The fuzzy logic toolbox is also provided in the Appendix for easy reference of the students and professionals. He received "Best Teacher Award" in 2001, "Dhakshina Murthy Award for Teaching Excellence" from PSG College of Technology, and "The Citation for Best Teaching and Technical Contribution" in 2002 from Government College of Technology, Coimbatore.
Introduction
Fuzzy Logic
Membership in a collection is thus found as binary, i.e. the element is a member of a collection or not. The fuzzy sets "long" and "short." The classification is subjective - it depends on which height is measured relatively.
Mat LAB – An Overview
Matlab is both an environment and a programming language, and the main advantage of the Matlab language is that it allows us to build our own reusable tools. Changing the features is easy, as most M files can be opened.
Classical Sets and Fuzzy Sets
Introduction
Classical Set
- Operations on Classical Sets
- Properties of Classical Sets
- Mapping of Classical Sets to a Function
- Solved Examples
It represents all those elements in the universeX that exist in both setsA and B at the same time (or belong to them). The complement of setAdenotedA is defined as the set of all elements in the universe that are not in setA.
- Fuzzy Sets
- Properties of Fuzzy Sets
- Solved Examples
The comparison of the Venn diagram for the classical groups and the fuzzy groups for the law of the excluded middle are shown in Fig. The properties of the classical set also fit the properties of fuzzy sets.
Summary
Review Questions
Write the expressions for the fuzzy set operation in set-theoretic form and function-theoretic form. How does the excluded middle law differ for the fuzzy set and the classical set.
Exercise Problems
The chips are sorted to meet certain maximum electrical characteristics, e.g. frequency and temperature classification, so the "best". Assume that each sample chip is screened and all chips are found to have a maximum operating frequency in the range of 7–15 MHz at 20◦C.
Classical and Fuzzy Relations
Introduction
Cartesian Product of Relation
Classical Relations
- Cardinality of Crisp Relation
- Operations on Crisp Relation
- Properties of Crisp Relations
- Composition
If the cardinality of X is nx and the cardinality of Y is ny, then the cardinality of the relation R, between these two universes is nx×y = nx×ny. The maximum-product composition is defined by the set-theoretic and membership-function-theoretic expressions:
Fuzzy Relations
- Cardinality of Fuzzy Relations
- Properties of Fuzzy Relations
As in the case of a fuzzy set, the excluded middle law and the law of contradiction do not hold in a fuzzy relation. Each fuzzy set could be thought of as a vector of membership values; each value is associated with a specific element in each set.
Tolerance and Equivalence Relations
- Crisp Relation
- Fuzzy Relation
The given matrix is reflexive, symmetry and transitive, therefore the given matrix is equivalence matrix. The given matrix is reflexive, symmetry and not transitive, therefore the given matrix is not an equivalence matrix. Example 3.16.Find the fuzzy relation using the fuzzy max–min method for the given using Matlab program.
Determine the soft membership of I. 7 to the universe of accelerations, A. Motor speed m degrees per second and voltage in volts. a) Find the ratio of the Cartesian productR. Generating another fuzzy set on the space V ∼. b) Use max–min composition to find V3. For each of the following relations on each set, indicate whether the relation is reflexive, symmetric, and transitive.
Membership Functions
Introduction
Features of Membership Function
If the region of the universe is characterized by a nonzero membership in the set A∼, this defines the support of a membership function for fuzzy set A. If the region of the universe has a nonzero membership but not a full membership, this defines the boundary of a membership; this defines the boundary of a membership function for fuzzy setA. The intersection of a membership function is the elements in the universe whose membership value is 0.5 µA.
Classification of Fuzzy Sets
Fuzzification
Membership Value Assignments
- Intuition
- Inference
- Rank Ordering
- Angular Fuzzy Sets
- Neural Networks
- Genetic Algorithm
- Inductive Reasoning
The membership for the other triangles can be given as a supplement to the logical union of the two already defined membership functions. Preferences are above for pairwise comparisons, and membership is ordered from here. Solution. The linguistic expressions regarding the direction of movement of the motor are given as.
Partial Clockwise (PC) – θ=−Π/4 Full Clockwise (FC) – θ=−Π/2 The angular soft set for this is shown in the figure. The complete membership mapping of different data points to different fuzzy classes can be determined using a neural network approach.
Solved Examples
Find the membership values using the angular fuzzy set approach for these linguistic labels and plot these values versus θ. Thus, the membership function can be formed using any of the previously discussed methods. How is the voting concept used in the ranking method to define the membership values.
Using the inference method described in this chapter, find the membership values for each of the triangular shapes (I. Find the membership values using the angular fuzzy set approach for these linguistic labels for the complement angles and plot these values against θ.
