Fuzzy reasoning

     In real world,there exists much fuzzy knowledge; Knowledge that is vague, imprecise, uncertain, ambiguous, inexact, or probabilistic in nature. Human thinking and reasoning frequently involve fuzzy information, originating from inherently inexact human concepts. Humans, can give satisfactory answers, which are probably true. However, our systems are unable to answer many questions. The reason is, most systems are designed based upon classical set theory and two-valued logic which is unable to cope with unreliable and incomplete information and give expert opinions. We want, our systems should also be able to cope with unreliable and incomplete information and give expert opinions. Fuzzy sets have been able provide solutions to many real world problems. Fuzzy Set theory is an extension of classical set theory where elements have degrees of membership.  

Classical Set Theory 

A Set is any well defined collection of objects. An object in a set is called an element or member of that set.Sets are defined by a simple statement describing whether a particular element having a certain property belongs to that particular set.

• Sets are defined by a simple statement describing whether a particular element having a certain property belongs to that particular set.

• Classical  set theory enumerates all its elements using 
  
  
  A={a1 , a2 , a3 , ....... , an}

Fuzzy Set

      The word "fuzzy" means "vagueness". Fuzziness occurs when the boundary of a piece of information is not clear-cut. Classical set theory allows the membership of the elements in the set in binary terms, a bivalent condition - an element either belongs or does not belong to the set.Fuzzy set theory permits the gradual assessment of the membership of elements in a set, described with the aid of a membership function valued in the real unit interval [0, 1]. 

Why Fuzzy Logic?

Fuzzy logic is useful for commercial and practical purposes.

• It can control machines and consumer products.
• It may not give accurate reasoning, but acceptable reasoning.
• Fuzzy logic helps to deal with the uncertainty in engineering.

Fuzzy Logic Systems Architecture

It has four main parts as shown −

• Fuzzification Module − It transforms the system inputs, which are crisp numbers, into fuzzy sets. It splits the input signal into five steps such as −

LP	x is Large Positive
MP	x is Medium Positive
S	x is Small
MN	x is Medium Negative
LN	x is Large Negative

• Knowledge Base − It stores IF-THEN rules provided by experts.
• Inference Engine − It simulates the human reasoning process by making fuzzy inference on the inputs and IF-THEN rules.
• Defuzzification Module − It transforms the fuzzy set obtained by the inference engine into a crisp value.

The membership functions work on fuzzy sets of variables.

Example:
 

Membership Function

Membership functions allow you to quantify linguistic term and represent a fuzzy set graphically. A membership function for a fuzzy set A on the universe of discourse X is defined as µA:X → [0,1].
Here, each element of X is mapped to a value between 0 and 1. It is called membership value or degree of membership. It quantifies the degree of membership of the element in X to the fuzzy set A.

• x axis represents the universe of discourse.
• y axis represents the degrees of membership in the [0, 1] interval.

There can be multiple membership functions applicable to fuzzify a numerical value. Simple membership functions are used as use of complex functions does not add more precision in the output.

All membership functions for LP, MP, S, MN, and LN are shown as below −
The triangular membership function shapes are most common among various other membership function shapes such as trapezoidal, singleton, and Gaussian.Here, the input to 5-level fuzzifier varies from -10 volts to +10 volts. Hence the corresponding output also changes.

Example of a Fuzzy Logic System

Let us consider an air conditioning system with 5-level fuzzy logic system. This system adjusts the temperature of air conditioner by comparing the room temperature and the target temperature value.

Algorithm

BUILD A FUZZY CONTROLLER
5 Steps

1. Pick the linguistic variable
Example: Let temperature (X) be input and motor speed (Y) be output

2. Pick the fuzzy sets
Define fuzzy subsets of the X and Y

3. Pick the fuzzy rules
Associate output to the input 

4.Obtain Fuzzy value

5.Perform Defuzzification
 
Goal: 
Design a motor speed controller for air conditioner
 
Step 1: assign input and output variables
 
Let X be the temperature in Fahrenheit
Let Y be the motor speed of the air conditioner


Step 2: Pick fuzzy sets

Define linguistic terms of the linguistic variables temperature (X) and motor speed (Y) and associate them with fuzzy sets

For example, 5 linguistic terms / fuzzy sets on X

Cold, Cool, Just Right, Warm, and Hot


 
Say 5 linguistic terms / fuzzy sets on Y

Stop, Slow, Medium, Fast, and Blast

 
 
Step 3: Assign a motor speed set to each temperature set
 
If temperature is cold then motor speed is stop
If temperature is cool then motor speed is slow
If temperature is just right then motor speed is medium
If temperature is warm then motor speed is fast
If temperature is hot then motor speed is blast


 

 

Step 4: Obtain fuzzy value

Fuzzy set operations perform evaluation of rules. The operations used for OR and AND are Max and Min respectively. Combine all results of evaluation to form a final result. This result is a fuzzy value.

Step 5: Perform defuzzification

Defuzzification is then performed according to membership function for output variable.