing is not physically possible for this machine or that expertise in this case. Cahier Technique Schneider no 191 / pp.13 3 A teaching application example 3.1 Introduction Most fuzzy logic achievements require preliminary following example is based on a fictitious specialist knowledge of the application area. In application and is designed to illustrate the order to be easily understood by the reader, the procedure for creating a fuzzy rule base. 3.2 Presentation of the example The example concerns a process for washing v Save water lettuce for the production of prepacked lettuce in v Save chlorine. the “fresh produce” counters of supermarkets. The operators manually controlling the process The lettuce is cut, washed and packed. The usually inspect the dirty water at the end of the purpose of washing is to remove earth from the tunnel washing. If the water is clear, they deduce lettuce as well as any micro-organisms which by experience that the lettuce will have a could proliferate during product shelf-life. The “clean” appearance. The decision is thus made manufacturer wishes to automate the washing to install an optic “turbidity” sensor designed to process. determine the degree of transparency of the Washing is a continuous process. The lettuce water. leaves are placed in “drums” which move Moreover, operators use once an hour a report through a “tunnel” fitted with nozzles spraying based on analysis conducted in the factory which chlorinated water. The water removes the earth, gives the ratio of micro-organisms and residual whereas the chlorine kills the micro-organisms chlorine found in washed and prewashed lettuce (see fig. 18 ). at the end of the line. The following priorities were formulated by the The aim is therefore to use the above marketing department and listed in the order of information to improve control of: their importance: c lettuce conveyor belt speed (in order to c With respect to the customer increase production output), v Guarantee quality c the amount of chlorine sprayed, - “Very clean” lettuce (appearance) - No taste of chlorine. c the amount of water sprayed. v Guarantee safety Limits are imposed: - Acceptable level of micro-organisms c on conveyor belt speed, by the mechanism, c With respect to profitability c on water flow to prevent damaging the lettuce v Maximise production leaves. Water flow Chlorine flow Tunnel Drum Measurement off line of: - chlorine ratio - micro-organism ratio Belt speed Turbidity measurement Waste water after washing Fig. 18 : lettuce washing process. Cahier Technique Schneider n o 191 / pp.14 3.3. Linguistic variables and terms The following variables will therefore be c outputs: chosen: v modification of water flow: Water_f_var c inputs: v modification of chlorine flow: Cl_f_var v micro-organism ratio: Micro_ratio v modification of speed: Speed-var v residual chlorine ratio: Cl_ratio A session with an experienced operator, a v turbidity of water: Turbidity microbiology specialist and a lettuce “taster” v conveyor belt speed: Speed produces the following membership functions v water flow: Water_f (see fig. 19 ): µ µ Negative Positive Positive Acceptable High big Cl_ratio Water_f_var µ µ Negative Positive Positive Low High big Turbidity CI_f_var µ µ Acceptable Negative Positive Low High TMicro_ratio Speed_var µ Not high High Water_f µ Not high High Speed Fig. 19 : piece-wise linear membership functions. 3.4. Rules and outputs Writing fuzzy rules Speed_var IS Positive AND Cl_f_var IS Positive A meeting with operators enables the seven AND Water_f_var IS Positive. rules below to be determined, each c Lettuce tastes of chlorine, but there are no corresponding to a specific case: micro-organisms c Lettuce badly washed IF Cl_ratio IS High AND Micro_ratio IS NOT High IF Turbidity IS High AND Water_f IS NOT High THEN Cl_f_var IS Negative. THEN Water_f_var IS Positive big. c Everything is fine and production is maximum: c Lettuce badly washed but high conveyor belt save water. speed IF Speed IS High AND Cl_ratio IS Acceptable IF Turbidity IS High AND Water_f IS High THEN AND Turbidity IS Low THEN Water_f_var IS Speed_var IS Negative. Negative. c Too many micro-organisms c No micro-organisms: save chlorine IF Micro_ratio IS High THEN Cl_f_var IS Positive If Micro_ratio IS Low THEN Cl_f_var IS Negative. big. c Everything is fine and production can be Defuzzification increased Insofar as the aim is progressive behaviour of IF Turbidity IS Low and Micro_ratio IS NOT High the rule base in all cases and an interpolation AND Speed IS NOT High and CL_ratio IS between the rules, the centre of gravity is chosen Acceptable AND Water_f IS NOT High THEN as the defuzzification operator. Cahier Technique Schneider no 191 / pp.15 4 Implementation 4.1 When can fuzzy rule bases be used? Fuzzy rule bases can be chosen to solve c the variables (inputs and outputs) can be application problems when the following measured or observed, (measurability), conditions are satisfied: c qualitative expertise (if it is mathematical, conventional automatic control should be c it is possible to act on the process preferred), (controllability), c gradual expertise (if it is boolean, expert c an expertise or knowhow exists, systems are more suitable). 