Category: fuzzy Logic

Fuzzy Logic – Quick Guide Fuzzy Logic – Introduction The word fuzzy refers to things which are not clear or are vague. Any event, process, or function that is changing continuously cannot always be defined as either true or false, which means that we need to define such activities in a Fuzzy manner. What is Fuzzy Logic? Fuzzy Logic resembles the human decision-making methodology. It deals with vague and imprecise information. This is gross oversimplification of the real-world problems and based on degrees of truth rather than usual true/false or 1/0 like Boolean logic. Take a look at the following diagram. It shows that in fuzzy systems, the values are indicated by a number in the range from 0 to 1. Here 1.0 represents absolute truth and 0.0 represents absolute falseness. The number which indicates the value in fuzzy systems is called the truth value. In other words, we can say that fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness. There can be numerous other examples like this with the help of which we can understand the concept of fuzzy logic. Fuzzy Logic was introduced in 1965 by Lofti A. Zadeh in his research paper “Fuzzy Sets”. He is considered as the father of Fuzzy Logic. Fuzzy Logic – Classical Set Theory A set is an unordered collection of different elements. It can be written explicitly by listing its elements using the set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set. Example A set of all positive integers. A set of all the planets in the solar system. A set of all the states in India. A set of all the lowercase letters of the alphabet. Mathematical Representation of a Set Sets can be represented in two ways − Roster or Tabular Form In this form, a set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas. Following are the examples of set in Roster or Tabular Form − Set of vowels in English alphabet, A = {a,e,i,o,u} Set of odd numbers less than 10, B = {1,3,5,7,9} Set Builder Notation In this form, the set is defined by specifying a property that elements of the set have in common. The set is described as A = {x:p(x)} Example 1 − The set {a,e,i,o,u} is written as A = {x:x is a vowel in English alphabet} Example 2 − The set {1,3,5,7,9} is written as B = {x:1 ≤ x < 10 and (x%2) ≠ 0} If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a member of set S, it is denoted by y∉S. Example − If S = {1,1.2,1.7,2},1 ∈ S but 1.5 ∉ S Cardinality of a Set Cardinality of a set S, denoted by |S||S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞∞. Example − |{1,4,3,5}| = 4,|{1,2,3,4,5,…}| = ∞ If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y. |X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when the number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y. |X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when the number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective. If |X| ≤ |Y| and |X| ≤ |Y| then |X| = |Y|. The sets X and Y are commonly referred as equivalent sets. Types of Sets Sets can be classified into many types; some of which are finite, infinite, subset, universal, proper, singleton set, etc. Finite Set A set which contains a definite number of elements is called a finite set. Example − S = {x|x ∈ N and 70 > x > 50} Infinite Set A set which contains infinite number of elements is called an infinite set. Example − S = {x|x ∈ N and x > 10} Subset A set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y. Example 1 − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X. Example 2 − Let, X = {1,2,3} and Y = {1,2,3}. Here set Y is a subset (not a proper subset) of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X. Proper Subset The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of set Y (Written as X ⊂ Y) if every element of X is an element of set Y and |X| < |Y|. Example − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y ⊂ X, since all elements in Y are contained in X too and X has at least one element which is more than set Y. Universal Set It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.

