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Stable Fuzzy Control of Single-Input-Single-Output Systems. Exponential . Chapters may be used for the fuzzy system half of the course. A Course in Fuzzy Systems and Control - Ebook download as PDF File .pdf), Text File .txt) or read book online. Provides a comprehensive, self-tutorial course in fuzzy logic and its increasing role in control theory. The book answers key questions about.

Primary terms. By the early s. Truth table for five operations that are frequently applied to propositions. The membership functions for P and F. If we use the Lukasiewicz implication 5. We now consider an example for computing the composition of fuzzy relations in continuous domains. In Chapters 10 and

A fuzzy set A is said to be convex if and only if its a-cut A. PA XZ ]. Since 2. Since a is an arbitrary point in 0. Let xa be an arbitrary element in A. The following lemma gives an equivalent definition of a convex fuzzy set. An intuitively appealing way of defining the union is the following: Definition We say B contains A. To show that this intuitively appealing definition is equivalent to 2.

More precisely.

The intersection as defined by 2. The equality. In this section.

In the sequel. Consider the two fuzzy sets D and F defined by 2. The membership functions for P and F. This makes sense because if a car is not a non-US car which is what the complement of F means intuitively. The De Morgan's Laws are true for fuzzy sets. With the operations of complement. As an example. Comparing 2. The complement of F.

We only prove 2. Figure 2. The membership function for F n D. Operations on Fuzzy Sets 31 Figure 2. The membership function for F U D. From the definitions 2. Consider the fuzzy sets F. The intuitive meaning of membership functions and how to determine intu- itively appealing membership functions for specific fuzzy descriptions. Exercise 2. Zadeh's original paper Zadeh [] is still the best source to learn fuzzy set and related concepts.

The paper was extremely well-written and the reader is encouraged to read it. Performing operations on specific examples of fuzzy sets and proving basic properties concerning fuzzy sets and their operations.

The basic operations and concepts associated with a fuzzy set were also introquced in Zadeh []. The definitions of fuzzy set. Determine reasonable membership functions for "short persons.

Model the following expressions as fuzzy sets: Show that the intersection of two convex fuzzy sets is also a convex fuzzy set. Show that the law of the excluded middle. G and H in Exercise 2. Determine the a-cuts of the fuzzy sets F. Prove the identity 2. Exercises 33 c F n G. What about the union? Other possibilities exist. Why do we need other types of operators? The main reason is that the operators 3.

We explained that the fuzzy set A U B defined by 3. In this chapter. But if we use the min operator of 3. Another reason is that from a theoretical point of view it is interesting to explore what types of operators are possible for fuzzy sets. For fuzzy sets there are other possibilities.

But what are they? What are the properties of these new operators? These are the questions we will try to answer in this chapter. We know that for nonfuzzy sets only one type of operation is possible for complement. The new operators will be proposed on axiomatic bases.

One class of fuzzy complements is the Sugeno class Sugeno defined by where X E Axiom c2 requires that an increase in membership value must result in a decrease or no change in membership value for the complement. Fuzzy Complement 35 3. Another type of fuzzy complement is the Yager class Yager [] defined by where w E 0.

It is easy to verify that 3. For all a. Axiom cl. For each value of the parameter A. It is a simple matter to check that the complement defined by 3. For each value of w. Axiom c l shows that if an element belongs to a fuzzy set to degree zero one. Any function c: Axiom c2. Definition 3. In order for the function c to be qualified In the case of 3.

In order for the function s to be qualified as an union. Figure 3. Sugeno class of fuzzy complements cx a for different values of A. S[PA X. With a particular choice of the parameters. It is straightforward to verify that 3. Axiom s l indicates what an union function should be in extreme cases. Axiom s3 shows a natural requirement for union: Axiom s3.

Axiom s2. Any function s: It is a simple matter to prove that the basic fuzzy union m a s of 3. Axiom s4. Axiom s4 allows us to extend the union operations to more than two fuzzy sets.

We now list three particular classes of s-norms: Dombi class Dombi []: We now list some of them below: Many other s-norms were proposed in the literature. These s-norms were obtained by generalizing the union operation for classical sets from different perspectives. Axiom s2 insures that the order in which the fuzzy sets are combined has no influence on the result. For any s-norm s. The practical reason is that some s-norms may be more meaningful than others in some applications.