4.2 Designing an application Choice of operators Methodology In most applications, “Mamdani” rule bases are Designing a fuzzy rule base is an interactive used. This choice is suitable except if the process. The largest portion of the task consists expertise contains indeterminations. of collecting knowledge. One of the advantages of fuzzy logic is the possibility of having the rule In most cases, the choice is also made to use base validated by the people who provided the “trapezoidal” membership functions as they expertise before testing it on a real system. are easier to implement and simplify the Figure 20 illustrates the procedure used. gathering of expertise. Output membership functions are often singletons, except when rules Collecting knowledge are concatenated. A triangular output This is a three-step process: membership function in fact implies an c listing the variables to be taken into account: uncertainty on the output to be applied, and does they will become the linguistic variables of the not have much effect on interpolation between rule base; the rules. c listing the qualitative quantities to be taken into Finally, defuzzification takes place using the account and specifying when they are true and “centre of gravity” for control (all active rules are false: these quantities will become the linguistic taken into account): the use of the “average of terms of the rule base; maxima” for decision-making problems enables c formulate how these concepts are manipulated: a decision to be made when rules are which cases should be considered, how they are “conflicting” and avoids intermediate decisions. characterised, how should you act in each case. Professional expertise level: Gathering - Expert knowledge - Operator - Designer Validation of principle Validation of operation Programming level: Interpretation in form of rules - Automatic control engineer and membership functions - Ladder / Grafcet « Open loop » Implementation tests Fig. 20 : design methodology. Cahier Technique Schneider n o 191 / pp.16 Transcription in fuzzy rule form is then straight c if the process can be simulated, closed loop forward. However as few membership functions simulations can also be performed. and rules as possible should be written in order to limit the number of parameters which will have Tuning to be tuned later on and to ensure legibility of the The rule bases written in this manner often give base. We observe that it is easier to add rules in satisfaction right away. However the rule base order to take new situations into account than to may need to be modified or tuned. The following remove them. principles will act as a guideline in searching for the probable cause of the deviation observed: Validating the knowledge base c if the behaviour of the closed loop controller is This takes place in a number of steps: the opposite to what you expected, some rules c presentation of the rule base to the experts have most likely been incorrectly written; who helped collect knowledge, and discussion. c if you wish to optimise performance, it is The aim of the discussion is to identify points normally preferable to properly tune the that have not been covered and to ensure that membership functions; the rules are understood by everyone; c if the system is not robust and works in some c “open loop” simulation: the experts compare cases but not all the time, it is likely that not all the behaviour of the rule base to the behaviour cases have been taken into account and that that they expect on cases chosen beforehand; rules must be added. 4.3 Using an application The function of the operators knowhow and to validate the resulting The degree of involvement of operators behaviour. controlling an application using fuzzy logic varies Production changes considerably. During an application, the rule base must be able The following cases can be observed: to be adapted to changes in the production c completely autonomous system: the end-user system and the products manufactured. These is not familiar with fuzzy logic and is not aware of changes can be of various kinds: its use, c objectives have changed (cooking c fuzzy logic is a “black box” which can be temperature, etc.), for example due to a change disconnected or changed to “manual mode” by in product manufactured. The