Fuzzy Logic – Introduction The word fuzzy refers to things which are not clear or are vague. Any event, process, or function that is changing continuously cannot always be defined as either true or false, which means that we need to define such activities in a Fuzzy manner. What is Fuzzy Logic? Fuzzy Logic resembles the human decision-making methodology. It deals with vague and imprecise information. This is gross oversimplification of the real-world problems and based on degrees of truth rather than usual true/false or 1/0 like Boolean logic. Take a look at the following diagram. It shows that in fuzzy systems, the values are indicated by a number in the range from 0 to 1. Here 1.0 represents absolute truth and 0.0 represents absolute falseness. The number which indicates the value in fuzzy systems is called the truth value. In other words, we can say that fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness. There can be numerous other examples like this with the help of which we can understand the concept of fuzzy logic. Fuzzy Logic was introduced in 1965 by Lofti A. Zadeh in his research paper “Fuzzy Sets”. He is considered as the father of Fuzzy Logic. Learning working make money

Fuzzy Logic – Quantification In modeling natural language statements, quantified statements play an important role. It means that NL heavily depends on quantifying construction which often includes fuzzy concepts like “almost all”, “many”, etc. Following are a few examples of quantifying propositions − Every student passed the exam. Every sport car is expensive. Many students passed the exam. Many sports cars are expensive. In the above examples, the quantifiers “Every” and “Many” are applied to the crisp restrictions “students” as well as crisp scope “(person who)passed the exam” and “cars” as well as crisp scope ”sports”. Fuzzy Events, Fuzzy Means and Fuzzy Variances With the help of an example, we can understand the above concepts. Let us assume that we are a shareholder of a company named ABC. And at present the company is selling each of its share for ₹40. There are three different companies whose business is similar to ABC but these are offering their shares at different rates – ₹100 a share, ₹85 a share and ₹60 a share respectively. Now the probability distribution of this price takeover is as follows − Price ₹100 ₹85 ₹60 Probability 0.3 0.5 0.2 Now, from the standard probability theory, the above distribution gives a mean of expected price as below − $100 × 0.3 + 85 × 0.5 + 60 × 0.2 = 84.5$ And, from the standard probability theory, the above distribution gives a variance of expected price as below − $(100 − 84.5)2 × 0.3 + (85 − 84.5)2 × 0.5 + (60 − 84.5)2 × 0.2 = 124.825$ Suppose the degree of membership of 100 in this set is 0.7, that of 85 is 1, and the degree of membership is 0.5 for the value 60. These can be reflected in the following fuzzy set − $$left { frac{0.7}{100}, : frac{1}{85}, : frac{0.5}{60}, right }$$ The fuzzy set obtained in this manner is called a fuzzy event. We want the probability of the fuzzy event for which our calculation gives − $0.7 × 0.3 + 1 × 0.5 + 0.5 × 0.2 = 0.21 + 0.5 + 0.1 = 0.81$ Now, we need to calculate the fuzzy mean and the fuzzy variance, the calculation is as follows − Fuzzy_mean $= left ( frac{1}{0.81} right ) × (100 × 0.7 × 0.3 + 85 × 1 × 0.5 + 60 × 0.5 × 0.2)$ $= 85.8$ Fuzzy_Variance $= 7496.91 − 7361.91 = 135.27$ Learning working make money

Fuzzy Logic – Approximate Reasoning Following are the different modes of approximate reasoning − Categorical Reasoning In this mode of approximate reasoning, the antecedents, containing no fuzzy quantifiers and fuzzy probabilities, are assumed to be in canonical form. Qualitative Reasoning In this mode of approximate reasoning, the antecedents and consequents have fuzzy linguistic variables; the input-output relationship of a system is expressed as a collection of fuzzy IF-THEN rules. This reasoning is mainly used in control system analysis. Syllogistic Reasoning In this mode of approximation reasoning, antecedents with fuzzy quantifiers are related to inference rules. This is expressed as − x = S1A′s are B′s y = S2C′s are D′s ———————— z = S3E′s are F′s Here A,B,C,D,E,F are fuzzy predicates. S1 and S2 are given fuzzy quantifiers. S3 is the fuzzy quantifier which has to be decided. Dispositional Reasoning In this mode of approximation reasoning, the antecedents are dispositions that may contain the fuzzy quantifier “usually”. The quantifier Usually links together the dispositional and syllogistic reasoning; hence it pays an important role. For example, the projection rule of inference in dispositional reasoning can be given as follows − usually( (L,M) is R ) ⇒ usually (L is [R ↓ L]) Here [R ↓ L] is the projection of fuzzy relation R on L Fuzzy Logic Rule Base It is a known fact that a human being is always comfortable making conversations in natural language. The representation of human knowledge can be done with the help of following natural language expression − IF antecedent THEN consequent The expression as stated above is referred to as the Fuzzy IF-THEN rule base. Canonical Form Following is the canonical form of Fuzzy Logic Rule Base − Rule 1 − If condition C1, then restriction R1 Rule 2 − If condition C1, then restriction R2 . . . Rule n − If condition C1, then restriction Rn Interpretations of Fuzzy IF-THEN Rules Fuzzy IF-THEN Rules can be interpreted in the following four forms − Assignment