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Fuzzy Systems and Control Günay Karlı, Ph.D.

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Before we begin… some clever people have said in the past…

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Precision is not truth. —Henri Matisse

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So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. —Albert Einstein

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As complexity rises, precise statements lose meaning and meaningful statements lose precision. Lotfi Zadeh

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What is Fuzzy Logic Fuzzy logic is a convenient way to map an input space to an output space.

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What is Fuzzy Logic Fuzzy logic is a convenient way to map an input space to an output space. * With information about how good your service was at a restaurant, a fuzzy logic system can tell you what the tip should be. * With your specification of how hot you want the water, a fuzzy logic system can adjust the faucet valve to the right setting. * With information about how far away the subject of your photograph is, a fuzzy logic system can focus the lens for you. * With information about how fast the car is going and how hard the motor is working, a fuzzy logic system can shift gears for you.

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What is Fuzzy Logic A graphical example of an input-output map

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What is Fuzzy Logic fuzzy systems, linear systems, neural networks, differential equations, interpolated multidimensional lookup tables, …. The black box can contain any number of things:

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Why use Fuzzy Logic "In almost every case you can build the same product without fuzzy logic, but fuzzy is faster and cheaper.“ -Lotfi Zadeh

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Why use Fuzzy Logic Here is a list of general observations about fuzzy logic: Fuzzy logic is conceptually easy to understand. Fuzzy logic is flexible Fuzzy logic is tolerant of imprecise data. Fuzzy logic can model nonlinear functions of arbitrary complexity. Fuzzy logic can be blended with conventional control techniques. Fuzzy logic is based on natural language.

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When NOT to Use Fuzzy Logic If you find Fuzzy Logic is not convenient, try something else. If a simpler solution already exists, use it.

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An Intoductory Example: Fuzzy vs. Non-Fuzzy What is the right amount to tip your waitperson? The Basic Tipping Problem. Given a number between 0 and 10 that represents the quality of service at a restaurant (where 10 is excellent), what should the tip be?

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An Intoductory Example: Tipping Problem The Non-Fuzzy Approach The tip always equals 15% of the total bill Tip=?

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An Intoductory Example: Tipping Problem The Non-Fuzzy Approach Because service is rated on a scale of 0 to 10, you might have the tip go linearly from 5% if the service is bad to 25% if the service is excellent. Tip=?

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An Intoductory Example: Tipping Problem The Non-Fuzzy Approach The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant, what should the tip be?

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An Intoductory Example: Tipping Problem The Non-Fuzzy Approach The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant, Suppose you want the service to be a more important factor than the food quality. Specify that service accounts for 80% of the overall tipping grade and the food makes up the other 20%. what should the tip be?

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An Intoductory Example: Tipping Problem The Fuzzy Approach capture the essentials of this problem leave aside all the factors that could be arbitrary what really matters in this problem The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant,

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An Intoductory Example: Tipping Problem The Fuzzy Approach The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant, Tipping Problem Rules — Service Factor If service is poor, then tip is cheap If service is good, then tip is average If service is excellent, then tip is generous

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An Intoductory Example: Tipping Problem The Fuzzy Approach The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant, Tipping Problem Rules — Service Factor If service is poor, then tip is cheap If service is good, then tip is average If service is excellent, then tip is generous Tipping Problem Rules — Food Factor If food is rancid, then tip is cheap If food is delicious, then tip is generous

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An Intoductory Example: Tipping Problem The Fuzzy Approach The Extended Tipping Problem: Given two sets of numbers between 0 and 10 (where 10 is excellent) that respectively represent the quality of the service and the quality of the food at a restaurant, Tipping Problem Rules — Service Factor If service is poor, then tip is cheap If service is good, then tip is average If service is excellent, then tip is generous Tipping Problem Rules — Food Factor If food is rancid, then tip is cheap If food is delicious, then tip is generous Tipping Problem — Both Service and Food Factors If service is poor OR the food is rancid, then tip is cheap If service is good, then tip is average If service is excellent OR food is delicious, then tip is generous

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An Intoductory Example: Tipping Problem The Fuzzy Approach Tipping Problem — Both Service and Food Factors If service is poor or the food is rancid, then tip is cheap If service is good, then tip is average If service is excellent or food is delicious, then tip is generous

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An Intoductory Example: Tipping Problem The Fuzzy Approach Tipping Problem — Both Service and Food Factors If service is poor or the food is rancid, then tip is cheap If service is good, then tip is average If service is excellent or food is delicious, then tip is generous # How are the rules all combined? # How do I define mathematically what an average tip is?

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Fuzzy Logic System

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Fuzzy Logic System Fuzzy Inference Diagram

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Fuzzy Logic System A 2 input 1 output FLS

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Implementation of FLS tip = 16.7%

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Building System with the MATLAB Fuzzy Logic Toolbox

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Fuzzy Sets Introduction/Overview Material for these slides obtained from: Modern Information Retrieval by Ricardo Baeza-Yates and Berthier Ribeiro-Neto.

Fuzzy Sets Introduction/Overview Material for these slides obtained from: Modern Information Retrieval by Ricardo Baeza-Yates and Berthier Ribeiro-Neto.

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