Tutorial: Fuzzy Logic – II

Membership Functions, Fuzzy Rules, Fuzzification & Defuzzification

Introduction

In Unit III, we studied the basic concepts of fuzzy sets, membership values, fuzzy union, fuzzy intersection, and fuzzy complement.

In this unit, we move one step further and learn how fuzzy systems represent linguistic concepts such as:

Low
Medium
High
Slow
Fast
Cold
Warm
Hot

The main topics covered in this unit are:

Membership Functions
Triangular Membership Function
Trapezoidal Membership Function
Gaussian Membership Function
Sigmoid Membership Function
Fuzzy Rules
Fuzzification
Fuzzy Inference
Defuzzification
Centroid Method

1. Membership Functions

A membership function determines the degree to which an element belongs to a fuzzy set.

For a fuzzy set $A$ , the membership function is written as:

\mu_A(x) : X \rightarrow [0, 1]

Therefore:

0 \leq \mu_A(x) \leq 1

where:

$x$ = input value
$A$ = fuzzy set
$\mu_A(x)$ = membership degree of $x$ in fuzzy set $A$

The membership value can be interpreted as:

Membership	Interpretation
$0$	Does not belong
$0.2$	Very low membership
$0.5$	Partial membership
$0.8$	High membership
$1$	Full membership

For example:

\mu_{\text{Hot}}(30) = 0.8

means that a temperature of $30^\circ$ C has a membership degree of $0.8$ in the fuzzy set Hot.

2. Common Membership Functions

Several mathematical functions can be used to define fuzzy membership.

Common membership functions include:

Triangular
Trapezoidal
Gaussian
Sigmoid

Each function produces a different shape for the membership curve.
The choice of membership function depends on the application and the shape of the concept being modeled.

3. Triangular Membership Function

The triangular membership function is one of the simplest and most commonly used membership functions.

It is defined using three parameters:

a,\quad b,\quad c

where:

$a$ = left endpoint
$b$ = peak
$c$ = right endpoint

The triangular membership function is:

\mu(x) = \begin{cases} 0, & x \leq a \\[4pt] \dfrac{x - a}{b - a}, & a < x \leq b \\[8pt] \dfrac{c - x}{c - b}, & b < x < c \\[8pt] 0, & x \geq c \end{cases}

At the peak:

\mu(b) = 1

Example

Suppose:

a = 10,\qquad b = 20,\qquad c = 30

Then:

\mu(x) = \begin{cases} 0, & x \leq 10 \\[4pt] \dfrac{x - 10}{10}, & 10 < x \leq 20 \\[8pt] \dfrac{30 - x}{10}, & 20 < x < 30 \\[8pt] 0, & x \geq 30 \end{cases}

At $x = 20$ :

\mu(20) = 1

At $x = 15$ :

\mu(15) = \dfrac{15 - 10}{10} = 0.5

Thus:

\mu(15) = 0.5

4. Trapezoidal Membership Function

A trapezoidal membership function is similar to the triangular function, but it contains a flat region where the membership value is $1$ .

It uses four parameters:

a,\quad b,\quad c,\quad d

The function is:

\mu(x) = \begin{cases} 0, & x \leq a \\[4pt] \dfrac{x - a}{b - a}, & a < x < b \\[8pt] 1, & b \leq x \leq c \\[4pt] \dfrac{d - x}{d - c}, & c < x < d \\[8pt] 0, & x \geq d \end{cases}

Here:

$a$ = beginning of the rising edge
$b$ = beginning of the plateau
$c$ = end of the plateau
$d$ = end of the falling edge

The membership is completely active between $b$ and $c$ :

\mu(x) = 1 \quad \text{for } b \leq x \leq c

5. Gaussian Membership Function

The Gaussian membership function produces a smooth bell-shaped curve.

It is commonly defined as:

\mu(x) = \exp\left( -\dfrac{(x - c)^2}{2\sigma^2} \right)

where:

$c$ = center of the Gaussian curve
$\sigma$ = standard-deviation-like width parameter
$x$ = input value

At the center ( $x = c$ ):

\mu(c) = \exp(0) = 1

As $x$ moves farther away from $c$ , the membership value decreases.

