Fuzzy logic provides a method to formalize reasoning with vague terms by allowing membership functions and degrees of truth rather than binary true/false values. It can be used to model problems involving linguistic variables like "poor", "good", and "excellent".
The document discusses a tipping example to demonstrate fuzzy logic. It defines fuzzy rules for tip amounts based on the quality of service and food. For example, one rule is that if service is poor or food is rancid, the tip should be cheap. Membership functions are then used to evaluate the fuzzy rules and determine appropriate tip amounts based on varying degrees of service and food quality.
Fuzzy logic provides a more intuitive way to model problems involving vague
- Fuzzy logic is an extension of classical logic that accounts for partial truth values between "true" and "false". It allows for gradual transitions between values in a membership function.
- Fuzzy logic has been applied to many areas including control systems, decision making, pattern recognition and other areas involving uncertainty. It uses fuzzy "if-then" rules to model imprecise human reasoning.
- The document discusses fuzzy sets, fuzzy relations, applications of fuzzy logic and provides biographical information about Lotfi Zadeh, the founder of fuzzy logic.
This document introduces fuzzy sets. It defines a fuzzy set as a set where elements have gradual membership rather than binary membership. Fuzzy sets allow membership values between 0 and 1. Operations on fuzzy sets like union, intersection, and complement are defined. An example fuzzy set distinguishes young and adult ages on a scale rather than a binary classification. Fuzzy sets permit ambiguous or uncertain boundaries unlike classical sets.
This document provides an introduction and overview of fuzzy logic, including:
- Fuzzy sets allow gradual membership rather than crisp membership in sets, addressing limitations of binary logic.
- A case study examines controlling the speed of a room cooler motor based on temperature and humidity using fuzzy logic rules and membership functions.
- Key fuzzy logic concepts covered include fuzzification, fuzzy rules and inference, and defuzzification to obtain a crisp output from fuzzy inputs and rules.
Fuzzy logic is a form of multivalued logic that allows intermediate values between conventional evaluations like true/false, yes/no, or 0/1. It provides a mathematical framework for representing uncertainty and imprecision in measurement and human cognition. The document discusses the history of fuzzy logic, key concepts like membership functions and linguistic variables, common fuzzy logic operations, and applications in fields like control systems, home appliances, and cameras. It also notes some drawbacks like difficulty in tuning membership functions and potential confusion with probability theory.
The document introduces fuzzy set theory as an extension of classical set theory that allows for elements to have varying degrees of membership rather than binary membership. It discusses key concepts such as fuzzy sets, membership functions, fuzzy logic, fuzzy rules, and fuzzy inference. Fuzzy set theory provides a framework for modeling imprecise and uncertain concepts that are common in human reasoning.
This presentation includes what is fuzzy logic, characteristics, membership function with example, fuzzy set theory, De-Morgans Law, Fuzzy logic V/S probability, advantages and disadvantages and application areas of fuzzy logic. This is a presentation is useful for IT students.
This document provides an overview of fuzzy logic, including its origins, key concepts, and applications. It discusses how fuzzy logic allows for approximate reasoning and decision making under uncertainty by using linguistic variables and fuzzy set theory. Membership functions are used to characterize fuzzy sets and assign degrees of truth between 0 and 1 rather than binary true/false values. Common fuzzy logic operations like intersection, union, and complement are also covered. Finally, some examples of fuzzy logic control systems are presented, including how they are designed using fuzzy rule bases and inference methods like Mamdani and Sugeno.
This document provides an overview of fuzzy logic. It begins by defining fuzzy as not being clear or precise, unlike classical sets which have clear boundaries. It then explains fuzzy logic allows for partial set membership rather than binary membership. The document outlines fuzzy logic's ability to model imprecise or nonlinear systems using natural language-based rules. It details the key concepts of fuzzy logic including linguistic variables, membership functions, fuzzy set operations, fuzzy inference systems and the 5-step fuzzy inference process of fuzzifying inputs, applying fuzzy operations and implications, aggregating outputs and defuzzifying results.
- Fuzzy logic is an extension of classical logic that accounts for partial truth values between "true" and "false". It allows for gradual transitions between values in a membership function.
- Fuzzy logic has been applied to many areas including control systems, decision making, pattern recognition and other areas involving uncertainty. It uses fuzzy "if-then" rules to model imprecise human reasoning.
- The document discusses fuzzy sets, fuzzy relations, applications of fuzzy logic and provides biographical information about Lotfi Zadeh, the founder of fuzzy logic.
This document introduces fuzzy sets. It defines a fuzzy set as a set where elements have gradual membership rather than binary membership. Fuzzy sets allow membership values between 0 and 1. Operations on fuzzy sets like union, intersection, and complement are defined. An example fuzzy set distinguishes young and adult ages on a scale rather than a binary classification. Fuzzy sets permit ambiguous or uncertain boundaries unlike classical sets.
This document provides an introduction and overview of fuzzy logic, including:
- Fuzzy sets allow gradual membership rather than crisp membership in sets, addressing limitations of binary logic.
- A case study examines controlling the speed of a room cooler motor based on temperature and humidity using fuzzy logic rules and membership functions.
- Key fuzzy logic concepts covered include fuzzification, fuzzy rules and inference, and defuzzification to obtain a crisp output from fuzzy inputs and rules.
Fuzzy logic is a form of multivalued logic that allows intermediate values between conventional evaluations like true/false, yes/no, or 0/1. It provides a mathematical framework for representing uncertainty and imprecision in measurement and human cognition. The document discusses the history of fuzzy logic, key concepts like membership functions and linguistic variables, common fuzzy logic operations, and applications in fields like control systems, home appliances, and cameras. It also notes some drawbacks like difficulty in tuning membership functions and potential confusion with probability theory.
The document introduces fuzzy set theory as an extension of classical set theory that allows for elements to have varying degrees of membership rather than binary membership. It discusses key concepts such as fuzzy sets, membership functions, fuzzy logic, fuzzy rules, and fuzzy inference. Fuzzy set theory provides a framework for modeling imprecise and uncertain concepts that are common in human reasoning.
This presentation includes what is fuzzy logic, characteristics, membership function with example, fuzzy set theory, De-Morgans Law, Fuzzy logic V/S probability, advantages and disadvantages and application areas of fuzzy logic. This is a presentation is useful for IT students.
This document provides an overview of fuzzy logic, including its origins, key concepts, and applications. It discusses how fuzzy logic allows for approximate reasoning and decision making under uncertainty by using linguistic variables and fuzzy set theory. Membership functions are used to characterize fuzzy sets and assign degrees of truth between 0 and 1 rather than binary true/false values. Common fuzzy logic operations like intersection, union, and complement are also covered. Finally, some examples of fuzzy logic control systems are presented, including how they are designed using fuzzy rule bases and inference methods like Mamdani and Sugeno.