setpoints or rule the operator, input membership functions must then be c the operator is able to modify (tune) the modified; membership functions according to the situation, c system dimensions have changed; the and he does this for a production change (for membership functions must then be modified; example); c the type of system has changed (e.g. portage c the operator is able to read the rules (e.g. their of the rule base from one machine to another); degree of activation): he understands and is able the rules and membership functions must then to interpret the actions of the rule base. For be modified. example he can control the rule base in exceptional situations; The most common changes are of the first type. c the operator is the main designer of the base: They can then be managed by qualified he has been given the means to record his own operators. 4.4 Choosing the implementation technology Most of today’s applications run on standard implementation of fuzzy rule bases without hardware platforms (micro-controller, micro- programming. processor, programmable controller, micro- Fuzzy inferences can be directly programmed computer, etc.). (assembler, C language, etc.). The disadvantage Many software programs designed to help of this solution is that it is slower in the prototype develop fuzzy rule bases and aimed at micro- phase and requires programming skills and controllers, programmable controllers and micro- command of fuzzy logic algorithms. computers (to name but a few), enable rapid Cahier Technique Schneider no 191 / pp.17 For applications with exacting response time now commonly integrated inside micro- demands or in order to obtain very low mass controllers, even low cost ones, where they are production cost prices, use of fuzzy logic ICs is used to accelerate fuzzy inferences. advantageous. Use of such electronic chips is Figure 21 shows as an example the increasing as: applicational needs that can be encountered in c the operations required to produce fuzzy number of rules (complexity of the application) inferences are elementary and feasible in and cycle time (rapidity) as well as the possible integers, technologies (1993 figures). The rules c some operations can be carried out in parallel, considered have one predicate and one c the calculation takes place in successive conclusion. steps, thereby enabling simple “pipeline” The necessary technical-economic choice is architectures to be made. thus a compromise between the flexibility In particular, numerous ASIC components provided by software solutions, scale economy designed for specific markets exist (car, and the performance of dedicated hardware electrical household appliances, etc.). They are solutions. Cycle time (s) 10-7 10-6 10-5 RISC 10-4 Image processing 32 bits 10-3 Control system, car -2 16 bits 10 8 bits Cameras 10-1 1 4 bits Control Washing 10 machines Financial analyses 2 10 Medical diagnosis 3 10 Number of rules 1 10 100 1 000 10 000 Micro-programming technology ASIC technology Analog technology Fig. 21 : performance of components and application areas. 4.5 Standards Components Today, a work group in which Schneider plays Absence of standards is one of the main an active part, has incorporated the “fuzzy logic” problems holding up the use of fuzzy logic chips. language standard into the language standard of This is because these components are not programmable controllers (first official draft of compatible with one another as each one is the standard IEC 61131-7 available in 1997). Other result of choices made by manufacturers. initatives in the field of fuzzy logic standardisation should spring from this. Software Regarding software, lack of portability has also slowed down widespread use of fuzzy logic in industry. Cahier Technique Schneider n o 191 / pp.18 5 Fuzzy applications 5.1 Application types Functions performed The following table shows the functions most often performed in industry by means of fuzzy Fuzzy Theory of systems (X means possible use, XX that the logic possibilities Probabilities technique is suitable for this type of problem). Rule bases excel in cases when interpolation and action are required, whereas classification Imprecision Uncertainty methods are suitable for evaluation and and graduality diagnosis tasks normally performed upstream. Applications sometimes combine several of these functions, while retaining the graduality of the information. Expertise Fuzzy rules Rule Classification bases algorithms Regulation, XX Neuron Conventional control network automatic control Data Model Automatic XX parameter setting Fig. 22 : comparing fuzzy logic with other control Decision-making help XX X techniques. Diagnosis X XX Quality XX control fuzzy logic may be preferred for the ease with which it is