6. Gaussian Membership Example

Suppose:

c = 50, \qquad \sigma = 10

The membership function becomes:

\mu(x) = \exp\left( -\dfrac{(x - 50)^2}{200} \right)

At $x = 50$ :

\mu(50) = 1

At $x = 60$ :

\mu(60) = \exp\left( -\dfrac{(60 - 50)^2}{200} \right) = e^{-0.5} \approx 0.607

Thus, $60$ has approximately $0.607$ membership in this fuzzy set.

7. Effect of $\sigma$

The parameter $\sigma$ controls the width of the Gaussian curve.

A smaller $\sigma$ produces a narrower curve.
A larger $\sigma$ produces a wider curve.

The Gaussian membership function remains:

\mu(x) = \exp\left( -\dfrac{(x - c)^2}{2\sigma^2} \right)

Therefore:

Small $\sigma$ → narrow membership region
Large $\sigma$ → wide membership region

8. Sigmoid Membership Function

The sigmoid membership function produces a smooth S-shaped curve.

A common sigmoid function is:

\mu(x) = \dfrac{1}{1 + e^{-a(x - c)}}

where:

$a$ controls the slope
$c$ controls the transition point

At $x = c$ :

\mu(c) = \dfrac{1}{1 + e^{0}} = \dfrac{1}{2} = 0.5

The sign of $a$ determines the direction of the transition.

9. Comparison of Membership Functions

Function	Shape	Parameters	Common Use
Triangular	Triangle	$a, b, c$	Simple fuzzy systems
Trapezoidal	Trapezoid	$a, b, c, d$	Concepts with a plateau
Gaussian	Bell-shaped

10. Fuzzy Rules

A fuzzy rule represents knowledge using linguistic variables.

A common fuzzy rule has the form:

IF temperature is HIGH THEN fan speed is FAST.

Mathematically, a fuzzy rule can be represented as:

\text{IF } A \text{ THEN } B

where:

$A$ = antecedent
$B$ = consequent

For example:

\text{IF Temperature is High THEN FanSpeed is Fast}

Here:

Temperature is High → antecedent
Fan Speed is Fast → consequent

11. Multiple Fuzzy Rules

A fuzzy system can contain multiple rules.

For example:

Rule 1: IF temperature is LOW THEN fan speed is SLOW.
Rule 2: IF temperature is MEDIUM THEN fan speed is MODERATE.
Rule 3: IF temperature is HIGH THEN fan speed is FAST.

Mathematically:

\begin{align*} R_1 &: \text{IF } T = \text{Low} \text{ THEN } F = \text{Slow} \\ R_2 &: \text{IF } T = \text{Medium} \text{ THEN } F = \text{Moderate} \\ R_3 &: \text{IF } T = \text{High} \text{ THEN } F = \text{Fast} \end{align*}

These rules allow the system to make decisions using linguistic concepts.

12. Fuzzy AND

Fuzzy rules may contain multiple conditions.

For example:

IF temperature is HIGH AND humidity is HIGH THEN fan speed is FAST.

Using the standard minimum operator, the firing strength is:

\alpha = \min\bigl( \mu_{\text{High}}(T),\ \mu_{\text{High}}(H) \bigr)

Suppose:

\mu_{\text{High}}(T) = 0.8, \qquad \mu_{\text{High}}(H) = 0.6

Then:

\alpha = \min(0.8, 0.6) = 0.6

The rule fires with strength $0.6$ .

13. Fuzzy OR

For an OR condition:

IF temperature is HIGH OR humidity is HIGH THEN fan speed is FAST.

Using the standard maximum operator:

\alpha = \max\bigl( \mu_{\text{High}}(T),\ \mu_{\text{High}}(H) \bigr)

If:

\mu_{\text{High}}(T) = 0.8, \qquad \mu_{\text{High}}(H) = 0.6

then:

\alpha = \max(0.8, 0.6) = 0.8

14. Fuzzification

Fuzzification is the process of converting a crisp numerical input into fuzzy membership values.

Suppose the input temperature is:

T = 30^\circ\text{C}

We may have three fuzzy sets: Cold, Warm, Hot.

The membership functions may produce:

\begin{align*} \mu_{\text{Cold}}(30) &= 0.0 \\ \mu_{\text{Warm}}(30) &= 0.7 \\ \mu_{\text{Hot}}(30) &= 0.4 \end{align*}

Therefore, the crisp input $T = 30$ is transformed into fuzzy values:

T \rightarrow \{\text{Cold}: 0.0,\ \text{Warm}: 0.7,\ \text{Hot}: 0.4\}

Notice that an input can have non-zero membership in multiple fuzzy sets.