This document provides an overview of fuzzy logic. It begins by defining fuzzy as not being clear or precise, unlike classical sets which have clear boundaries. It then explains fuzzy logic allows for partial set membership rather than binary membership. The document outlines fuzzy logic's ability to model imprecise or nonlinear systems using natural language-based rules. It details the key concepts of fuzzy logic including linguistic variables, membership functions, fuzzy set operations, fuzzy inference systems and the 5-step fuzzy inference process of fuzzifying inputs, applying fuzzy operations and implications, aggregating outputs and defuzzifying results.
Best-first search is a heuristic search algorithm that expands the most promising node first. It uses an evaluation function f(n) that estimates the cost to reach the goal from each node n. Nodes are ordered in the fringe by increasing f(n). A* search is a special case of best-first search that uses an admissible heuristic function h(n) and is guaranteed to find the optimal solution.
The Fuzzy Logic is discussed with three simple example problems all solved in MATLAB
1. Restaurant Problem
2. Temperature Controller
3. Washing Machine Problem
This document discusses the application of fuzzy logic to optimal capacitor placement in distribution systems. It begins with definitions of fuzzy logic and fuzzy sets. It then describes the key components of a fuzzy logic system including fuzzification, fuzzy inference rules, and defuzzification. It proposes using power loss reduction index and bus voltage as input variables, and capacitor placement suitability index as the output variable, to determine the optimal locations and sizes of capacitors. The goal is to minimize power losses and maximize annual savings using fuzzy logic techniques.
Fuzzy logic is a method of reasoning that resembles human decision making by allowing for intermediate possibilities between yes and no or true and false. It is used in control systems like temperature controllers, anti-lock braking systems, washing machines, and air conditioners. Fuzzy logic applications can be found in areas like aerospace, automotive, defense, electronics, mining, robotics, securities, and industrial processes. The field of fuzzy logic continues to grow and provide opportunities to develop effective controllers for complex systems across many domains.
The document discusses fuzzy logic and its applications in control systems. It begins with definitions of fuzzy logic and fuzzy sets. It then discusses the history and applications of fuzzy logic, including ABS brakes, expert systems, and control units. The document outlines the formal definitions, operations, and structure of fuzzy logic controllers. It provides examples of membership functions, rule bases, fuzzification, inference engines, and defuzzification. It concludes with an example of a fuzzy logic air conditioner controller.
---TABLE OF CONTENT---
Introduction
Differences between crisp sets & Fuzzy sets
Operations on Fuzzy Sets
Properties
MF formulation and parameterization
Fuzzy rules and Fuzzy reasoning
Fuzzy interface systems
Introduction to genetic algorithm
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
This document is an introduction to combinatorics presented by A.B. Benedict Balbuena from the University of the Philippines. It discusses fundamental combinatorics concepts like the addition rule, product rule, and inclusion-exclusion principle. Examples of counting problems are provided to illustrate how to use these rules to calculate the number of possible outcomes in situations involving sets, permutations, and combinations.
This document discusses fuzzy systems and their applications. It introduces fuzzy logic as an extension of Boolean logic that allows for partial set memberships and uncertainties. It provides examples of fuzzy systems in washing machines, vacuum cleaners, rice cookers, and cars. Fuzzy logic is used in washing machines to adjust operations based on sensor readings. Vacuum cleaners use fuzzy logic to control motor speed based on distance sensors. Rice cookers employ neuro-fuzzy systems for precise heat adjustment. Cars can use fuzzy logic for automatic transmissions to shift gears like an experienced human driver.
Mathematical Optimisation - Fundamentals and ApplicationsGokul Alex
My Session on Mathematical Optimisation Fundamentals and Industry applications for the Academic Knowledge Refresher Program organised by Kerala Technology University and College of Engineering Trivandrum, Department of Interdisciplinary Studies.
Fuzzy inference systems use fuzzy logic to map inputs to outputs. There are two main types:
Mamdani systems use fuzzy outputs and are well-suited for problems involving human expert knowledge. Sugeno systems have faster computation using linear or constant outputs.
The fuzzy inference process involves fuzzifying inputs, applying fuzzy logic operators, and using if-then rules. Outputs are determined through implication, aggregation, and defuzzification. Mamdani systems find the centroid of fuzzy outputs while Sugeno uses weighted averages, making it more efficient.
LINEAR RECURRENCE RELATIONS WITH CONSTANT COEFFICIENTSAartiMajumdar1
This document discusses linear recurrence relations with constant coefficients. It covers homogeneous solutions, particular solutions, and the total solution. It also discusses solving recurrence relations using generating functions. Key points:
- Homogeneous solutions are found by solving the characteristic equation.
- Particular solutions are found for homogeneous and non-homogeneous equations.
- The total solution is the sum of the homogeneous and particular solutions.
- Generating functions can also be used to solve recurrence relations.
Fuzzy logic is a form of logic that deals with reasoning that is approximate rather than precise. It allows intermediate values to be defined between conventional evaluations like true/false, and uses a continuum of truth values between 0 and 1. Fuzzy logic is useful for problems with imprecise or uncertain data, and can represent human reasoning that uses approximate terms like "warm" or "fast". It has been applied in various systems to control variables like temperature, speed, and focus based on fuzzy linguistic rules.
This document discusses soft computing and fuzzy set theory. It explains that fuzzy set theory allows for uncertain or vague knowledge to be represented using propositions and rules. Operations on fuzzy sets like intersection, union, and complement are defined using characteristic functions in a similar way to classical set theory. Fuzzy sets have applications in areas like artificial intelligence, control engineering, and decision making. Fuzzy rule-based systems and fuzzy control use fuzzy knowledge bases and inference to derive conclusions. Fuzzy data mining and fuzzy optimization apply fuzzy set concepts to improve existing techniques for data analysis and constrained optimization problems.
Fuzzy sets allow for gradual membership of elements in a set, rather than binary membership as in classical set theory. Membership is described on a scale of 0 to 1 using a membership function. Fuzzy sets generalize classical sets by treating classical sets as special cases where membership values are restricted to 0 or 1. Fuzzy set theory can model imprecise or uncertain information and is used in domains like bioinformatics. Examples of fuzzy sets include sets like "tall people" where membership in the set is a matter of degree.
Fuzzy logic is a flexible machine learning technique that mimics human thought by allowing intermediate values between true and false. It provides a mechanism for interpreting and executing commands based on approximate or uncertain reasoning. Unlike binary logic which can only have true or false values, fuzzy logic uses linguistic variables and degrees of membership to represent concepts that may have a partial truth. Fuzzy systems find applications in automatic control, prediction, diagnosis and user interfaces.