Statements These kinds of statements use “=” (equal to sign) for the purpose of assignment. They are of the following form − a = hello climate = summer Conditional Statements These kinds of statements use the “IF-THEN” rule base form for the purpose of condition. They are of the following form − IF temperature is high THEN Climate is hot IF food is fresh THEN eat. Unconditional Statements They are of the following form − GOTO 10 turn the Fan off Linguistic Variable We have studied that fuzzy logic uses linguistic variables which are the words or sentences in a natural language. For example, if we say temperature, it is a linguistic variable; the values of which are very hot or cold, slightly hot or cold, very warm, slightly warm, etc. The words very, slightly are the linguistic hedges. Characterization of Linguistic Variable Following four terms characterize the linguistic variable − Name of the variable, generally represented by x. Term set of the variable, generally represented by t(x). Syntactic rules for generating the values of the variable x. Semantic rules for linking every value of x and its significance. Propositions in Fuzzy Logic As we know that propositions are sentences expressed in any language which are generally expressed in the following canonical form − s as P Here, s is the Subject and P is Predicate. For example, “Delhi is the capital of India”, this is a proposition where “Delhi” is the subject and “is the capital of India” is the predicate which shows the property of subject. We know that logic is the basis of reasoning and fuzzy logic extends the capability of reasoning by using fuzzy predicates, fuzzy-predicate modifiers, fuzzy quantifiers and fuzzy qualifiers in fuzzy propositions which creates the difference from classical logic. Propositions in fuzzy logic include the following − Fuzzy Predicate Almost every predicate in natural language is fuzzy in nature hence, fuzzy logic has the predicates like tall, short, warm, hot, fast, etc. Fuzzy-predicate Modifiers We discussed linguistic hedges above; we also have many fuzzy-predicate modifiers which act as hedges. They are very essential for producing the values of a linguistic variable. For example, the words very, slightly are modifiers and the propositions can be like “water is slightly hot.” Fuzzy Quantifiers It can be defined as a fuzzy number which gives a vague classification of the cardinality of one or more fuzzy or non-fuzzy sets. It can be used to influence probability within fuzzy logic. For example, the words many, most, frequently are used as fuzzy quantifiers and the propositions can be like “most people are allergic to it.” Fuzzy Qualifiers Let us now understand Fuzzy Qualifiers. A Fuzzy Qualifier is also a proposition of Fuzzy Logic. Fuzzy qualification has the following forms − Fuzzy Qualification Based on Truth It claims the degree of truth of a fuzzy proposition. Expression − It is expressed as x is t. Here, t is a fuzzy truth value. Example − (Car is black) is NOT VERY True. Fuzzy Qualification Based on Probability It claims the probability, either numerical or an interval, of fuzzy proposition. Expression − It is expressed as x is λ. Here, λ is a fuzzy probability. Example − (Car is black) is Likely. Fuzzy Qualification Based on Possibility It claims the possibility of fuzzy proposition. Expression − It is expressed as x is π. Here, π is a fuzzy possibility. Example − (Car is black) is Almost Impossible. Learning working make money

Fuzzy Logic – Database and Queries We have studied in our previous chapters that Fuzzy Logic is an approach to computing based on “degrees of truth” rather than the usual “true or false” logic. It deals with reasoning that is approximate rather than precise to solve problems in a way that more resembles human logic, hence database querying process by the two valued realization of Boolean algebra is not adequate. Fuzzy Scenario of Relations on Databases The Fuzzy Scenario of Relations on Databases can be understood with the help of the following example − Example Suppose we have a database having the records of persons who visited India. In simple database, we will have the entries made in the following way − Name Age Citizen Visited Country Days Spent Year of Visit John Smith 35 U.S. India 41 1999 John Smith 35 U.S. Italy 72 1999 John Smith 35 U.S. Japan 31 1999 Now, if anyone queries about the person who visited India and Japan in the year 99 and is the citizen of US, then the output will show two entries having the name of John Smith. This is simple query generating simple output. But what if we want to know whether the person in the above query is young or not. According to the above result, the age of the person is 35 years. But can we assume the person to be young or not? Similarly, same thing can be applied on the other fields like days spent, year of visit, etc. The solution of the above issues can be found with the help of Fuzzy Value sets as follows − FV(Age){ very young, young, somewhat old, old } FV(Days Spent){ barely few days, few days, quite a few days, many days } FV(Year of Visit){distant past, recent past, recent } Now if any query will have the fuzzy value then the result will also be fuzzy in nature. Fuzzy Query System A fuzzy query system is an interface to users to get information from the database using (quasi) natural language sentences. Many fuzzy query implementations have been proposed, resulting in slightly different languages. Although there are some variations according to the particularities of different implementations, the answer to a fuzzy query sentence is generally a list of records, ranked by the degree of matching. Learning working make money

