The following points highlight the four main applications of fuzzy logic. The applications are: 1. Fuzzy Expert Systems 2. Fuzzy Control 3. Fuzzy Data Bases and Information Retrievel Systems 4. Fuzzy Survey.
Data bases associated with the real world problems are invariably contaminated with imprecision and inconsistency of data. Besides, the knowledge base is found to contain pieces of knowledge with doubtful certainty.
Application # 1. Fuzzy Expert Systems:
These are the expert system using fuzzy theory.
They work on the following procedure:
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Engage experts.
1. Have them write their knowledge in a natural language.
2. Convert the language into “if… then” or “if… then do…” type of rules.
3. Let experts by setting the words in the rules as fuzzy sets, give subjectively the corresponding membership functions to the words.
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4. Invoke the fuzzy inference.
5. Fix the membership functions, if the result is not as expected.
Some algorithms are available to identify an optimal membership function.
Example:
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Suppose we are steering a car, for simplicity, we consider only the direction of a car. Seeing the direction of the car we control the angle of the steering wheel. Let us generate the fuzzy rules.
If x and y are the angles of the car and of the steering wheel respectively then we can frame the following three rules based on the knowledge of the car driving:
R1:
If x is right then y is turned to the left.
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R2:
If x is left then y is turned to the right.
R3:
If x is straight, then y is kept the same.
Accordingly we will have six fuzzy sets that is the three, for angles of the car; left, right and straight and the three for movements of the steering wheel; turning to the left, right and keep the same. These fuzzy sets will correspond to membership function defined in Fig. 13.11. and Fig. 13.12. respectively. Let us identify fuzzy rule 1.
Universal set of angles of the car, U = ranging from -15 to +15 degree
U = {-15, -10, -5, 0, +5, +10 +15} and Universal set of angles of the steering (in degree)
V = {-45, -30, -15, 0, +15, +30, +45}
Other degrees can be linearly interpolated between two points which are closest to each other.
Under this assumption, we have fuzzy rule 1 represented by the fuzzy relation R1, in table 13.7. We are using the simplest method, truncation method, when the direction of the car is definitely veering to right, the membership value “the right” of 10° is 1 (Fig. 13.11a) gives the entire fuzzy set “turning to the left of V”.
However when the membership value is small, say 0.5 with regards to 5° in Fig. 13.11., truncate the fuzzy set of “turning to the left” with the value of 0.5. Similarly the fuzzy relations R2 and R3 are given in the tables 13.8 and 13.9., respectively. The reader may verify the results.
Let us say that the angle of the car is veering +50. The output of the fuzzy relations R1 and R2 are shown in Fig. 13.13, by membership function. A (dotted) and membership function B (regular) respectively R3 gives nothing. The result of fuzzy in the expert system is defined by the union set of all fuzzy sets generated by each rule. The rules are examined independently one at a time. This is different than the conventional expert system. In the fuzzy expert system an inference is done roughly and superficially.
Interpretation of the Resulting Fuzzy Rules:
When the car is veering slightly to the right (5°), apply both the rule which gives the steering wheel to the left and the rule which keeps the wheel at the same angle, which results in the middle angle between the ‘left’ and the ‘same’. In order to actually drive a car we have to decide on a certain angle for the steering wheel.
Thus, instead of the resulting fuzzy set we should some how extract a single crisp value from the fuzzy set. This process is called defuzzifying. There are many proposals for it, but here, we compute the mean of points which give the highest membership value in the fuzzy set.
In the Fig. 13.13, we see – 45 to + 15° have membership value of 0.5, thus we have a mean -15 which makes us steer the wheel to the left slightly. Here in the example an input is given by a crisp value. What will happen if we give fuzzy input? Thus, the fuzzy inference easily covers this case as well.
For example, Given fuzzy input that “the car is veering slightly right” represented in Fig. 13.13(a). The output shall be shown in Fig. 13.13(b). The reader can verify the result. Thus, the fuzzy inference easily covers this case as well.
These simple rules make it possible to drive a car. In fuzzy rules, there is no consideration for the exact values of weight and speed of the car, which are essential for the conventional control method. Thus, fuzzy expert system, using fuzzy rules are able to deal with a complicated system, in as good way as a human being can do, and are not possible with exact methods.
Application # 2. Fuzzy Control:
Fuzzy control is a methodology for constructing control systems in which mapping between real-valued input and output parameters is represented by fuzzy rules. In fact fuzzy control is an application of fuzzy inference. It is exactly like the fuzzy expert system. In fuzzy inference rules are described in natural language using fuzzy sets and converted to fuzzy relations.
One difference between fuzzy inference and fuzzy control is its output. Fuzzy inference takes fuzzy or crisp input and generates a fuzzy set, fuzzy controls has a single valued output. The input to the fuzzy control is usually given by a numerical value which may come from sensors in a system, which may be multiple.
We have seen that fuzzy inference can be implemented by a fuzzy relation. In fuzzy control, fuzzy rules are used to control the target.