Fuzzy logic is a form of logic that accounts for partial truth and vagueness. It is used in control systems and decision support systems. The document discusses the history of fuzzy logic and its applications in areas like automotive, robotics, manufacturing, medical, and more. Fuzzy logic controllers combine fuzzy linguistic variables and rules to automate tasks like speed control in vehicles and temperature control in air conditioners and washing machines.
This document discusses NP-complete problems and their properties. Some key points:
- NP-complete problems have an exponential upper bound on runtime but only a polynomial lower bound, making them appear intractable. However, their intractability cannot be proven.
- NP-complete problems are reducible to each other in polynomial time. Solving one would solve all NP-complete problems.
- NP refers to problems that can be verified in polynomial time. P refers to problems that can be solved in polynomial time.
- A problem is NP-complete if it is in NP and all other NP problems can be reduced to it in polynomial time. Proving a problem is NP-complete involves showing
The document discusses several algorithms for pattern matching in strings:
1) Brute-force algorithm compares the pattern to every substring of the text, running in O(nm) time where n and m are the lengths of the text and pattern.
2) Boyer-Moore algorithm uses heuristics like the last occurrence function to skip comparisons, running faster in O(nm+s) time where s is the alphabet size.
3) Knuth-Morris-Pratt algorithm builds a failure function to determine the maximum shift of the pattern after a mismatch, running optimally in O(n+m) time.
- Fuzzy logic was developed by Lotfi Zadeh to address applications involving subjective or vague data like "attractive person" that cannot be easily analyzed using binary logic. It allows for partial truth values between completely true and completely false.
- Fuzzy logic controllers mimic human decision making and involve fuzzifying inputs, applying fuzzy rules, and defuzzifying outputs. This allows systems to be specified in human terms and automated.
- Fuzzy logic has many applications from industrial process control to consumer products like washing machines and microwaves. It offers an intuitive way to model real-world ambiguities compared to mathematical or logic-based approaches.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Best-first search is a heuristic search algorithm that expands the most promising node first. It uses an evaluation function f(n) that estimates the cost to reach the goal from each node n. Nodes are ordered in the fringe by increasing f(n). A* search is a special case of best-first search that uses an admissible heuristic function h(n) and is guaranteed to find the optimal solution.
The Fuzzy Logic is discussed with three simple example problems all solved in MATLAB
1. Restaurant Problem
2. Temperature Controller
3. Washing Machine Problem
This document discusses the application of fuzzy logic to optimal capacitor placement in distribution systems. It begins with definitions of fuzzy logic and fuzzy sets. It then describes the key components of a fuzzy logic system including fuzzification, fuzzy inference rules, and defuzzification. It proposes using power loss reduction index and bus voltage as input variables, and capacitor placement suitability index as the output variable, to determine the optimal locations and sizes of capacitors. The goal is to minimize power losses and maximize annual savings using fuzzy logic techniques.
Fuzzy logic is a method of reasoning that resembles human decision making by allowing for intermediate possibilities between yes and no or true and false. It is used in control systems like temperature controllers, anti-lock braking systems, washing machines, and air conditioners. Fuzzy logic applications can be found in areas like aerospace, automotive, defense, electronics, mining, robotics, securities, and industrial processes. The field of fuzzy logic continues to grow and provide opportunities to develop effective controllers for complex systems across many domains.
The document discusses fuzzy logic and its applications in control systems. It begins with definitions of fuzzy logic and fuzzy sets. It then discusses the history and applications of fuzzy logic, including ABS brakes, expert systems, and control units. The document outlines the formal definitions, operations, and structure of fuzzy logic controllers. It provides examples of membership functions, rule bases, fuzzification, inference engines, and defuzzification. It concludes with an example of a fuzzy logic air conditioner controller.
---TABLE OF CONTENT---
Introduction
Differences between crisp sets & Fuzzy sets
Operations on Fuzzy Sets
Properties
MF formulation and parameterization
Fuzzy rules and Fuzzy reasoning
Fuzzy interface systems
Introduction to genetic algorithm
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
This document is an introduction to combinatorics presented by A.B. Benedict Balbuena from the University of the Philippines. It discusses fundamental combinatorics concepts like the addition rule, product rule, and inclusion-exclusion principle. Examples of counting problems are provided to illustrate how to use these rules to calculate the number of possible outcomes in situations involving sets, permutations, and combinations.
This document discusses fuzzy systems and their applications. It introduces fuzzy logic as an extension of Boolean logic that allows for partial set memberships and uncertainties. It provides examples of fuzzy systems in washing machines, vacuum cleaners, rice cookers, and cars. Fuzzy logic is used in washing machines to adjust operations based on sensor readings. Vacuum cleaners use fuzzy logic to control motor speed based on distance sensors. Rice cookers employ neuro-fuzzy systems for precise heat adjustment. Cars can use fuzzy logic for automatic transmissions to shift gears like an experienced human driver.
Mathematical Optimisation - Fundamentals and ApplicationsGokul Alex
My Session on Mathematical Optimisation Fundamentals and Industry applications for the Academic Knowledge Refresher Program organised by Kerala Technology University and College of Engineering Trivandrum, Department of Interdisciplinary Studies.
Fuzzy inference systems use fuzzy logic to map inputs to outputs. There are two main types:
Mamdani systems use fuzzy outputs and are well-suited for problems involving human expert knowledge. Sugeno systems have faster computation using linear or constant outputs.
The fuzzy inference process involves fuzzifying inputs, applying fuzzy logic operators, and using if-then rules. Outputs are determined through implication, aggregation, and defuzzification. Mamdani systems find the centroid of fuzzy outputs while Sugeno uses weighted averages, making it more efficient.
LINEAR RECURRENCE RELATIONS WITH CONSTANT COEFFICIENTSAartiMajumdar1
This document discusses linear recurrence relations with constant coefficients. It covers homogeneous solutions, particular solutions, and the total solution. It also discusses solving recurrence relations using generating functions. Key points:
- Homogeneous solutions are found by solving the characteristic equation.
- Particular solutions are found for homogeneous and non-homogeneous equations.
- The total solution is the sum of the homogeneous and particular solutions.
- Generating functions can also be used to solve recurrence relations.
Fuzzy logic is a form of logic that deals with reasoning that is approximate rather than precise. It allows intermediate values to be defined between conventional evaluations like true/false, and uses a continuum of truth values between 0 and 1. Fuzzy logic is useful for problems with imprecise or uncertain data, and can represent human reasoning that uses approximate terms like "warm" or "fast". It has been applied in various systems to control variables like temperature, speed, and focus based on fuzzy linguistic rules.
This document discusses soft computing and fuzzy set theory. It explains that fuzzy set theory allows for uncertain or vague knowledge to be represented using propositions and rules. Operations on fuzzy sets like intersection, union, and complement are defined using characteristic functions in a similar way to classical set theory. Fuzzy sets have applications in areas like artificial intelligence, control engineering, and decision making. Fuzzy rule-based systems and fuzzy control use fuzzy knowledge bases and inference to derive conclusions. Fuzzy data mining and fuzzy optimization apply fuzzy set concepts to improve existing techniques for data analysis and constrained optimization problems.