understood by operators. Fuzzy logic and other techniques Hybridation of techniques Fuzzy logic is above all an extension and a Fuzzy logic is often used in combination with generalisation of boolean logic. It enables other techniques. These combinations are graduality to be introduced into notions which advantageous when each approach make use of were previously either true or false. its own strong points. Probabilities, without challenging the binary c Learning fuzzy rules or neurofuzzy nature of events (either true or false) enable the Fuzzy rule bases can be modified using learning uncertainty of these events to be managed. methods. On the boundary between these two The first methods known as “self-organizing approaches, the theory of possibilities (invented controller” were developed as early as 1974 and by Lotfi Zadeh) enables both graduality and aimed at heuristically modifying the content of uncertainty to be taken into account (see fig. 22 ). fuzzy rules belonging to a “rules table”. The Fuzzy base rules are often compared for control/ actual expertise is modified by the learning, but regulation applications to neuron networks and the membership functions remain the same. conventional automatic control. These three A second approach, consists of modifying approaches require respectively, in order to be parameters representative of the membership applied, an expertise, data for learning purposes, functions. Unlike the first method, the rules and and a dynamic model of the process. structure of the expertise are not altered. The These approaches can only be compared when membership function parameters are modified all three are available at the same time, which is using optimisation methods, for example often the case in theoretical studies but rare in gradient methods, or global optimisation practice. If all three are available, practical methods such as genetic algorithms or simulated considerations often take priority. In particular, annealing. This approach is often referred to as Cahier Technique Schneider no 191 / pp.19 “neurofuzzy”, in particular when the gradient is c Using fuzzy logic in association with automatic used. Use of the gradient to optimise these control parameters is likened to “retropropagation” used A fuzzy rule base is sometimes part of a in neuron networks known as “multi-layer controller. Use of fuzzy logic to simulate a perceptrons” in order to optimise weights proportional term allows all kinds of non- between neuron network layers. linearities. Specific cases of downgraded A third approach (that can be qualified as operation such as overloads, maintenance or structural optimisation of the rule base) aims at partial failures are easily integrated. simultaneously determining rules and membership A fuzzy rule base is used to greater advantage functions by learning. The learning process then outside the control loop, to supervise a normally takes place without referring to an controller. It then replaces an operator in order to expertise. The resulting rules can then tune controller parameters according to control theoretically be used to help build an expertise. system operating conditions. 5.2 Examples of industrial achievements Today fuzzy logic is accepted as being one of costs, air flow is kept to a minimum compatible the methods commonly used to control industrial with the biological process. processes. Added to these requirements is the consideration Although PID controllers still suffice for most of some specific operating cases, such as for applications, fuzzy logic is increasingly example a very high upstream flow, which is an recognised and used for its differentiating extreme circumstance under which parameters advantages, particularly for controlling quality of are seriously modified and sewage capacity production and costs. Due to the competitive affected. advantages offered by fuzzy logic in some applications, the integrator or end-user do not Although partial mathematical models of plants normally wish to mention the subject. These are available, there are no complete models, and applications benefit from extensive acquisition of the overall control strategy must often be knowhow or use of a crafty technical short-cut. heuristically developed. Confidentiality is then essential. This explains Use of fuzzy logic is relatively common why it was not possible to describe in a detailed nowadays in sewage plants. The plant shown in way all the examples given below. figure 23, based in Germany, has been in operation since 1994. Fuzzy logic was produced Sewage plant on a Schneider Modicon programmable Most modern sewage plants use biological controller by means of its standard fuzzy control processes (development of bacteria in ventilated functional modules. tanks) to