For example:
If the temperature x, is high = A
and the difference of the temperature y, is small = B
then close a valve, z slightly = C
Here, A, B, and C are fuzzy sets. Such rules for the fuzzy control are also sometime, called fuzzy control rules.
Let us have input x = a and y = b and fuzzy rules Ai, Bi, and Ci‘ (Fig. 13.15). First compare the given inputs with the conditional part of the i th rule and obtain degree of I th consistency. We have the minimum values of Ai‘s consistency, which is membership value Ai of a, and Bi‘s consistency which is membership value Bi of b.
The reason why do we adopt the minimum is that the rule states the case when both conditions high (Ai) AND small (Bi) must be satisfied. Write a0and b0 being the consistencies of Ai and Bi respectively, total consistency C0 = min (a0 ,b0), shown in Fig. 13.15. The i th output is the fuzzy set Ci in the latter part of the rule whose membership value are limited upto C0.
Computing for each of the n rules we have n outputs. Now the question is which outputs should be chosen as the answer?
Even if we pickup the largest fuzzy set as the answer, two fuzzy sets are not always comparable since fuzzy sets cannot be put in numerical order because one fuzzy set might contain a smaller fuzzy set. Alternatively, we determine the output by the union of all fuzzy sets whose membership function has the highest value for each n value.
This is illustrated in Fig. 13.16. In this figure the result of the union is the fuzzy set labeled by fuzzy output. The same thing was done when we studied the car driving using fuzzy output in fuzzy expert systems. Fuzzy relations represented in tables 13.7-13.9 were used. Fig. 13.16, illustrates the truncated algorithm. This algorithm is not the only one but is the mostly used for fuzzy inference.
Finally, we extract one value from the union fuzzy set. In the example of fuzzy expert system we took the mean of only the highest points. In general with membership value as its weight all R points are used to compute the mean value. This method is called the centre of gravity method. The final output is in the form of crisp value.
The conversion of fuzzy sets into crisp value is called defuzzifying, the inverse of fuzzifying. Fuzzy controllers vary substantially according to the nature of the control problems they are supposed to solve. Control problems range from complex tasks, typical in robotics, which require a multiple of co-ordinated actions, to simple goals, such as maintaining a prescribed state of a single variable.
Traditional control systems are in general based on mathematical models, which describe the control system using one or more differential equations which define the system’s response to its inputs. Such systems are often implemented as so called ‘PID’ (proportional-integral-derivative) controllers.
Such controllers are the products of decades of development and theoretical work and are highly effective. Even though PID and other traditional control systems are well developed fuzzy control can be useful, because it has some advantages.
In many cases, the mathematical model of the control process may not exist or may be too expensive in terms of computer processing power and memory-and a system based on empirical rules may be more effective. Furthermore, fuzzy logic is well-suited to low-cost implementations based on cheap sensors, low-resolution analogue-to-digital (AD) converters, and 4-digit or 8-digit one-chip micro controllers, and such systems can be easily upgraded by adding new rules to improve performance or add new features. In many cases, fuzzy control can be used to improve existing controller systems by adding an extra layer of intelligence to the current control method.
Application # 3. Fuzzy Data Bases and Information Retrievel Systems:
The motivation for the application of fuzzy set theory to the design of databases and information storage and retrieval systems lies in the need to handle imprecise information. The database which can accommodate imprecise information can store and manipulate not only precise facts, but also subjective expert opinions, judgements and values which can be specified in linguistic terms.
This type of information can be quite useful when the database is to be used as a decision aid in areas where subjective and imprecise data are not only common but also quite valuable. In addition, it is also desirable to relieve the user of constraint of having to formulate queries to the database in precise terms.
How do such fuzzy database systems work?
In conventional database systems we can only retrieve the information about some articles and reference which are exactly matched with the given key word. Some systems have a relevance table among key words, which is used to give some related keywords to the original keywords and shows broader articles than others, though this is not common.
The conventional database is based on binary logic, where the result of matching is forced to be either a success or failure. There is no degree of meaning. The relevance table can be considered as a method to convey the semantics. Nevertheless, a search for an uncertain request is not accepted in conventional database.
This limitation of the conventional database can be overcome if we introduce fuzzy logic into database system. Let us consider a simple example, a list of names and their occupation, Fig. 13.17. According to the conventional database, it is possible to retrieve the age and occupation from a name; and also to retrieve the list of names, given age and occupation.
Can we look up a young and intelligent person in the system? The pat answer is no, because the key words, young and intelligent person cannot be used, as these key words are not present in the database. Further, the adjectives young and intelligent depend on the individual concept.
Fig. 13.18., shows a simple structure of a fuzzy database; the basic data is the same as given in Fig. 13.17. The additional query: “Who is young and Intelligent person”.
This type of database linked to the management system is used for the interpretation of fuzzy request. When the query is made the management system identifies that the keywords young and intelligent occupation are the fuzzy sets of age and occupation. The syntax is like
age = young
and occupation = intellectual
Next the system retrieves the semantics of the keywords from the interpretation of the database. The interpretation data is given in Fig. 13.19.