Fuzzy sets allow for gradual membership of elements in a set, rather than binary membership as in classical set theory. Membership is described on a scale of 0 to 1 using a membership function. Fuzzy sets generalize classical sets by treating classical sets as special cases where membership values are restricted to 0 or 1. Fuzzy set theory can model imprecise or uncertain information and is used in domains like bioinformatics. Examples of fuzzy sets include sets like "tall people" where membership in the set is a matter of degree.
Fuzzy logic is a flexible machine learning technique that mimics human thought by allowing intermediate values between true and false. It provides a mechanism for interpreting and executing commands based on approximate or uncertain reasoning. Unlike binary logic which can only have true or false values, fuzzy logic uses linguistic variables and degrees of membership to represent concepts that may have a partial truth. Fuzzy systems find applications in automatic control, prediction, diagnosis and user interfaces.
Fuzzy logic is a form of logic that accounts for partial truth and vagueness. It is used in control systems and decision support systems. The document discusses the history of fuzzy logic and its applications in areas like automotive, robotics, manufacturing, medical, and more. Fuzzy logic controllers combine fuzzy linguistic variables and rules to automate tasks like speed control in vehicles and temperature control in air conditioners and washing machines.
This document discusses NP-complete problems and their properties. Some key points:
- NP-complete problems have an exponential upper bound on runtime but only a polynomial lower bound, making them appear intractable. However, their intractability cannot be proven.
- NP-complete problems are reducible to each other in polynomial time. Solving one would solve all NP-complete problems.
- NP refers to problems that can be verified in polynomial time. P refers to problems that can be solved in polynomial time.
- A problem is NP-complete if it is in NP and all other NP problems can be reduced to it in polynomial time. Proving a problem is NP-complete involves showing
The document discusses several algorithms for pattern matching in strings:
1) Brute-force algorithm compares the pattern to every substring of the text, running in O(nm) time where n and m are the lengths of the text and pattern.
2) Boyer-Moore algorithm uses heuristics like the last occurrence function to skip comparisons, running faster in O(nm+s) time where s is the alphabet size.
3) Knuth-Morris-Pratt algorithm builds a failure function to determine the maximum shift of the pattern after a mismatch, running optimally in O(n+m) time.
- Fuzzy logic was developed by Lotfi Zadeh to address applications involving subjective or vague data like "attractive person" that cannot be easily analyzed using binary logic. It allows for partial truth values between completely true and completely false.
- Fuzzy logic controllers mimic human decision making and involve fuzzifying inputs, applying fuzzy rules, and defuzzifying outputs. This allows systems to be specified in human terms and automated.
- Fuzzy logic has many applications from industrial process control to consumer products like washing machines and microwaves. It offers an intuitive way to model real-world ambiguities compared to mathematical or logic-based approaches.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Fuzzy Logic And Application Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains an exam for a course on Fuzzy Logic and Applications. It includes 8 questions covering topics such as operations on crisp and fuzzy sets using Venn diagrams, fuzzy relations, membership functions, fuzzy logic connectives, defuzzification methods, and decision making under fuzzy conditions. Students are instructed to answer any 5 of the 8 questions.
This document provides an overview of Adaptive Neural Fuzzy Inference Systems (ANFIS). It discusses how ANFIS aims to integrate the benefits of fuzzy systems and neural networks by using neural network learning methods to determine the parameters of fuzzy inference systems. The document outlines ANFIS architecture and computational complexity. It also describes how ANFIS uses a hybrid learning algorithm with a least squares estimate to identify linear parameters and backpropagation to adjust nonlinear parameters.
Fuzzy relations, fuzzy graphs, and the extension principle are three important concepts in fuzzy logic. Fuzzy relations generalize classical relations to allow partial membership and describe relationships between objects to varying degrees. Fuzzy graphs describe functional mappings between input and output linguistic variables. The extension principle provides a procedure to extend functions defined on crisp domains to fuzzy domains by mapping fuzzy sets through functions. These concepts form the foundation of fuzzy rules and fuzzy arithmetic.
This document provides examples of propositional and predicate logic. It introduces basic logical concepts like propositions, truth values, connectives like "if...then" and "and", and quantifiers like "all" and "some". Propositional logic examples use letters like p and q to represent simple statements without internal structure. Predicate logic examples assign predicates like "loves" and "crazy" to arguments to represent statements with meaning, like "Loves(John, Mary)". The document explains how to represent quantified statements using predicates, arguments, and quantifiers.
This document discusses fuzzy logic systems and fuzzy control. It begins by explaining why fuzzy logic is useful for control systems where simplicity and speed of implementation are important. It then provides examples of commercial applications of fuzzy control in various industries. The document goes on to describe the engineering motivation for fuzzy logic, and how to implement a basic fuzzy logic system using fuzzification, fuzzy decision blocks, and defuzzification. It also discusses developing fuzzy logic control rules based on common sense "if-then" statements. Finally, it briefly discusses using fuzzy control in feedback systems to mimic the control procedures of skilled human operators.
This document provides an overview of fuzzy logic and its applications. It begins with motivations for fuzzy logic by discussing limitations of crisp sets and fuzzy sets as an alternative approach. It then defines fuzzy sets and fuzzy logic operations. It describes how fuzzy logic systems work by combining fuzzy sets and logic operations. Several example applications are mentioned, including industrial control systems and modeling human decision making. The document concludes by noting fuzzy logic has been applied in many domains and there are ongoing developments in fuzzy logic approaches.
How can you deal with Fuzzy Logic. Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree
between 0 and 1
Presentation is about genetic algorithms. Also it includes introduction to soft computing and hard computing. Hope it serves the purpose and be useful for reference.
This document describes a study conducted by undergraduate students at Uva Wellassa University of Sri Lanka on applying fuzzy logic to aircraft landing control. It provides background on fuzzy logic and fuzzy set theory. It then presents the students' simulation of an aircraft's final descent and landing approach, where fuzzy logic is used to control the aircraft's vertical velocity based on its current height above ground. Over multiple cycles, the simulation demonstrates how the fuzzy logic system gradually reduces the aircraft's velocity as it gets closer to landing for a soft touchdown.
Fingerprint identification is commonly used for biometric identification. Fingerprints are analyzed by their patterns (loops, whorls, arches) and ridge characteristics (minutiae). Most automatic fingerprint identification systems (AFIS) match minutiae features. The document proposes using soft computing techniques like neural networks and fuzzy logic to improve an AFIS. It describes a 5-layer network that uses minutiae extraction from divided fingerprint blocks to classify fingerprints hierarchically and reduce errors. Potential applications include security access, mobile unlock systems, driver's licenses, and national ID programs.