Defuzzification
- Introduction
- Lambda Cuts for Fuzzy Sets
- Lambda Cuts for Fuzzy Relations
- Defuzzification Methods
- Solved Examples
The lambda cut process for relations is similar to that for sets lambda cut. In addition to lambda-cut sets and relations, which convert fuzzy sets or relations into crisp sets or relations, there are other various defuzzification methods used to convert fuzzy quantities into crisp quantities. Example 5.7. Use Matlab to find the crisp relations of the lambda set for λ = 0.2, the fuzzy matrix is given by.
State the method of lambda cuts used to convert the fuzzy set to salted. Using the Matlab program find the lambda crisp cut group relations for λ= 0.4, the fuzzy matrix is given by:.
Fuzzy Rule-Based System
Introduction
Formation of Rules
In this statement, some specific conditions are mentioned, if the conditions are met, then it enters the following statements, called constraints. These statements can be stated as vague conditional statements, such as If conditionC Then constraintF. There is no specific condition that must be met in this form of declaration.
Both conditional and unconditional sentences place restrictions on the consequences of a rule-based procedure due to certain conditions. A rule-based system with a set of conditional rules (canonical rule form) is shown in Table 6.1.
Decomposition of Rules
A rule-based system with a set of conditional rules (the canonical form of the rules) is shown in the Fuzzy Rule Based System table. Therefore, the rule is easy.
Aggregation of Fuzzy Rules
Properties of Set of Rules
A set of IF-THEN rules is continuous if it has no adjacent rules with output fuzzy sets with an empty intersection.
Fuzzy Inference System
- Construction and Working of Inference System
- Fuzzy Inference Methods
- Mamdani’s Fuzzy Inference Method
- Takagi–Sugeno Fuzzy Method (TS Method)
- Comparison Between Sugeno and Mamdani Method
- Advantages of Sugeno and Mamdani Method Advantages of the Sugeno MethodAdvantages of the Sugeno Method
Finding the consequence of the rule by combining the strength of the rule and the output membership function. The results of all the fuzzy rules must now be combined to obtain a fuzzy output distribution. The output membership functions on the right side of the figure are combined using fuzzy OR to obtain the output distribution shown in the lower right corner of Fig.
Finally, it uses the maximum operator to calculate the fuzzy OR for combining the results of the two rules. For a zero-order Sugeno model, the output level is a constant (a=b = 0). The output level of each rule is weighted by the firing power of the rule.
Solved Examples
The number of input fuzzy sets and fuzzy rules required by Sugeno fuzzy systems depends on the number and locations of extrema of the function to be approximated. The comparison and merits of both Mamdani and Sugeno model methods are also discussed. How fuzzy rules play a key role in determining the output of a system.
Discuss in detail all the methods used to decompose the compound rules. Write a FIS for controlling the temperature in an air conditioner using one of the inference methods.
Fuzzy Decision Making
- Introduction
- Fuzzy Ordering
- Individual Decision Making
- Multi-Person Decision Making
- Multi-Objective Decision Making
- Fuzzy Bayesian Decision Method
It is common that the fuzzy sets of impressive goals and constraints in this formulation are not defined directly on the set of actions, but by the other sets that characterize relevant states of nature. The individual decision makers have access to different information on which to base their decision. Classical Bayesian decision-making methods assume that the future states of nature can be characterized as probabilistic events.
It is found that the decision-making situation differs between individual decision-making and multi-person decision-making. In the case of multi-objective decision making, one alternative is found to be chosen from among many alternatives.
Applications of Fuzzy Logic
Fuzzy Logic in Power Plants
Fuzzy logic control uses regular measurements of process conditions to detect the addition of plastic to the coal. The specification of the control task led to the preliminary concept of a fuzzy logic controller. The knowledge of the operator was then used to determine the control strategy of the fuzzy logic system.
Therefore, the communication setup between the distributed process control systems (DCS) and the fuzzy logic controller was important. The WIN95 computer is also used for online optimization and visualization of fuzzy logic controller.
Fuzzy Logic Applications in Data Mining
- Adaptive Fuzzy Partition in Data Base Mining
In almost all situations the performance of the fuzzy controller was much higher than the manual control. The following parameters were used in the data processing of the data set of 412 odor compounds:. The number of rules applied to each relationship depended on the complexity of the distribution of the composition with respect to a given odor.
A possible explanation can be found in the fact that only complex combinations of molecular descriptors can represent the distribution of the ethereal compounds, thus requiring a large number of rules. Then new tools must be developed to provide a user-friendly representation of the composite distribution in the descriptor hyperspace.
Fuzzy Logic in Image Processing
- Fuzzy Image Processing IntroductionIntroduction
Step: fuzzification of the current pixel (membership in groups of dark, gray and light pixels) (Fig. 8.14). In image processing, some objective quality criteria are usually used to ascertain the goodness of the results (eg the image is good if it has a low amount of blur indicating high contrast). In general, smoothing algorithms distribute a portion of the intensity of a pixel in the image to adjacent pixels.
After one application of the enhancement, i.e. the INT operator on the above range of pixels we get the following results. The scaled membership values of the pixels can be obtained by dividing the pixel intensity by 255.