If we use the Yager s-norm 3. Fkom the nondecreasing condition Axiom s3 and the boundary condition Axium s l. Consider the fuzzy sets D and F defined in Example 2. Example 3. The theoretical reason is that they become identical when the membership values are restricted to zero or one. We first prove max a.

Comparing Figs. In general. If we use the algebraic sum 3. Theorem 3. Next we prove s a. Membership function of D u F using the al- gebraic sum 3. Membership function of DUF using the Yager s-norm 3. We first prove 3. Lemma 3. Let sx a. By commutativity. I -'] x. By the commutative condition Axiom s2 we have s a. These axioms can be justified in the same way as for Axioms sl-s4. Any function t: We can verify that the basic fuzzy intersection min of 3.

We can verify that 3. In order for the func- tion t to be qualified as an intersection. Axiom t4: Fuzzy Intersection-The T-Norms 41 3. For any t-norm. Dubois-Prade and Yager classes. Axiom t2: Axiom t3: Yager class Yager []: Associated with the particular s-norms 3.

PA XI. Dubois-Prade class Dubois and Prade []: Axiom t 1: Dubois-Prade and Yager classes 3. The proof of this theorem is very similar to that of Theorem 3.

Similar to Lemma 3. If we use the algebraic product 3.

P x which is plotted in Fig. For any t-norm t. If we use the Yager t-norm 3. Let tx a. Membership function of D n F using the Yager t-norm 3. Comparing 3. Membership function of D nF using the al- gebraic product 3. Similar to the s-norms and t-norms. To show this.

I] to [0. Here we list four of them: But what does this "associated" mean? It means that there exists a fuzzy complement such that the three together satisfy the DeMorgan's Law. The algebraic sum 3. The Yager s-norm sw a. Many averaging operators were proposed in the literature.

Yager t-norm tw a. The operators that cover the interval [min a. I] x [0. Max-min averages: The full scope of fuzzy aggregation operators. Averaging Operators 45 I nimum j maximum drastlc sum Einsteln sum algebraic sum b. Show that the Yager s2norm 3. The "fuzzy and" covers the range from min a.

The equilibrium of a fuzzy complement c is defined as a E [O. It also can be shown that the generalized means cover the whole range from min a. Exercise 3. Dubois and Prade [I provided a very good review of fuzzy union. How to prove various properties of some particular fuzzy complements. Some specific classes of fuzzy complements. Let the fuzzy sets F and G be defined as in Exercise 2. The axiomatic definitions of fuzzy complements.

Prove Theorem 3. The materials in this chapter were extracted from Klir and Yuan [I where more details on the operators can be found. Determine s-norm s. Exercises 47 Exercise 3. Prove that the following triples form an associated class with respect to any fuzzy complement c: Show that the operator u: Prove that the generalized means 3.

Chapter 4 Fuzzy Relations and the Extension Principle 4. The Cartesian product of U and V. A relation between U and V is a subset of U x V. A nonfuzzy relation among nonfuzzy sets Ul.

Example 4. Un is a subset of the Cartesian product Ul x U2 x. Un t o denote a relation among Ul. V be a relation named "the first element is no smaller than the second element.. The relation Q U. How- ever With the representation scheme 2. For certain relationships. Definition 4. A classical relation represents a crisp relationship among sets. If we use a number in the interval [O. A fuzzy relation is a fuzzy set defined in the Cartesian product of crisp sets Ul.

LLQinto a relational matriq see the following example. The concept of fuzzy relation was thus introduced.

V defined over U x V which contains finite elements. CO c R2. These concepts can be extended to fuzzy relations. As a special case. Let Q be a fuzzy relation in Ul x. A binary relation on a finite Cartesian product is usually represented by a fuzzy relational matrix. As a special case..

Let U and V be the set of real numbers. A fuzzy relation "x is approximately equal to y. According to 4. Note that 4. Projections and cylindric extensions of a re- lation. The projection constrains a fuzzy relation to a subspace.

Consider the projections Q1 and Q2 in Example 4. The Cartesian product of Al. Lemma 4. To characterize this property formally.

If Q is a fuzzy relation in Ul x. Let A Boston Let Q p be a fuzzy relation in Uil x.. Relation between the Cartesian product and intersection of cylindric sets.

Substituting 4. Compositions o f Fuzzy Relations 53 Figure Using the membership function representation of relations see 4. The composition of P and Q. W be two crisp binary relations that share a common set V For x. We first show that if P o Q is the composition according to the definition. The max-min composition of fuzzy relations P U.