1. The document describes an expert system and its components.
2. It defines an expert system as an intelligent computer program that uses knowledge and reasoning to solve problems that usually require human expertise.
3. The key components of an expert system are the knowledge base, inference engine, explanation facility, and knowledge acquisition facility.
Fuzzy logic is a form of logic that accounts for partial truth and intermediate values between true and false. It is used in control systems to mimic how humans apply fuzzy concepts like "cold" or "hot" temperature. Some key applications of fuzzy logic include temperature controllers, washing machines, air conditioners, and anti-lock braking systems. Fuzzy logic controllers use if-then rules to determine outputs based on fuzzy inputs and degrees of membership rather than binary logic.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a Boolean expression or a truth table. There are two main methods to convert between these representations: (1) using a sum of products to get a Boolean expression from a truth table by including all variable combinations that evaluate to 1, and (2) using a product of sums to get a Boolean expression by including all variable combinations that evaluate to 0.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a logical expression using AND, OR, and NOT operators or as a truth table. There are standard methods to convert between these representations, such as using sums of minterms or products of maxterms. Boolean algebra postulates and theorems allow logical expressions to be simplified in ways that result in equivalent but simpler circuits.
Fuzzy set theory is an extension of classical set theory that allows for partial membership in a set rather than crisp boundaries. In fuzzy set theory, elements have degrees of membership in a set represented by a membership function between 0 and 1. This allows for modeling of imprecise concepts like "young" where the boundary is ambiguous. Fuzzy set theory is useful for modeling human reasoning and systems that can handle unreliable or incomplete information. Key concepts include fuzzy rules in an if-then format and fuzzy inference using methods like Mamdani inference involving fuzzification, rule evaluation, aggregation, and defuzzification.
Fuzzy set theory is an extension of classical set theory that allows for partial membership in a set rather than crisp boundaries. In fuzzy set theory, elements have a degree of membership in a set defined by a membership function ranging from 0 to 1 rather than simply belonging or not belonging to a set. Fuzzy sets and logic can model imprecise concepts and are used in applications involving uncertain or ambiguous information like fuzzy controllers.
Fuzzy set theory is an extension of classical set theory that allows for partial membership in a set rather than crisp boundaries. In fuzzy set theory, elements have a degree of membership in a set ranging from 0 to 1 rather than simply belonging or not belonging to the set. This allows fuzzy set theory to model imprecise concepts more accurately. Fuzzy sets use membership functions to define the degree of membership for each element. Common membership functions include triangular, trapezoidal, and Gaussian functions. Fuzzy set theory is useful for modeling human reasoning and systems that involve imprecise or uncertain information.
- Boolean algebra uses binary values (1/0) to represent true/false in digital circuits.
- The basic Boolean operations are AND, OR, and NOT. Truth tables and Boolean expressions can both be used to represent the functions of circuits.
- Boolean expressions can be simplified using algebraic rules like commutative, distributive, DeMorgan's, and absorption laws. This allows simpler circuit implementations.
This document provides an introduction to fuzzy logic and fuzzy sets. It discusses key concepts such as fuzzy sets having degrees of membership between 0 and 1 rather than binary membership, and fuzzy logic allowing for varying degrees of truth. Examples are given of fuzzy sets representing partially full tumblers and desirable cities to live in. Characteristics of fuzzy sets such as support, crossover points, and logical operations like union and intersection are defined. Applications mentioned include vehicle control systems and appliance control using fuzzy logic to handle imprecise and ambiguous inputs.
It is known as two-valued logic because it have only two values Ramjeet Singh Yadav
This document discusses fuzzy logic and its applications. It begins by explaining classical logic and crisp sets, which have binary membership. It then introduces fuzzy logic, which was developed by Lotfi Zadeh in 1965 and allows partial set membership between 0 and 1. This allows fuzzy logic to handle concepts involving degrees of truth. The key concepts of fuzzy logic discussed include fuzzy sets and membership functions, fuzzy operations like union and intersection, fuzzy relations, fuzzy rules, and fuzzy inference systems. Real-world applications of fuzzy inference systems are also mentioned, such as automatic control, expert systems, and medical engineering.
This document provides an introduction to fuzzy logic and fuzzy set theory. It defines key concepts such as membership functions, fuzzy sets over discrete and continuous universes, operations on fuzzy sets like intersection and union using t-norms and t-conorms, and linguistic variables and terms. It also discusses fuzzy rules and binary fuzzy relations used to represent fuzzy rules.
This document provides an overview of fuzzy logic and fuzzy set theory. It discusses how fuzzy sets allow for partial membership rather than crisp binary membership in sets. Various membership functions are described, including triangular, trapezoidal, Gaussian, and sigmoidal functions. Properties of fuzzy sets like support, convexity, and symmetry are defined. Finally, fuzzy set operations analogous to classical set operations are mentioned.
The document provides an overview of fuzzy logic and fuzzy sets. It discusses how fuzzy logic can handle imprecise data unlike classical binary sets. Membership functions assign degrees of membership values between 0 and 1. Fuzzy logic systems use if-then rules and linguistic variables. An example shows how fuzzy logic is used to estimate project risk levels based on funding and staffing levels. Fuzzy logic has been applied in various domains due to its ability to model human reasoning.
The document provides an overview of the structure and content covered on the AP Calculus AB exam, including:
- The exam is 3 hours 15 minutes long and divided into multiple choice and free response sections testing limits, derivatives, integrals, and applications of calculus.
- Content topics covered include limits of functions, continuity, derivatives and their applications (related rates, max/min problems), integrals, and differential equations.
- Formulas and strategies are provided for evaluating limits, finding derivatives using various rules, applying derivatives to sketch curves, solve optimization problems, and solve motion problems using related rates.
Linear regression is a supervised learning technique used to predict continuous valued outputs. It fits a linear equation to the training data to find the relationship between independent variables (x) and the dependent variable (y).
Gradient descent is an algorithm used to minimize the cost function in linear regression. It works by iteratively updating the parameters (θ values) in the direction of the steepest descent as calculated by the partial derivatives of the cost function with respect to each parameter. The learning rate (α) controls the step size in each iteration.
Multivariate linear regression extends the technique to problems with multiple independent variables by representing the hypothesis and parameters as vectors and matrices, allowing gradient descent to optimize the parameters to fit the linear model
Fuzzy logic was introduced in 1965 by Lofti Zadeh based on fuzzy set theory. It allows for intermediate values between 0 and 1, unlike boolean logic which only considers true or false. A fuzzy logic system uses fuzzification to convert crisp inputs to fuzzy values, applies a rule base and inference engine to the fuzzy values, and then uses defuzzification to convert the fuzzy output to a crisp value. Fuzzy logic is useful for approximate reasoning and has applications in areas like control systems, decision making, and pattern recognition.