purify sewage water before discharging The designer highlights the advantage of using it into the natural environment. The organic fuzzy logic in control: exceptions, i.e. situations matter contained in the waste water is used by when sewage capacity is partially downgraded, the bacteria to create its cellular components. are treated simply and without discontinuity. The bacteria discharges carbon dioxide (CO2) The method chosen to introduce these and nitrogen (N2). Air is blown into the tanks. exceptional states into a control loop is The energy used for ventilation purposes described below: frequently accounts for more than half the global energy consumed by the plant. In order to A proportional term which must adapt to the ensure correct development of bacteria and exceptional circumstances is identified in the sewage, the NH4 and O2 concentrations in the control loop: this term is first transcribed in fuzzy ventilation tanks must be carefully controlled, all logic, then this fuzzy logic element is inserted in the more so since in order to reduce energy the control loop. Cahier Technique Schneider n o 191 / pp.20 Precipitant tanks for phosphates Control station and Blower operating building Recirculation Grid building 10 11 4 3 2 1 7 6 5 1 - Sewage water supply 2 - Inlet mechanism lifting 3 - Ventilated sediment removal basin 4 - Venturi drain 5 - Excess sludge 8 7 6 - Recycled sludge 9 7 - Sludge scraper 8 - Final purification I 9 - Final purification II Outlet 10 - Nitrification channels 11 - Denitrification basin Fig. 23 : block diagram of the sewage plant. Once the membership functions have been IF average input THEN average output suitably tuned, two rules are sufficient to (see fig. 24 ). describe the proportional controller: Once the proportional term has been simulated, IF low input THEN low output. the exceptions are introduced in the form of IF high input THEN high output. other rules depending on other input variable A third rule is added at the operators’ request as combinations. they find it improves their understanding of the A simple example of this possibility is illustrated operation: in figure 25 . µ Controlled output z Low Average High Area corresponding to the Exception input input input proportional controller influencing area µ Low Average High output output output Input variable x Exception y Fig. 24 : simulating a controller proportional term. Fig. 25 : introducing an exception into a proportional term. Cahier Technique Schneider no 191 / pp.21 The table in figure 26 lists the rules for input variables “nitri O2 content” and “denitri O2 recirculation. The proportional term is created content” define an exceptional situation in the from the input variable “NOx content”. The two first rule. IF nitro O2 content AND denitri O2 content AND NOx content THEN recirculation quantity Not low Greater than 0 Low Low Low Normal Normal High High Fig. 26 : recirculation function rules table. Below is another treatment using fuzzy logic: part The exceptional condition is detected by the of the sludge deposited in the downstream basin strong turbidity, as sludge deposits minimum is recycled and re-injected upstream. The table in sediment due to the excessively high flow. figure 27 lists the rules for sludge recycling. The For information, other installation functions use first rule expresses an exception due to an fuzzy logic: excessively high upstream flow. In these conditions, a high degree of recycling would c air injection, result in increased overload of the installation. c management of excess sludge. IF turbity AND drained off quantity AND sludge level THEN quantity of discharged water of recycled sludge High Low Low Normal Low Low High Low Normal Low Normal High Normal Normal Normal High Normal High Low High Normal Normal High High Fig. 27 : sludge recycling function rules table. Food produce The chosen example is an aperitive biscuit Automation of industrial oven production lines production line. used for cooking biscuits interests biscuit A French group contacted Schneider who then, manufacturers both in France and Germany. For in co-operation with ENSIA (French Higher this control type, a conventional solution is not Institute of Agricultural and Food Industries) satisfactory due to the non-linearities, multiplicity worked out an automated solution. and heterogeneity of sensitive parameters. Modelling of the cooking process is both The main characteristics that can be measured complex and uncomplete. However, experienced in a biscuit are its colour, humidity and operators are perfectly able to control cooking dimensions. These characteristics can be using their empirical knowledge. affected by variations in quality of pastry Cahier Technique Schneider n o 191 / pp.22 ingredients, environmental