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**Fuzzy Logic & Intelligent Control Systems Lecture 2**

ASSLAMU ALIKUM From Muhammad Khurram Shaikh BE (Elect) , NEDUET MC(CS) , Bradley Univ, Peoria, IL , USA

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**Historical Background of FL**

400 B.C : Classical logic of Aristotle : Law of Bivalence (which was philosophically correct for more than 2000 yrs) “Every Proposition is either True or False (no middle)” 1900 : Jan Lukasiewicz proposed three-valued logic : True , False and Possible 1965 : Lofti Asker Zadeh published “Fuzzy sets” 1979 : S. Hacck “ Do we need Fuzzy Logic”

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Boolean vs. Fuzzy Aristotle came up with binary logic(0,1), the principle foundation of mathematics. Every statement is true or false or has the truth value 1 or 0. But two centuries before Aristotle, Buddha’s belief was a rose could be to a certain degree completely Red but at the same time could also be at a certain degree not Red. Boolean Logic states A glass can be full or not full. Suppose a glass was only halfway filled which means glass can be half-full and half-not-full. Now this disapproves Aristotle’s law of bivalence. This concept of certain degree or multivalence is the fundamental concept stated by Dr. Lofti which introduced fuzzy logic.

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**Essential Characteristics of Fuzzy Logic laid down by Dr. Lofti A**

Essential Characteristics of Fuzzy Logic laid down by Dr. Lofti A. Zadeh Exact Reasoning is viewed as a limiting case of approximate reasoning. In FL. Everyting is a matter of degree. Any logical system can be fuzzified. In FL, knowledge is interpreted as a collection of elastic or equvalently , fuzzy constraint on collection of variables. Taught at the UC Berkely since 1959. Inference is viewed as a process of propagation of elastic constraints. Boolean logic can be defined as a subset of Fuzzy Logic.

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**Basis on which Fuzzy Logic Built**

“As the complexity of a system increases, it becomes more difficult and eventually impossible to make a precise statement about its behavior, eventually arriving at a point of complexity where the fuzzy logic method born in humans is the only way to get at the problem.”

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Introduction

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Intro Contd. The objective is to control the valve V1, such that the tank is refilled when the level is as low as LL, and stop the refilling when the level is as high as LH. The sensor LL is 1 when the level is above the mark, and 0 when the level is below; likewise with the sensor LH. The valve opens when V1 is set to 1, and it closes when V1 is set to 0.

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**Intro Contd. In Boolean logic, the controller can be**

V1= 1, if LL switches from 1 to 0 0, if LH switches from 0 to 1. In Fuzzy Logic, If the level is low then open V1 If the level is high then close V1 Boolean logic can be implemented using PLC FL can be implemented using Fuzzy logic controller.

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**Intro Contd. Membership**

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**The Problem: Real-World Vagueness**

Natural language abounds with vague and imprecise concepts “Sally is tall” or “It is very hot day” Suppose you are driving down a typical, two way, 6 lane street in a large city, one mile between signal lights. The speed limit is posted at 45 Mph. It is usually optimum and safest to "drive with the traffic," which will usually be going about 48 Mph. How do you define with specific, precise instructions "driving with the traffic?" It is difficult. But, it is the kind of thing humans do every day and do well. There will be some drivers weaving in and out and going more than 48 Mph and a few drivers driving exactly the posted 45 Mph. But, most drivers will be driving 48 Mph. They do this by exercising "fuzzy logic" - receiving a large number of fuzzy inputs, somehow evaluating all the inputs in their human brains and summarizing, weighting and averaging all these inputs to yield an optimum output decision.

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FL Basic Concepts Fuzzy logic is the way the human brain works, and we can mimic this in machines so they will perform somewhat like humans (not to be confused with Artificial Intelligence, where the goal is for machines to perform EXACTLY like humans). Fuzzy logic control and analysis systems may be electro-mechanical in nature, or concerned only with data, for example economic data, in all cases guided by "If-Then rules" stated in human language.

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**FL Basic Concepts Contd.**

The central idea to fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. For example, let us take the statement: "Jane is old." If Jane's age was 75, we might assign the statement the truth value of 0.80. The statement could be translated into set terminology as follows: "Jane is a member of the set of old people." This statement would be rendered symbolically with fuzzy sets as: mOLD(Jane) = 0.80 where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 1.0.