Normalization is the process of organizing data in a database to reduce data redundancy and improve data integrity. It involves separating relations into smaller relations and linking them through relationships. The normal forms, such as first normal form, second normal form, etc. are used to reduce redundancy and anomalies like insertion, update and deletion anomalies. Some key aspects are that first normal form disallows multi-valued attributes and composite attributes. Second normal form eliminates non-prime attributes in relations that depend on part of a composite primary key.
This document discusses soft computing and fuzzy sets. It begins by defining soft computing as being tolerant of imprecision and focusing on approximation rather than precise outputs. Fuzzy sets are introduced as a tool of soft computing that allow for graded membership in sets rather than binary membership. Key concepts regarding fuzzy sets are explained, including fuzzy logic operations, fuzzy numbers, and fuzzy variables. Linear programming problems are discussed and how they can be modeled as fuzzy linear programming problems to account for imprecision in the coefficients and constraints.
Probability theory provides a framework for quantifying and manipulating uncertainty. It allows optimal predictions given incomplete information. The document outlines key probability concepts like sample spaces, events, axioms of probability, joint/conditional probabilities, and Bayes' rule. It also covers important probability distributions like binomial, Gaussian, and multivariate Gaussian. Finally, it discusses optimization concepts for machine learning like functions, derivatives, and using derivatives to find optima like maxima and minima.
Fuzzy logic is a form of logic that deals with reasoning that is approximate rather than fixed and exact. It was introduced in 1965 with the proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic uses fuzzy sets and membership functions to deal with imprecise or uncertain inputs and allows for reasoning that allows for partial truth of inputs between fully true and fully false. Fuzzy controllers combine fuzzy logic with control theory to control complex systems. They involve fuzzification of inputs, applying fuzzy rules through inference, and defuzzification of outputs to obtain a crisp control action.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Determination of Equivalent Circuit parameters and performance characteristic...pvpriya2
Includes the testing of induction motor to draw the circle diagram of induction motor with step wise procedure and calculation for the same. Also explains the working and application of Induction generator
Impartiality as per ISO /IEC 17025:2017 StandardMuhammadJazib15
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Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Build the Next Generation of Apps with the Einstein 1 Platform.
Rejoignez Philippe Ozil pour une session de workshops qui vous guidera à travers les détails de la plateforme Einstein 1, l'importance des données pour la création d'applications d'intelligence artificielle et les différents outils et technologies que Salesforce propose pour vous apporter tous les bénéfices de l'IA.
OOPS_Lab_Manual - programs using C++ programming language
Fuzzy logicintro by
1. 1
Fuzzy Logic and Fuzzy Inference
• Why use fuzzy logic?
• Tipping example
• Fuzzy set theory
• Fuzzy inference
2. 2
What is fuzzy logic?
• A super set of Boolean logic
• Builds upon fuzzy set theory
• Graded truth. Truth values between True and False. Not
everything is either/or, true/false, black/white, on/off etc.
• Grades of membership. Class of tall men, class of far cities, class of
expensive things, etc.
• Lotfi Zadeh, UC/Berkely 1965. Introduced FL to model
uncertainty in natural language. Tall, far, nice, large, hot, …
• Reasoning using linguistic terms. Natural to express expert
knowledge.
If the weather is cold then wear warm clothing
3. 3
Fuzzy logic – A Definition
• Fuzzy logic provides a method to formalize reasoning when dealing
with vague terms.
• Traditional computing requires finite precision which is not always
possible in real world scenarios. Not every decision is either true or
false, or as with Boolean logic either 0 or 1.
• Fuzzy logic allows for membership functions, or degrees of
truthfulness and falsehoods. Or as with Boolean logic, not only 0
and 1 but all the numbers that fall in between.
4. 4
Why use fuzzy logic?
Pros:
• Conceptually easy to understand w/ “natural” maths
• Tolerant of imprecise data
• Universal approximation: can model arbitrary nonlinear functions
• Intuitive
• Based on linguistic terms
• Convenient way to express expert and common sense knowledge
Cons:
• Not a cure-all
• Crisp/precise models can be more efficient and even convenient
• Other approaches might be formally verified to work
5. 5
Tipping example
• The Basic Tipping Problem: Given a number between 0 and 10
that represents the quality of service at a restaurant what should
the tip be?
Cultural footnote: An average tip for a meal in the U.S. is 15%,
which may vary depending on the quality of the service provided.
6. 6
Tipping example: The non-fuzzy approach
• Tip = 15% of total bill
• What about quality of service?
7. 7
Tipping example: The non-fuzzy approach
• Tip = linearly proportional to service from 5% to 25%
tip = 0.20/10*service+0.05
• What about quality of the food?
8. 8
Tipping example: Extended
• The Extended Tipping Problem: Given a number between 0 and
10 that represents the quality of service and the quality of the food,
at a restaurant, what should the tip be?
How will this affect our tipping formula?
9. 9
Tipping example: The non-fuzzy approach
• Tip = 0.20/20*(service+food)+0.05
• We want service to be more important than food quality. E.g., 80% for
service and 20% for food.
10. 10
Tipping example: The non-fuzzy approach
• Tip = servRatio*(.2/10*(service)+.05) + servRatio = 80%
(1-servRatio)*(.2/10*(food)+0.05);
• Seems too linear. Want 15% tip in general and deviation only for
exceptionally good or bad service.
11. 11
Tipping example: The non-fuzzy approach
if service < 3,
tip(f+1,s+1) = servRatio*(.1/3*(s)+.05) + ...
(1-servRatio)*(.2/10*(f)+0.05);
elseif s < 7,
tip(f+1,s+1) = servRatio*(.15) + ...
(1-servRatio)*(.2/10*(f)+0.05);
else,
tip(f+1,s+1) = servRatio*(.1/3*(s-7)+.15) + ...
(1-servRatio)*(.2/10*(f)+0.05);
end;
12. CS 561, Sessions 20-21 12
Tipping example: The non-fuzzy approach
Nice plot but
• ‘Complicated’ function
• Not easy to modify
• Not intuitive
• Many hard-coded parameters
• Not easy to understand
13. 13
Tipping problem: the fuzzy approach
What we want to express is:
1. If service is poor then tip is cheap
2. If service is good the tip is average
3. If service is excellent then tip is generous
4. If food is rancid then tip is cheap
5. If food is delicious then tip is generous
or
1. If service is poor or the food is rancid then tip is cheap
2. If service is good then tip is average
3. If service is excellent or food is delicious then tip is generous
We have just defined the rules for a fuzzy logic system.