The max-product composition of fuzzy relations P U. P o Q is the composition of P U. The composition of fuzzy relations P U. Because the t-norm in 4.

Ev t[pp x. E V t[pp x. W is a fuzzy relation P o Q in U x W defined by the membership function where 2. If P o Q is the composition. Now we generalize the concept of composition to fuzzy relations. From Lemma 4. The two most commonly used compositions in the literature are the so-called max-min composition and max-product composition. PQ HK. NYC and Tokyo. Beijing 0. Beijing ]. Define the fuzzy relation "very near'' in V x W. Let P U.

Using the max-min composition 4. Boston 0. NYC ]. Beijing 4. Let U and V be defined as in Example 4. V denote the fuzzy relation "very far" defined by 4. UQ Boston. We now consider two examples for how to compute the compositions.

NYC 0. We now consider an example for computing the composition of fuzzy relations in continuous domains. In Example 4. V and W contain a finite number of ele- ments. Specifi- cally. For max-product composition. In most engineering applications. We now want to determine the composition A E o ML. V and W are real-valued spaces that contain an infinite number of elements.

For max-min composition. We now check that 4. Using the max-product composition. Consider the fuzzy relation AE ap- proximately equal and ML much larger defined by 4. The identity 4. If f is not one-to-one.. In practice. The necessary condition for such y is Because it is impossible to obtain a closed form solution for 4. If f is an one-to-one mapping. Comparing this example with Example 4. The Extension Principle 57 To compute the right hand side of 4. Exercise 4.

These original papers of Zadeh were very clearly written and are still the best sources to learn these fundamental concepts. Consider the fuzzy relation Q defined in Ul x. The max-min and max-product compositions of fuzzy relations.

The concepts of fuzzy relations. Ul x U3 and U4. The extension principle and its applications. Determine the fuzzy set f A using the extension principle. The basic ideas of fuzzy relations. Given an n-ary relation. Consider the three binary fuzzy relations defined by the relational matrices: Z in Example 4.

We now define three fuzzy sets "slow. Definition 5. Example 5. But when a variable takes words as its values. In the fuzzy theory literature. If we view x as a linguistic variable.

Now the question is how to formulate the words in mathematical terms? Here we use fuzzy sets to characterize the words. In order to provide such a formal framework. The speed of a car is a variable x that takes values in the interval [0. When a variable takes numbers as its values. Roughly speaking. If a variable can take words in natural languages as its values. Why is the concept of linguistic variable important? Because linguistic variables are the most fundamental elements in human knowledge representation.

M is a semantic rule that relates each linguistic value in T with a fuzzy set in U. This definition is given below. From these definitions we see that linguistic variables are extensions of numerical variables in the sense that they are allowed to take fuzzy sets as their values. U is the actual physical domain in which the linguistic variable X takes its quantitative crisp values. Comparing Definitions 5. T is the set of linguistic values that X can take.

The speed of a car as a linguistic variable that can take fuzzy sets "slow. When we use sensors to measure a variable. A linguistic variable is characterized by X. M relates "slow.

X is the name of the linguistic variable. The terms "not. These atomic terms may be classified into three groups: Primary terms. From numerical variable to linguistic variable. Our task now is to characterize hedges. This is the first step to incorporate human knowledge into engineering systems in a systematic and efficient manner.

In our daily life. Linguistic Hedges 61 numerical variable linguistic variable Figure An atomic fuzzy proposition is a single statement. In this spirit. Let A be a fuzzy set in U. M and F denote the fuzzy sets "slow. Note that in a compound fuzzy proposition. How to determine the membership functions of these fuzzy relations? For connective "and" use fuzzy intersections. A compound fuzzy proposition is a composition of atomic fuzzy propositions using the connectives "and.

For connective "or" use fuzzy unions. A is a fuzzy set defined in the physical domain of x. V and A represent classical logic operations "not. V and A operators in 5. From Table 5. The fuzzy proposition 5. In classical propositional calculus. Since there are a wide variety of fuzzy complement.

We list some of them below. Dienes-Rescher Implication: If we replace the logic operators. By generalizing it to fuzzy propositions. Giidel Implication: The Godel implication is a well-known implication for- mula in classical logic.. Y and max[l. Another question is: Are 5. Since 0 I 1. For all x. When p and q are crisp propositions that is.