Fuzzy Logic – Classical Set Theory A set is an unordered collection of different elements. It can be written explicitly by listing its elements using the set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set. Example A set of all positive integers. A set of all the planets in the solar system. A set of all the states in India. A set of all the lowercase letters of the alphabet. Mathematical Representation of a Set Sets can be represented in two ways − Roster or Tabular Form In this form, a set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas. Following are the examples of set in Roster or Tabular Form − Set of vowels in English alphabet, A = {a,e,i,o,u} Set of odd numbers less than 10, B = {1,3,5,7,9} Set Builder Notation In this form, the set is defined by specifying a property that elements of the set have in common. The set is described as A = {x:p(x)} Example 1 − The set {a,e,i,o,u} is written as A = {x:x is a vowel in English alphabet} Example 2 − The set {1,3,5,7,9} is written as B = {x:1 ≤ x < 10 and (x%2) ≠ 0} If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a member of set S, it is denoted by y∉S. Example − If S = {1,1.2,1.7,2},1 ∈ S but 1.5 ∉ S Cardinality of a Set Cardinality of a set S, denoted by |S||S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞∞. Example − |{1,4,3,5}| = 4,|{1,2,3,4,5,…}| = ∞ If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y. |X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when the number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y. |X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when the number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective. If |X| ≤ |Y| and |X| ≤ |Y| then |X| = |Y|. The sets X and Y are commonly referred as equivalent sets. Types of Sets Sets can be classified into many types; some of which are finite, infinite, subset, universal, proper, singleton set, etc. Finite Set A set which contains a definite number of elements is called a finite set. Example − S = {x|x ∈ N and 70 > x > 50} Infinite Set A set which contains infinite number of elements is called an infinite set. Example − S = {x|x ∈ N and x > 10} Subset A set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y. Example 1 − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X. Example 2 − Let, X = {1,2,3} and Y = {1,2,3}. Here set Y is a subset (not a proper subset) of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X. Proper Subset The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of set Y (Written as X ⊂ Y) if every element of X is an element of set Y and |X| < |Y|. Example − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y ⊂ X, since all elements in Y are contained in X too and X has at least one element which is more than set Y. Universal Set It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U. Example − We may define U as the set of all animals on earth. In this case, a set of all mammals is a subset of U, a set of all fishes is a subset of U, a set of all insects is a subset of U, and so on. Empty Set or Null Set An empty set contains no elements. It is denoted by Φ. As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero. Example – S = {x|x ∈ N and 7 < x < 8} = Φ Singleton Set or Unit Set A Singleton set or Unit set contains only one element. A singleton set is denoted by {s}. Example − S = {x|x ∈ N, 7 < x < 9} = {8} Equal Set If two sets contain the same elements, they are said to be equal. Example − If A = {1,2,6} and B = {6,1,2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A. Equivalent Set If the cardinalities of two sets are same, they are called equivalent sets. Example − If

Fuzzy Logic – Decision Making It is an activity which includes the steps to be taken for choosing a suitable alternative from those that are needed for realizing a certain goal. Steps for Decision Making Let us now discuss the steps involved in the decision making process − Determining the Set of Alternatives − In this step, the alternatives from which the decision has to be taken must be determined. Evaluating Alternative − Here, the alternatives must be evaluated so that the decision can be taken about one of the alternatives. Comparison between Alternatives − In this step, a comparison between the evaluated alternatives is done. Types of Decision Making We will now understand the different types of decision making. Individual Decision Making In this type of decision making, only a single person is responsible for taking decisions. The decision making model in this kind can be characterized as − Set of possible actions Set of goals $G_ileft ( i : in : X_n right );$ Set of Constraints $C_jleft ( j : in : X_m right )$ The goals and constraints stated above are expressed in terms of fuzzy sets. Now consider a set A. Then, the goal and constraints for this set are given by − $G_ileft ( a right )$ = composition$left [ G_ileft ( a right ) right ]$ = $G_i^1left ( G_ileft ( a right ) right )$ with $G_i^1$ $C_jleft ( a right )$ = composition$left [ C_jleft ( a right ) right ]$ = $C_j^1left ( C_jleft ( a right ) right )$ with $C_j^1$ for $a:in :A$ The fuzzy decision in the above case is given by − $$F_D = min[iin X_{n}^{in}fG_ileft ( a right ),jin