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Probability vs. FL At this point, it is important to point out the distinction between fuzzy systems and probability. Both operate over the same numeric range, and at first glance both have similar values: 0.0 representing False (or non-membership), and 1.0 representing True (or membership). However, there is a distinction to be made between the two statements: The probabilistic approach yields the natural-language statement, "There is an 80% chance that Jane is old," while the fuzzy terminology corresponds to "Jane's degree of membership within the set of old people is 0.80.“ The semantic difference is significant: the first view supposes that Jane is or is not old (still caught in the Law of the Excluded Middle); it is just that we only have an 80% chance of knowing which set she is in. By contrast, fuzzy terminology supposes that Jane is "more or less" old, or some other term corresponding to the value of 0.80.

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**FL Control Analysis Method**

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**FL Terms Found in Books and Articles**

Fuzzy Sets - A fuzzy set is almost any condition for which we have words: short men, tall women, hot, cold, new buildings, accelerator setting, ripe bananas, high intelligence, speed, weight, spongy, etc., where the condition can be given a value between 0 and Example: A woman is 6 feet, 3 inches tall. In my experience, I think she is one of the tallest women I have ever met, so I rate her height at .98. This line of reasoning can go on indefinitely rating a great number of things between 0 and 1.

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**FL Terms Found in Books and Articles**

Degree of Membership - The degree of membership is the placement in the transition from 0 to 1 of conditions within a fuzzy set. If a particular building's placement on the scale is a rating of .7 in its position in newness among new buildings, then we say its degree of membership in new buildings is .7. In fuzzy logic method control systems, degree of membership is used in the following way. A measurement of speed, for example, might be found to have a degree of membership in "too fast of" .6 and a degree of membership in "no change needed" of .2. The system program would then calculate the center of mass between "too fast" and "no change needed" to determine feedback action to send to the input of the control system.

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**FL Terms Found in Books and Articles**

Fuzzy Variable - Words like red, blue, etc., are fuzzy and can have many shades and tints. They are just human opinions, not based on precise measurement in angstroms. These words are fuzzy variables. If, for example, speed of a system is the attribute being evaluated by fuzzy, "fuzzy" rules, then "speed" is a fuzzy variable. Linguistic Variable - Linguistic means relating to language, in our case plain language words. Speed is a fuzzy variable. Accelerator setting is a fuzzy variable. Examples of linguistic variables are: somewhat fast speed, very high speed, real slow speed, excessively high accelerator setting, accelerator setting about right, etc.

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**FL Terms Found in Books and Articles**

A fuzzy variable becomes a linguistic variable when we modify it with descriptive words, such as somewhat fast, very high, real slow, etc. The main function of linguistic variables is to provide a means of working with the complex systems mentioned above as being too complex to handle by conventional mathematics and engineering formulas. Linguistic variables appear in control systems with feedback loop control and can be related to each other with conditional, "if-then" statements. Example: If the speed is too fast, then back off on the high accelerator setting.

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**FL Terms Found in Books and Articles**

Universe of Discourse - Let us make women the object of our consideration. All the women everywhere would be the universe of women. If we choose to discourse about (talk about) women, then all the women everywhere would be our Universe of Discourse. Universe of Discourse then, is a way to say all the objects in the universe of a particular kind, usually designated by one word, that we happen to be talking about or working with in a fuzzy logic solution. A Universe of Discourse is made up of fuzzy sets. Example: The Universe of Discourse of women is made up of professional women, tall women, Asian women, short women, beautiful women, and on and on. Fuzzy Algorithm - An algorithm is a procedure, such as the steps in a computer program. A fuzzy algorithm, then, is a procedure, usually a computer program, made up of statements relating linguistic variables. Examples: If "green x" is very large, then make "tall y" much smaller. If the rate of change of temperature of the steam engine boiler is much too high then turn the heater down a lot.

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**ALLAH HAFIZ SEE U SOON Muhammad Khurram Shaikh BE(Elect) NEDUET**

MS(CS) Bradley Peoria, IL, USA

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