15. 15
Tipping problem: fuzzy solution
• Before we have a fuzzy solution we need to find out
a) how to define terms such as poor, delicious, cheap, generous etc.
b) how to combine terms using AND, OR and other connectives
c) how to combine all the rules into one final output
16. 16
Fuzzy sets
• Boolean/Crisp set A is a mapping for the elements of S to
the set {0, 1}, i.e., A: S {0, 1}
• Characteristic function:
A(x) = {1 if x is an element of set A
0 if x is not an element of set A
• Fuzzy set F is a mapping for the elements of S to the interval
[0, 1], i.e., F: S [0, 1]
• Characteristic function: 0 F(x) 1
• 1 means full membership, 0 means no membership and anything in
between, e.g., 0.5 is called graded membership
17. 17
Fuzzy Sets (contd.)
• fuzzy set A
• A = {(x, µA(x))| x Є X} where µA(x) is called the membership
function for the fuzzy set A. X is referred to as the universe of
discourse.
• The membership function associates each element x Є X with a
value in the interval [0,1].
18. 18
Example: Crisp set Tall
• Fuzzy sets and concepts are commonly used in natural language
John is tall
Dan is smart
Alex is happy
The class is hot
• E.g., the crisp set Tall can be defined as {x | height x > 1.8 meters}
But what about a person with a height = 1.79 meters?
What about 1.78 meters?
…
What about 1.52 meters?
19. 19
Example: Fuzzy set Tall
• In a fuzzy set a person with a height of 1.8 meters would be
considered tall to a high degree
A person with a height of 1.7 meters would be considered tall to a
lesser degree etc.
• The function can change
for basketball players,
Danes, women,
children etc.
20. 20
Membership functions: S-function
• The S-function can be used to define fuzzy sets
• S(x, a, b, c) =
• 0 for x a
• 2(x-a/c-a)2 for a x b
• 1 – 2(x-c/c-a)2 for b x c
• 1 for x c
a b c
21. 21
Membership functions of one dimension
• These membership functions are some of the commonly used
membership functions in the fuzzy inference systems.
• Triangle(x; a, b, c) = 0 if x a;
= (x-a)/(b-a) if a x b;
= (c-b)/(c-b) if b x c;
= 0 if c x.
• Trapezoid(x; a, b, c, d) = 0 if x a;
= (x-a)/(b-a) if a x b;
= 1 if b x c;
= (d-x)/(d-c) 0 if c x d;
= 0, if d x.
• Sigmoid(x; a, c) = 1/(1 + exp[-a(x-c)]) where a controls slope at
the crossover point x = c.
22. 22
Membership functions of two dimensions
• One dimensional fuzzy set can be extended to form its cylindrical
extension on second dimension
• Fuzzy set A = “(x,y) is near (3,4)” is
• µA(x,y) = exp[- ((x-3)/2)2 -(y-4)2 ]
= exp[- ((x-3)/2)2 ] exp -(y-4)2 ]
=gaussian(x;3,2)gaussian(y;4,1)
• This is a composite MF since it can be decomposed into two
gaussian MFs
23. 23
Membership functions: P-Function
• P(x, a, b) =
• S(x, b-a, b-a/2, b) for x b
• 1 – S(x, b, b+a/2, a+b) for x b
E.g., close (to a)
b-a b+a/2b-a/2 b+a
a
a
25. 25
Other representations of fuzzy sets
• A finite set of elements:
F = 1/x1 + 2/x2 + … n/xn
+ means (Boolean) set union
• For example:
TALL = {0/1.0, 0/1.2, 0/1.4, 0.2/1.6, 0.8/1.7, 1.0/1.8}
26. 26
Fuzzy sets with a discrete universe
• Let X = {0, 1, 2, 3, 4, 5, 6} be a set of numbers of children a family
may possibly have.
• fuzzy set A with “sensible number of children in a family” may be
described by
• A = {(0, 0.1), (1, 0.3), (2, 0.7), (3, 1), (4, 0.7), (5, 0.3), (6, 0.1)}
27. 27
Fuzzy sets with a continuous universe
• X = R+ be the set of possible ages for human beings.
• fuzzy set B = “about 50 years old” may be expressed as
• B = {(x, µB(x)) | x Є X}, where
• µB(x) = 1/(1 + ((x-50)/10)4)
28. 28
We use the following notation to describe fuzzy sets.
• A = Σ xi Є X µA(xi)/ xi, if X is a collection of discrete objects,
• A = ∫X µA(x)/ x, if X is a continuous space.
29. 29
Fuzzy set operators
• Equality
A = B
A (x) = B (x) for all x X
• Complement
A’
A’ (x) = 1 - A(x) for all x X
• Containment
A B
A (x) B (x) for all x X
• Union
A B
A B (x) = max(A (x), B (x)) for all x X
• Intersection
A B
A B (x) = min(A (x), B (x)) for all x X
30. 30
• Support(A) is set of all points x in X such that
• {(x, µA(x)) | µA(x) > 0 }
• core(A) is set of all points x in X such that
• {(x, µA(x)) | µA(x) =1 }
• Fuzzy set whose support is a single point in X with µA(x) =1 is
called fuzzy singleton
31. 31
• Crossover point of a fuzzy set A is a point x in X such that
• {(x, µA(x)) | µA(x) = 0.5 }
• α-cut of a fuzzy set A is set of all points x in X such that
• {(x, µA(x)) | µA(x) ≥ α }
33. 33
Linguistic Hedges
• Modifying the meaning of a fuzzy set using hedges such as very,
more or less, slightly, etc.
• Concentration or Con operator
• Very F = F2
• Dilation or Dil operator
• More or less F = F1/2
• more or less tall
= DIL(old);
• extremely tall
= CON(CON(CON(old)))
• etc.
tall
More or less tall
Very tall
35. 35
Fuzzy logical operations
• AND, OR, NOT, etc.
• NOT A = A’ = 1 - A(x)
• A AND B = A B = min(A (x), B (x))
• A OR B = A B = max(A (x), B (x))
A B A and B
0 0 0
0 1 0
1 0 0
1 1 1
A B A or B
0 0 0
0 1 1
1 0 1
1 1 1
A not A
0 1
1 0
1-Amax(A,B)min(A,B)
From the following
truth tables it is
seen that fuzzy
logic is a superset
of Boolean logic.
36. 36
If-Then Rules
• Use fuzzy sets and fuzzy operators as the subjects and verbs of
fuzzy logic to form rules.
if x is A then y is B
where A and B are linguistic terms defined by fuzzy sets on the sets
X and Y respectively.
This reads
if x == A then y = B
37. 37
Evaluation of fuzzy rules
• In Boolean logic: p q
if p is true then q is true
• In fuzzy logic: p q
if p is true to some degree then q is true to some degree.
0.5p => 0.5q (partial premise implies partially)
• How?
38. CS 561, Sessions 20-21 38
Evaluation of fuzzy rules (cont’d)
• Apply implication function to the rule
• Most common way is to use min to “chop-off” the consequent
(prod can be used to scale the consequent)
39. 39
Summary: If-Then rules
1. Fuzzify inputs
Determine the degree of membership for all terms in the premise.