15. Fuzzification Example

Suppose the fuzzy sets for temperature are Cold, Warm, Hot.

For an input $T = 25$ , the membership functions may produce:

\begin{align*} \mu_{\text{Cold}}(25) &= 0.1 \\ \mu_{\text{Warm}}(25) &= 0.8 \\ \mu_{\text{Hot}}(25) &= 0.2 \end{align*}

The fuzzified representation is:

T = 25 \rightarrow \begin{cases} \text{Cold}: 0.1 \\ \text{Warm}: 0.8 \\ \text{Hot}: 0.2 \end{cases}

This is different from crisp classification, where $25$ would typically be assigned to only one category.

16. Fuzzy Inference

After fuzzification, the fuzzy rules are evaluated.

The general process is:

\text{Crisp Input} \rightarrow \text{Fuzzification} \rightarrow \text{Rule Evaluation} \rightarrow \text{Aggregation} \rightarrow \text{Defuzzification}

For example, $T = 30$ may produce:

\mu_{\text{Warm}}(30) = 0.7, \qquad \mu_{\text{Hot}}(30) = 0.4

Suppose we have:

IF temperature is WARM THEN fan speed is MEDIUM.
IF temperature is HOT THEN fan speed is FAST.

The corresponding rule strengths are:

\alpha_1 = 0.7, \qquad \alpha_2 = 0.4

These rule strengths are then used to determine the fuzzy output.

17. Defuzzification

The output of a fuzzy inference system is often a fuzzy set.
However, many real-world applications require a single numerical value.

Defuzzification converts the fuzzy output into a crisp numerical value.

For example:

\text{Fuzzy Fan Speed} \rightarrow \text{Crisp Fan Speed}

If the fuzzy system determines that the appropriate fan speed is represented by several membership values, defuzzification can produce a value such as:

\text{Fan Speed} = 72

18. Centroid Method

The centroid method (also called the center of gravity method) is one of the most common defuzzification techniques.

For discrete values, the centroid is:

x^* = \dfrac{\displaystyle\sum_i x_i \mu(x_i)}{\displaystyle\sum_i \mu(x_i)}

where:

$x_i$ = possible output value
$\mu(x_i)$ = membership value at $x_i$

The formula calculates a weighted average of the output values.

19. Centroid Example

Suppose the output membership values are:

$x_i$	$\mu(x_i)$
20	0.2
40	0.5
60	0.8
80	0.4

The centroid is:

x^* = \dfrac{20(0.2) + 40(0.5) + 60(0.8) + 80(0.4)}{0.2 + 0.5 + 0.8 + 0.4}

Numerator:

20(0.2) + 40(0.5) + 60(0.8) + 80(0.4) = 4 + 20 + 48 + 32 = 104

Denominator:

0.2 + 0.5 + 0.8 + 0.4 = 1.9

Therefore:

x^* = \dfrac{104}{1.9} \approx 54.74

The final crisp output is approximately 54.74.

20. Continuous Centroid Formula

For a continuous output membership function, the centroid can be written as:

x^* = \dfrac{\displaystyle\int x\,\mu(x)\,dx}{\displaystyle\int \mu(x)\,dx}

The numerator represents the weighted first moment of the membership function, while the denominator represents the total membership area.

For a general output domain:

x^* = \dfrac{\displaystyle\int_X x\,\mu(x)\,dx}{\displaystyle\int_X \mu(x)\,dx}

21. Complete Fuzzy Logic System

A complete fuzzy logic system can be represented as:

\text{Crisp Input} \rightarrow \text{Fuzzification} \rightarrow \text{Fuzzy Rules} \rightarrow \text{Inference} \rightarrow \text{Defuzzification} \rightarrow \text{Crisp Output}

For example, in a fan-control system:

\text{Temperature} \rightarrow \text{Fuzzification} \rightarrow \text{Rules} \rightarrow \text{Inference} \rightarrow \text{Defuzzification} \rightarrow \text{Fan Speed}

22. Complete Example: Temperature-Controlled Fan

Suppose a fuzzy system controls fan speed based on temperature.