conditions and the c Subjective evaluation time the biscuit remains in the oven... These Most quality defining notions depend on a influences must be compensated by oven setting number of variables. One of the factors for and conveyor belt speed. Control of production evaluating quality is colour which is three- quality of this kind of food process can be broken dimensional: hence the interest of defining down into the following functional steps: membership functions upon several variables. c conditioning and merging of data, Classification algorithms, based on the input c evaluation of subjective quantities (linked to variables perform a gradual evaluation of such quality), qualitative variables (top of biscuit well cooked, c diagnosis of quality deviations, over cooked,...). c decision-making, c Diagnosis c subjective evaluation The fuzzy ladder was used to diagnose quality Fuzzy logic enables qualitative variables to be deviations observed on biscuits (see fig. 29 ). taken into consideration and existing The oven has 3 sections. “professional” expertise to be used. Fuzzy rule The overall operating evaluation is satisfactory. bases have been used associated with other techniques (see. fig. 28 ). Other examples c Automation systems G.P.C.s (Global Predictive Controllers) are Functions Associated techniques extremely effective, but require the setting of 4 parameters: N1, N2, Nu, l (2 control horizons, Sensor melting prediction horizon, weighting factor). Such Subjective evaluation Fuzzy classification setting is both lengthy and difficult and normally Diagnosis Fuzzy ladder requires an expert. Schneider’s NUM subsidiary is currently developing numerical controls and Decision making Fuzzy rule bases would like to use G.P.C.s in future productions. Fig. 28 : functions and associated techniques. High biscuit Bottom of biscuit Top of biscuit Section 1 humidity well cooked well cooked temperature too low Bottom of biscuit Top of biscuit a little over cooked a little over cooked Bottom of biscuit Top of biscuit far too cooked far too cooked Bottom of biscuit Top of biscuit undercooked undercooked Fig. 29 : fuzzy ladder for quality deviation diagnosis. Cahier Technique Schneider no 191 / pp.23 Schneider has thus developed for NUM a the Danish company, F.L. Smidth Automation, to method for automatically setting the parameters control cement kilns. This process takes many for such controllers by means of fuzzy rule variables into account, and in particular the bases. Some twenty rules suffice for rapid, climatic influences on the kiln which is several reliable parameter setting. Moreover, the dozen metres high. presence of a monitoring and control specialist, hard to find in numerical control installations, is c General public electrical and electronic no longer necessary. household appliances A large number of applications are now available c Car industry to the general public, especially in Japan. For Renault and Peugeot (PSA) have announced an example, compact size digital camcorders are automatic gear box which uses fuzzy logic to highly sensitive to movement. Fuzzy logic adapt to the type of driving of the person behind controls the stadyshop image stabilization of the wheel. these devices. c Cement plants The first industrial application of fuzzy logic, then copied by other manufacturers, was produced by 6 Conclusion c Classed as an artificial intelligence technique, (see fig. 30 ), and offer simple evaluation fuzzy logic is used to model and replace process possibilities. control expertise and designer/operator expertise. c Evaluation limited to competition with the other c A tool for enhancing quality and increasing conventional control tools is not productive as productivity, fuzzy logic offers competitive such tools (e.g. PID controllers) continue to be advantages to industrial firms seeking technical- useful in most application areas. economic optimisation. c Fuzzy logic has its own special areas in which c This Cahier Technique specifies the areas in it works wonders: these are areas involving which this interesting approach can be used to expertise, nuanced decision-making, advantage. consideration of non-linear phenomena and subjective parameters, not to mention c Thanks to suitable programmable controllers contradictory decision-making factors. Contact and user-friendly tools, fuzzy logic is now with Schneider specialists will enable users and accessible to all automatic control engineers designers to find a suitable answer to their wishing to increase the scope of their skills and perfectly understandable question: the performance of their achievements. These tools are available in the development “What decisive advantages can fuzzy logic offer environment of some programmable controllers me in my application?”