If there is one term then this is the degree of support for the
consequent.
2. Apply fuzzy operator
If there are multiple parts, apply logical operators to determine the
degree of support for the rule.
3. Apply implication method
Use degree of support for rule to shape output fuzzy set of the
consequent.
How do we then combine several rules?
40. 40
Multiple rules
• We aggregate the outputs into a single fuzzy set which combines their
decisions.
• The input to aggregation is the list of truncated fuzzy sets and the
output is a single fuzzy set for each variable.
• Aggregation rules: max, sum, etc.
• As long as it is commutative then the order of rule exec is irrelevant.
41. 41
max-min rule of composition
• Given N observations Ei over X and hypothesis Hi over Y we have N
rules:
if E1 then H1
if E2 then H2
if EN then HN
• H = max[min(E1), min(E2), … min(EN)]
42. 42
Defuzzify the output
• Take a fuzzy set and produce a single crisp number that represents
the set.
• Practical when making a decision, taking an action etc.
I x
I
Center of largest area
Center of gravity
I=
43. CS 561, Sessions 20-21 43
Tip = 16.7 %
Result of defuzzification
(centroid)
Fuzzyinferenceoverview
44. 44
Limitations of fuzzy logic
• How to determine the membership functions? Usually requires fine-
tuning of parameters
• Defuzzification can produce undesired results
47. 47
Specific Fuzzified Applications
• Otis Elevators
• Vacuum Cleaners
• Hair Dryers
• Air Control in Soft Drink Production
• Noise Detection on Compact Disks
• Cranes
• Electric Razors
• Camcorders
• Television Sets
• Showers
Japan, Korea, China were the early adapters of
Fuzzy Logic into industrial applications
48. 48
Expert Fuzzified Systems
• Medical Diagnosis
• Legal
• Stock Market Analysis
• Mineral Prospecting
• Weather Forecasting
• Economics
• Politics
49. 49
References
• Slides from CS 561 (Intro to AI) course, Sessions 20-21 of Paolo Pirjanian
• Slides from OperMgt 345 course of Mitch Pence, Boise State University
• Slides from CS623-Lec11-4Sept06, S.G.Sanjeevi of IIT Bombay.
• Fuzzy Logic. Fuzzy Logic - a powerful new technology.
http://www.austinlinks.com/Fuzzy/
• FuzzyNet On-line. Automatic Transmission
http://www.aptronix.com/fuzzynet/applnote/transmis.htm
• Garner, Martin. Weird Water and Fuzzy Logic: More notes of a Fringe
Watcher.
• Generation 5. An Introduction to Fuzzy Logic.
http://www.generation5.org/fuzzyintro.shtml
• Sowell, Thomas . FUZzy Logic For “Just Plain Folks” .
• You Fuzzin’ with me?
http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/sbaa/article1.html
• Zadeh, Lotfi A.
http://http.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
• Zadeh, L. (1965), "Fuzzy sets", Information and Control, 8: 338-353
• Jang J.S.R., (1997): ANFIS architecture. In: Neuro-fuzzy and Soft
Computing (J.S. Jang, C.-T. Sun, E. Mizutani, Eds.), Prentice Hall, New
50. 50
An Example
The following, illustrates a basic “fuzzy” automatic transmission
system. The transmission uses four fuzzy sensor inference
inputs to control the best gear selection for the given conditions.
The inputs are throttle, vehicle speed, engine speed and engine
load.
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An Example (cont.)
Using the labels as defined in the previous slides, rules can be
written for the fuzzy interface unit shown earlier. The rules provide
a tangible knowledge base required for the decision process and
are represented as English like if-then statements.
57. 57
An Example (cont.)
To create the fuzzy interface unit, rules such as the following would
be developed to facilitate the automatic shifting of the vehicle.
If vehicle speed is low, variation of vehicle speed is small, slope
resistance is positive large and accelerator is medium then mode is
steep uphill mode.
If vehicle speed is medium, variation of vehicle speed is small, slope
resistance is negative large and accelerator is small then mode is
gentle downhill mode.
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An Example (cont.)
If Mode is Steep uphill mode, the Shift is No. 2
If Mode is Gentle downhill mode, then Shift is No. 3
The previous slides illustrate how fuzzy logic can provide a powerful
tool when addressing complex situations that are not feasible using
conventional approaches. By employing fuzzy logic, we have the
ability to include additional variables and rules to take into account
factors that might improve the behavior of the control system.
* See reference (see notes):
http://www.aptronix.com/fuzzynet/applnote/transmis.htm
59. 59
Binary fuzzy relation
• A binary fuzzy relation is a fuzzy set in X × Y which maps each
element in X × Y to a membership value between 0 and 1. If X and
Y are two universes of discourse, then
• R = {((x,y), R(x, y)) | (x,y) Є X × Y } is a binary fuzzy relation in X
× Y.
• X × Y indicates cartesian product of X and Y
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Fuzzy Rules
• Fuzzy rules are useful for modeling human thinking, perception and
judgment.
• A fuzzy if-then rule is of the form “If x is A then y is B” where A
and B are linguistic values defined by fuzzy sets on universes of
discourse X and Y, respectively.
• “x is A” is called antecedent and “y is B” is called consequent.
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Fuzzy relations
• A fuzzy relation for N sets is defined as an extension of the crisp
relation to include the membership grade.
R = {R(x1, x2, … xN)/(x1, x2, … xN) | xi X, i=1, … N}
which associates the membership grade, R , of each tuple.
• E.g.
Friend = {0.9/(Manos, Nacho), 0.1/(Manos, Dan),
0.8/(Alex, Mike), 0.3/(Alex, John)}
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Fuzzy intersection and Union
• AB = T(A(x), B(x)) where T is T-norm operator. There are some
possible T-Norm operators.
• Minimum: min(a,b)=a ٨ b
• Algebraic product: ab
• Bounded product: 0 ٧ (a+b-1)
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• C(x) = AB = S(A(x), B(x)) where S is called S-norm operator.
• It is also called T-conorm
• Some of the T-conorm operators
• Maximum: S(a,b) = max(a,b)
• Algebraic sum: a+b-ab
• Bounded sum: = 1 ٨(a+b)
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Examples, for such a rule are
• If pressure is high, then volume is small.
• If the road is slippery, then driving is dangerous.
• If the fruit is ripe, then it is soft.
65. 65
• The fuzzy rule “If x is A then y is B” may be abbreviated as A→ B
and is interpreted as A × B.
• A fuzzy if then rule may be defined (Mamdani) as a binary fuzzy
relation R on the product space X × Y.
• R = A→ B = A × B =∫X×Y A(x) T-norm B(y)/ (x,y).