PQL x. Lemma 5. This is an important question and we will discuss it in Chapters PFPz Y. In logic terms. THEN resistance is high. PQD x. PFPI x. We now try to answer this question. So a question arises: Based on what criteria do we choose the combination of fuzzy complements. The fol- lowing lemma shows that the Zadeh implication is smaller than the Dienes-Rescher implication. THEN resistance is low.

THEN y is large 5. Let xl be the speed of a car. We now consider some examples for the computation of Q D. If we use algebraic product for the t-norm in 5. Figure 5. Division of the domains of Zadeh and Mamdani implications. A way to resolve this complexity is to use a single smooth function t o approximate the nonsmooth functions. IF x is large. If we use the Lukasiewicz implication 5.

Suppose we know that x E U is somewhat inversely propositional to y E V. To formulate this knowledge. Suppose we use to approximate the pslOw xl of 5. Now if we use Mamdani product implication 5.

Godel and Mamdani implications. This is consistent with our earlier discussion that Dienes-Rescher. This paper is another piece of art and the reader is highly recommended to study it. Zadeh and Godel implications are global. Summary and Further Readings 71 For the Zadeh implication 5.

The concept of linguistic variables and the characterization of hedges.. Linguistic variables were introduced in Zadeh's seminal paper Zadeh []. QZ and QG give full membership value to them. Properties and computation of these implications. The comprehensive three-part paper Zadeh [I summarized many concepts and principles associated with linguistic variables.

Show that Exercise 5. Combine these lin- guistic variables into a compound fuzzy proposition and determine its membership function. Give three examples of linguistic variables. Exercise 5.

Let QL. Use basic fuzzy operators 3. Let Q be a fuzzy relation in U x U. Plot these membership functions.

Q is called reflexive if pQ u. Consider some other linguistic hedges than those in Section 5. Use m i n for the t-norm in 5.

Show that if Q is reflexive. The fundamental truth table for conjunction V. By combining. In classical logic. The most commonly used complete set of primitives is negation-. Given n basic propositions p l. Chapter 6 Fuzzy Logic and Approximate Reasoning 6. This generalization allows us to perform approximate reasoning.

Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any number in the interval [0. V and A in appropriate algebraic expressions. Since n propositions can assume 2n possible combinations of truth values. The new proposition is usually called a logic function.

Table 6. If p is a proposition. Truth table for five operations that are frequently applied to propositions.

Logic formulas are defined recursively as follows: The truth values 0 and 1 are logic formulas. The following logic formulas are tautologies: To prove 6. The three most commonly used inference rules are: The only logic formulas are those defined by a - c. When the proposition represented by a logic formula is always true regardless of the truth values of the basic propositions participating in the formula.

Various forms of tautologies can be used for making deductive inferences. Example 6. If p and q are logic formulas. They are referred to as inference rules. The ultimate goal of fuzzy logic is t o provide foundations for approximate reasoning with imprecise propositions using fuzzy set theory as the principal tool.

They are the fundamental principles in fuzzy logic. To achieve this goal. Intuitive criteria relating Premise 1 and the Conclusion for given Premise 2 in generalized modus ponens.

ELSE y is not B. Criterion p7 is interpreted as: Generalized M o d u s Ponens: Premise 1: We note that if a causal relation between "x is A" and "y is B" is not strong in Premise 2. B' and B are fuzzy sets.

Similar to the criteria in Table 6. B and B' are fuzzy sets. C and C' are fuzzy sets. Intuitive criteria relating Premise 1 and the Conclusion for given Premise 2 in generalized modus tollens.

Other criteria can be justified in a similar manner. By applying the hedge more or less to cancel the very. Criteria s2 is obtained from the following intuition: Intuitive criteria relating y is B' in Premise 2 and z is C' in the Conclusion in generalized hypothetical syllogism.

We have now shown the basic ideas of three fundamental principles in fuzzy logic: Going one step further in our chain of generalization. Although these criteria are not absolutely correct. The next question is how to determine the membership functions of the fuzzy propositions in the conclusions given those in the premises.

Then we project I on V yielding the interval b. Figure 6. To find the interval b which is inferred from a and f x. They should be viewed as guidelines or soft constraints in designing specific infer- ences. The compositional rule of inference was proposed to answer this question. Let us generalize the above procedure by assuming that a is an interval and f x is an interval-valued function as shown in Fig. Inferring fuzzy set B' from fuzzy set A' and fuzzy relation Q.

