Saturday, 28 December 2013

Introduction to Fuzzy Set

Introduction

Fuzzy logic has rapidly become one of the most successful of today's technologies for developing sophisticated control systems. The reason for which is very simple. Fuzzy logic addresses such applications perfectly as it resembles human decision making with an ability to generate precise solutions from certain or approximate information. It fills an important gap in engineering design methods left vacant by purely mathematical approaches (e.g. linear control design), and purely logic-based approaches (e.g. expert systems) in system design.

While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models.

What does it offer?
The first applications of fuzzy theory were primaly industrial, such as process control for cement kilns. However, as the technology was further embraced, fuzzy logic was used in more useful applications. In 1987, the first fuzzy logic-controlled subway was opened in Sendai in northern Japan. Here, fuzzy-logic controllers make subway journeys more comfortable with smooth braking and acceleration. Best of all, all the driver has to do is push the start button! Fuzzy logic was also put to work in elevators to reduce waiting time. Since then, the applications of Fuzzy Logic technology have virtually exploded, affecting things we use everyday.
Take for example, the fuzzy washing machine . A load of clothes in it and press start, and the machine begins to churn, automatically choosing the best cycle. The fuzzy microwave, Place chili, potatoes, or etc in a fuzzy microwave and push single button, and it cooks for the right time at the proper temperature. The fuzzy car, manuvers itself by following simple verbal instructions from its driver. It can even stop itself when there is an obstacle immedeately ahead using sensors. But, practically the most exciting thing about it, is the simplicity involved in operating it.


What do ya mean fuzzy ??!!
Before illustrating the mechanisms which make fuzzy logic machines work, it is important to realize what fuzzy logic actually is. Fuzzy logic is a superset of conventional(Boolean) logic that has been extended to handle the concept of partial truth- truth values between "completely true" and "completely false". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.
The essential characteristics of fuzzy logic as founded by Zader Lotfi are as follows.


  • In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.
  • In fuzzy logic everything is a matter of degree.
  • Any logical system can be fuzzified
  • In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables
  • Inference is viewed as a process of propagation of elastic constraints.
The third statement hence, define Boolean logic as a subset of Fuzzy logic.
Fuzzy Sets

Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965. What Zadeh proposed is very much a paradigm shift that first gained acceptance in the Far East and its successful application has ensured its adoption around the world.
A paradigm is a set of rules and regulations which defines boundaries and tells us what to do to be successful in solving problems within these boundaries. For example the use of transistors instead of vacuum tubes is a paradigm shift - likewise the development of Fuzzy Set Theory from conventional bivalent set theory is a paradigm shift.
Bivalent Set Theory can be somewhat limiting if we wish to describe a 'humanistic' problem mathematically. For example, Fig 1 below illustrates bivalent sets to characterise the temperature of a room.


The most obvious limiting feature of bivalent sets that can be seen clearly from the diagram is that they are mutually exclusive - it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transiton from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth (unnoticeable) drift from warm to hot would occur.

This natural phenomenon can be described more accurately by Fuzzy Set Theory. Fig.2 below shows how fuzzy sets quantifying the same information can describe this natural drift.
The whole concept can be illustrated with this example. Let's talk about people and "youthness". In this case the set S (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, which answers the question "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. The easiest way to do this is with a membership function based on the person's age.
young(x) = { 1, if age(x) <= 20,

(30-age(x))/10, if 20 < age(x) <= 30,

0, if age(x) > 30 }

A graph of this looks like:


Given this definition, here are some example values:
Person    Age    degree of youth
--------------------------------------
Johan     10        1.00 
Edwin     21        0.90
Parthiban 25        0.50
Arosha    26        0.40
Chin Wei  28        0.20
Rajkumar  83        0.00 

So given this definition, we'd say that the degree of truth of the statement "Parthiban is YOUNG" is 0.50.
Note: Membership functions almost never have as simple a shape as age(x). They will at least tend to be triangles pointing up, and they can be much more complex than that. Furthermore, membership functions so far is discussed as if they always are based on a single criterion, but this isn't always the case, although it is the most common case. One could, for example, want to have the membership function for YOUNG depend on both a person's age and their height (Arosha's short for his age). This is perfectly legitimate, and occasionally used in practice. It's referred to as a two-dimensional membership function. It's also possible to have even more criteria, or to have the membership function depend on elements from two completely different universes of discourse.

Fuzzy Set Operations.

Union

The membership function of the Union of two fuzzy sets A and B with membership functions and respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion.

The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra.

Intersection

The membership function of the Intersection of two fuzzy sets A and B with membership functions and respectively is defined as the minimum of the two individual membership functions. This is called the minimum criterion.
The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra.

Complement

The membership function of the Complement of a Fuzzy set A with membership function is defined as the negation of the specified membership function. This is caleed the negation criterion.
The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.
The following rules which are common in classical set theory also apply to Fuzzy set theory.

De Morgans law

,

Associativity

Commutativity

Distributivity

Glossary
Universe of Discourse
The Universe of Discourse is the range of all possible values for an input to a fuzzy system.
Fuzzy Set
A Fuzzy Set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1].
Support
The Support of a fuzzy set F is the crisp set of all points in the Universe of Discourse U such that the membership function of F is non-zero.
Crossover point
The Crossover point of a fuzzy set is the element in U at which its membership function is 0.5.
Fuzzy Singleton
A Fuzzy singleton is a fuzzy set whose support is a single point in U with a membership function of one.