For each person in the basic database, we are to estimate the degree of satisfaction to the request of young and intellectual occupation. From the tables in Figs. 13.17 and 13.19 and we draw another table shown in Fig. 13.20, which serves as sample answer to our query exhibiting the occupation, age membership function, occupation membership function, and degree of satisfaction (Using AND operation between µage and µoccupation.
When we input the fuzzy input request and mention the threshold value <x, greater than 0.5 as value of degree of satisfaction, the result of the ordered list would be exhibited as:
Here, we used a crisp basic database, which could be extended to a fuzzy database so that instead of specifying an exact value such as 30 years old fuzzy labels such as young could be inserted. The degree of satisfaction is computed by the agreement of the two fuzzy sets. Such systems with a fuzzy database with features specifying fuzzy labels have actually been developed.
Application # 4. Fuzzy Survey:
Fuzzy logic is also of immense help in surveying the stock market or framing a public opinion about a particular issue, say as the growing number of divorces in the Indian society. This topic is illustrated, as to how subjectivity of the subjects creeps in giving the opinion. Suppose a questionnaire is based on the following statement, adopted from the magazine, “Nikki Electronics”.
S1 The PC is my friend.
S2 I always wait until the last minute to do something.
S3 I am never frugal with money for my favourite dishes.
S4 I like mathematics.
S5 I prefer regular coffee.
S6 I never sacrifice pleasure for study.
S7 I am polite.
S8 I always meet my obligations to attend ceremonies.
S9 I hate a job which requires human communication.
These nine statements are not likely to be answered either yes or no. Suppose Arisha and Bobby answered these statements, as given in table 13.10., max-min operation, to show their degree of consensus is also shown in the table. The maximum value, 0.6, of the degree of consensus of the two sets, is reflected by the two statements, S3 and S6.
To determine the fuzzy survey, we can adopt measures other than that of degree of consensus. This is called the degree of agreement; 0.4 and 0.2 and 0.8 and 0.6 have the same degree of agreement based on degree of difference.
For this purpose, we are going to have the measure of Arisha’s agreement relation among the restricted questions within S1, S4, S5, S8. The result is shown by the matrix in the table 13.11. The matrix is, as we can see symmetric since difference between S. and S. is equal to that of Sj and Si Moreover, all diagonal values in the matrix are 1.
These properties are called symmetric and reflective. The matrix is illustrated visually in Fig. 13.21, some times called fuzzy graph. In this figure we have the degree of agreement of 0.9 between S1 and S4 and 0.6 between S4 and S5. Then what degree shall be assigned to the agreement between S1 and S5 by way of S4. According the principle of simplicity in fuzzy theory, we choose the minimum value in 0.9 and 0.6 for assigned degree. However, something strange would happen in table 13.10.
The direct agreement degree between S1 and S5 given as 0.5 is less than that 0.6 got from the indirect agreement via S4. Between S1 and S5 there are a total of five routes: S1– S5, S1 – S8 – S5 – S1– S4 – S5, S1 – S4 – S5 and S1– S4 – S8 – S5. To define the consistent degree, we take the maximum value for all routes. This is also a max-min operation. Finally we have new fuzzy relation which satisfies the transitivity. Transitivity can be explained from an example in real life.
I have two friends Sharma and Chopra. They may not be friendly with each other, but we assume that they are friends (through me). This is an example of transitivity.
A relationship which is symmetric, reflective and transitive is called a similarity relation, this is shown in table 13.12.
A similarity relation has interesting property. For example, in table 13.12, if we collect elements which have values greater than 0.6 then all elements are collected into a single set. If we choose elements greater than 0.7, a pair of S1 and S4 and a pair S5 and S6 are put together independently. Similarly, the threshold values of 0.9 and 1.0 partition the set into three and four disjoint subsets respectively; as shown in Fig. 13.22.
Here, a, indicates the strength of association. The partitioned subsets correspond to the modified fuzzy graph in which any branch less than alpha are eliminated.
Fig. 13.22, shows that if the threshold (or) is 0.7, the relationship among statements in the questionnaire are as follows:
Arisha’s feelings that she likes PC is close to that she likes mathematics and preference for regular coffee is as important to her in the obligation for ceremony is while the two likes have nothing to do with each other, the value of this information is only to show how important these things are to the subject.
This overall shows that fuzzy theory can be used for classification of questions in a survey and is also good for estimation of relationship among classes.
The application of fuzzy theory to even economics, the Social Science management and psychology have become, established. In these applications some common approaches, such as decision making in uncertain environments, fuzzy modeling and uncertain identification are reflected.
Thus, as we are closely involved in computer systems, the requirement for our uncertainties to be recognized by computers becomes significant, and fuzzy theory can play an important role in the human-computer interface. In this sense, fuzzy theory exceeds conventional theories, though the meta language of fuzzy theory is defined in binary logic. It can appear at almost any place where computers and modern control theory are overly precise. In tasks requiring delicate human intuition and experience based knowledge fuzzy knowledge can be highly beneficial.