Inferring interval b from interval a and interval-valued function f x. The Compositional Rule of Inference 79 Figure 6. Different implication principles give different fuzzy relations. In summary. In the literature. For generalized hypothetical syllogism. In Chapter 5. Our task is to determine the corresponding B'.

Suppose we use min for the t-norm and Mamdani's product impli- cation 5. These results show the properties of the implication rules. We consider the generalized modus ponens. Consider four cases of A': Properties of the Implication Rules 81 is C.

We now study some of these properties. UA XO. From 6. In this example. Y in the generalized modus ponens 6. To summarize. This approximate reasoning is truely approximate! Similar to Example 6. Consider four cases of B': Properties of the Implication Rules 85 Finally. Similar to Examples 6.

Zadeh [I and other papers of Zadeh in the s. The idea and applications of the compositional rule of inference. Basic inference rules Modus Ponens. A comprehensive treatment of many-valued logic was prepared by Rescher []. The generalizations of classical logic principles to fuzzy logic were proposed in Zadeh []. Exercise 6. The compositional rule of inference also can be found in these papers of Zadeh. THEN y is B" is given. Modus Tollens.

Using truth tables to prove the equivalence of propositions. Generalized Modus Tollens. Use the truth table method to prove that the following are tautologies: Determining the resulting membership functions from different implication rules and typical cases of premises. For any two arbitrary propositions. Show exactly how they are restricted. A and B. Imposing such requirement means that pairs of truth values of A and B become restricted to a subset of [O.

Repeat Exercise 6. Consider a fuzzy logic based on the standard operation min. With min as the t-norm and Mamdani minimum implication 5.

Let U. Use min for the t-norm and Lukasiewicz implication 5. Use min for the t-norm and Dienes-Rescher implication 5. Now given a fact "y is B'. Exercises 87 Exercise 6. Y in the generalized modus tollens 6. In Chapter 9. We will see how the fuzzy mathematical and logic principles we learned in Part I are used in the fuzzy systems. In Chapters 10 and In Chapter 7.

In this part Chapters We will derive the compact mathematical formulas for different types of fuzzy systems and study their approximation properties. Dispatched from the UK in 3 business days When will my order arrive? Home Contact us Help Free delivery worldwide.

Free delivery worldwide. Bestselling Series. Harry Potter. Popular Features. New in Description Provides a comprehensive, self-tutorial course in fuzzy logic and its increasing role in control theory. The book answers key questions about fuzzy systems and fuzzy control. It introduces basic concepts such as fuzzy sets, fuzzy union, fuzzy intersection and fuzzy complement.

Learn about fuzzy relations, approximate reasoning, fuzzy rule bases, fuzzy inference engines, and several methods for designing fuzzy systems. For professional engineers and students applying the principles of fuzzy logic to work or study in control theory. Table of contents 1. Why Fuzzy Systems? What Are Fuzzy Systems? Summary and Further Readings. From Classical Sets to Fuzzy Sets.

Operations on Fuzzy Sets. Further Operations on Fuzzy Sets. Fuzzy Complement. Fuzzy Union -- The S-Norms. Fuzzy Intersection -- The T-Norms. Averaging Operators. Fuzzy Relations and the Extension Principle. From Classical Relations to Fuzzy Relations. Compositions of Fuzzy Relations. The Extension Principle. From Numerical Variables to Linguistic Variables. Linguistic Hedges. Fuzzy Logic and Approximate Reasoning. From Classical Logic to Fuzzy Logic. The Compositional Rule of Inference.

Properties of the Implication Rules. Fuzzy Rule Base. Fuzzy Inference Engine. Fuzzifiers and Defuzzifiers. Fuzzy Systems as Nonlinear Mappings. Fuzzy Systems As Universal Approximators. Approximation Properties of Fuzzy Systems I.

Preliminary Concepts. Design of A Fuzzy System. Approximation Accuracy of the Fuzzy System. Application to Truck Backer-Upper Control. Application to Time-Series Prediction. Exercises and Projects. Choosing the Structure of Fuzzy Systems. Designing the Parameters by Gradient Descent. Application to Nonlinear Dynamic System Identification. Design of the Fuzzy System. Derivation of the Recursive Least Squares Algorithm.

Application to Equalization of Nonlinear Communication Channels. Design of Fuzzy Systems Using Clustering. An Optimal Fuzzy System. Design of Fuzzy Systems By Clustering. Application to Adaptive Control of Nonlinear Systems. Fuzzy Control Versus Conventional Control. Case Study I: