Jiet Setg, Jodhpur

JIET SETG, JODHPUR

[Type text] Page 1 JIET SETG, JODHPUR

ABSTRACT

FL is a problem-solving control system methodology that lends itself to implementation in systems ranging from simple, small, embedded micro-controllers to large, networked, multi- channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. FL provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. FL's approach to control problems mimics how a person would make decisions, only much faster.

FL incorporates a simple, rule-based IF X AND Y THEN Z approach to a solving control problem rather than attempting to model a system mathematically. The FL model is empirically-based, relying on an operator's experience rather than their technical understanding of the system. For example, rather than dealing with temperature control in terms such as "SP =500F", "T <1000F", or "210C

FL requires some numerical parameters in order to operate such as what is considered significant error and significant rate-of-change-of-error, but exact values of these numbers are usually not critical unless very responsive performance is required in which case empirical tuning would determine them. For example, a simple temperature control system could use a single temperature feedback sensor whose data is subtracted from the command signal to compute "error" and then time-differentiated to yield the error slope or rate-of-change-of-error, hereafter called "error-dot". Error might have units of degree F and a small error considered to be 2F while a large error is 5F. The "error-dot" might then have units of degree/min with a small error-dot being 5F/min and a large one being 15F/min. These values don't have to be symmetrical and can be "tweaked" once the system is operating in order to optimize performance. Generally, FL is so forgiving that the system will probably work the first time without any tweakin

INTRODUCTION

The science or art of exact reasoning, or of pure and formal thought, or of the laws according to which the processes of pure thinking should be conducted; the science of the formation and application of general notions; the science of generalization, judgment, classification, reasoning, and systematic arrangement; correct reasoning is called logic.

Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor for computer science at the University of California in Berkeley. Basically, Fuzzy Logic (FL) is a multi valued logic that allows intermediate values to be defined between conventional evaluations like true/false, yes/no, high/low, etc. Notions like rather tall or very fast can be formulated mathematically and processed by computers, in order to apply a more human-like way of thinking in the programming of computers. Fuzzy systems are an alternative to traditional notions of set membership and logic that has its origins in ancient Greek philosophy. The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him. In their efforts to devise a concise theory of logic, and later mathematics, the so- called”Laws of Thought” were posited. One of these, the”Law of the Excluded Middle,” states that every proposition must either be true or false. Even when Parminedes proposed the first version of this law, there were strong and immediate objections: for example, Heraclitus proposed that things could be simultaneously true and not true. It was Plato who laid the foundation for what would become fuzzy logic, indicating that there was a third region (beyond True and False) where these opposites”tumbled about.” Other, more modern philosophers echoed his sentiments, notably Hegel, Marx, and Engels. But it was Lukasiewicz who first proposed a systematic alternative to the bi–valued logic of Aristotle.

Fuzzy Logic has emerged as a profitable tool for the controlling and steering of of systems and complex industrial processes, as well as for household and entertainment electronics, as well as for other expert systems and applications like the classification of SAR data.

NEED FOR FUZZY LOGIC

Fuzzy Logic is a paradigm for an alternative design methodology which can be applied in developing both linear and non-linear systems for embedded control. By using Fuzzy Logic designers can realize lower development costs, superior features, and better end product performance. Furthermore, products can be brought to market faster and more cost-effectively.

In order to appreciate why a fuzzy based design methodology is very attractive in embedded control applications let us examine a typical design flow. Using the conventional approach our first step is to understand the physical system and its control requirements. Based on this understanding, our second step is to develop a model which includes the plant, sensors and actuators. The third step is to use linear control theory in order to determine a simplified version of the controller, such as the parameters of a PID controller. The fourth step is to develop an algorithm for the simplified controller. The last step is to simulate the design including the effects of non-linearity, noise, and parameter variations. If the performance is not satisfactory we need to modify our system modeling, re-design the controller, re-write the algorithm and re-try.

With Fuzzy Logic the first step is to understand and characterize the system behavior by using our knowledge and experience. The second step is to directly design the control algorithm using fuzzy rules, which describe the principles of the controller's regulation in terms of the relationship between its inputs and outputs. The last step is to simulate and debug the design. If the performance is not satisfactory we only need to modify some fuzzy rules and re-try.

Although the two design methodologies are similar, the fuzzy-based methodology substantially simplifies the design loop. This results in some significant benefits, such as reduced development time, simpler design and faster time to market.

For example, a simple temperature regulator that uses a fan might look like this:

IF temperature IS<100 C THEN stop fan IF temperature IS<100 C and >200 C THEN turn down fan IF temperature IS<200 C and >250 C THEN maintain level IF temperature IS<250 C and >320 C THEN speed up fan

The same temperature regulator that uses a fan might look like this when using fuzzy logic:

IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan

FUZZY SETS AND CRISP SETS

The very basic notion of fuzzy systems is a fuzzy (sub) set. In classical mathematics we are familiar with what we call crisp sets. For example, the possible interferometric coherence g values are the set X of all real numbers between 0 and 1. From this set X a subset A can be defined, (e.g. all values 0 <=g >= 0.2). A characteristic function of A assigns a number 1 or 0 to each element in X, depending on whether the element is in the subset A or not. The elements which have been assigned the number 1 can be interpreted as the elements that are in the set A and the elements which have assigned the number 0 as the elements that are not in the set.

This concept is sufficient for many areas of applications, but it can easily be seen, that it lacks in flexibility for some applications like classification of remotely sensed data analysis. For example it is well known that water shows low interferometric coherence g in SAR images. Since g starts at 0, the lower range of this set ought to be clear. The upper range, on the other hand, is rather hard to define. As a first attempt, we set the upper range to 0.2. Therefore we get B as a crisp interval B= [0, 0.2]. But this means that a g value of 0.20 is low but a g value of 0.21 not. Obviously, this is a structural problem, for if we moved the upper boundary of the range from g =0.20 to an arbitrary point we can pose the same question. A more natural way to construct the set B would be to relax the strict separation between low and not low. This can be done by allowing not only the (crisp) decision Yes/No, but more flexible rules like ” fairly low”. A fuzzy set allows us to define such a notion. The aim is to use fuzzy sets in order to make computers more ’intelligent’, therefore, the idea above has to be coded more formally. In the example, all the elements were coded with 0 or 1. A straight way to generalize this concept is to allow more values between 0 and 1. In fact, infinitely many alternatives can be allowed between the boundaries 0 and 1, namely the unit interval I = [0, 1]. The interpretation of the numbers, now assigned to all elements is much more difficult. Of course, again the number 1 assigned to an element means that the element is in the set B and 0 means that the element is definitely not in the set B. All other values mean a gradual membership to the set B.

The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. The membership function, operating in this case on the fuzzy set of interferometric coherence g, returns a value between 0.0 and 1.0. For example, an interferometric coherence g of 0.3 has a membership of 0.5 to the set low coherence.

It is important to point out the distinction between fuzzy logic and probability. Both operate over the same numeric range, and have similar values: 0.0 representing False (or non- membership), and 1.0 representing True (or full-membership). However, there is a distinction to be made between the two statements: The probabilistic approach yields the natural-language statement,”There is a 50% chance that g is low,” while the fuzzy terminology corresponds to”g’s degree of membership within the set of low interferometric coherence is 0.50.” The semantic difference is significant: the first view supposes that g is or is not low; it is just that we only have a 50% chance of knowing which set it is in. By contrast, fuzzy terminology supposes that g is”more or less” low, or in some other term corresponding to the value of 0.50.

FUZZY SETS AND MEMBERSHIP FUNCTION

Zadeh introduced the term fuzzy logic in his seminal work “Fuzzy sets,” which described the mathematics of fuzzy set theory (1965). Plato laid the foundation for what would become fuzzy logic, indicating that there was a third region beyond True and False. It was Lukasiewicz who first proposed a systematic alternative to the bivalued logic of Aristotle. The third value Lukasiewicz proposed can be best translated as “possible,” and he assigned it a numeric value between True and False. Later he explored four-valued logic and five-valued logic, and then he declared that, in principle, there was nothing to prevent the derivation of infinite-valued logic. FL provides the opportunity for modeling conditions that are inherently imprecisely defined. Fuzzy techniques in the form of approximate reasoning provide decision support and expert systems with powerful reasoning capabilities. The permissiveness of fuzziness in the human thought process suggests that much of the logic behind thought processing is not traditional two valued logic or even multi valued logic, but logic with fuzzy truths, fuzzy connectiveness, and fuzzy rules of inference. A fuzzy set is an extension of a crisp set. Crisp sets allow only full membership or no membership at all, whereas fuzzy sets allow partial membership. In a crisp set, membership or non membership of element x in set A is described by a characteristic

function µA(x), where µA(x)=1 if x∈A and µA(x)=0 if x∉ A. Fuzzy set theory extends this concept by defining partial membership. A fuzzy set A on a universe of discourse U is characterized by a membership function µA(x) that takes values in the interval [0,1]. Fuzzy sets represent commonsense linguistic labels like slow, fast, small, large, heavy, low, medium, high, tall, etc. A given element can be a member of more than one fuzzy set at a time. A fuzzy set A in U may be represented as a set of ordered pairs. Each pair consists of a generic element x and

its grade of membership function; that is, i A={(x, µA(x))|x∈U}, x is called a support value if

µA(x)>0. A linguistic variable x in the universe of discourse U is characterized by

1 2 n 1 2 n T(x)={Tx ,Tx ,……,Tx } and µ(x)={ µx , µx ,….., µx }, where T(x) is the term set of x- that is,

i the set of names of linguistic values of x, with each Tx being a fuzzy number with membership

i function µx defined on U. For example, if x indicates height, then may refer to sets such as short, medium, or tall. A membership function is essentially a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. As an example, consider a fuzzy set tall. Let the universe of discourse be heights from 40 inches to 90 inches. With a crisp set, all people with height 72 or more inches are considered tall, and all people with height of less than 72 inches are considered not tall. The curve below defines the transition from not tall and shows the degree of membership for a given height. We can extend this concept to multiple sets. If we consider a universe of discourse from 40 inches to 90 inches, then, to describe height, we can use three term values such as short, average, and tall. In practice, the terms short, medium, and tall are not used in the strict sense. Instead, they imply a smooth transition.

LOGICAL OPERATIONS AND IF–THEN RULES

Fuzzy set operations are analogous to crisp set operations. The important thing in defining fuzzy set logical operators is that if we keep fuzzy values to the extremes 1 (True) or 0 (False), the standard logical operations should hold. In order to define fuzzy set logical operators, let us first consider crisp set operators. The most elementary crisp set operations are union, intersection, and complement, which essentially correspond to OR, AND, and NOT operators, respectively. Let A and B be two subsets of U. The union of A and B, denoted AUB, contains all

elements in either A or B; that is, (x)=1 if x∈A or x∈B. The intersection of A and B,

denoted A B contains all the elements that are simultaneously in A and B; that is, (x)=1

if x∈A and x∈B. The complement of A is denoted by , and it contains all elements that are not

in A; that is (x)=1 if x∉ A, and =0 if x∈A.

In FL, the truth of any statement is a matter of degree. In order to define FL operators, we have to find the corresponding operators that preserve the results of using AND, OR, and NOT operators. The answer is min, max, and complement operations. These operators are defined, respectively, as

(x)=max[µA(x), µB(x)]

(x)=min[µA(x), µB(x)]

(x)=1- µA(x)

The formulas for AND, OR, and NOT operators are useful for proving other mathematical properties about sets; however, min and max are not the only ways todescribe the intersection and union of two sets.

FUZZIFICATION AND DEFUZZIFICATION PROCESS

Steps:  Fuzzy Inputs: The first step is to take the crisp inputs, x1 and y1 and determine the degree to which these inputs belong to each of the appropriate fuzzy sets.  Apply Fuzzy Operators: The second step is to take the fuzzified inputs and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. This number (the truth value) is then applied to the consequent membership function. To evaluate the disjunction of the rule antecedents, operation OR fuzzy operation, typically, fuzzy expert systems make use of the classical fuzzy operation, union. Similarly, in order to evaluate the conjunction of the rule antecedents, we apply the

AND fuzzy operation intersection. Now the result of the antecedent evaluation can be applied to the membership function of the consequent. The most common method of correlating the rule consequent with the truth value of the rule antecedent is to cut the consequent membership function at the level of the antecedent truth. This method is called clipping. Since the top of the membership function is sliced, the clipped fuzzy set loses some information. However, clipping is still often preferred because it involves less complex and faster mathematics, and generates an aggregated output surface that is easier to defuzzify.  Aggregate All Outputs: Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set. The input of the aggregation process is the list of clipped or scaled consequent membership functions and the output is one fuzzy set for each output variable.  Defuzzify: The last step in the fuzzy inference process is defuzzification. Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number. The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number.

FUZZY INFERENCE SYSTEM

A fuzzy inference system (FIS) essentially defines a nonlinear mapping of the input data vector into a scalar output, using fuzzy rules. The mapping process involves input/output membership functions, FL operators, fuzzy if–then rules, aggregation of output sets, and defuzzification. An FIS with multiple outputs can be considered as a collection of independent multiinput, single output systems. The FLS maps crisp inputs into crisp outputs. It can be seen from the figure that the FIS contains four components: the fuzzifier, inference engine, rule base, and defuzzifier. The rule base contains linguistic rules that are provided by experts. It is also possible to extract rules from numeric data. Once the rules have been established, the FIS can

[Type text] Page 16 JIET SETG, JODHPUR be viewed as a system that maps an input vector to an output vector. The fuzzifier maps input numbers into corresponding fuzzy memberships. This is required in order to activate rules that are in terms of linguistic variables. The fuzzifier takes input values and determines the degree to which they belong to each of the fuzzy sets via membership functions. The inference engine defines mapping from input fuzzy sets into output fuzzy sets. It determines the degree to which the antecedent is satisfied for each rule. If the antecedent of a given rule has more than one clause, fuzzy operators are applied to obtain one number that represents the result of the antecedent for that rule. It is possible that one or more rules may fire at the same time. Outputs for all rules are then aggregated. During aggregation, fuzzy sets that represent the output of each rule are combined into a single fuzzy set. Fuzzy rules are fired in parallel, which is one of the important aspects of an FIS. In an FIS, the order in which rules are fired does not affect the output. The defuzzifier maps output fuzzy sets into a crisp number. Given a fuzzy set that encompasses a range of output values, the defuzzifier returns one number, thereby moving from a fuzzy set to a crisp number. Several methods for defuzzification are used in practice, including the centroid, maximum, mean of maxima, height, and modified height defuzzifier. The most popular defuzzification method is the centroid, which calculates and returns the center of gravity of the aggregated fuzzy set.

A fuzzy rule base contains a set of fuzzy rules R. A single if–then rule assumes the form “if x is Tx then y is Ty.” An example of a rule might be “if education is high and experience is high, then salary is very high.” For a multi input, multi output system,

where the ith fuzzy rule is

The p preconditions of Ri form a fuzzy set Tx1 X Tx2 X……….X Txp

R1: if education is low and experience is low, then salary is very low

R2: if education is low and experience is medium, then salary is low

R3: if education is low and experience is high, then salary is medium

R4: if education is medium and experience is low, then salary is low

R5: if education is medium and experience is medium, then salary is medium

R6: if education is medium and experience is high, then salary is high

R7: if education is high and experience is low, then salary is medium

R8: if education is high and experience is medium, then salary is high

R9: if education is high and experience is high, then salary is very high

Interpreting an if–then rule is a three part process: (a) Resolve all fuzzy statements in the antecedent to a degree of membership between 0 and 1; (b) if there are multiple parts to the antecedent, apply fuzzy logic operators and resolve the antecedent to a single number between 0 and 1, is the result being the degree of support for the rule; and (c) apply the implication method, using the degree of support for the entire rule to shape the output fuzzy set. If the rule has more than one antecedent, the fuzzy operator is applied to obtain one number that represents the result of applying that rule.

DEFUZZIFICATION

A fuzzy inference system maps an input vector to a crisp output value. In order to obtain a crisp output, we need a defuzzification process. The input to the defuzzification process is a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzification process is a single number. Many defuzzification techniques have been proposed in the literature. The most commonly used method is the centroid. Other methods include the maximum, the means of maxima, height, and modified height method.

Let us look at an example which will make the process of fuzzification and defuzzification clearer.

OTHER SYSTEMS OF FUZZY LOGIC

 Pavelka's logic. (Łukasiewicz with rational truth constants; see Pavelka 1979, Novak et al. 2000; V. Novak systematically develops this logic as a logic with evaluated syntax (working with pairs (formula, truth value)), fuzzy theories (sets of evaluated formulas) and fuzzy modus ponens [from (φ,u), (φ→ψ,v) derive (ψ,u*v) where * is Łukasiewicz t- norm]. This has excellent properties thanks to the fact that Łukasiewicz t-norm is the only continuous t-norm whose residuum is continuous.

 Expansions of basic logic BL by aditional connectives. These include logics with an additional involutive negation (Esteva et al. 2000), and logics putting Łukasiewicz and product logic together.

 The monoidal t-norm based logic MTL. Introduced in Esteva & Godo 2001 as well as

its predicate variant MTL∀. This is a generalization of the logic BL — a logic of left

continuous t-norms. It has stronger variants IMTL and ΠMTL generalizing the Łukasiewicz and product logic. These logics are (strongly) complete with respect to corresponding algebras.

 The monoidal t-norm based logic MTL. Introduced in Esteva & Godo 2001 as well as

its predicate variant MTL∀. This is a generalization of the logic BL — a logic of left

continuous t-norms. It has stronger variants IMTL and ΠMTL generalizing the Łukasiewicz and product logic. These logics are (strongly) complete with respect to corresponding algebras.

 Fuzzy logics with non-commutative conjunction. (φ&ψ not necessarily equivalent to ψ&φ).

 Fuzzy logic and vagueness. Is fuzzy logic, logic of vague notions? This is discussed; there are two monographs on vagueness written by philosophers, Shapiro 2006 and Smith 2008. They also discuss the relation of vagueness to truth degrees (fuzziness).

 Axiomatic fuzzy set theory. Let us mention two important approaches: first, an axiomatic theory (over a fuzzy predicate logic) which should be analogous to the classical Zermelo-Fraenkel set theory. This is well possible - see Hajek & Hanikova 2003. Another very interesting approach is to have a theory (over Łukasiewicz predicate logic) which would have full comprehension — each formula determines a set of all elements satisfing the formula. Over classical logic this is contradictory (Russel's paradox), but over Łukasiewicz it is consistent (Cantor-Łukasiewicz set theory), as was proved by White.

COMPLEXITY OF FUZZY LOGIC

For propositional logics it is always a natural question whether a logic is decidable, i.e., whether its set of tautologies is recursive, and if it is, whether it is in co-NP (its complement being non-neterministically computable in polynomial time). Similarly for the set of satisfiable formulas and NP (Also sets of positive tautologies, i.e. formulas having a positive value in each evaluation and positively satisfiable formulas are discussed.) It has been shown that for our logics tautologies are co-NP-complete (of maximal complexity in co-NP) and satisfiable formulas are NP-complete. The corresponding predicate logics are undecidable (as is the classical predicate logic) but of various degree of undecidability in the sense of so-called arithmetical hierarchy of Σn-sets and Πn-sets. For the reader knowing this hierarchy we mention that for example the set of standard predicate tautologies of Gödel logic is Σ1-complete, for

Łukasiewicz it is Π2-complete and for product logic it is non-arithmetical (outside the arithmetical hierarchy). Not surprisingly, the set of general predicate tautologies of each of these logics is Σ1-complete (due to completeness theorem).

IMPORTANT CHARACTERISTICS OF FUZZY LOGIC

 Exact reasoning is considered as a limiting case of approximate reasoning.  In fuzzy logic, everything is a matter of degree  Any logical system can be fuzzified means Boolean logic is a subset of fuzzy logic

Fuzzy logic is a very useful method of reasoning when mathematical models are not available and large amount of input is present. It provides a simple way to arrive at a definite conclusion based upon ambiguous, imprecise, noisy or missing input information. It is quiet helpful in accelerating the speed of decision making. When true or false is inadequate to describe human reasoning, fuzzy logic is used.

ADVANTAGES OF FUZZY LOGIC

 It requires little amount of data

 It is applicable for all kinds of uncertainty

 It is completely comprehensive in nature

 It is quiet fast and the process of computation involved is also relatively easier

 One need not give prior information about correlations

 Though it is somewhat conservative in its structure but not hyper conservative

 The result lies in between worst case and probability

 Back calculations are easier to solve than in traditional mathematics

DRAWBACKS OF FUZZY LOGIC

The rules of combining membership functions discussed above are known as the minmax rule for conjunctive (AND) and disjunctive (OR) reasoning. These rules have a major drawback: They are not robust at all. If we try to imitate the way human’s reason, the minmax rule is definitely not the way.

Many researchers have proposed different rules of combining conjunctive or disjunctive clauses: for example, instead of taking the minimum or the maximum of the membership functions, they take the arithmetic or the geometric mean. These rules are arbitrary, and there are lots of them. It is possible, if we have enough training data, i.e. conditions and class assignments by the experts, to train our system so that it chooses the best rule that fits the way of reasoning of the expert that did the classification.