Input variable: $T = \text{Temperature}$
Output variable: $F = \text{Fan Speed}$

Temperature fuzzy sets: $\{\text{Cold},\ \text{Warm},\ \text{Hot}\}$
Fan-speed fuzzy sets: $\{\text{Slow},\ \text{Medium},\ \text{Fast}\}$

Suppose $T = 30^\circ$ C. After fuzzification:

\mu_{\text{Warm}}(30) = 0.7, \qquad \mu_{\text{Hot}}(30) = 0.5

Rules:

IF temperature is WARM THEN fan speed is MEDIUM.
IF temperature is HOT THEN fan speed is FAST.

Rule firing strengths:

\alpha_1 = 0.7, \qquad \alpha_2 = 0.5

The fuzzy inference system combines these outputs.
Finally, centroid defuzzification produces a crisp fan speed:

F^* = \dfrac{\displaystyle\sum_i x_i \mu(x_i)}{\displaystyle\sum_i \mu(x_i)}

The resulting value can then be used to control the actual fan.

23. Important Formulas for Unit IV

Membership Function

\mu_A(x) : X \rightarrow [0, 1]

Triangular Membership

\mu(x) = \begin{cases} 0, & x \leq a \\ \dfrac{x-a}{b-a}, & a < x \leq b \\ \dfrac{c-x}{c-b}, & b < x < c \\ 0, & x \geq c \end{cases}

Trapezoidal Membership

\mu(x) = \begin{cases} 0, & x \leq a \\ \dfrac{x-a}{b-a}, & a < x < b \\ 1, & b \leq x \leq c \\ \dfrac{d-x}{d-c}, & c < x < d \\ 0, & x \geq d \end{cases}

Gaussian Membership

\mu(x) = \exp\left( -\dfrac{(x-c)^2}{2\sigma^2} \right)

Sigmoid Membership

\mu(x) = \dfrac{1}{1 + e^{-a(x-c)}}

Fuzzy Rule

\text{IF } A \text{ THEN } B

Fuzzy AND

\mu_{A \cap B}(x) = \min\bigl( \mu_A(x),\ \mu_B(x) \bigr)

Fuzzy OR

\mu_{A \cup B}(x) = \max\bigl( \mu_A(x),\ \mu_B(x) \bigr)

Centroid Defuzzification (Discrete)

x^* = \dfrac{\sum_i x_i \mu(x_i)}{\sum_i \mu(x_i)}

Continuous Centroid

x^* = \dfrac{\int x\,\mu(x)\,dx}{\int \mu(x)\,dx}

24. Summary

In Unit IV, we learned how fuzzy systems transform numerical inputs into linguistic concepts and then produce useful numerical outputs.

A membership function maps an input to a membership degree:

\mu_A(x) \in [0, 1]

The major membership functions are:

Triangular
Trapezoidal
Gaussian
Sigmoid

A fuzzy rule follows the structure:

\text{IF } A \text{ THEN } B

Fuzzification converts a crisp input into fuzzy membership values:

\text{Crisp Input} \rightarrow \text{Membership Values}

Fuzzy inference evaluates the rules and produces a fuzzy output.
Defuzzification converts the fuzzy output into a crisp numerical value.

The complete fuzzy-system pipeline is:

\text{Input} \rightarrow \text{Fuzzification} \rightarrow \text{Rule Evaluation} \rightarrow \text{Inference} \rightarrow \text{Defuzzification} \rightarrow \text{Output}

The centroid method is one of the most widely used defuzzification methods:

x^* = \dfrac{\sum_i x_i \mu(x_i)}{\sum_i \mu(x_i)}

These concepts form the foundation for designing practical fuzzy-control systems such as temperature controllers, washing machines, automatic braking systems, air conditioners, and decision-support systems.

Tutorial: Fuzzy Logic – II

Membership Functions, Fuzzy Rules, Fuzzification & Defuzzification

Introduction

1. Membership Functions

2. Common Membership Functions

3. Triangular Membership Function

Example

4. Trapezoidal Membership Function

5. Gaussian Membership Function

6. Gaussian Membership Example

7. Effect of σ\sigmaσ

8. Sigmoid Membership Function

9. Comparison of Membership Functions

10. Fuzzy Rules

11. Multiple Fuzzy Rules

12. Fuzzy AND

13. Fuzzy OR

14. Fuzzification

15. Fuzzification Example

16. Fuzzy Inference

17. Defuzzification

18. Centroid Method

19. Centroid Example

20. Continuous Centroid Formula

21. Complete Fuzzy Logic System

22. Complete Example: Temperature-Controlled Fan

23. Important Formulas for Unit IV

24. Summary

7. Effect of $\sigma$