. Cahier Technique Schneider n o 191 / pp.24 a - configuration of the fuzzy logic module c - writing of rules b - definition of membership functions d - simulation - validation Fig. 30 : for fuzzy logic, the Schneider programmable controllers are equipped with user-friendly development tools on PC. Cahier Technique Schneider no 191 / pp.25 Appendix Operators between fuzzy sets The table in figure 31 shows the ZADEH operators. ZADEH Logic operator operation A∩B A B µA µB Intersection µA∩B = MIN (µA, µB) AND µA∩B A∪B A B µA µB Union µA∪B = MAX (µA, µB) OR µA∪B _ A µA µA _ Negation A µA _ = 1 - µA NOT A Fig. 31 : operators between fuzzy sets. Singleton output membership functions µ “Singleton” membership functions are often 1 used as output membership functions for fuzzy Low rules. This is because they allow the same Average interpolation effect between rules as for triangular membership functions (for example) High for far simpler calculations. There is no need to calculate the maximum of output membership Output functions (aggregation), and the centre of gravity Action is also simplified. Figure 32 illustrates this Fig. 32 : defuzzification of singleton membership calculation. functions. Cahier Technique Schneider n o 191 / pp.26 Fuzzy inferences: fuzzy implication and Generalised Modus Ponens Rules (implications) As shown in figure 33 , the conventional forward inference mechanism “from the front” or “modus ponens” consists of using rules, also known as Facts implications, and a deduction mechanism (the observed Modus Ponens Conclusions modus ponens) to deduce conclusions from observed facts. Fig. 33 : principle of inference from the front. The implication “A ⇒ B” is considered to be true as long as it is not invalidated (A true and A A' B false): see figure 34 . With knowledge whether the implication is true or false, the modus ponens A⇒B 0 1 B' 0 1 enables a conclusion B’ to be deduced from an 0 1 0 0 0 0 B A⇒B observation A’. 1 1 1 1 0 1 The same theoretical principle can be Implication Modus Ponens generalised in fuzzy logic. The general diagram is given in figure 35 . Fig. 34 : principle of implication and Modus Ponens. Rules (fuzzy implications) Inputs Fuzzification Generalised Modus Ponens Defuzzification Outputs Fig. 35 : principle of fuzzy inferences. The mechanism generalising the implication is The Lukeziewicz operator behaves like the known as the “fuzzy implication”. There are conventional implication when we limit ourselves several fuzzy implication operators, including to boolean values. This is not the case for the those mentioned below: Larsen and Mamdani operators used in the MAMDANI: µA ⇒B = MIN (µA, µB) Mamdani rule bases. These operators are the LARSEN: µA ⇒B = µA . µB most extensively used as: LUKASIEWICZ: µA ⇒B = MIN (1,1 - µA + µB ) c they offer a high degree of robustness in The fuzzy implication works like a conventional applications. implication, where A and B are fuzzy sets. c calculations are considerably simplified and The mechanism generalising the modus ponens allow simple graphic interpretation (see section is known as the “generalised modus ponens”. It 2.4.). Calculations on input x and output y are obeys the following formula, and is used to decoupled, as the formula below shows: determine a B’ conclusion fuzzy set. In most µB’(y) = MAXx (Min (µA’ (x), µA(x), µB(y)) ) cases the operator T used is the Minimum (known as the Zadeh operator). = Min (µB(y), MAXx (Min (µA’(x), µA(x)) ) µB’ (y) = MAXx (T(µA’(x), µA⇒B (x,y)) ) where T: modus ponens operator (t - standard), Cahier Technique Schneider no 191 / pp.27 Bibliography Standards IEC 61131-7: Programmable Controllers part 7 Fuzzy Control Programming. Miscellaneous works c Fuzzy models for pattern recognition. IEEE Press, 1992. James C. BEZDEK & Sanker K. PAL. c Fuzzy sets and systems: Theory and applications. Academic Press 1980, Mathematics in Sciences and Engineering vol. 144. D. DUBOIS, H. PRADE. c Evaluation subjective ; méthodes, applications et enjeux. Les cahiers des clubs CRIN, club CRIN logique floue. c A.I. and expert system myths, legends and facts. IEEE Expert 02/90, pp 8-20, 29 réf. M.S. FOX. c La logique floue et ses applications. Addison-Wesley, 1995. Bernadette BOUCHON-MEUNIER. Internet c http://www-isis.ecs.soton.ac.uk/research/nfinfo/ fuzzy.html c http://www.ortech-engr.com/fuzzy/ reservoir.html c http://www-cgi.cs.cmu.edu/afs/cs.cmu.edu/ project/ai-repository/ai/areas/fuzzy/0.html Cahier Technique Schneider n o 191 / pp.28 © 1998 Schneider Schneider Direction Scientifique et Technique, Edition: Schneider Service Communication Technique Real: AXESS - Saint-Péray (07). F-38050 Grenoble cedex 9 Printing: CLERC-Fontaine-France-1000 Télécopie : (33) 04 76 57 98 60 -100FF- 007431 12-98