Another disadvantage of the rules discussed earlier is that they give the same importance to all factors that are to be combined. For example, it is possible that the role of soil depth or rock

[Type text] Page 27 JIET SETG, JODHPUR permeability is not of the same importance to soil erosion as the role of slope. This issue can be resolved if we do not insist on all membership functions taking values between 0 and 1. APPLICATIONS OF FUZZY LOGIC

 Automatic control of dam gates for hydroelectric-power plants  Simplified control of robots  Camera aiming for the telecast of sporting events  Substitution of an expert for the assessment of stock exchange activities  Efficient and stable control of car-engines  Efficient and stable control of car-engines  Cruise-control for automobiles  Improved efficiency and optimized function of industrial control applications  Positioning of wafer-steppers in the production of semiconductors  Optimized planning of bus time-tables  Archiving system for documents  Prediction system for early recognition of earthquakes  Medicine technology: cancer diagnosis  Combination of Fuzzy Logic and Neural Nets  Recognition of handwritten symbols with pocket computers  Recognition of motives in pictures with video cameras  Automatic motor-control for vacuum cleaners with recognition of surface condition and degree of soiling  Back light control for camcorders  Single button control for washing-machines  Recognition of handwriting, objects, voice  Flight aid for helicopters  Simulation for legal proceedings  Software-design for industrial processes  Controlling of machinery speed and temperature for steel-works

 Controlling of subway systems in order to improve driving comfort, precision of halting and power economy  Improved fuel-consumption for automobiles  Improved sensitiveness and efficiency for elevator control  Improved safety for nuclear reactors  Compensation against vibrations in camcorder

FUTURE OF FUZZY LOGIC

Just from the examples given previously, it is clear that fuzzy logic can be used in numerous applications. It can appear almost any place where computers and modern control theory are overly precise as well as in tasks requiring delicate human intuition and experience-based knowledge. Consider the example below which is currently undergoing intensive research in OMRON Research Center, Japan.

It may seem obvious that babies nowadays don't drink the way it is described in child care books. They may drink a little or a lot depending on their physical condition, mood, and other factors. But if a fuzzy-logic program can be created that would recommend how much to feed the baby, mothers would have an easier time raising the child. The basis of the research is to develop a program to determine the appropriate amount of milk to feed the child according to a knowledge base that includes the child's personality, physical condition, and some environmental factors. This can prevent the child from being fed unnecessarily. Now although adapting fuzzy logic to babies may seem silly, one can easily imagine using it to control the feeding of animals in captivity, for instance.

Here are some of the future fuzzy uses:  Vast expert decision makers, theoretically able to distill the wisdom of every document ever written.  Sex robots with a humanlike repertoire of behavior.  Computers that understand and respond to normal human language.

 Machines that write interesting novels and screenplays in a selected style, such as Hemingway's.  Molecule-sized soldiers of health that will roam the blood-stream, killing cancer cells and slowing the aging process.

Hence, it can be seen that with the enormous reseach currently being done in Japan and many other countries whose eyes have opened, the future of fuzzy logic is undetermined. There is no limit to where it can go. The future is bright. The future is fuzzy.

CONCLUSION

Fuzzy Logic provides a different way to approach a control or classification problem. This method focuses on what the system should do rather than trying to model how it works. One can concentrate on solving the problem rather than trying to model the system mathematically, if that is even possible. On the other hand the fuzzy approach requires a sufficient expert knowledge for the formulation of the rule base, the combination of the sets and the defuzzification. In General, the employment of fuzzy logic might be helpful, for very complex processes, when there is no simple mathematical model (e.g. Inversion problems), for highly nonlinear processes or if the processing of (linguistically formulated) expert knowledge is to be performed. According to literature the employment of fuzzy logic is not recommendable, if the conventional approach yields a satisfying result, an easily solvable and adequate mathematical model already exists, or the problem is not solvable.

BIBLIOGRAPHY

[1] L.A. Zadeh, Fuzzy Sets, Information and Control, 1965 [2] L.A. Zadeh, Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes, 1973 [3] L.A. Zadeh,”Fuzzy algorithms,” Info. & Ctl., Vol. 12, 1968, pp. 94-102 [4] S. Haack, ”Do we need fuzzy logic?” Int. Jrnl. of Man-Mach. Stud., Vol. 11, 1979, pp.437- 445. [5] R. Kruse, J. Gebhardt, F. Klawon, ”Foundations of Fuzzy Systems”, Wiley, Chichester 1994

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