Fuzzy Operation

 

Fuzzy Logic

 

Fuzzy logic is a form of many-valued logic in which the true value of variables may be any real number between 0 and 1, both being inclusive.

 

Fuzzy Systems as a subject was developed to model the uncertainty and vagueness present in the human thought process.

Degree of membership function

A fuzzy set is denoted as a tuple that includes the element and a value indicating its degree of membership to the universal set.

Gaussian function

Derived from the normal distribution, this function works in most scenarios, where information about the data is unknown.

Mathematically, it is given as follows:

G(x,μ,σ)=exp(−(x−μ)^2/2σ^2)

 Types of operations in fuzzy logic

Union

Union operation concatenates the two sets and produces a new set.

The degree of membership for this value is the maximum value the element has.

The union of two fuzzy sets A and B is a fuzzy set C , written as C = A ∪ B

 

C = A ∪ B = {(x, μA _∪ B (x)) | ∀x ∈ X}

 

μC(x) = μA _∪ B (x) = μA(x) ∨ μB(x)

 

= max(μA(x), μB(x)), ∀x ∈ X

C = A ∪ B = {(x, μA ∪ B (x)) | ∀x ∈ X}

A = { (x1, 0.2), (x2, 0.5), (x3, 0.6), (x4, 0.8), (x5, 1.0) }

B = { (x1, 0.8), (x2, 0.6), (x3, 0.4), (x4, 0.2), (x5, 0.1) }

A ∪ B = { (x1, 0.8), (x2, 0.6), (x3, 0.6), (x4, 0.8), (x5, 1.0) }

Intersection

The intersection operation finds the common elements between the two sets.

the element with the least degree or minimum of membership is retained in the new set.

 

The intersection of two fuzzy sets A and B is a fuzzy set C , written as C = A ∩ B

 

C = A ∩ B = {(x, μA _∩ B (x)) | ∀x ∈ X}

 

μC(x) = μA _∩ B (x) = μA(x) ⋀ μB(x)

 

= min(μA(x), μB(x)), ∀x ∈ X

C = A ∩ B = {(x, μA ∩ B (x)) | ∀x ∈ X}

A = {(x1, 0.2), (x2, 0.5), (x3, 0.6), (x4, 0.8), (x5, 1.0) }

B = {(x1, 0.8), (x2, 0.6), (x3, 0.4), (x4, 0.2), (x5, 0.1) }

So,

A ∩ B = {(x1, 0.2), (x2, 0.5), (x3, 0.4), (x4, 0.2), (x5, 0.1) }

Complementation

The degree of membership for a fuzzy set can range between 0 & 1. Given a set of elements with a degree of memberships,

The inverted degree of membership for element x is defined as follows:

A’(x)=1−degree of membership(x)

The complement of fuzzy set A, is denoted by A^C, is defined as

A^C = {(x, μ_AC (x)) | ∀x ∈ X}

A^C (x) = 1 – μ_A(x)

A’ (x) = 1 – μA(x)

A = {(x1, 0.2), (x2, 0.5), (x3, 0.6), (x4, 0.8), (x5, 1.0)}

A’ = {(x1, 0.8), (x2, 0.5), (x3, 0.4), (x4, 0.2), (x5, 0.0)}

A ⋃ A^C = {(x₁, 0.8), (x₂, 0.5), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0)} ≠ X

A ∩ A^C = {(x₁, 0.2), (x₂, 0.5), (x₃, 0.4), (x₄, 0.2), (x₅, 0.0) } ≠ Φ

 

Architecture of Fuzzy Logic Control

 

Steps in Designing FLC

1.  Identification of variables − the input, output and state variables must be identified of the plant which is under consideration.
2.  Fuzzy subset configuration − The universe of information is divided into number of fuzzy subsets and each subset is assigned a linguistic label.
3.  Obtaining membership function −   membership function for each fuzzy subset
4.  Fuzzy rule − formulate the fuzzy rule base by assigning relationship between fuzzy input and output.
5.  Fuzzification − Combining fuzzy outputs, locate the fuzzy output and merge into set.
6. Defuzzification − initiate defuzzification process to form a crisp output.

Advantages of Fuzzy Logic Control

1. Cheaper − Developing a FLC is comparatively cheaper than developing model controller in terms of performance. 
2. Robust − FLCs are more robust controllers because of their capability to cover a huge range of operating conditions.
3. Customizable − FLCs are customizable. 
4. Emulate human deductive thinking − Basically FLC is designed to emulate human deductive thinking 
5. Reliability − FLC is more reliable than conventional control system. 
6. Efficiency − Fuzzy logic provides more efficiency when applied in control system.

 

Disadvantages of Fuzzy Logic Control

 

1. Requires lots of data − FLC needs lots of data to be applied.
2. Needs high human expertise − This is one drawback as the accuracy of the system depends on the knowledge and expertise of human beings.
3. Needs regular updating of rules − The rules must be updated with time. 

 Happy Learning 

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Milan Tomic

Hi. I’m Designer of Blog Magic. I’m CEO/Founder of ThemeXpose. I’m Creative Art Director, Web Designer, UI/UX Designer, Interaction Designer, Industrial Designer, Web Developer, Business Enthusiast, StartUp Enthusiast, Speaker, Writer and Photographer. Inspired to make things looks better.

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