X_{m}^{in}fC_jleft ( a right )]$$ Multi-person Decision Making Decision making in this case includes several persons so that the expert knowledge from various persons is utilized to make decisions. Calculation for this can be given as follows − Number of persons preferring $x_i$ to $x_j$ = $Nleft ( x_i, : x_j right )$ Total number of decision makers = $n$ Then, $SCleft ( x_i, : x_j right ) = frac{Nleft ( x_i, : x_j right )}{n}$ Multi-objective Decision Making Multi-objective decision making occurs when there are several objectives to be realized. There are following two issues in this type of decision making − To acquire proper information related to the satisfaction of the objectives by various alternatives. To weigh the relative importance of each objective. Mathematically we can define a universe of n alternatives as − $A = left [ a_1, :a_2,:…, : a_i, : …, :a_n right ]$ And the set of “m” objectives as $O = left [ o_1, :o_2,:…, : o_i, : …, :o_n right ]$ Multi-attribute Decision Making Multi-attribute decision making takes place when the evaluation of alternatives can be carried out based on several attributes of the object. The attributes can be numerical data, linguistic data and qualitative data. Mathematically, the multi-attribute evaluation is carried out on the basis of linear equation as follows − $$Y = A_1X_1+A_2X_2+…+A_iX_i+…+A_rX_r$$ Learning working make money

Fuzzy Logic – Inference System Fuzzy Inference System is the key unit of a fuzzy logic system having decision making as its primary work. It uses the “IF…THEN” rules along with connectors “OR” or “AND” for drawing essential decision rules. Characteristics of Fuzzy Inference System Following are some characteristics of FIS − The output from FIS is always a fuzzy set irrespective of its input which can be fuzzy or crisp. It is necessary to have fuzzy output when it is used as a controller. A defuzzification unit would be there with FIS to convert fuzzy variables into crisp variables. Functional Blocks of FIS The following five functional blocks will help you understand the construction of FIS − Rule Base − It contains fuzzy IF-THEN rules. Database − It defines the membership functions of fuzzy sets used in fuzzy rules. Decision-making Unit − It performs operation on rules. Fuzzification Interface Unit − It converts the crisp quantities into fuzzy quantities. Defuzzification Interface Unit − It converts the fuzzy quantities into crisp quantities. Following is a block diagram of fuzzy interference system. Working of FIS The working of the FIS consists of the following steps − A fuzzification unit supports the application of numerous fuzzification methods, and converts the crisp input into fuzzy input. A knowledge base – collection of rule base and database is formed upon the conversion of crisp input into fuzzy input. The defuzzification unit fuzzy input is finally converted into crisp output. Methods of FIS Let us now discuss the different methods of FIS. Following are the two important methods of FIS, having different consequent of fuzzy rules − Mamdani Fuzzy Inference System Takagi-Sugeno Fuzzy Model (TS Method) Mamdani Fuzzy Inference System This system was proposed in 1975 by Ebhasim Mamdani. Basically, it was anticipated to control a steam engine and boiler combination by synthesizing a set of fuzzy rules obtained from people working on the system. Steps for Computing the Output Following steps need to be followed to compute the output from this FIS − Step 1 − Set of fuzzy rules need to be determined in this step. Step 2 − In this step, by using input membership function, the input would be made fuzzy. Step 3 − Now establish the rule strength by combining the fuzzified inputs according to fuzzy rules. Step 4 − In this step, determine the consequent of rule by combining the rule strength and the output membership function. Step 5 − For getting output distribution combine all the consequents. Step 6 − Finally, a defuzzified output distribution is obtained. Following is a block diagram of Mamdani Fuzzy Interface System. Takagi-Sugeno Fuzzy Model (TS Method) This model was proposed by Takagi, Sugeno and Kang in 1985. Format of this rule is given as − IF x is A and y is B THEN Z = f(x,y) Here, AB are fuzzy sets in antecedents and z = f(x,y) is a crisp function in the consequent. Fuzzy Inference Process The fuzzy inference process under Takagi-Sugeno Fuzzy Model (TS Method) works in the following way − Step 1: Fuzzifying the inputs − Here, the inputs of the system are made fuzzy. Step 2: Applying the fuzzy operator − In this step, the fuzzy operators must be applied to get the output. Rule Format of the Sugeno Form The rule format of Sugeno form is given by − if 7 = x and 9 = y then output is z = ax+by+c Comparison between the two methods Let us now understand the comparison between the Mamdani System and the Sugeno Model. Output Membership Function − The main difference between them is on the basis of output membership function. The Sugeno output membership functions are either linear or constant. Aggregation and Defuzzification Procedure − The difference between them also lies in the consequence of fuzzy rules and due to the same their aggregation and defuzzification procedure also differs. Mathematical Rules − More mathematical rules exist for the Sugeno rule than the Mamdani rule. Adjustable Parameters − The Sugeno controller has more adjustable parameters than the Mamdani controller. Learning working make money